Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures

HOMOLOGICAL ALGEBRA OF SEMIMODULES AND SEMICONTRAMODULES SEMI-INFINITE HOMOLOGICAL ALGEBRA OF ASSOCIA TIVE ALGEBRAIC STRUCTURES LEONID POSITSELSKI With app endices coa utho r ed b y S. Arkhip ov and D. Rum ynin Abstract. W e dev elop the basic constructions of homolo gical algebra in the (ap- propriately deﬁned) unbounded der ived categorie s of mo dules ov er alg ebras over coalgebr as o ver noncomm utative rings (whic h we call semialgebr as ov er c orings ). W e deﬁne double- s ided derived functors SemiT or and SemiEx t of the functor s of semitensor pro duct and semihomomo rphisms, and construct an equiv alence b e- t ween the exo tic der ived categor ies of semimo dules and se micontramodules. Certain (co)ﬂatness and/or ( co)pro jectivit y conditions ha ve to be imp osed on the coring and semialgebra to make the mo dule categor ies abelian (and the cotensor pro duct as s o ciative). Bes ides, for a num b er o f technical reaso ns w e mostly have to assume tha t the basic ring has a ﬁnite homolo gical dimension (no s uch assumptions ab out the coring and semialg e bra a re made). In the ﬁnal sections we constr uct mo del ca tegory structures o n the categor ie s of complexes of s emi(contra)modules, and develop relative no nhomogeneous K oszul duality theory for ﬁltered semialge br as a nd qua si-diﬀerential cor ings. Our motiv a ting examples come fro m the semi-inﬁnite cohomolo gy theory . Com- parison with the semi-inﬁnite (co)homology o f T ate Lie algebr a s and g raded as so- ciative algebr as is established in a pp endices; and the semi-inﬁnite homology of a lo cally compact topolog ical group relativ e to an open proﬁnite subgroup is deﬁned. 1 T o the memory of my father Contents In tro duction 3 0. Preliminaries and Summary 12 1. Semialgebras and Semitensor Pro duct 36 2. Deriv ed F unctor SemiT or 50 3. Semicon tramo dules and Semihomomorphisms 67 4. Deriv ed F unctor SemiExt 84 5. Como dule-Con tramo dule Corresp ondence 96 6. Semimo dule-Semicon tramo dule Corresp ondence 112 7. F unctoriality in the Coring 129 8. F unctoriality in the Semialgebra 145 9. Closed Mo del Category Str uctures 169 10. A Construction of Semialgebras 182 11. Relativ e Nonhomogeneous K o szul Dualit y 191 App endix A. Con tramo dules o v er Coalgebras o v er Fields 222 App endix B. Comparison with Arkhip o v’s Ext ∞ / 2+ ∗ and Sevost y anov ’s T o r ∞ / 2+ ∗ 229 App endix C. Semialgebras Asso ciated to Harish-Chandra Pairs 238 App endix D. T a te Harish-Chandra P airs and T ate Lie Algebras 253 App endix E. Groups with Op en Proﬁnite Subgroups 290 App endix F. Algebraic G r o up oids with Closed Subgroup o ids 301 References 308 2 Intr oduction This mono graph grew out o f the aut ho r’s att empts to understand the deﬁnitions of semi-inﬁnite (co)homology of asso ciativ e algebras that had b een prop osed in the literature and particularly in t he w orks of S. Arkhip o v [1, 2] (see also [1 3, 43]). Roughly sp eaking, the semi-inﬁnite cohomology is deﬁned for a Lie or asso ciat ive algebra-lik e ob ject whic h is split in tw o halve s; the semi-inﬁnite cohomolo gy has the features of a homology theory (left deriv ed functor) alo ng o ne half of the v ariables and a cohomology theory (rig ht deriv ed functor) along the ot her half. In the Lie a lgebra case, the splitting in tw o halv es only has to b e c hosen up to a ﬁnite-dimensional space ; in particular, the homology of a ﬁnite-dimensional Lie algebra only diﬀers from its cohomology b y a shift of the homological degree and a t wist of the mo dule of co eﬃcien ts. So one can deﬁne the semi-inﬁnite homology of a T ate (lo cally linearly compact) Lie algebra [7] (see also [3]); it dep ends, to b e precise, on the c hoice of a compact o p en v ector subspace in t he Lie alg ebra, but when the subspace changes it undergo es o nly a dimensional shift and a determinantal t wist. Let us emphasize that what is often called the “semi-inﬁnite cohomology” of Lie algebras should b e though t of as their semi-inﬁnite hom olo gy , fro m our p o int o f view. What we call the semi-inﬁnite c ohom olo gy of T at e Lie algebras is a diﬀeren t and dual functor, deﬁned in this b o ok (see App endix D) . In the ass o ciative case, p eople us ually consid ered an algebra A with t w o subalgebras N and B suc h t ha t N ⊗ B ≃ A and there is a grading o n A for which N is p ositiv ely graded a nd lo cally ﬁnite-dimensional, while B is nonp ositive ly g raded. W e show that b oth the g r a ding and the second subalgebra B are redundant; all one needs is an asso ciativ e algebra R , a subalgebra K in R , and a coalgebra C dual to K . Certain ﬂatness/pro jectivit y a nd “in tegrability” conditions hav e to b e imp osed on t his data. If they are satisﬁed, the tensor pro duct S = C ⊗ K R has a semialgebr a structure and all the mac hinery describ ed b elo w can b e applied. F urthermore, we prop ose the following general setting for semi-inﬁnite (co)homol- ogy of asso ciativ e algebraic structures. Let C be a coalgebra o v er a ﬁeld k . Then C - C -bicomo dules form a tensor category with resp ect to the op erat ion of cotensor pro duct o v er C ; the categories of left and right C -como dules a r e mo dule categories o v er this tensor category . Let S b e a ring ob j ect in this tensor cat ego ry; we call suc h an ob ject a semialgebr a ov er C (due to it b eing “an algebra in half of the v ariables and a coalgebra in the other half ”). One can consider module ob jects o v er S in the mo dule categories of left and righ t C -como dules; these are called left and righ t S -semimo dules . The categories of left and right sem imo dules ar e only ab elian if S is an injective right and left C -como dule, resp ectiv ely; let us supp ose that it is. There is a na t ur a l op eration of se m itensor pr o duct of a right semimo dule and a left semimo dule o v er S ; it can b e though t of as a mixture of the cotensor pro duct in the 3 direction of C a nd the tensor pro duct in the direction of S relativ e to C . This functor is neither left, no r righ t exact. Its double-sided deriv ed functor SemiT or is suggested as the asso ciativ e version o f semi-inﬁnite homolo gy theory . Before describing the functor SemiHom (whose deriv ed functor SemiExt prov ides the asso ciativ e v ersion of semi-inﬁnite c ohom olo gy ), let us discuss a little bit of ab- stract nonsense. L et E b e an (asso ciativ e, but noncomm utativ e) tensor category , M b e a left mo dule categor y o v er it, N b e a right mo dule category , a nd K b e a category suc h that there is a pairing b et w een the mo dule categories M and N ov er E taking v al- ues in K . This means that there are m ultiplication functors E × E → E , E × M → M , N × E → N , and N × M → K and asso ciativity constrain ts f o r ternary m ultiplications E × E × E → E , E × E × M → M , N × E × E → N , and N × E × M → K satisfying the appropriate p en tagona l diagram equations. Let A b e a ring o b ject in E . The n one can consider t he category A E A of A - A -bimo dules in E , the categor y A M of left A -mo dules in M , and the cat ego ry N A of righ t A -mo dules in N . If the categories E , M , N , and K are ab elian, there a re functors of tensor pro duct ov er A , making A E A in to a tensor category , A M and N A in to left and right mo dule catego r ies o v er A E A , and providing a pairing N A × A M − → K . These new t ensor structures ar e asso ciativ e whenev er the origina l multiplication functors w ere right exact. Supp ose that w e w a n t to iterate this construction, considering a coring ob ject C in A E A , the categories of C - C -bicomo dules in A E A and C -como dules in A M and N A , etc. Since the functors of tensor pro duct o v er A are not left exact in general, the cotensor pro ducts ov er C will b e only asso ciativ e under certain (co)ﬂatness conditions. If o ne mak es t he next step and considers a ring ob ject S in the category of C - C -bicomo dules in A E A , one disco v ers that the functors of tensor pro ducts ov er S are only partially deﬁned. C onsidering pa r tially deﬁned tensor structures, one can indeed build this to w er of mo dule-como dule categories and tensor-cotensor pro ducts in them as high a s one wishes. In this b o ok, we restrict ourselv es to 3 -story tow ers of semialgebr as ov er c orings ov er (ordinary) rings, mainly b ecause w e don’t kno w how to deﬁne unbounded (co)deriv ed categories of (co) mo dules for an y higher lev els (see b elow). No w let us in tro duce c ontr amo dules . The functor ( V , W ) 7− → Hom k ( V , W ) mak es the categor y opp osite to the category of v ector spaces into a mo dule category ov er the tensor category of v ector spaces. A con tramo dule ov er an algebra R or a coalgebra C is an ob ject of the category opp o site to the category of mo dules or como dules in k – vect op o v er the ring ob jec t R or the coring ob jec t C in k – vect . One can easily see that an R -contramo dule is just a n R -mo dule, while the v ector space of k - linear maps from a C - como dule to a k -ve ctor space prov ides a t ypical example of C -contr a - mo dule. Setting E = M = k – v ect and N = K = k – vect op in the ab o v e construction, one obtains a r ig h t mo dule category C – contra op o v er the tensor category C – como d – C together with a pairing Cohom op C : C – como d × C – contra op − → k – vect op . G iv en a semialgebra S ov er C , o ne can apply the construction again and obtain the category of 4 S -semic ontr amo dules and the functor SemiHom op S : S – simo d × S – sicntr op − → k – vect op assigning a v ector space to an S -semimo dule and an S -semicontramo dule. Th ough como dules and contramodules are quite diﬀeren t, there is a strong dua lity-analogy b et w een them on the one hand, and an equiv alence of their appropriately deﬁned (exotic) un b ounded deriv ed categories on the other hand. Let us explain ho w w e deﬁne double-sided deriv ed functors. While the author kno ws of no natural wa y to deﬁne a deriv ed functor of one argumen t that would not b e either a left or a right deriv ed f unctor, suc h a deﬁnition of deriv ed functor of two a r guments do es exist in the balanced case. Namely , let Θ : H 1 × H 2 − → K b e a functor a nd S i ⊂ H i b e lo calizing classes of morphisms in categories H 1 and H 2 . W e w ould like to deﬁne a derive d functor D Θ : H 1 [ S − 1 1 ] × H 2 [ S − 1 2 ] − → K . Let F 1 b e the full sub category of “ ﬂat ob jects in H 1 relativ e to Θ” consisting of all ob jects F ∈ H 1 suc h that the morphism Θ( F , s ) is an isomorphism in K f or any morphism s ∈ S 2 . Let F 2 b e the full sub category in H 2 deﬁned in the analog ous wa y . Suppose tha t the natural functors F i [( S i ∩ F i ) − 1 ] − → H i [ S − 1 i ] are equiv alences o f categories. Then the restriction of the functor Θ to the subcategor y F 1 × H 2 of the Carthesian pro duct H 1 × H 2 factorizes through F 1 [( S 1 ∩ F 1 ) − 1 ] × H 2 [ S − 1 2 ] and therefore deﬁnes a f unctor on the category H 1 [ S − 1 1 ] × H 2 [ S − 1 2 ]. The same deriv ed functor can b e o bta ined by restricting the functor Θ to the sub category H 1 × F 2 of H 1 × H 2 . This construction can b e ev en extended to par t ia lly deﬁned functors of t w o argumen ts Θ (see 2.7). F or this deﬁnition of the do uble-sided derive d functor to w ork prop erly , the lo- calizing classes in the ho mo t o p y categories ha v e t o b e carefully c hosen (see 0.2.3). That is wh y our deriv ed functors SemiT or and SemiExt are not deﬁned on the con- v en tional deriv ed categories o f semimo dules and semicon tramo dules, but on their semiderive d c ate gories . The semideriv ed catego ry of S -semi(con t r a )mo dules is a mix- ture of the usual deriv ed catego r y in the mo dule direction (relative to C ) and the c o / c ontr aderive d category in the C -co/con tramo dule direction. The co deriv ed cate- gory o f C -como dules is equiv alen t to the ho motop y category of complexes of injec- tiv e C - como dules, and analogously , the con traderiv ed category of C -contra mo dules is equiv alen t to the homot o p y category of complexes of pro j ective C -contra mo dules. So the distinction b etw een the deriv ed and co/con traderiv ed categories is only relev an t for un b ounded complexes and only in the case o f inﬁnite ho mological dimension. A notable attempt to dev elop a general theory of semi-inﬁnite homological algebra w as undertak en b y A. V oronov in [46]. Let us p oin t out the diﬀerenc es b et w een our approac hes. First of all, V orono v only considers the semi-inﬁnite homology of Lie algebras, while w e work with asso ciative algebraic structures. Secondly , V orono v constructs a double-sided deriv ed functor of a functor of one a rgumen t a nd the c hoice of a class of resolutions b ecomes an additional ing redien t of his construction, while we deﬁne double-sided deriv ed functors of functors of t w o argumen ts and the conditio ns imp osed on resolutions are determined b y the functors themselv es. Thirdly , V oronov 5 w orks with graded L ie algebras and his functor of semiv a rian ts is obtained as the image of the inv a rian ts with resp ect to one half of the Lie algebra in the coin v ariants with resp ect to the other half, while w e consider ungra ded T ate Lie alg ebras with only one subalgebra c ho sen, and our functor of sem i i n variants is constructed in a more delicate w a y (see b elo w). Finally , no exotic deriv ed categories app ear in [46]. The co derive d category of C -como dules and the contraderiv ed category of C - con- tramo dules turn out to b e naturally equiv alen t. This equiv alence can b e thought of as a cov ar ia n t analogue of the con trav aria n t functor R Hom( − , R ) : D ( R – mo d ) − → D ( mo d – R ) on the deriv ed category of mo dules ov er a ring R . Moreo v er, there is a na t ur a l equiv alence b etw een the semideriv ed categories of S -semimo dules a nd S -semicontramo dules. The functors R Ψ S : D si ( S – simo d ) − → D si ( S – sicntr ) and L Φ S : D si ( S – sicntr ) − → D si ( S – simo d ) pro viding this equiv alence ar e deﬁned in terms of the spaces of homorphisms in the catego ry of S -semimo dules and the op eration of c ontr a tens o r pr o d uct of an S -semimo dule and an S -semicontramodule. The latter is a righ t exact functor whic h r esem bles the functor of tensor pro duct of mo dules o v er a ring. This equiv alence of triangulated categories tranforms the functor SemiExt S in to the functors Ext in either of the semideriv ed categories (and the functor SemiT or S in to the left deriv ed f unctor CtrT or S of t he functor of contratens or pro duct). W e call this kind of equiv a lence of triangulated categories the c omo dule-c ontr am o dule c orr e sp onden c e or the sem imo dule-sem ic ontr amo dule c orr esp ond e n c e . The dualit y-analogy b etw een semimo dules and semicon tramo dules partly breaks do wn when one passes fro m homological algebra to t he structure theory . Como dules o v er a coa lg ebra ov er a ﬁeld are simplistic creatures; con tramo dules are quite a bit more complicated, though still m uc h simpler than mo dules ov er a r ing , the structure theory of a coalgebra o v er a ﬁeld b eing m uc h simpler than that of an algebra or a ring. W e construct some relev a nt coun terexamples. There is an analog ue of Nak ay ama’s Lemma for con tramo dules, a description of con tramo dules ov er an inﬁnite dir ect sum of coalgebras, etc. These results can b e extended to contramo dules ov er certain top o- logical rings (m uc h mo r e general than the top ological algebras dual to coalg ebras). Con tramo dules o v er to p ological Lie alg ebras can also b e deﬁned; and a n isomorphism of the categories o f con tramo dules ov er a top olo g ical Lie a lgebra a nd its top ological en v eloping algebra can b e pro v en under certain assumptions. A c oring C ov er a ring A is a coring ob ject in the tensor category of bimo dules o v er A . (In a diﬀeren t terminology , this is called a c o algebr oid .) A semialgebr a S o v er a coring C is a ring ob ject in the tensor category of bicomo dules ov er C ; for this deﬁnition to make sense, certain (co)ﬂatness conditions hav e to b e imp osed on C and S to mak e the cotensor pro duct of bicomo dules w ell-deﬁned and asso ciativ e. Throughout this monograph (with the exception of Section 0 and the app endices) w e w ork with corings C ov er noncommu tative rings A and semialgebras S o v er C . Mostly w e hav e to assume that the ring A ha s a ﬁnite homolog ical dimension—for 6 a num b er of reasons, the most imp ortan t one b eing that otherwise w e don’t know ho w to deﬁne appropriately the un b ounded (co)deriv ed category of C -como dules. No assumptions ab out the homological dimension of the coring and the semialgebra a r e made. Besides, w e mostly hav e to supp ose t ha t C is a ﬂat left and righ t A -mo dule and S is a coﬂat left and right C -como dule, and ev en certain (co)pro jectivity conditions ha v e to b e imp osed in order to work with con tramo dules. Nonhomogeneous quadratic dualit y [39, 40] establishes a correspo ndence b etw een nonhomogeneous Koszul algebras and Koszul CDG-algebras. This dua lity has a rel- ativ e v ersion with a base ring , a ssigning, e. g., the de R ham DG-alg ebra to the ﬁltered algebra of diﬀeren tial o p erators (the ba se ring b eing the ring of functions, in this case). F or a n um b er of reasons, it is a dvisable to av oid passing to the dual v ector space/mo dule in this construction, workin g with CDG- coalgebras instead o f CDG-algebras; in particular, this allows to include inﬁnitely-(co)generated Koszul al- gebras and coalgebras. In the relativ e case, this means considering the graded coring of p olyv ector ﬁelds, rather tha n the graded a lgebra of diﬀeren tial forms, as t he dual ob ject to t he diﬀerential op erators. The relev ant additional structure on the p olyve c- tor ﬁelds (corresp onding to the de Rham diﬀeren tial on the diﬀeren t ial forms) is that of a quasi- d i ﬀer ential c oring . Another imp ortant v ersion of relativ e no nhomogeneous quadratic duality uses base coalgebras in place of base rings. This situation is sim- pler in some r esp ects, since one still obtains CDG-coa lg ebras as the dual ob jects. As a generalization o f these tw o dualities, one can consider nonhomogeneous Koszul semialgebras ov er corings and a ssign Koszul quasi-diﬀeren tial corings ov er corings to them. The P oincare-Birkhoﬀ-Witt theorem for Koszul semialgebras claims that this corresp ondence is a n equiv alence of categories. Relativ e nonhomog eneous K o szul duality theorem provide s an equiv alence b etw een the semideriv ed category of semimo dules o v er a nonhomogeneous Koszul semialge- bra and the co derive d category of quasi-diﬀeren tial como dules ov er the corresp ond- ing quasi-diﬀeren tial coring, and an ana lo gous equiv alence b et w een the semideriv ed category of semicon tramo dules and the con t r aderiv ed catego ry of quasi-diﬀeren tial con tramo dules. In particular, for a smo oth algebraic v ariet y M and a v ector bun- dle E ov er M with a glo bal connection ∇ , there is an equiv alence b et w een the de- riv ed category of mo dules o v er t he algebra/sheaf of diﬀeren tial op erato rs acting in the sections of E and the co deriv ed category (and also the con traderiv ed category , when M is aﬃne) of CDG-mo dules ov er the CDG -algebra Ω( M , End ( E )) o f diﬀeren- tial forms with co eﬃcien ts in the v ector bundle End( E ) , where the diﬀeren tial d in Ω( M , End( E )) is the de Rha m diﬀeren tial dep ending on ∇ a nd the curv ature elemen t h ∈ Ω 2 ( M , End( E )) is the curv ature of ∇ . Natural examples of semialgebras and semimo dules come fro m Lie theory . Namely , let ( g , H ) b e an alg ebraic Harish-Chandra pair, i. e., g is a Lie algebra o v er a ﬁeld k and H is a smo oth a ﬃne a lgebraic group corresp onding to a ﬁnite-dimensional Lie 7 subalgebra h ⊂ g . Let C ( H ) b e the coalgebra of functions on H . Then the category O ( g , H ) of Harish-Chandra mo dules is isomorphic to t he category of left semimo dules o v er the semialgebra S ( g , H ) = U ( g ) ⊗ U ( h ) C ( H ). If the group H is unimo dular, the semialgebra S = S ( g , H ) has an inv olutive an ti-auto morphism. In general, the o pp o- site semialgebras S and S op are Morita-equiv alent in some sense; mor e precisely , there is a canonical left S ⊗ k S -semimo dule E = E ( g , H ) suc h that the semitensor pro duct with E provides an equiv alence b etw een the categories o f rig h t and left S -semimo d- ules. Geometrically , E ( g , H ) is the bimo dule of distributions on an alg ebraic group G supp orted in its subgroup H and regular alo ng H . So the semitensor pro duct of S -semimodules can b e considered as a functor on the category O ( g , H ) × O ( g , H ). This functor factorizes through the functor of tensor pro duct in the catego r y O ( g , H ) and is closely related to the functor of ( g , H ) -semiinvariants M 7− → M g ,H on the category of ( g , H )-mo dules. The semiin v ariants are a mixture of inv arian ts ov er H and coinv arian ts along g / h . More generally , let ( g , H ) b e a T ate Harish-Chandr a p air , that is g is a T ate Lie algebra and H is an aﬃne proalgebraic group cor r esp o nding to a compact op en subalgebra h ⊂ g . Let κ : ( g ′ , H ) − → ( g , H ) b e a morphism o f T ate Harish-Chandra pairs with the same proalgebraic group H suc h that the Lie algebra map g ′ − → g is a cen tral extension whose k ernel is iden tiﬁed with k ; assume also that H acts trivially in k ⊂ g ′ . One example of suc h a cen tral extension of T ate Harish-Chandra pa irs comes from the canonical cen tral extension g ∼ of g ; w e denote the corresp onding morphism b y κ 0 . There is a semialgebra S κ ( g , H ) = U κ ( g ) ⊗ U ( h ) C ( H ) o v er the coalgebra C ( H ) suc h that the category of left semimo dules ov er S κ = S κ ( g , H ) is isomorphic to the category o f discrete ( g ′ , H )-mo dules where the unit central elemen t of g ′ acts b y the identit y (Harish-Chandra mo dules with the cen tral charge κ ). Left semicon tra mo dules o v er the opp osite semialgebra S op κ can b e desc rib ed in terms of compatible structures of g ′ -contramo dules and C ( H )-contramo dules. These are called Harish-Chandr a c ontr amo dules with the cen tral ch arge κ ; the dual v ector spaces to Harish-Chandra mo dules with the cen tral c harge − κ can b e f o und among them. The semialgebras S κ and S op − κ 0 − κ are naturally isomorphic, at least, when the pairing U ( h ) ⊗ k C ( H ) − → k is nondegenerate in C ( H ). In view of the semimo dule- semicon tra mo dule corresp ondence theorem, it follo ws tha t the semideriv ed categories of Harish-Chandra mo dules with the cen tr a l charge κ and Harish-Chandra con tra- mo dules with the cen tral c harg e κ + κ 0 o v er ( g , H ) are natura lly equiv alen t. So the w ell-kno wn phenomenon of corresp ondence b etw een complexes of mo dules with complemen tar y cen tral c harges o v er certain inﬁnite-dimensional Lie algebras can b e form ulated as an equiv alence of triangulated categories using the notions of contra- mo dules and semideriv ed catego ries. Besides, it follow s that the category of righ t semimo dules o v er S κ is isomorphic to the category of Harish-Chandra mo dules with the cen tral c harg e − κ − κ 0 . When the proalgebraic group H is pr o unip oten t (and 8 h is exactly the Lie algebra of H ), the o b ject SemiT or S κ ( N • , M • ) of the derive d cat- egory of k -v ector spaces is represen ted b y the complex of semi-inﬁnite forms ov er g with co eﬃcien ts in N • ⊗ k M • . This provides a comparison of our theory of SemiT or with the semi-inﬁnite homology of T at e Lie algebras. Semi-inﬁnite cohomolo gy o f Lie algebras, whose co eﬃcien ts are contramo dules ov er (the canonical cen tr al extensions of ) T ate Lie algebras, is related to SemiExt in the analogous w a y . T o a top o logical group G with an op en proﬁnite subgroup H and a comm utativ e ring k one can asso ciate a semialgebra S k ( G, H ) o v er the coring C k ( H ) of k -v alued lo cally constant functions on H suc h that the categories o f left and rig h t semimo dules o v er S k ( G, H ) are isomorphic to the category of smo oth G -mo dules ov er k . So the category of semimo dules ov er S k ( G, H ) do es not dep end on H , neither do es t he category of semicon tramo dules ov er S k ( G, H ); all the semialgebras S k ( G, H ) with a ﬁxed G and v arying H are naturally Morita equiv alen t. The semideriv ed categories of semimo dules and semicon tramo dules ov er S k ( G, H ) do depend on H essen tially , ho w ev er, as do the functors SemiT or a nd SemiExt o v er S k ( G, H ). These double-sided deriv ed functors ma y b e called t he semi-inﬁnite (co)homology o f a gro up with an op en proﬁnite subgroup. The semi-inﬁnite homolo g y of to p ological groups is a mixture of the discrete group homology a nd the proﬁnite group cohomology . Examples of corings C o v er comm utativ e rings A for whic h the left and the rig h t actions of A in C are diﬀeren t come from the alg ebraic group o ids theory , a nd examples of semialgebras ov er such corings come from Lie theory of gro up oids. Namely , let ( M , H ) b e a smo oth aﬃne group oid, i. e., M a nd H are smooth aﬃne algebraic v arieties, there are t w o smo oth morphisms s H , t H : H ⇒ M of source and target, and there are unit, m ultiplication, and inv erse elemen t morphisms satisfying the usual group oid axioms. Let A = A ( M ) be the ring of functions on M and C = C ( H ) b e the ring of functions on H . Then C is a coring ov er A . Moreo v er, supp ose that ( M , H ) is a closed subgroup oid o f a group oid ( M , G ). Let g and h b e the Lie algebroids ov er the ring A corr esp o nding to the group oids ( M , G ) and ( M , H ), and let U A ( g ) and U A ( h ) b e their env eloping alg ebras. Then there is a semialgebra S = S M ( G, H ) = U A ( g ) ⊗ U A ( h ) C ( H ) o v er the coring C and a canonical left S ⊗ k S -semi- mo dule E = E M ( G, H ) pro viding an equiv alence b et w een the categories of righ t and left S -semimo dules. The semimo dule E consists of a ll distributions on G t wisted with the line bundle s ∗ G (Ω − 1 M ) ⊗ t ∗ G (Ω − 1 M ), supp orted in H a nd regular alo ng H (where Ω M denotes the bundle of top f orms on M ). Examples of c orings o ve r noncomm uta t ive rings come from Noncomm uta tiv e Geometry [33]. Noncomm utativ e stac ks a r e represen ted as quotients of noncom- m utativ e aﬃne sc hemes corr esp o nding to rings A by actions of corings C ov er A . The cotensor pro duct of C -como dules can b e understo o d a s the g r oup of globa l sections of the tensor pro duct of a right and a left sheaf o v er a noncommutativ e stac k, while the tensor pro duct o f shea v es itself do es no t exist. 9 Notice that the roles of the ring and coring structures in our constructions are not symmetric; in particular, we hav e to consider conv en tional deriv ed cat ego ries along the algebra v ariables and co/contraderiv ed categories along the coalgebra v ariables. The cause o f this diﬀerence is that the tensor pro duct of mo dules comm utes with the inﬁnite direct sums, but not with t he inﬁnite pro ducts. This can b e c hanged b y pass- ing to pro-o b jects; consequen tly o ne can still deﬁne v ersions of deriv ed functors Cotor and Co ext o v er a coring C without making an y homological dimension assumptions at a ll b y considering pro - and ind-mo dules (see Remarks 2.7 and 4 .7 ). A problem remains to construct the como dule-contramo dule corresp ondence without any ho mo- logical dimension assumptions on the ring A . Here w e only manage to weak en the ﬁnite homological dimension assumption to the Gorensteinness a ssumption. Algebras/coalgebras o v er ﬁelds and semialgebras o v er coalgebras o v er ﬁelds are brieﬂy discussed in Section 0. Semialgebras o v er corings and the functors of semi- tensor pro duct ov er them are in tro duced in Section 1, and imp ortant constructions of ﬂat como dules and coﬂat semimo dules are presen ted there. The deriv ed functor SemiT or is deﬁned in Section 2. Contramo dules ov er corings and semicon tramo d- ules ov er semialgebras are in tro duced in Section 3, and t he deriv ed functor SemiExt is deﬁned in Section 4. Equiv alence of exotic deriv ed categories of como dules and con tramo dules is prov en in Section 5; and the same for semimo dules and semicon tra- mo dules is done in Section 6. F unctors of change of r ing and coring for the categor ies of como dules and con tramo dules are introduced in Section 7; functors of c hange of coring and se mialgebra for the categories of semimo dules a nd semicon tra mo dules are constructed in Sec tion 8. Closed mo del category structures on the categories of complexes of semimo dules and semicon tramo dules are deﬁned in Section 9. The construction of a semialgebra dep ending on t hr ee em b edded rings and a coring dual to the middle ring is considered in Section 10. The Poincare–Birkhoﬀ–Witt theorem and the Ko szul dualit y theorem for nonhomogeneous Koszul semialgebras are pro v en in Section 11. The basic structure theory o f con tramo dules ov er a coalgebra o v er a ﬁeld is dev elop ed in App endix A. W e compare our theory of SemiExt and SemiT or with Arkhip ov’s and Sev osty anov ’s semi-inﬁnite Ext and T or in App endix B. Semial- gebras corresp onding to Har ish-Chandra pairs and their Hopf alg ebra ana lo gues are discusse d in App endix C. T ate Harish-Chandra pairs are considered in App endix D, and the t heorem o f comparison with semi-inﬁnite cohomology of T ate Lie algebras is prov en there. Semialgebras corresp o nding to top o lo gical groups are discussed in App endix E. Pairs of algebraic g roup oids are considered in App endix F. App endix C w as written in collab oratio n with Dmitriy Rum ynin. App endix D w as written in collab ora tion with Sergey Arkhip ov. One terminological note: w e will generally use the w ords the homotopy c ate g ory of (an additiv e category A ) a nd the homotopy c ate gory of c omplex e s of ( ob jects from A ) 10 as synon ymous. Analogously , the homotopy c ate gory of c omplexes (with a particular prop erty ) over A is a full sub category of the homotopy category of A . Ac kno wledgemen ts. I am grateful to B. F eigin fo r p osing the problem of deﬁning the semi-inﬁnite cohomology of asso ciativ e algebras back in the ﬁrst half of 199 0 ’s. Ev en earlier, I learned ab out the problem of constructing a deriv ed equiv alence b e- t w een mo dules with complemen tary cen tral c harges from M. Ka prano v’s handwrit- ten notes on Ko szul duality . S. Arkhip ov patiently explained me his ideas ab out the semi-inﬁnite cohomolo g y many times ov er the y ears, contributing to my eﬀorts to understand the sub ject. In the Summer of 200 0 , this work was stim ulated by dis- cussions with S. Arkhip ov and R. Bezruk avnik ov , and m y gra titude go es to b o th of them. I wish to tha nk J. Bernstein, B. F eigin, B. Keller, V. Lunts, and V. V ologo d- sky for helpful con v ersations, and I. Mirk o vic for stim ulating interest. P arts of the mathematical conten t of this monograph w ere w orke d out when the a uthor w as vis- iting Sto ck holm Univ ersit y , W eizmann Institute, Indep enden t Univ ersit y of Mosco w, Max-Planc k-Institut in Bonn, and W arwick Univ ersity; I was supp o r t ed b y the Eu- rop ean P ost-Do ctoral Institute during a par t of that time. The author was partially supp orted by grants from CRDF, INT AS, and P . Deligne’s 2004 Balzan prize while writing the manu script up. 11 0. Preliminaries and Summar y This section contains some kno wn results and some results deemed to b e new, but no pro ofs. Its goal is t o prepare the reader for the more tec hnically in v olv ed constructions o f the main b o dy of the monograph (where the pro of s are give n). In particular, w e don’t ha v e to w orry ab out nonasso ciativit y of the cotensor pro duct and pa rtial deﬁnition of the semitensor pro duct here, distiguish b et w een the myriad of no t io ns o f absolute/relative coﬂatness/copro jectivit y/injectivit y of como dules and analogously for con tramo dules, etc., b ecause w e only consider coalgebras ov er ﬁelds. 0.1. Unbounded T or and Ext . Let R b e an alg ebra ov er a ﬁeld k . 0.1.1. W e would like to extend the familiar deﬁnition of the deriv ed functor o f tensor pro duct T o r R : D − ( mo d – R ) × D − ( R – mo d ) − → D − ( k – vect ) on the Carthesian pro duct of the deriv ed categories of right and left R - mo dules b ounded from ab ov e, so as to obtain a functor on the Cart hesian pro duct of un b ounded deriv ed catego ries. As alwa ys, the tensor pro duct of a complex of r ig h t R -mo dules N • and a complex of left R -mo dules M • is deﬁned as the tota l complex of the bicomplex N i ⊗ R M j , constructed by taking inﬁnite direct sums along the diagonals. This pro vides a functor Hot ( mo d – R ) × Hot ( R – mo d ) − → Hot ( k – vect ) on the Carthesian pro duct of un b ounded homotop y categories of R -mo dules. The mo st straightforw a r d wa y to deﬁne the ob ject T or R ( N • , M • ) of D ( k – vect ) is to represen t it by the total complex of the bicomplex · · · − − → N • ⊗ k R ⊗ k R ⊗ k M • − − → N • ⊗ k R ⊗ k M • − − → N • ⊗ k M • , constructed by taking inﬁnite direct sums along the diago na ls. One can c hec k that this bar construction indeed deﬁnes a functor T or R : D ( mo d – R ) × D ( R – mo d ) − − → D ( k – vect ) . The un b ounded deriv ed functor T or R can b e also deﬁned by restricting the functor of tensor pr o duct to appropriate sub categories of complexes adjusted to the functor of tensor pro duct in the un b ounded ho mo t o p y categor ies of R -mo dules. Namely , let us call a complex of left R -mo dules M • ﬂat if the complex of k - v ector spaces M • ⊗ R N • is acyclic whenev er a complex of righ t R -mo dules N • is acyclic. Not every c omplex of ﬂat R -mo dules is a ﬂat c ompl e x of R -mo d ules ac c or ding to this deﬁnition. In particular, an acyclic complex of left R -mo dules is ﬂat if and o nly if it is pur e , i. e., it remains acyclic after taking the tensor pro duct with a ny righ t R - mo dule. So an acyclic complex of ﬂat R -mo dules is ﬂa t if and only if a ll of its mo dules of co cycles are ﬂat. On the other hand, any complex of ﬂat R -mo dules b ounded from ab o v e is ﬂat. If the ring R has a ﬁnite w eak homolog ical dimension, then an y complex of ﬂat R -mo dules is ﬂat . F or example, t he acyclic complex M • of free mo dules ov er the ring of dual n um b ers R = k [ ε ] / ε 2 whose ev ery t erm is equal to R 12 and eve ry diﬀeren tial is the op erator of m ultiplication with ε is not ﬂat. Indeed, let N • = ( · · · → k [ ε /ε 2 ] → k → 0 → · · · ) b e a free resolution of the R - mo dule k ; then the complex N • ⊗ R M • is quasi-isomorphic to k ⊗ R M • and ha s a one-dimensional cohomology space in ev ery degree, ev en though the complex N • is acyclic. An y complex o f R - mo dules is quasi-isomorphic to a ﬂat complex, and moreov er, the quotien t category of the homoto p y category Hot ﬂ ( R – mo d ) of ﬂat complexes of R -mo dules b y the thic k sub category of acyclic ﬂat comple xes Acycl ( R – mo d ) ∩ Hot ﬂ ( R – mo d ) is equiv alent to the deriv ed category D ( R – mo d ). This result holds for an arbitrary ring [44], a nd ev en for an arbitrary DG -ring [3 1, 12]. The derive d functor T or R can b e deﬁned b y restricting the functor of t ensor pro duct o v er R to either of the full sub categories Hot ( mo d – R ) × Hot ﬂ ( R – mo d ) or Hot ﬂ ( mo d – R ) × Hot ( R – mo d ) of the category Hot ( mo d – R ) × Hot ( R – mo d ). 0.1.2. The functor Ho m R : Hot ( R – mo d ) op × Hot ( R – mo d ) − → Hot ( k – vect ) and its deriv ed f unctor Ext R : D ( R – mo d ) op × D ( R – mo d ) − → D ( k – v ect ) need no sp ecial deﬁ- nition: once the unbounded homoto p y and deriv ed categories are deﬁned, so are the spaces of homomorphisms in them. F or an y (un b ounded) complexes of left R -mo dules L • and M • , the total complex o f the cobar bicomplex Hom k ( L • , M • ) − − → Hom k ( R ⊗ k L • , M • ) − − → Hom k ( R ⊗ k R ⊗ k L • , M • ) − − → · · · , constructed b y taking inﬁnite direct pro ducts along the diagonals, represen ts the ob ject Ext R ( L • , M • ) in D ( k – v ect ). The unbounded derive d functor Ext R can b e also computed b y restricting the functor Hom R to appropriate sub categories in the Carthesian pro duct of homotop y categories of R -mo dules. L et us call a complex of left R -mo dules L • pr oje ctive if the complex Hom R ( L • , M • ) is acyclic f or any acyclic complex of left R -mo dules M • . Analogously , a complex of left R - mo dules M • is called inje ctive if the com- plex Hom R ( L • , M • ) is acyclic for any acyclic complex of left R -mo dules L • . An y pro jectiv e complex of R -mo dules is ﬂat. Any complex of pro jec tiv e R -mo dules b ounded from ab ov e is pro jectiv e, and an y complex o f injectiv e R -mo dules b ounded from b elow is injectiv e. If the ring R has a ﬁnite left homological dimension, then an y complex of pro jectiv e left R -mo dules is pro jectiv e and an y complex of injectiv e left R -mo dules is injectiv e. A complex of R - mo dules is pro jectiv e if and only if it b elongs to the minimal trian- gulated sub category of the homotop y category of R -mo dules containing the complex · · · → 0 → R → 0 → · · · a nd closed under inﬁnite direct sum s. Analog o usly , a complex of R -mo dules is injective if and only if up to the ho motop y equiv a lence it can b e obtained from the complex · · · → 0 → Hom k ( R, k ) → 0 → · · · using the op erations of shift, cone, and inﬁnite direct pro duct. The homot op y category Hot p roj ( R – mo d ) of pro jective complexes of R -mo dules and the homotop y category Hot inj ( R – mo d ) of injective complexes of R - mo dules are equiv alen t to the un b ounded 13 deriv ed category D ( R – mo d ). The results mentioned in this paragraph ev en hold fo r an arbitrary DG- ring [31, 12]. The functor Ext R can b e obtained by restricting the functor Hom R to either of the f ull sub categories Hot p roj ( R – mo d ) op × Hot ( R – mo d ) or Hot ( R – mo d ) op × Hot inj ( R – mo d ) of the category Hot ( R – mo d ) op × Hot ( R – mo d ). 0.1.3. The deﬁnitions of unbounded T or and Ext in terms of (co)bar constructions w ere kno wn at least since the 1960’s. The notions of ﬂat, pro jective , and injec- tiv e (un b ounded) complexes of R -mo dules were intro duced b y N. Spaltenstein [4 4] (who attributes the idea to J. Bernstein). Suc h complexes we re called “ K -ﬂat”, “ K -pro jectiv e”, and “ K -injectiv e” in [44]; they a r e often called “ H -pro jectiv e” or “homotop y pro jec tiv e” etc. no w ada ys. 0.2. Coalgebras o v er ﬁelds; Cotor and Co ext. The notion of a coalgebra o v er a ﬁeld is obtained from that o f an algebra by for mal dualization. Since any coasso- ciativ e coalgebra is t he union of its ﬁnite-dimensional sub coalgebras, the category o f coalgebras is anti-equiv alent to the category of proﬁnite-dimensional a lgebras. There are tw o w ay s of dualizing the notio n of a mo dule o v er an alg ebra: one can con- sider c omo dules and c ontr amo dules ov er a coalgebra. Como dules can b e tho ugh t of as discrete mo dules whic h a re unions of their ﬁnite-dimensional submo dules, while con tramo dules are mo dules where certain inﬁnite summation op erations are deﬁned. Dualizing the constructions of the tensor pro duct of mo dules and the space of ho- momorphisms b et w een mo dules, one obtains the functors of cotensor pro duct and cohomomorphisms. Their derive d functors are called Cotor and Co ext . 0.2.1. A coasso ciativ e c o a lgebr a with counit o v er a ﬁeld k is a k -v ector space C en- do w ed with a c om ultiplic ation map µ C : C − → C ⊗ k C and a c ounit map ε C : C − → k satisfying the equations dual to the asso ciativit y a nd unity equations on the multipli- cation and unit maps of an assotiativ e algebra with unit. More precisely , one should ha v e ( µ C ⊗ id C ) ◦ µ C = (id C ⊗ µ C ) ◦ µ C and ( ε C ⊗ id C ) ◦ µ C = id C = (id C ⊗ ε C ) ◦ µ C . A left c omo dule M ov er a coalg ebra C is a k -vec tor space endo w ed with a left c o action map ν M : M − → C ⊗ k M satisfying the equations dual to the asso ciativity and unit y equations on the action map of a mo dule ov er an asso ciativ e algebra with unit. More precisely , one should hav e ( µ C ⊗ id M ) ◦ ν M = (id C ⊗ ν M ) ◦ ν M and ( ε C ⊗ id M ) ◦ ν M = id M . A right c omo dule N ov er a coalg ebra C is a k -v ector space endo w ed with a right c o action map ν N : N − → N ⊗ k C satisfying the coasso ciativity and counit y equations ( ν N ⊗ id C ) ◦ ν N = (id N ⊗ µ C ) ◦ ν N and (id N ⊗ ε C ) ◦ ν N = id N . F or example, t he coalgebra C ha s natural structures of a left a nd a r ig h t como dule ov er itself. The categories o f left and right C - como dules are ab elian. W e will denote them b y C – como d and como d – C , resp ectiv ely . Both inﬁnite direct sums and inﬁnite pro ducts exist in the categor y of C -como dules, but only inﬁnite direct sums are preserv ed b y the f orgetful functor C – como d − → k – vect ( while the inﬁnite products ar e not 14 ev en exact in C – como d ) . A c ofr e e C -como dule is a C -como dule of the form C ⊗ k V , where V is a k -v ector space. The space of como dule homomorphisms into the cofree C -como dule is describ ed b y the form ula Hom C ( M , C ⊗ k V ) ≃ Hom k ( M , V ). The category of C -como dules ha s enough injectiv es; b esides, a left C -como dule is injective if and only if it is a direct summand of a cofree C -como dule. The c otensor pr o duct N  C M of a right C - como dule N and a left C -como dule M is deﬁned as the ke rnel of t he pair of maps ( ν N ⊗ id M , id N ⊗ ν M ) : N ⊗ k M ⇒ N ⊗ k C ⊗ k M . This is the dual construc tion to the tensor pro duct of a righ t mo dule and a left mo dule ov er an asso ciative algebra. There are natur a l isomorphisms N  C C ≃ N and C  C M ≃ M . The functor of cotensor pro duct o v er C is left exact. 0.2.2. The cotensor pro duct N •  C M • of a complex of right C - como dules N • and a complex of left C -como dules M • is deﬁned as the total complex of the bicomplex N i  C M j , constructed by taking inﬁnite direct sums along the diagonals. W e would lik e to deﬁne the deriv ed functor Cotor C of the functor of cotensor pro d- uct in suc h a w ay that it could b e obtained b y restricting the functor  C to appropria te sub categories of the Carthesian pro duct of homotopy categories Hot ( como d – C ) and Hot ( C – como d ). In addition, w e w ould like the o b ject Cotor C ( N • , M • ) of D ( k – vect ) to b e represen ted b y t he total complex of the cobar bicomplex (1) N • ⊗ k M • − − → N • ⊗ k C ⊗ k M • − − → N • ⊗ k C ⊗ k C ⊗ k M • − − → · · · , constructed b y ta king inﬁnite direct sums along the diagonals. It turns out that a functor Cotor C with these prop erties do es exist, but it is not deﬁne d o n the Carthesian pr o duct of c onventiona l unb ounde d derive d c ate gories D ( como d – C ) and D ( C – como d ). F or example, let C b e the coalgebra dual to the algebra of dual n um b ers C ∗ = k [ ε ] /ε 2 , so that C -como dules are just k [ ε ] /ε 2 -mo dules. L et M • b e the acyclic complex of cofree C -como dules whose ev ery term is equal to C and ev ery diﬀeren t ial is t he op erator of m ultiplication with ε , and let N • b e the complex of C -como dules whose only nonzero term is the C -como dule k . Then the cobar complex that w e w an t to compute Cotor C ( N • , M • ) is quasi-isomorphic to the complex N •  C M • and has a one-dimensional cohomology space in ev ery degree, ev en thoug h M • represen ts a zero ob ject in D ( C – como d ). Therefore, a more r eﬁned v ersion o f un b ounded derive d category of C -como dules has to b e considered. A complex o f left C - como dules is called c o acyclic if it b elongs to the minimal triangulated sub cat ego ry of the homotopy category Hot ( C – como d ) con taining the total complexes of exact triples ′ K • → K • → ′′ K • of complexes of left C -como dules and closed under inﬁnite direct sums. (By the total complex of an exact triple o f complexes w e mean the tota l complex of the corresp onding bicomplex with three ro ws.) An y coacyclic complex is acyclic; an y acyclic complex b ounded from b elow 15 is coacyclic. The complex M • from the ab ov e example is acyclic, but not coacyclic. (Indeed, the total complex of t he cobar bicomplex (1) is acyc lic whenev er M • is coacyclic.) The c o derive d c a te gory of left C -como dules D co ( C – como d ) is deﬁned as the quotien t categor y of the homoto p y category Hot ( C – como d ) b y the thick sub category of coacyclic complexes Acycl co ( C – como d ). In the same w a y one can deﬁne the co deriv ed category of any ab elian category with exact functors o f inﬁnite direct sum. Over a category of ﬁnite homological dimension, ev ery acyclic complex b elongs to the minimal tr ia ngulated sub category of the homoto p y category con taining the tot a l complexes of exact triples of complexes, ev en without the inﬁnite direct sum closure. The cotensor pro duct N •  C M • of a complex of righ t C -como dules M • and a com- plex of left C -como dules N • is acyclic whenev er one of the complexes M • and N • is coacyclic and the other one is a complex of injectiv e C -como dules. Besides , the co de- riv ed category D co ( C – como d ) is equiv alen t to the homotopy category Hot ( C – como d inj ) of injectiv e C -como dules. Th us one can deﬁne t he un b ounded deriv ed functor Cotor C : D co ( como d – C ) × D co ( C – como d ) − − → D ( k – vect ) b y restricting t he functor of cotensor pro duct to either of the full sub categories Hot ( como d – C ) × Hot ( C – como d inj ) or Hot ( como d inj – C ) × Hot ( C – como d ) of the cate- gory Hot ( como d – C ) × Hot ( C – como d ). 0.2.3. If one attempts to construct a derived functor of cotensor pro duct on the Carthesian pro duct of con v entional un b ounded derived categories of como dules in the w a y ana logous to 0.1.1, the result may not lo ok lik e what one exp ects. Consider the example of a ﬁnite-dimensional coalgebra C dual to a F rob enius alge- bra C ∗ = F . Let us assume the con v ention that left C - como dules are left F -mo dules and r ig h t C -como dules a re righ t F -mo dules. Then the functor  C is left exact and the functor ⊗ F is right exact, but the diﬀerence b et w een them is still rather small: if either a left (co)mo dule M , or a right (co)mo dule N is pro jectiv e-injectiv e, then there is a natural isomorphism N  C M ≃ ( N  C F ) ⊗ F M , a nd after o ne c ho oses a n isomorphism b etw een the left mo dules F and C , the right mo dules N and N  C F will only diﬀer b y the F rob enius a utomorphism of the F r ob enius algebra F . So if one deﬁnes “coﬂat ” complexes of C -como dules as the complexes whose coten- sor pro duct with acyclic complexes is acyclic, then the quotien t category of the ho- motop y category of “coﬂat ” complexes b y the thic k sub category of acyclic “coﬂat” complexes will b e indeed equiv alen t to the derive d category of como dules, and one will b e able to deﬁne a “deriv ed functor of cotensor pro duct ov er C ” in this w a y , but the resulting deriv ed functor will coincide, up to the F rob enius t wist, with the functor T or F . (Indeed, an y ﬂat complex of ﬂat mo dules will b e “coﬂat”.) When the complexes this functor is a pplied t o are concen trat ed in degree 0, this functor will 16 pro duce a complex situated in the negative cohomological degrees, a s is c haracteristic of T or F , and not in the p ositiv e ones, as one would exp ect of Cotor C . Lik ewise, if one attempts to construct a deriv ed f unctor of tensor pro duct on the Carthesian pro duct of co derive d catego r ies of mo dules in the wa y analog o us to 0.2.2, one will ﬁnd, in the F r o b enius algebra case, that the tensor pro duct o f a complex of pro j ective F -mo dules with a coacyclic complex is acyclic, the homotop y catego r y of complexes o f pro jectiv e modules is indeed equiv alen t to the co derive d categor y of F -mo dules, and one can deﬁne a “derive d functor of tensor pro duct ov er F ” by restricting to this sub category , but the resulting deriv ed functor will coincide, up to the F rob enius t wist, with the functor Cotor C . Nev ertheless, it is w ell kno wn how to deﬁne a deriv ed functor of cotensor pro d- uct on the con v en tional un b ounded deriv ed categor ies of como dules (see 0.2.10, cf. Remark 2.7). 0.2.4. The category k – vect op opp osite to the category of ve ctor spaces ha s a natural structure of a mo dule c ate gory o v er the tensor category k – vect with the a ction f unc- tor k – vect × k – vect op − → k – vect op deﬁned b y the rule ( V , W op ) 7− → Hom k ( V , W ) op . More precisely , there ar e t w o mo dule catego ry structures asso ciated with this func- tor: the left mo dule category with the asso ciativity constrain t Hom k ( U ⊗ k V , W ) ≃ Hom k ( U, Hom k ( V , W ) ) and the righ t mo dule category with the a sso ciativit y con- strain t Hom k ( U ⊗ k V , W ) ≃ Hom k ( V , Hom k ( U, W )). The catego ry of left c ontr amo d - ules o v er a coalgebra C is the o pp osite catego r y to the category of como dule ob jec ts in the right mo dule category k – vect op o v er the coring ob j ect C in the tensor category k – vect . Explicitly , a C -contramo dule P is a k -ve ctor space endow ed with a c on- tr aaction map π P : Hom k ( C , P ) − → P satisfying the c on tr aasso ciativity a nd c ounity equations π P ◦ Hom(id C , π P ) = π P ◦ Hom( µ C , id P ) a nd π P ◦ Hom( ε C , id P ) = id P . F or any r ig h t C -como dule N and an y k -vec tor space V the space Hom k ( N , V ) has a na t ura l structure of left C -contramo dule. The category of left C - contramo dules is ab elian. W e will denote it b y C – contra . Both inﬁnite direct sums and inﬁnite pro ducts exist in the category of con tramo dules, but only the inﬁnite pro ducts are preserv ed b y the forgetful functor C – contra − → k – vect (while the inﬁnite direct sums a re not ev en exact in C – contra ) . The category of con tramo dules has enough pro jectiv es. Besides, a C -contramo dule is pro jectiv e if a nd only if it is a direct summand of a fr e e C -contramo dule of the fo rm Hom k ( C , V ) for some vec tor space V . The space of contramodule ho momorphisms f r o m the free C -contramo dule is describ ed by the form ula Hom C (Hom k ( C , V ) , P ) ≃ Hom k ( V , P ). 17 Let M b e a left C -como dule and P b e a left C -contramo dule. The sp ac e of c oho- momorphisms Cohom C ( M , P ) is deﬁned as the cokerne l of the pair of maps (Hom( ν M , id P ) , Hom(id M , π P )) : Hom k ( C ⊗ k M , P ) = Hom k ( M , Hom k ( C , P )) ⇒ Hom k ( M , P ) . This is the dua l construction to that of the space of homomorphisms b et w een t w o mo dules o v er a ring. There are natural isomorphisms Cohom C ( C , P ) ≃ P and Cohom C ( M , Hom k ( N , V ) ) ≃ Hom k ( N  C M , V ). The f unctor of cohomomor phisms o v er C is right exact. 0.2.5. The complex of coho momo r phisms Cohom C ( M • , P • ) from a complex of left C -como dules M • to a complex of left C -contramo dules P • is deﬁne d as the tota l complex of the bicomplex Cohom C ( M i , P j ), constructed b y taking inﬁnite pro ducts along the diagonals. Let us deﬁne the deriv ed functor Co ext C of the functor of cohomomorphisms. A complex of C -contra mo dules is called c ontr aacyclic if it b elongs to the min- imal tria ngulated sub catego r y of the ho mo t op y category Hot ( C – contra ) con taining the tot al complexes of exact triples ′ K • → K • → ′′ K • of complexes of C -contramo d- ules and closed under inﬁnite pro ducts. Any contraacyclic complex is acyclic; an y acyclic complex b ounded from ab o v e is contraacyclic . The c ontr a d erive d c ate gory of C -contramo dules D ctr ( C – contra ) is deﬁned as the quotien t cat ego ry of the homo- top y category Hot ( C – contra ) b y the thic k sub category of contraacyc lic complexes Acycl ctr ( C – contra ). The complex of cohomomorphisms Cohom C ( M • , P • ) is acyclic whenev er either M is a complex of inj ective C -como dules and P is con traacyclic, or M is coacyclic and P is a complex of pro jectiv e C -contramo dules. Besides, the contraderiv ed catego ry D ctr ( C – contra ) is equiv alent to the homotopy category o f pro jectiv e C -contramo dules Hot ( C – contra p roj ). Th us one can deﬁne the deriv ed functor Co ext C : D co ( C – como d ) op × D ctr ( C – contra ) − − → D ( k – vect ) b y restricting the functor Cohom C to either of the sub categories Hot ( C – como d inj ) op × Hot ( C – contra ) or Hot ( C – como d ) op × Hot ( C – contra p roj ) of the Carthesian pro duct Hot ( C – como d ) op × Hot ( C – contra ). The con tramo dule v ersion o f bar construction pro vides a f unctorial complex com- puting Co ext C . Namely , fo r an y complex o f left C -como dules M • and complex o f left C -contramo dules P • the total complex of the bicomplex · · · − − → Hom k ( C ⊗ k C ⊗ k M • , P • ) − − → Hom k ( C ⊗ k M • , P • ) − − → Hom k ( M • , P • ) , constructed b y taking inﬁnite pro ducts a long the diag onals, represen ts the ob ject Co ext C ( M • , P • ) in D ( k – v ect ). 18 0.2.6. The categories of left C -como dules and left C -contramo dules are isomorphic if the coalg ebra C is ﬁnite-dimensional, but in general they are quite diﬀeren t. How ev er, the co deriv ed category of left C - como dules is naturally equiv alen t to the con traderiv ed category of left C -contramo dules, D co ( C – como d ) ≃ D ctr ( C – contra ). Indeed, the co deriv ed category D co ( C – como d ) is equiv alen t to the homotop y cat- egory Hot ( C – como d inj ) and the contraderiv ed cat ego ry D ctr ( C – contra ) is equiv alen t to the homotopy category Hot ( C – contra p roj ). F urthermore, the additiv e category o f injectiv e C -como dules is the idemp oten t closure of the category of cofree C - como dules and the additiv e category o f pro jectiv e C -contramo dules is the idemp otent clo- sure o f the category of free C - contramo dules. One has Hom C ( C ⊗ k U, C ⊗ k V ) = Hom k ( C ⊗ k U, V ) = Hom k ( U, Hom k ( C , V )) = Hom C (Hom k ( C , U ) , Hom k ( C , V )), so the categories of cofree como dules and free con tramo dules are equiv alen t. T o describ e this equiv alence of additive categor ies in a more in v ariant w ay , let us deﬁne the op erat io n o f con tratensor pro duct of a como dule and a contramodule. Let N b e a right C -como dule and P b e a left C -con tramo dule. The c ontr atensor pr o duct N ⊙ C P is deﬁned as the cok ernel of the pair of maps ((id N ⊗ ev C ) ◦ ( ν N ⊗ id Hom k ( C , P ) ) , id N ◦ π P ) : N ⊗ k Hom k ( C , P ) ⇒ N ⊗ k P , where ev C denotes the ev aluation map C ⊗ k Hom k ( C , P ) → P . The contratensor pro duct functor is not a par t o f any tensor or mo dule category structure; instead, it is dual t o the functors Hom in the categories of C - como dules and C - contramo dules. The functor of con tratensor pro duct ov er C is right exact. There are na t ural isomorphisms N ⊙ C Hom k ( C , V ) ≃ N ⊗ k V and Hom k ( N ⊙ C P , V ) ≃ Hom C ( P , Hom k ( N , V ) ). The desired equiv alence b et w een the a dditiv e categories of injectiv e left C - como d- ules and pro jectiv e left C -contramo dules is pro vided b y the pair of adjoin t func- tors Ψ C ( M ) = Hom C ( C , M ) and Φ C ( P ) = C ⊙ C P b et w een t he categor ies of left C -como dules and left C -contramo dules. Here the space Hom C ( C , M ) is endo w ed with a C -contra mo dule structure as the k ernel of a pa ir of contramodule morphisms Hom k ( C , M ) ⇒ Hom k ( C , C ⊗ k M ) (where the con tramo dule structure on Hom k ( C , M ) and Hom k ( C , C ⊗ k M ) comes from the rig h t C -como dule structure o n C ), while the space C ⊙ C P is endow ed with a left C -como dule structure as the coke rnel of a pair of como dule morphisms C ⊗ k Hom k ( C , P ) ⇒ C ⊗ k P . 0.2.7. The followin g examples illustrate the necessit y of considering the exotic de- riv ed categor ies in t he ab ov e construction of t he deriv ed como dule-con tramo dule corresp ondence. Let W b e a v ector space and C b e the symmetric coalgebra of W . One can construct C as the sub coalgebra of the tensor coalg ebra L ∞ n =0 W ⊗ n formed b y the symmetric tensors. Consider the trivial left C -contramo dule k ; it has a left pro jectiv e C - contr a mo dule resolution of the f o rm · · · − − → Hom k ( C , ( V 2 k W ) ∗ ) − − → Hom k ( C , W ∗ ) − − → Hom k ( C , k ) . 19 Applying t he functor Φ C to t he ab o v e complex of con tramo dules, one obtains the complex of injectiv e left C - como dules (2) · · · − − → C ⊗ k ( V 2 k W ) ∗ − − → C ⊗ k W ∗ − − → C . When W is ﬁnite-dimensional, the complex (2 ) has its o nly nonv anishing cohomol- ogy in the degree − dim W ; this cohomology a trivial one-dimensional C -como dule naturally isomorphic to det( W ) ∗ = ( V dim W k W ) ∗ as a v ector space. When W is inﬁnite-dimensional, the complex (2) is acyc lic; one can think of it as an “injec- tiv e resolution o f a o ne- dimensional C -como dule concen trated in the degree −∞ ”. So when dim W = ∞ the equiv alence of categories D co ( C – como d ) ≃ D ctr ( C – contra ) transforms the acyclic complex o f C -como dules (2) in to the trivial C -contra mo dule k considered as a complex concen trated in degree 0, and back. Analogously , consider the trivial left C -como dule k ; it has a right injectiv e C -como dule resolution o f the form C − − → C ⊗ k W − − → C ⊗ k V 2 k W − − → · · · Applying the functor Ψ C to this complex of como dules, one obtains the complex of pro jectiv e left C -contramo dules Hom k ( C , k ) − − → Ho m k ( C , W ) − − → Hom k ( C , V 2 k W ) − − → · · · When W is ﬁnite-dimensional, the latter complex has its only non v anishing coho- mology in the degree dim W ; the cohomology a trivial one-dimens ional C -contra- mo dule natura lly isomorphic to det( W ) as a vec tor space. When W is inﬁnite- dimensional, this complex is acyclic; one can think of it as a “pro jectiv e resolution of a one-dimensional C -contra mo dule concen trated in the degree + ∞ ”. In this case, the equiv alence of categories D co ( C – como d ) ≃ D ctr ( C – contra ) t ransforms the trivial C -como dule k considered as a complex concentrated in degree 0 into this acyclic complex of C -contramo dules, and bac k. The cohomolog y computations ab ov e ar e v ery similar to computing Ext R ( k , R ) for the algebra R of p olynomials in a ﬁnite or inﬁnite num b er of v ariables ov er a ﬁeld k . 0.2.8. The functor Ext C : D co ( C – como d ) op × D co ( C – como d ) − → D ( k – vect ) of ho- momorphisms in t he co derived category D co ( C – como d ) can b e computed b y r e- stricting the functor Hom C : Hot ( C – como d ) op × Hot ( C – como d ) − → Hot ( k – v ect ) of homomorphisms in t he homotop y category Hot ( C – como d ) to the full sub category Hot ( C – como d ) op × Hot ( C – como d inj ) of the category Hot ( C – como d ) op × Hot ( C – como d ). The complex Hom C ( L • , M • ) is acyclic whenev er L • is a coacyclic complex of left C -como dules and M • is a complex of injectiv e left C -como dules. Analogously , the functor Ext C : D ctr ( C – contra ) op × D ctr ( C – contra ) − → D ( k – vect ) of homomorphisms in the con traderiv ed category D ctr ( C – contra ) can b e computed by re- stricting the functor Hom C : Hot ( C – contra ) op × Hot ( C – contra ) − → Hot ( k – vect ) to the 20 full sub category Hot ( C – contra p roj ) op × Hot ( C – contra ) of the category Hot ( C – contra ) op × Hot ( C – contra ). The complex Hom C ( P • , Q • ) is acyclic whenev er P • is a complex o f pro jectiv e C - contr a mo dules and Q • is a con traacyclic complex of C -contramo dules. The con tratensor pro duct N • ⊙ C P • of a complex of right C -como dules N • and a complex of left C -contramo dules P • is deﬁned as the total complex of the bicomplex N i ⊙ C P j , constructed by taking inﬁnite direct sums a lo ng the dia g onals. The complex N • ⊙ C P • is a cyclic whenev er N • is a coacyclic complex of righ t C - como dules and P • is a complex of pro jectiv e left C -contra mo dules. The left deriv ed functor Ctrtor C of the functor of contratensor pro duct, Ctrtor C : D co ( como d – C ) × D ctr ( C – contra ) − − → D ( k – vect ) , is deﬁned by restricting the functor of contratens or pro duct to the full sub categor y Hot ( como d – C ) × Hot ( C – contra p roj ) of the category Hot ( como d – C ) × Hot ( C – contra ). Notice that the (ab elian or homotopy ) category of r ig h t C -como dules do es not con tain enough ob jects a djusted to con tratensor pro duct. The equiv alence of tr ia ngulated categories D co ( C – como d ) ≃ D ctr ( C – contra ) trans- forms the f unctor Co ext C in to either of the functors Ext C or Ext C and the f unctor Cotor C in to the functor Ctrtor C . 0.2.9. A left C -como dule M is called c o ﬂat if the functor N 7− → N  C M is exact on the category o f righ t C -como dules. A left C -como dule M is called c opr oj e ctive if the functor P 7− → Cohom C ( M , P ) is exact on the category of left C -contramo dules. It is easy to see that an injectiv e como dule is copro jec tiv e and a copro jectiv e como dule is coﬂat. Using the fact that an y como dule is a union of its ﬁnite-dimensional sub- como dules, one can show that an y coﬂat como dule is injectiv e. Th us all the three conditions are equiv alen t. A left C -contramo dule P is called c ontr a ﬂat if the functor N 7− → N ⊙ C P is exact on the category of right C -como dules. A left C -contramo dule P is called c oinje ctive if the functor M 7− → Cohom C ( M , P ) is exact on the category of left C -como dules. It is easy to see that a pro j ective contramo dule is coinjectiv e and a coinj ective con tramo dule is con traﬂat. W e will sho w in 5.2 tha t an y coinjectiv e contramodule is pro j ectiv e and in A.3 that any contraﬂat con tramo dule is pro jectiv e. Th us all the three conditions are equiv alent. 0.2.10. Our deﬁnition of the deriv ed functor of cotensor pro duct for un b o unded complexes diﬀers from the most traditional one, whic h w as in tro duced (in the greater generalit y of DG -coalgebras a nd DG- como dules) b y Eilen b erg and Mo ore [21]. Husemoller, Mo o re, and Stasheﬀ [27] call the functor deﬁned by Eilen b erg–Mo ore the diﬀer ential derive d functor of c o tens o r pr o duct of the ﬁrst kind and denote it by 21 Cotor C ,I or simply Cotor C , while the functor Cotor C deﬁned in 0.2.2 is (the nondif- feren tial part icular case of ) what t hey call the diﬀer ential derive d functor o f c otens or pr o duct of the se c ond kind and denote b y Cotor C ,I I . The functor Cotor C ,I is computed by the to t al complex of t he cobar bicomplex (1), constructed by taking inﬁnite pr o ducts along the diagonals ( while the tensor pro duct complexes N • ⊗ C ⊗ · · · ⊗ C ⊗ M • constituting the cobar bicomplex are still deﬁned a s inﬁnite direct sums). It is indeed a functor on the Carthesian pro duct of conv en tional un b ounded derive d categories D ( como d – C ) a nd D ( C – como d ). The unbounded deriv ed functor T or R deﬁned in 0.1.1 is a deriv ed functor o f the ﬁrst kind in this terminology . Roughly , deriv ed functors of the ﬁrst kind corresp ond to the con v en tional deriv ed categories D (which can b e therefore called derive d c a t- e gories of the ﬁrst kind ), while deriv ed functors of the second kind correspo nd to the co deriv ed and con traderive d categories D co and D ctr (whic h can b e called derive d c ate go ries of the se c ond kind ). The distinction, whic h is only relev an t for un b ounded complexes of mo dules (como dules, or contramodules), manifests itself also f or quite ﬁnite-dimensional DG - mo dules (DG-como dules, o r DG-contramo dules). The co deriv ed cat ego ries of como dules w ere in tro duced by K. Lef ` evre-Hasega w a [35, 32] in the con text of Koszul dualit y; o ur deﬁnition is equiv alen t to (the nondif- feren tial case of ) his one. They ﬁrst app eared in the a utho r’s o wn research in the v ery same con text. An elab orate discussion of the t w o kinds o f deriv ed categories and their ro les in Koszul duality can b e found in [4 1]; a pro of of the equiv alence of the tw o deﬁnitions is also give n there. Con tramo dules w ere deﬁned b y Eilen b erg and Mo ore in [20] and studied b y Barr in [4]. All the most imp orta nt results of this subsection can b e extended straig htforw ardly to DG -coalgebras and ev en CDG- coalgebras (see 0.4.3 and 1 1.2.2 f o r the deﬁnition). Generally , the constructions of deriv ed categories and functors o f the ﬁrst kind can b e generalized to A ∞ -algebras, while the constructions o f deriv ed categories and functors of the second kind can b e naturally extended to CDG-coalgebras. 0.3. Semialgebras ov er coalgebras ov er ﬁelds. The notion of a semialgebra o v er a coalgebra is dual to that of a coring o v er a noncomm utativ e ring. The similarit y b et w een the tw o theories only go es so far, ho w ev er. 0.3.1. Let C and D b e tw o coalg ebras ov er a ﬁeld k . A C - D -b ic omo dule K is k -ve ctor space endow ed with a left C -como dule and a righ t D - como dule structures suc h that the left C -coa ction map ν ′ K : K − → C ⊗ k K is a morphism of right D -como dules, or, equiv alen tly , the right D -coa ctio n map ν ′′ K : K − → K ⊗ k D is a morphism o f left C -como dules. A bicomo dule can b e also deﬁned as a v ector space endow ed with a bi- c o action map K − → C ⊗ k K ⊗ k D satisfying the coasso ciativity and counity equations. The cat ego ry of C - D -bicomo dules is ab elian. W e will denote it b y C – como d – D . 22 Let C , D , and E b e three coalgebras, N b e a C - D - bicomo dule, a nd M b e a D - E -bi- como dule. Then the cotensor pro duct N  D M is endo w ed with a C - E -bicomo dule structure as the k ernel of a pair of bicomo dule morphisms N ⊗ k M ⇒ N ⊗ k D ⊗ k M . The cotensor pro duct of bicomo dules is asso ciativ e: for an y coalgebras C and D , an y righ t C -como dule N , left D - como dule M , and C - D -bicomo dule K there is a natura l isomorphism N  C ( K  D M ) ≃ ( N  C K )  D M . 0.3.2. In particular, the category of C - C - bicomo dules is an asso ciativ e t ensor cate- gory with the unit o b ject C . A semialgebr a S o v er C is an asso ciativ e ring ob ject with unit in this tensor catego r y; in ot her w ords, it is a C - C - bicomo dule endo w ed with a semimultiplic a tion ma p m S : S  C S − → S and a semiunit map e S : C − → S whic h ha v e to b e C - C -bicomo dule morphisms satisfying the asso ciativit y and unity equations m S ◦ ( m S  id S ) = m S ◦ (id S  m S ) and m S ◦ ( e S  id S ) = id S = m S ◦ (id S  e S ). The categor y of left C -como dules is a left mo dule category ov er the tensor cat- egory C – como d – C , and the cat ego ry of rig h t C -comodules is a right mo dule cate- gory o v er it. A le f t semimo dule M o v er S is a mo dule ob ject in this left mo d- ule category ov er the ring ob ject S in this tensor category; in other w ords, it is a left C - como dule endow ed with a left s e m iaction map n M : S  C M − → M , whic h has to b e a morphism of left C -como dules satisfying the asso ciativity and unit y equations n M ◦ ( m S  id M ) = n M ◦ (id S  n M ) and n M ◦ ( e S  id M ) = id M . A right semimo d ule N o v er S is a rig h t C -como dule endo w ed with a right sem i a c- tion morphism of right C - como dules n N : N  C S − → N satisfying the equations n N ◦ ( n N  id S ) = n N ◦ (id N  m S ) a nd n N ◦ (id N  e S ) = id N . F or any left C - como dule L , t he cotensor pro duct S  C L has a natural left semimo d- ule structure. It is called the left S -semimo dule induc e d from a left C -como dule L . The space o f semimo dule homomorphisms from the induced semimo dule is describ ed b y the formula Hom S ( S  C L , M ) ≃ Hom C ( L , M ). W e will denote the category of left S -semimo dules by S – simo d a nd the catego r y of r ig h t S -semimo dules by simo d – S . The category of left S -semimo dules is ab elian pro vided that S is an injectiv e right C -como dule. Moreo v er, S is a n injective right C -como dule if and only if the category S – simo d is ab elian and the forgetful functor S – simo d − → C – como d is exact. The op eration of cotensor pro duct o v er C pro vides a pairing functor como d – C × C – como d − → k – vect compatible with the right mo dule catego r y structure o n como d – C and the left mo dule category structure on C – como d ov er the tensor cat- egory C – como d – C . The semitensor pr o duct N ♦ S M of a righ t S - semimo dule N and a left S -semimo dule M is deﬁned as the cok ernel o f the pair of maps ( n N  id M , id N  n M ) : N  C S  C M ⇒ N  C M . There a r e natural isomorphisms N ♦ S ( S  C L ) ≃ N  C L and ( R  C S ) ♦ S M ≃ R  C M . The f unctor of semitensor pro duct is neither left, no r rig h t exact. 23 0.3.3. The semitensor pro duct N • ♦ S M • of a complex of right S -semimo dules N • and a complex of left S -semimo dules M • is deﬁned as the t o tal complex of the bicomplex N i ♦ S M j , constructed b y taking inﬁnite direct sums a long the dia gonals. Assume that S is an injectiv e left and righ t C -como dule. W e w ould like to deﬁne the double-sided deriv ed functor SemiT or S of the functor of semitensor pro duct. The semiderive d c a te gory of left S -semimo dules D si ( S – simo d ) is deﬁned as the quotien t category of the homotop y category Hot ( S – simo d ) by the thic k sub catego ry Acycl co - C ( S – simo d ) of complexes of S -semimo dules that are c o acyclic as c om p lexes of C -c omo dules . F or example, if the coalgebra C coincides with the gr o und ﬁeld k , and S = R is just a k -algebra, then the semideriv ed category D si ( S – simo d ) coincides with the deriv ed category D ( R – mo d ), while if the semialgebra S coincides with t he coalgebra C , then the semideriv ed category D si ( S – simo d ) coincides with the co deriv ed category D co ( C – como d ). A complex of left C -como dules M • is called semiﬂat if the semitensor pro duct N • ♦ S M • is acyclic for an y C - coacyclic complex of rig ht S -semimo dules N • . F or example, the complex of S -semimo dules S  C L • induced from a complex of injectiv e C -como dules L • is semiﬂat. The quotien t category of the homotop y category Hot siﬂ ( S – simo d ) o f semiﬂat com- plexes o f S -semimo dules b y the thic k sub categor y of C -coacyclic semiﬂat complexes Acycl co - C ( S – simo d ) ∩ Hot siﬂ ( S – simo d ) is equiv alen t to the semideriv ed category of S -semimodules. The deriv ed functor SemiT or S : D si ( simo d – S ) × D si ( S – simo d ) − − → D ( k – vect ) is deﬁned by restricting the functor of semitensor pro duct o v er S to either of the full sub categories Hot ( simo d – S ) × Hot siﬂ ( S – simo d ) or Hot siﬂ ( simo d – S ) × Hot ( S – simo d ) o f the category Hot ( simo d – S ) × Hot ( S – simo d ). 0.3.4. Let C and D b e tw o coalgebras, K b e a C - D -bicomo dule, and P b e a left C -contramo dule. Then the space of cohomomorphisms Cohom C ( K , P ) is endo w ed with a left D -contramo dule structure as the cok ernel of a pair of D - contramo dule morphisms Hom k ( C ⊗ k K , P ) = Hom k ( K , Hom k ( C , P )) ⇒ Hom k ( K , P ). F o r any left D -como dule M , left C -contramo dule P , and C - D -bicomo dule K there is a natural isomorphism Cohom C ( K  D M , P ) ≃ Cohom D ( M , Cohom C ( K , P )). 0.3.5. Therefore, the category opp osite to the category of left C -contramo dules is a righ t mo dule category ov er the tensor categor y of C - C -bicomo dules with re- sp ect to the a ction functor Cohom C . The category of left S - s emic ontr amo dules is the opp osite catego r y to the category of mo dule o b jects in the rig ht mo dule cat- egory C – contra op o v er the ring ob ject S in the tensor category C – como d – C . In other words, a left semicon tramo dule P o v er S is a left C -contramo dule endo w ed with a le f t se m ic ontr aaction map p P : P − → Cohom C ( S , P ), whic h has to be a 24 morphism of left C - contramo dules satisfying the asso ciativity and unity equations Cohom(id S , p P ) ◦ p P = Cohom( m S , id P ) ◦ p P and Cohom ( e S , id P ) ◦ p P = id P . F or example, if the coa lg ebra C coincides with the ground ﬁeld k , and S = R is just a k -a lgebra, then left S -semicontramo dules are simply left R -mo dules. F or any r ig h t S -semimo dule N and an y k - v ector space V the space Hom k ( N , V ) has a natural structure of left S -semicontramo dule. F o r any left C - contramo dule Q , the space of cohomomorphisms Cohom C ( S , Q ) ha s a natural structure of left semicon- tramo dule. It is called the left S -semicontramo dule c o i n duc e d from a left C -contra- mo dule Q . The space of semicon t r a mo dule homomorphisms into the coinduced semi- con tramo dule is described by the form ula Hom S ( P , Cohom C ( S , Q )) ≃ Hom C ( P , Q ). W e will denote the category of left S -semicontramo dules b y S – sicntr . The category of left S -semicontramo dules is ab elian provide d tha t S is an injectiv e left C - como dule. Moreo v er, S is an injectiv e left C -como dule if and only if the category S – sicntr is ab elian and the for g etful f unctor S – sicntr − → C – contra is exact. The functor Cohom op C : C – como d × C – contra op − → k – vect op is a pairing compatible with the left mo dule category structure on C – como d and the righ t mo dule category structure on C – contra op o v er the tensor category C – como d – C . Th us one can deﬁne the sp ac e of semi h omomorphisms SemiHom S ( M , P ) from a left S -semimo dule M to a left S -semicontramo dule P as the k ernel of the pair of maps (Cohom( n M , id P ) , Cohom(id M , p P )) : Cohom C ( M , P ) ⇒ Cohom C ( S  C M , P ) = Cohom C ( M , Cohom C ( S , P )) . There are natural isomorphisms SemiHom S ( S  C L , P ) ≃ Cohom C ( L , P ) and SemiHom S ( M , Cohom C ( S , Q )) ≃ Cohom C ( M , Q ). The functor of semihomomor- phisms is neither left, nor r igh t exact. 0.3.6. The complex of semihomomorphisms SemiHom S ( M • , P • ) from a complex of left S -semimo dules M • to a complex of left S -semicontramo dules P • is deﬁned as the to tal complex of the bicomplex SemiHom S ( M i , P j ), constructed by ta king inﬁnite pro ducts along the diago nals. Assume that S is a n injectiv e left and righ t C -como dule. Let us deﬁne the double-sided derive d functor SemiExt S of the functor of semihomomorphisms. The semiderive d c a te gory D si ( S – sicntr ) o f left S - semicontramo dules is deﬁned as the quotien t category of the homotopy category Hot ( S – sicntr ) b y t he thic k sub category Acycl ctr - C ( S – sicntr ) of comple xes of S -semicontramo dules that are c ontr aacyclic as c omplex e s of C -c ontr amo dules . A complex of left S -semimodules M • is called semipr oje ctive if the complex SemiHom S ( M • , P • ) is acyclic for an y C -contr a acyclic complex of left S -semicontra- mo dules P • . A complex of left S -semicontramo dules P • is called semiinje ctive if the 25 complex SemiHom S ( M • , P • ) is a cyclic for an y C -coacyclic complex of left S -semi- mo dules M • . F or example, the complex of S -semimo dules S  C L • induced from a complex of injectiv e C -como dules L • is semipro jectiv e. An y semipro j ective complex of semimo dules is semiﬂat. The complex of S -semicontramo dules Cohom C ( S , Q • ) coinduced from a complex o f pro jectiv e C -contr a mo dules Q • is semiinjectiv e. The quotien t category of the ho motop y category Hot sip r ( S – simo d ) of semipro jec- tiv e complexes o f S -semimo dules b y the thic k sub category of C -coacyclic semipro jec- tiv e complexes Acycl co - C ( S – simo d ) ∩ Hot sip r ( S – simo d ) is equiv alen t to the semideriv ed category of S -semimo dules. Analogously , the quotien t category of the homot o p y category Hot siin ( S – sicntr ) of semiinjectiv e complexes of S -semicontramo dules b y the thic k sub category of C -contra acyclic semiinjectiv e complexes Acycl ctr - C ( S – sicntr ) ∩ Hot siin ( S – sicntr ) is equiv alen t to the semideriv ed category of S -semicontramo dules. The derived functor SemiExt S : D si ( S – simo d ) op × D si ( S – sicntr ) − − → D ( k – vect ) is deﬁned by restricting the functor of semihomomorphisms to either of the full sub- categories Hot sip r ( S – simo d ) op × Hot ( S – sicntr ) o r Hot ( S – simo d ) op × Hot siin ( S – sicntr ) of the category Hot ( S – simo d ) op × Hot ( S – sicntr ). 0.3.7. Assume that S is an injectiv e left a nd righ t C -como dule. One can c hec k t ha t the adjoin t functors Ψ C : C – como d − → C – contra and Φ C : C – contra − → C – como d transform left C - como dules with an S - semimo dule structure in to left C -contramo d- ules with an S -semicontramo dule structure and vice vers a. Th erefore, there is a pair of adjoin t functors Ψ S : S – simo d − → S – sicntr and Φ S : S – sicntr − → S – simo d agreeing with the functors Ψ C and Φ C and pro viding an equiv alence b et w een the exact categories o f C - injectiv e left S -semimo dules a nd C -pro jective left S -semicontra- mo dules. T o construct this pair of adjoint functors in a natural wa y , let us deﬁne the op er- ation of con tratensor pro duct of a semimo dule a nd a semicon tramo dule. Let N b e a rig h t S -semimo dule and P b e a left S -semicontramo dule. The c ontr a- tensor p r o duct N ⊚ S P is deﬁned as the cokerne l of the pair of maps ( n N ⊙ id P , η S ◦ (id N  C S ⊙ p P )) : ( N  C S ) ⊙ C P ⇒ N ⊙ C P where the natural “ev aluation” map η K : ( N  C K ) ⊙ D Cohom C ( K , P ) − → N ⊙ C P exists for any r igh t C -como dule N , left C -contramo dule P , and C - D -bicomo dule K and is dual to the map Hom k ( η K , V ) = Cohom C ( K , − ) : Hom C ( P , Hom k ( N , V )) − − → Hom D (Cohom C ( K , P ) , Cohom C ( K , Hom k ( N , V ) )) 26 for any k - vector space V . There are natura l isomorphisms ( R  C S ) ⊚ S P ≃ R ⊙ C P and Hom k ( N ⊚ S P , V ) ≃ Hom S ( P , Hom k ( N , V )). The functor o f contratens or pro duct o v er S is rig h t exact whenev er S is a n injectiv e left C - como dule. The adjoin t functors Ψ S and Φ S can be deﬁned b y the form ulas Ψ S ( M ) = Hom S ( S , M ) and Φ S ( P ) = S ⊚ S P . Here the space Hom S ( S , M ) is endo w ed with a left S -semicontramo dule structure as a subsemicon tra mo dule of the semicon tra - mo dule Hom k ( S , M ), while the space S ⊚ S P is endo w ed with a left S -semimo dule structure as a quotien t semimo dule of the semimo dule S ⊗ k P . The quotien t category of the homot op y catego r y of C -inj ective S -semimo dules Hot ( S – simo d inj - C ) b y the thick sub category of C -contractible comple xes of C -injec- tiv e S -semimodules is equv alen t t o the semide riv ed category of S -semimodules. Analogously , the quotien t category of the homotopy category Hot ( S – sicntr p roj - C ) of C -pro jectiv e S -semicontramo dules b y the thic k sub catego r y of C -contractible com- plexes of C -projectiv e S -semicontramo dules is equiv alen t to the semideriv ed cat ego ry of S -semicontramo dules. Th us the semideriv ed categories of left S -semimo dules and left S -semicontramo dules are equiv alen t , D si ( S – simo d ) ≃ D si ( S – sicntr ). When S is not an injectiv e left or rig h t C -como dule, the exact categories of C -injectiv e S -semimo dules and C -projectiv e S -semicontramo dules are still equiv alen t, ev en though the functors Ψ S and Φ S are not deﬁned on the whole categories of all como dules and contramodules. 0.3.8. The functor Ext S : D si ( S – simo d ) op × D si ( S – simo d ) − → D ( k – vect ) of homo- morphisms in the semideriv ed category D si ( S – simo d ) can be computed by restrict- ing the functor Hom S : Hot ( S – simo d ) op × Hot ( S – simo d ) − → Hot ( k – v ect ) of homo- morphisms in the homotop y category Hot ( S – simo d ) to an appropria te sub category of the Carthesian pro duct Hot ( S – simo d ) op × Hot ( S – simo d ). Namely , a complex of left S -semimodules L • is called pr oje ctive r elative to C ( S / C - pro jectiv e) if the complex Hom S ( L • , M • ) is acyclic f o r an y C -contractible complex o f C -injectiv e left S -semimodules M • . F or example, the complex of S -semimo dules S  C L • induced from a complex of C - como dules L • is pro j ective relative to C . The quotient cate- gory of the homotop y category Hot p roj - S / C ( S – simo d ) of S / C -projectiv e complexes of S -semimodules by the thick sub category Acycl co - C ( S – simo d ) ∩ Hot p roj - S / C ( S – simo d ) of C -coacyclic S / C -projectiv e complexes is equiv a len t to the semideriv ed category of S - semimo dules. The functor Ext S can b e obtained b y restricting the functor Hom S to the full sub category Hot p roj - S / C ( S – simo d ) op × Hot ( S – simo d inj - C ) of the cat- egory Hot ( S – simo d ) op × Hot ( S – simo d ). Analogously , the functor Ext S : D si ( S – sicntr ) op × D si ( S – sicntr ) − → D ( k – vect ) of homomorphisms in the semideriv ed category D si ( S – sicntr ) can b e computed b y re- stricting the functor Hom S : Hot ( S – sicntr ) op × Hot ( S – sicntr ) − → Hot ( k – vect ) to an appropriate sub categor y of the Carthesian pro duct Hot ( S – sicntr ) op × Hot ( S – sicntr ). 27 A complex of S - semicontramo dules Q • is called inje ctive r elative to C ( S / C -injectiv e) if the complex Hom S ( P • , Q • ) is acyclic f or an y C -contractible complex of C -projec- tiv e S -semicontramo dules P • . F or example, the complex of S -semicontramo dules Cohom C ( S , Q • ) coinduced from a complex of C - contr a mo dules Q • is S / C -injectiv e. The quotien t category of the homoto p y category Hot inj - S / C ( S – sicntr ) of S / C -injectiv e complexes of S -semicontramo dules b y the thic k sub category Acycl ctr - C ( S – sicntr ) ∩ Hot inj - S / C ( S – sicntr ) o f C -contraacyclic S / C -injectiv e complexes is equiv alent to the semideriv ed category of S -semicontramodules. The functor Ext S can b e obtained b y restricting the functor Hom S to the full subcat ego ry Hot ( S – sicntr p roj - C ) op × Hot inj - S / C ( S – sicntr ) of the category Hot ( S – sicntr ) op × Hot ( S – sicntr ). The con tratensor pro duct N • ⊚ S P • of a complex of righ t S -semimo dules N • and a complex of left S -semicontramo dules P • is deﬁned as the to tal complex o f the bicom- plex N i ⊚ S P j , constructed by taking inﬁnite direct sums along the diagonals. Let us deﬁne the left derived functor CtrT o r S of the functor of contratens or pro duct ov er S . A complex o f righ t S -semimo dules N • is called c ontr a ﬂat r elative to C ( S / C -contraﬂat) if the complex N • ⊚ S P • is acyclic for any C -contractible complex of C -projective S -semicontramo dules P • . F or example, the complex of S - semimo dules R •  C S in- duced from a complex o f right C -como dules R • is contraﬂat relativ e to C . A complex of righ t S -semimo dules N • is con tr a ﬂat relativ e to C if and only if the complex of left S -semimo dules Hom k ( N • , k ) is injectiv e relativ e to C . The quotien t category of the homotop y category Hot ctrﬂ - S / C ( simo d – C ) of S / C -contraﬂat complexes o f right S -semimodules b y the thic k sub category Acycl co - C ( simo d – S ) ∩ Hot ctrﬂ - S / C ( simo d – C ) of C -coacyclic S / C -con traﬂat complexes is equiv alen t to the semideriv ed category of righ t S -semimo dules. The left deriv ed functor CtrT or S : D si ( simo d – S ) × D si ( S – sicntr ) − − → D ( k – v ect ) is deﬁned by restricting the functor of contratens or pro duct to the full sub categor y Hot ctrﬂ - S / C ( simo d – S ) × Hot ( S – sicntr p roj - C ) of the category Hot ( simo d – S ) × Hot ( S – sicntr ). The equiv alence of t r ia ngulated cat ego ries D si ( S – simo d ) ≃ D si ( S – sicntr ) transforms the double-sided deriv ed functor SemiExt S in to t he functor Ext in either of the semi- deriv ed categories and the double-sided deriv ed functor SemiT or S in to the left derive d functor CtrT or S . 0.3.9. An y semipro jectiv e complex o f S -semimo dules is S / C -proj ectiv e. An S / C -pro- jectiv e complex of C -injectiv e S -semimo dules is semipro jectiv e. The homo t op y cat- egory of semipro jectiv e complexes of C -injective S -semimo dules is equiv alen t to the semideriv ed category of S - semimo dules. Analogously , any semiinjectiv e complex of S -semicontramo dules is S / C -injectiv e. An S / C -injective complex of C -projective S -semicontramo dules is semiinjectiv e. The 28 homotop y category o f semiinjectiv e complexes of C -injectiv e S -semicontramo dules is equiv alen t to the semideriv ed category of S - semicontramo dules. Our deﬁnitions of S / C -projective and S / C -injectiv e complexes diﬀer f r om the tra- ditional ones; cf. B.3 and Remark 9.2.1. 0.4. N onhomogeneous Koszul dualit y o v er a base ring. This subsection is in tended to supply preliminary material for Section 11 a nd App endix D. 0.4.1. A graded ring S = S 0 ⊕ S 1 ⊕ S 2 ⊕ · · · is called quadr atic if it is generated b y S 1 o v er S 0 with relations o f degree 2 only . In o ther w ords, this means that if one considers the graded ring freely generated by the S 0 - S 0 -bimo dule S 1 (the “tensor ring” of the S 0 - S 0 -bimo dule S 1 ), i. e., the graded ring T S 0 ,S 1 with comp onen ts S ⊗ S 0 n 1 = S 1 ⊗ S 0 S 1 ⊗ S 0 · · · ⊗ S 0 S 1 , then the ring S should b e isomorphic to the quotien t ring of T S 0 ,S 1 b y the ideal generated by a certain subbimodule I S in S 1 ⊗ S 0 S 1 . A quadratic ring S is called 2 -left ﬁnitely pr oje ctive if b oth left S 0 -mo dules S 1 and S 2 are pro jectiv e and ﬁnitely generated. A quadratic ring is called 3 -left ﬁnitely pr oje ctive if the same applies to S 1 , S 2 , and S 3 . F urther conditions of this kind are not very sensible to consider for general quadratic rings. Analogo usly one deﬁnes 2 -right ﬁ nitely pr oje ctive and 3 -right ﬁnitely pr oje ctive quadratic rings. There is a n anti-equiv alence b et w een the category of 2-left ﬁnitely pro jectiv e qua- dratic rings and the category of 2-r ig h t ﬁnitely pro jectiv e quadrat ic rings, called the quadr atic duality . The dualit y functors are deﬁned b y the formulas R 0 = S 0 , R 1 = Ho m S 0 ( S 1 , S 0 ), R 2 = Ho m S 0 ( I S , S 0 ), I R ≃ Hom S 0 ( S 2 , S 0 ), a nd con v ersely , S 1 = Hom R op 0 ( R 1 , R 0 ), S 2 = Hom R op 0 ( I R , R 0 ), I S ≃ Hom R op 0 ( R 2 , R 0 ). Here w e use the natural isomorphism Hom S 0 ( N , S 0 ) ⊗ S 0 Hom S 0 ( M , S 0 ) ≃ Hom S 0 ( M ⊗ S 0 N , S 0 ) for S 0 - S 0 -bimo dules M and N that a r e pro jectiv e and ﬁnitely g enerated left S 0 -mo dules, and the a nalogous isomorphism Hom R op 0 ( N , R 0 ) ⊗ R 0 Hom R op 0 ( M , R 0 ) ≃ Hom R op 0 ( M ⊗ R 0 N , R 0 ) for R 0 - R 0 -bimo dules M and N that are pro jective and ﬁnitely generated right R 0 -mo dules. The dualit y functor sends 3-left ﬁnitely pro jectiv e quadratic ring s to 3- righ t ﬁnitely pro jectiv e quadratic rings and vice ve rsa. Indeed, set J S = I S ⊗ S 0 S 1 ∩ S 1 ⊗ S 0 I S ⊂ S 1 ⊗ S 0 S 1 ⊗ S 0 S 1 ; then 0 − − → I S − − → I A ⊗ S 0 S 1 ⊕ S 1 ⊗ S 0 I A − − → S 1 ⊗ S 0 S 1 ⊗ S 0 S 1 − − → S 3 − − → 0 is an exact sequence of ﬁnitely generated pro jectiv e left S 0 -mo dules, a nd R 3 ≃ Hom S 0 ( J S , S 0 ), since the sequence remains exact after applying Hom S 0 ( − , S 0 ). 29 0.4.2. A g raded ring S = S 0 ⊕ S 1 ⊕ S 2 ⊕ · · · is called left ﬂat Koszul if it is ﬂat as a left S 0 -mo dule and o ne has T o r S i,j ( S 0 , S 0 ) = 0 for i 6 = j . Here S 0 is endo w ed with the righ t and left S -mo dule structures via the augmentation map S − → S 0 and the second grading j on the T or is induced b y the grading o f S . R ight ﬂat Koszul graded rings are deﬁned in the ana lo gous wa y . A left/righ t ﬂat Koszul ring is called left / right ( ﬁnitely ) pr oje ctive Ko s zul , if it is a pro jectiv e (with ﬁnitely generated grading comp onen ts) left/righ t S 0 -mo dule. Notice that when S is a ﬂa t left S 0 -mo dule, the reduced relativ e bar construction · · · − − → S ⊗ S 0 S/S 0 ⊗ S/S 0 − − → S ⊗ S/S 0 − − → S is a ﬂat resolution of the left S -mo dule S 0 , so one can use it to compute T or S ( S 0 , S 0 ). When S is a pro jectiv e left S -mo dule, the same resolution can b e used to compute Ext S ( S 0 , S 0 ). Assume that t he grading comp onen ts o f S ar e ﬁnitely generated pro- jectiv e left S 0 -mo dules; t hen it follow s that S is left ﬁnitely pro j ectiv e Koszul if and only if Ext i,j S ( S 0 , S 0 ) = 0 for i 6 = j and Ext i,i S ( S 0 , S 0 ) a re pro jectiv e righ t S 0 -mo dules. Assume that a g raded ring S is a ﬂat left S 0 -mo dule. Then S is left ﬂat Koszul if and only if it is quadrat ic and for eac h degree n the lattice of subbimo dules in S ⊗ S 0 n 1 generated b y the n − 1 subbimo dules S ⊗ S 0 i − 1 1 ⊗ S 0 I S ⊗ S 0 S ⊗ S 0 n − i − 1 1 is distributiv e. This means that fo r a ny three subbimo dules X , Y , Z that can b e obtained from the generating subbimo dules b y applying the op erations of sum and in tersection one should hav e ( X + Y ) ∩ Z = X ∩ Z + Y ∩ Z . F urthermore, if S is a left ﬁnitely pro jectiv e Koszul ring, then the ring R quadratic dual to S is right ﬁnitely pro jectiv e K oszul, and vise vers a; b esides, in this case the graded ring Ext S ( S 0 , S 0 ) is isomorphic to R op and t he graded ring Ext R op ( R 0 , R 0 ) is isomorphic t o S . 0.4.3. Let S b e a 3-left ﬁnitely pro jectiv e quadratic ring. Supp ose tha t w e are given a ring S ∼ endo w ed with an increasing ﬁltration F 0 S ∼ ⊂ F 1 S ∼ ⊂ F 2 S ∼ ⊂ · · · suc h that S = S n F n S ∼ and the asso ciated graded ring gr F S ∼ is iden tiﬁed with S . Suc h a ring S ∼ will b e called a 3 - left ﬁni tely pr oje ctive nonhomo gene ous quadr a tic ring . If the graded ring S is left ﬁnitely pro jectiv e Koszul, the ﬁltered ring S ∼ is called a left ﬁnitely pr oje ctive non h omo gen e ous Koszul ring . Let R b e the 3 - righ t ﬁnitely pro jectiv e quadratic ring dual to S . W e w ould like to describe the a dditional structure on the ring R corresp onding to the data of a ﬁltered ring S ∼ endo w ed with an isomorphism gr F S ∼ ≃ S . A CDG-ring ( curve d diﬀer ential gr ade d ring ) is a g raded ring R = L n R n endo w ed with an o dd deriv ation d of degree 1 and a “curv ature elemen t” h ∈ R 2 suc h that d 2 ( x ) = [ h, x ] for all x ∈ R a nd d ( h ) = 0. A morphism of CDG-rings ′ R − → ′′ R is a pa ir ( f , a ), where f : ′ R − → ′′ R is a mo r phism of graded rings and a is an elemen t in ′′ R 1 suc h that f ( d ′ ( x )) = d ′′ ( f ( x )) + [ a, f ( x )] (the sup ercomm utator) for all x ∈ ′ R 30 and f ( h ′ ) = h ′′ + d ′′ ( a ) + a 2 . Comp osition of morphisms is deﬁned b y the rule ( g , b )( f , a ) = ( g f , b + g ( a )). Iden tity morphisms are the morphisms (id , 0). So the c ate gory of CDG-ring s is deﬁned. Notice that the natural functor from the category of DG- r ings to the category of CDG-rings is f aithful, but not fully faithful. In other w ords, tw o DG-rings ma y b e isomorphic in the category of CDG-rings without b eing isomorphic as DG -rings. F urthermore, t w o CDG-r ings of the f orm ( R , d + [ a, · ] , h + d ( a ) + a 2 ) and ( R , d, h ) are alw a ys naturally isomorphic, the isomorphism b eing give n by the pair (id , a ). There is a fully fa ithful con trav a rian t functor from the category of 3-left ﬁnitely pro jectiv e nonhomogeneous quadratic r ing s S ∼ with a ﬁxed ring F 0 S ∼ to the category of CDG- rings ( R , d, h ) with the same comp onen t R 0 = F 0 S ∼ suc h that the underlying graded ring R of the CDG -ring ( R, d, h ) corr esp o nding to S ∼ is the 3-right ﬁnitely pro jectiv e quadratic ring dual to the ring S = gr F S ∼ (in the grading R i = R i ). This functor is constructed a s follow s. F or eac h 3-left ﬁnitely pr o jectiv e nonho- mogeneous quadrat ic ring S ∼ c ho ose a complemen tary left S 0 = F 1 S ∼ -submo dule V to the submo dule F 0 S ∼ in the left S 0 -mo dule F 1 S ∼ . This can b e done, b ecause the quotien t mo dule S 1 = F 1 S ∼ /F 0 S ∼ is pro jectiv e. Since V maps isomorphically to S 1 = F 0 S ∼ /F 1 S ∼ , it is endow ed with a structure of an S 0 - S 0 -bimo dule. The em b ed- ding V − → F 0 S ∼ is only a morphism of left S 0 -mo dules, how ev er; the right actions of S 0 in V and F 1 S ∼ are compatible mo dulo F 0 S ∼ . Put q ( v , s ) = m ( v , s ) − v s for v ∈ V , s ∈ S 0 , where m ( v , s ) is the pro duct in S ∼ and v s denotes t he right action of S 0 in V . This deﬁnes a map q : V ⊗ Z S 0 − → S 0 . Let I ∼ b e the full preimage of the subbimo dule I S ⊂ S 1 ⊗ S 0 S 1 under the surjectiv e map S 1 ⊗ Z S 1 − → S 1 ⊗ S 0 S 1 . Using the iden tiﬁcation of V with S 1 , w e will consider I ∼ as the full preimage of F 1 S ∼ under the m ultiplication map m : V ⊗ Z V − → S ∼ . Let us split the map m : I ∼ − → F 1 S ∼ in to t w o comp onents ( g , − h ) according to the direct sum decomp osition F 1 S ∼ ≃ V ⊕ S 0 , so that g : I ∼ − → V and h : I ∼ − → S 0 . The diﬀerentials d 0 : R 0 − → R 1 and d 1 : R 1 − → R 2 are deﬁned in terms of the maps q and g b y the formulas h v , d 0 ( s ) i = q ( v , s ) , h i, d 1 ( r ) i = h g ( ˜ ı ) , r i − q ( ˜ ı 1 , h ˜ ı 2 , r i ) , where h , i denotes the pairing o f V with R 1 and of I S with R 2 , and ˜ ı is any preimage of i in I ∼ , written also a s ˜ ı = ˜ ı 1 ⊗ ˜ ı 2 . The map h factorizes through I S , pro viding the curv ature elemen t in R 2 = Hom S 0 ( I S , S 0 ). Finally , to a morphism of nonhomogeneous quadratic rings f : S ′′ ∼ − → S ′ ∼ with c hosen complemen tary submo dules V ′′ ⊂ F ′′ 1 S ′′ ∼ and V ′ ⊂ F ′ 1 S ′ ∼ one assigns a morphism of dual CDG -rings ( g , a ) : ( ′ R, d ′ , h ′ ) − → ( ′′ R, d ′′ , h ′′ ) deﬁned a s fo llows. The morphism of quadratic rings g : ′ R − → ′′ R is the quadratic dual map to the a sso ciated graded morphism gr f : S ′′ − → S ′ , while the elemen t a ∈ ′′ R 1 = Hom S 0 ( S ′′ 1 , S 0 ) is equal t o minus the comp osition V ′′ − → F ′′ 1 S ′′ ∼ − → F ′ 1 S ′ ∼ − → S 0 of the embedding 31 V ′′ − → F ′′ 1 S ′′ ∼ , the map f , and the pro jection F ′ 1 S ′ ∼ − → S 0 along V ′ . In particular, for a give n nonhomogeneous quadratic ring S ∼ c hanging the splitting of F 1 S ∼ b y the rule V ′′ = { v ′ − a ( v ′ ) | v ′ ∈ V ′ } leads to a natural mo r phism of CDG-ring s (id , a ) : ( R , d ′ , h ′ ) − → ( R, d ′′ , h ′′ ). One has to mak e quite some computations in or der to c hec k that ev erything is w ell-deﬁned and compatible in this construction. In particular, the 3-left pro j ectivity is actually used in the form of the dualit y b et w een J S (where some self-consistency equations on the deﬁning relations of S ∼ liv e) and R 3 (where the equations d ( e ) = 0 for e ∈ I R , d 2 ( r ) = [ h, r ], and d ( h ) = 0 hav e to b e v eriﬁed). The nonhomogeneous quadratic duality functor restricted to the categories of left ﬁnitely pro jectiv e nonhomogeneous Koszul rings and righ t ﬁnitely pro jectiv e Koszul CDG-rings becomes a n equiv alence of categories. In other w ords, any CDG-ring whose underlying graded ring is right ﬁnitely pro jectiv e Koszul corresp onds t o a left ﬁnitely pro jectiv e nonhomog eneous Koszul ring. This is the statemen t of the P oincare–Birkhoﬀ–Witt theorem for ﬁnitely pro jectiv e nonhomogeneous Koszul ring s. 0.4.4. A quasi-diﬀer ential ring R ∼ is a graded ring R ∼ = L n R n ∼ endo w ed with an o dd deriv atio n ∂ of degree − 1 with zero square suc h that the cohomology of ∂ v a nish (equiv alen tly , the unit elemen t of R ∼ lies in the imag e of ∂ ). A quasi-diﬀer ential structur e on a gr aded ring R is the data of a quasi-diﬀeren tial ring ( R ∼ , ∂ ) together with an isomorphism o f g r aded rings k er ∂ ≃ R . The category of quasi-diﬀeren tial rings is equiv alen t to the category of CDG-ring s. This equiv alence assigns to a CDG-ring ( R, d, h ) the quasi-diﬀerential ring R ∼ = R [ δ ] with an added generator δ of degree 1, the relations [ δ, x ] = d ( x ) (the sup ercomm u- tator) for x ∈ R and δ 2 = h , and the deriv ation ∂ = ∂ /∂ δ (the partial deriv ative in δ , meaning the unique o dd deriv ation ∂ of R ∼ for whic h ∂ ( R ) = 0 a nd ∂ ( δ ) = 1). Con v ersely , to construct a CDG-ring structure on the k ernel R of the deriv ation ∂ of a quasi-diﬀeren t ia l ring R ∼ , it suﬃces to c ho ose an elemen t δ ∈ R 1 ∼ suc h that ∂ ( δ ) = 1 a nd set d ( x ) = [ δ, x ], h = δ 2 . Cho osing t w o diﬀeren t elemen ts δ leads to t w o naturally isomorphic CDG-rings. A left CD G-mo dule M o v er a CDG- ring ( R , d, h ) is a graded left R -mo dule endo w ed with a d -deriv ation d M (that is a homogeneous map M − → M of degree 1 for whic h d M ( r x ) = d ( r ) x + ( − 1) | r | r d ( x ) f o r r ∈ R , x ∈ M , where | r | denotes the degree o f a homogeneous elemen t r ) suc h tha t d 2 M ( x ) = hx . A quasi-d i ﬀ er ential left mo dule o v er a quasi-diﬀerential ring R ∼ is just a gra ded left R ∼ -mo dule (without an y diﬀeren tial). The category o f left CDG -mo dules ov er a CDG -ring ( R, d, h ) is isomorphic to the category o f quasi-diﬀeren tial left mo dules ov er the quasi-diﬀeren tial ring R ∼ corresp onding to ( R, d, h ); this isomorphism of categories assigns to a graded R ∼ -mo dule structure o n a graded left R -mo dule M the deriv ation d M ( x ) = δ x on M . 32 Analogously , a right CDG-m o dule N o v er ( R, d , h ) is a graded righ t R -mo dule endo w ed with a d -deriv a tion d N (that is a homogeneous map N − → N of degree 1 for whic h d N ( xr ) = d N ( x ) r + ( − 1) | x | xd ( r ) f or x ∈ N , r ∈ R ) suc h tha t d 2 N ( x ) = − xh . A quasi-di ﬀ er ential right mo dule ov er a quasi-diﬀeren t ia l ring R ∼ is just a gra ded left R ∼ -mo dule. The category of rig h t CDG-mo dules ov er ( R, d, h ) is isomorphic to the category of quasi-diﬀerential R ∼ -mo dules when R ∼ corresp onds to ( R , d, h ); this isomorphism of categories assigns to a graded R ∼ -mo dule structure o n a gra ded r ig h t R -mo dule N the deriv atio n d N ( x ) = ( − 1) | x | +1 xδ on N . 0.4.5. CDG-mo dules ov er a CDG-r ing fo rm a DG-c ategory , i. e., a category where for an y tw o give n ob jects there is a complex o f morphisms b etw een them. W e will consider the cases of left and rig h t CDG-mo dules separately . Let L and M b e t w o left CDG-mo dules o ve r a CDG-ring ( R, d, h ). The com- plex Hom • R ( L, M ) is deﬁned as follo ws. The comp onen t Hom n R ( L, M ) consists o f all homo g eneous maps L − → M of degree n sup ercommuting with the R -mo dule structures in L and M . This means that fo r f ∈ Hom n R ( L, M ) and r ∈ R , x ∈ L one should hav e f ( r x ) = − 1 n | r | r f ( x ). The diﬀeren tial is deﬁned b y the form ula ( d f )( x ) = d M f ( x ) − ( − 1) | f | f d L ( x ). One has d 2 ( f ) = 0, b ecause f ( hx ) = hf ( x ). Let K and N b e tw o r igh t CDG-mo dules o v er ( R, d , h ). The comp onen t Hom n R ( K , N ) of the complex Hom • R ( K , N ) consists o f all homogeneous maps K − → N of degree n comm uting with the R -mo dule structures in L and M (without a n y signs). The diﬀerential is deﬁned by the f o rm ula ( d f )( x ) = d N f ( x ) − ( − 1) | f | f d K ( x ). One can see that shifts and cones exist in the DG-cat ego ries of (left or righ t) CDG-mo dules, and moreo v er, a CDG - mo dule structure can b e twisted with an y co c hain in the complex of endomorphisms satisfying the Maurer–Cartan equation [14 ]. It follows that the homo t o p y categories of CDG-mo dules, deﬁned as the categories of zero cohomology of the D G-categories of CDG-mo dules, are t r iangulated. F urthermore, one can sp eak ab o ut the total CDG-mo dules of complex es of CDG-mo dules, constructed b y taking inﬁnite direct sums or inﬁnite pro ducts along the diagonals. In particular, there are total CDG-mo dules of exact triples of CDG-mo dules. This allows one to deﬁne the c o derive d and c ontr ad e rive d c ate g o ries of CDG-mo dules ov er ( R, d, h ) as the quotien t categories of t he homotopy cate- gories of CDG- mo dules by the minimal triangulated sub categories con taining the total CDG-mo dules of exact triples of CDG-mo dules a nd closed under inﬁnite direct sums and inﬁnite pro ducts, respective ly . Notice that one c annot deﬁne the con v en tional deriv ed categor y of CD G -mo dules, as CDG -mo dules don’t hav e an y cohomology g r oups. 0.4.6. Let S ∼ b e a left ﬁnitely pr o jectiv e nonhomogeneous Koszul ring and ( R, d, h ) b e the dual CDG-r ing. Assume t hat the ring S 0 has a ﬁnite right homolog ical di- mension. Then the Koszul dualit y theorem claims that the deriv ed category of righ t 33 S ∼ -mo dules is equiv alen t to the co deriv ed category of righ t CDG-mo dules N ov er ( R, d, h ) suc h that eve ry elemen t o f N is annihilated b y R n for n ≫ 0. Assuming that S 0 has a ﬁnite left homo lo gical dimension, the deriv ed catego r y of left S ∼ -mo dules is also describ ed as b eing equiv alent to the con traderiv ed category o f left CDG-mo dules o v er ( R, d, h ) in whic h certain inﬁnite summation op era t io ns are deﬁned. One can drop the homolo gical dimension assumptions, replacing the deriv ed cat - egories of S ∼ -mo dules in the formulations of these r esults with certain semideriv ed categories relativ e to S 0 (see Theorem 11.8 and Remark 11.7.3). And the con v en- tional deriv ed category of righ t S ∼ -mo dules without the homological dimension as- sumption on S 0 is equiv alen t to the quotien t category of t he co derive d category of lo cally nilp oten t (in the ab ov e sense) right CDG-mo dules o v er ( R, d, h ) by its min- imal tria ngulated sub catego r y closed under inﬁnite direct sums and containing all the CDG-mo dules N , where R n act by zero for all n > 0 and whic h are acyclic with resp ect to d N (one has d 2 N = 0, since N h = 0). The latter result has an o b vious analogue in the case of left CDG -mo dules with inﬁnite summation op erations. 0.4.7. The followin g example is t hematic. Let M b e a smo oth aﬃne algebraic v a- riet y and E b e a vec tor bundle ov er M . Let Diﬀ M ,E denote the ring of diﬀerential op erators acting in the sections of E . The natura l ﬁltra t io n of Diﬀ M ,E b y the order of diﬀerential op erato r s ma kes it a left (and righ t) ﬁnitely pro j ective nonhomoge- neous Koszul ring. T o construct the dual CDG -ring, c ho ose a global connection ∇ E in E . Let Ω( M , End( E )) b e the g r a ded algebra o f diﬀeren tial fo rms with co eﬃcien t s in the v ector bundle End( E ) of endomorphisms of E , endo w ed with the de Rham diﬀeren tial d ∇ dep ending on the connection ∇ End( E ) corresp onding t o ∇ E and the elemen t h ∇ ∈ Ω 2 ( M , End( E )) equal to the curv ature of ∇ E . The Ko szul dua lity the- orem prov ides an equiv alence b et w een the deriv ed category of right Diﬀ M ,E -mo dules and the co deriv ed cat ego ry of right CDG -mo dules ov er Ω( M , End( E )). The pro of of this result giv en in 11.8 generalizes easily to no na ﬃne v arieties (the a pproac h with quasi-diﬀeren tial structures a llows to get rid of the c hoice o f a global connection). These results a r e ev en v alid in prime c haracteristic, describing the deriv ed cat ego ry of mo dules ov er the ring/sheaf of crystalline diﬀeren tial op erato rs (tho se generated b y endomorphisms and v ector ﬁelds with comm utation relations analogous to the zero c haracteristic case). F urthermore, it is not diﬃcult to see tha t the quotien t category of the homotopy category of ﬁnitely g enerated right CDG-mo dules o v er Ω( M , End( E )) b y its minimal t hick sub category containing the total CDG- mo dules of exact tr iples of ﬁnitely generated CDG-mo dules is a full sub category of the co deriv ed category o f CDG-mo dules. This full sub catego ry is equiv alen t t o the b ounded deriv ed category of ﬁnitely g enerated (coheren t ) righ t D iﬀ M ,E -mo dules. All o f this is applicable to any smo oth v arieties, not necessarily aﬃne. 34 F or a smo oth aﬃne v ariet y M , the deriv ed categor y of left Diﬀ M ,E -mo dules is equiv alen t to the contraderiv ed category of left CDG-mo dules o v er Ω( M , End( E )) . 0.4.8. Koszul algebras w ere introduced b y S. Priddy; the standard con temp orary sources are [10 , 40]. Nonhomogeneous quadratic duality (the equiv alence of categories of nonhomogeneous Ko szul algebras and Koszul CDG- algebras) w as deve lop ed in [39, 40]. Homogeneous Ko szul dualit y (the equiv alence of deriv ed categories of graded mo dules ov er dual Koszul algebras) w as established in [10]. K oszul duality in the con text of A ∞ -algebras and DG- coalgebras w a s w ork ed out in [35]. All of t hese pap ers only consider duality ov er the g round ﬁeld (or, in the case of [10], a semisimple algebra) rather than ov er an arbit r a ry ring, as ab ov e. Notable attempts to deﬁne a vers ion o f deriv ed catego ry of DG - mo dules o v er the de Rham complex so t hat the derive d category of mo dules o v er the diﬀeren tial op erators w ould b e equiv alen t to it w ere undertake n in [3 0] and [8, subsection 7.2]. They were not entirely succes sful, in the presen t author’s view, in that in [3 0] the a nalytic top ology and analytic functions w ere used in the deﬁnition of an essen tially purely algebraic catego r y , while in [8] the r igh t ha nd side of the purp oted equiv alence of categories is to a certain exten t deﬁned in terms of the left hand side. The latter problem is also presen t in Lef ` evre-Hasega w a’s Koszul dualit y [35, 32]. 35 1. Semialge bras and Semitensor Product Through t Sections 1–11 , k is a comm utativ e ring. All our rings, bimo dules, ab elian groups . . . will b e k -mo dules; a ll additive categories will b e k -linear. 1.1. Cor ings and como dules. Let A b e an asso ciative k - a lgebra (with unit). 1.1.1. A c oring C ov er A is a coring ob ject in the tensor category of A - A -bimo dules; in o t her w ords, it is a k - mo dule endo w ed with an A - A -bimo dule structure and tw o A - A -bimo dule maps of c omultiplic ation C − → C ⊗ A C a nd c ounit C − → A satisfying the coasso ciativit y and counity equations: tw o comp o sitions of the com ultiplication map C − → C ⊗ A C with the maps C ⊗ A C ⇒ C ⊗ A C ⊗ A C induced by the com ultiplication map should coincide with eac h other and t w o comp o sitions C − → C ⊗ A C ⇒ C of the com ultiplication map with the maps C ⊗ A C ⇒ C induced b y the counit map should coincide with the iden tity map of C . A left c omo dule M o v er a coring C is a como dule ob j ect in the left mo dule catego ry of left A - mo dules ov er the coring ob ject C in the tensor category of A - A -bimo dules; in other w ords, it is a left A -mo dule endo w ed with a left A -mo dule map of lef t c o action M − → C ⊗ A M satisfying the coassociativity and counit y equations: tw o comp ositions of the coaction map M − → C ⊗ A M with the maps C ⊗ A M ⇒ C ⊗ A C ⊗ A M induced b y the com ultiplication and coaction maps should coincide with eac h other and the comp osition M − → C ⊗ A M − → M of the coaction map with the map C ⊗ A M − → M induced b y the counit map should coincide with the identit y map of M . A rig h t c omo dule N o v er C is a como dule ob ject in the right mo dule category of right A -mo dules o v er the coring ob ject C in the tensor category of A - A -bimo dules; in other w ords, it is a righ t A -mo dule endo w ed with a right A -mo dule map of right c o action N − → N ⊗ A C satisfying the coasso ciativit y and counity equations f or the comp ositions N − → N ⊗ A C ⇒ N ⊗ A C ⊗ A C and N − → N ⊗ A C − → N . 1.1.2. If V is a left A - mo dule, then the left C - como dule C ⊗ A V is called the left C -como dule c oi n duc e d from an A -mo dule V . The k -mo dule of como dule homomor- phisms from an arbit r a ry C -como dule in to the coinduced C -como dule is describ ed by the form ula Hom C ( M , C ⊗ A V ) ≃ Hom A ( M , V ). This is an instance of the following general f a ct, whic h we prefer to f o rm ulate in the tensor (monoidal) category langua g e, though it can b e also f o rm ulated in the monad language. Lemma. L et E b e a ( not ne c essarily additive) asso ciative tensor c ate gory w ith a unit obje ct, M b e a le ft mo dule c ate gory over it, R b e a ring obje ct with unit in E , and R M b e the c ate gory of R -m o dule obje cts in M . Then the induction functor M − → R M deﬁne d by the rule V 7− → R ⊗ V is left adjoint to the f or getful functor R M − → M . Pr o of . F or an y ob ject V and a n y R -mo dule M in M , the map Hom M ( V , M ) − → Hom M ( R ⊗ V , M ) is a split equalizer (see [3 6]) of the pair of maps Hom M ( R ⊗ 36 V , M ) ⇒ Hom M ( R ⊗ R ⊗ V , M ) in t he category of sets, with the splitting maps Hom M ( V , M ) ← − Hom M ( R ⊗ V , M ) ← − Hom M ( R ⊗ R ⊗ V , M ) induced b y the unit morphism of R (applied at the rightm ost fa cto r R ).  W e will denote the category o f left C -como dules b y C – como d a nd the category of righ t C -como dules b y como d – C . The category of left C -como dules is ab elian whenev er C is a ﬂa t rig ht A -mo dule. Moreo v er, the righ t A - mo dule C is ﬂat if and only if the category C – como d is ab elian and the forgetful functor C – como d − → A – mo d is exact. This is an instance of a general fact applicable t o an y monad ov er a n a b elian category . The “only if ” assertion is straigh tforw ardly che c k ed, while the “if ” part is deduced from the observ at io ns that the coinduction functor V 7− → C ⊗ A V is right adjoint to the forgetful functor and a righ t adjoin t functor is left exact. A t the same time, for an y coring C there are four natural exact categories of left como dules: the exact category of A - pro jectiv e C -como dules, the exact category of A -ﬂat C -como dules, the exact category of arbit r a ry C -como dules with A -split exact triples, and the exact category of ar bit r a ry left C - como dules with A -pur e exact triples, i. e., the exact t riples whic h as triples of left A - mo dules remain exact a f ter the tensor pro duct with an y righ t A - mo dule. Besides, an y morphism o f C -como dules has a cok ernel a nd the forgetful functor C – como d − → A – mo d preserv es cokernels . When a mor phism of C -como dules has the prop ert y t ha t its k ernel in the categor y o f A -mo dules is preserv ed by the functors of t ensor pro duct with C a nd C ⊗ A C ov er A , this k ernel has a natura l C -como dule structure, which make s it the k ernel of that morphism in the category of C - como dules. Inﬁnite direct sums alw a ys exist in the category of C -como dules and the fo rgetful functor C – como d − → A – mo d preserv es them. The coinduction functor A – mo d − → C – como d pre serv es b oth inﬁnite direct sums and inﬁnite pro ducts. T o construct pro ducts of C - como dules, o ne can presen t them as k ernels of mor phisms of coinduced como dules, so the category o f C - como dules has inﬁnite pro ducts if it has k ernels. If C is a pro jectiv e righ t A -mo dule, or C is a ﬂat righ t A -mo dule and A is a left No etherian ring, then an y left C -como dule is a union of its sub como dules that are ﬁnitely generated as A -mo dules [16]. 1.1.3. Assume that the coring C is a ﬂat left and righ t A -mo dule and the ring A has a ﬁnite w eak homological dimension (T or-dimension). Lemma. Ther e exists a (not a lways additive) functor assigning to any C -c omo dule a surje ctive map onto it fr om an A -ﬂat C -c omo dule. Pr o of . Let G ( M ) − → M b e a surjectiv e map on to a n A -mo dule M from a ﬂat A -mo dule G ( M ) f unctoria lly dep ending on M . F or example, one can ta k e G ( M ) to b e the direct sum of copies of the A -mo dule A o v er all elemen ts of M . Let M b e a left C -como dule. Consider the coaction map M − → C ⊗ A M ; it is an injectiv e 37 morphism o f left C - como dules; let K ( M ) denote it s coke rnel. Let Q ( M ) b e the k ernel of the comp osition C ⊗ A G ( M ) − → C ⊗ A M − → K ( M ). Then the comp osition of maps Q ( M ) − → C ⊗ A G ( M ) − → C ⊗ A M factorizes through the injection M − → C ⊗ A M , so there is a natural sujectiv e morphism o f C - como dules Q ( M ) − → M . Let us sho w that the ﬂat dimension df A Q ( M ) of the A -mo dule Q ( M ) is smaller than that of M . Indeed, the A - mo dule C ⊗ A G ( M ) is ﬂat, hence df A Q ( M ) = df A K ( M ) − 1 6 df A ( C ⊗ A M ) − 1 6 df A M − 1, b ecause the A -mo dule K ( M ) is a direct summand ot the A -mo dule C ⊗ A M and a ﬂat resolution of the A -mo dule C ⊗ A M can b e constructed by taking the tensor pro duct of a ﬂat resolution of the A -mo dule M with the A - A -bimo dule C . It remains to iterate the functor M 7− → Q ( M ) suﬃcien tly man y t imes. Notice that the como dule Q ( M ) is an extension of M by a coinduced como dule C ⊗ A k er( G ( M ) → M ).  1.2. Cot ensor pro duct. 1.2.1. The c otensor pr o duct N  C M of a righ t C -como dule N and a left C -como dule M is a k -mo dule deﬁned as the kerne l of the pair of maps N ⊗ A M ⇒ N ⊗ A C ⊗ A M one of whic h is induced b y the C -coaction in N and the ot her b y the C -coactio n in M . The functor o f cotensor pro duct is neither left, nor righ t exact in general; it is left exact if the ring A is absolutely ﬂat. F or an y right A -mo dule V and any left C -como dule M there is a natural isomorphism ( V ⊗ A C )  C M ≃ V ⊗ A M . This is a n instance of the follow ing general fact. Lemma. L et E b e a tensor c ate gory, M b e a left mo dule c ate gory ove r it, N b e a right mo dule c ate gory, K b e an additive c a te gory, and ⊗ : N × M − → K b e a p ai rin g functor c omp atible with the mo d ule c ate gory structur es on M and N . L et R b e a ring obj e ct with unit in E , M b e a n R - m o dule o b j e ct in M , and V b e an obje ct of N . Then the morphism V ⊗ R ⊗ M − → V ⊗ M induc e d by the action of R in M is a c o k ernel of the p air of morphisms V ⊗ R ⊗ R ⊗ M ⇒ V ⊗ R ⊗ M , one of which is induc e d by the multiplic ation in R and the other by the R -action in M . Pr o of . The whole bar complex · · · − → V ⊗ R ⊗ R ⊗ M − → V ⊗ R ⊗ M − → V ⊗ M − → 0 is con tractible with contracting homoto p y · · · ← − V ⊗ R ⊗ R ⊗ M ← − V ⊗ R ⊗ M ← − V ⊗ M induced by the unit morphism of R (applied at the leftmost fa cto r R ).  1.2.2. Assume that C is a ﬂat right A -mo dule. A righ t como dule N ov er C is called c oﬂat if the functor of cotensor pro duct with N is exact on the category of left C -como dules. It is easy to see that any coﬂat C -como dule is a ﬂat A -mo dule. The C -como dule coinduced from a ﬂat A - mo dule is coﬂat. A left como dule M ov er C is called c oﬂat r ela tive to A ( C / A -coﬂat) if its cotensor pro duct with an y exact triple of A -ﬂat righ t C -como dules is an exact triple. An y coinduced C -como dule is C / A - coﬂat. The deﬁnition of a relativ ely coﬂat C -como dule do es not really dep end on the ﬂatness assumption on C , but app ears to b e useful when this assumption holds. 38 Lemma. The classes of c oﬂat right C -c om o dules and C / A -c o ﬂat left C -c omo dules ar e close d under extensio n s. The quotient c omo dule of a C / A -c oﬂat left C -c omo d ule by a C / A -c oﬂat sub c omo dule is C / A -c oﬂat; an A -ﬂat quotient c om o dule of a c oﬂat right C -c omo dule by a c oﬂat sub c omo dule is c oﬂat. The c otensor pr o duct of an exa c t triple of c o ﬂat right C - c omo d ules with an y left C -c omo dule i s a n exact triple and the c otensor pr o duct of an A -ﬂat right C -c omo dule with an exact triple of C / A -c oﬂat left C -c omo dules is an exact triple. Pr o of . All of these r esults follow from the standard prop erties of the right deriv ed functor of the left exact functor of cotensor pro duct on the Carthesian pro duct of the exact category of A -ﬂat righ t C -como dules and the ab elian category of left C -como dules. One can simply deﬁne t he k -mo dules Cotor C i ( N , M ), i = 0, − 1, . . . as the cohomology of the coba r complex N ⊗ A M − → N ⊗ A C ⊗ A M − → N ⊗ A C ⊗ A C ⊗ A M − → · · · for a n y A -ﬂat right C - como dule N and any left C -como dule M . Then Cotor C 0 ( N , M ) ≃ N  C M , and there are long exact sequences of Cotor C ∗ asso ciated with exact tr iples of C -como dules in either argumen t, since in b ot h cases the cobar complexes form an exact triple. Now an A - ﬂat righ t C -como dule N is coﬂat if and only if Cotor C i ( N , M ) = 0 for an y left C -como dule M a nd all i < 0. Indeed, the “if ” assertion follows f r om the homological exact sequence, and “only if ” holds since the cobar complex is the cotensor pro duct of the como dule N with the cobar resolution of the como dule M , which is exact except in degree 0. Analogously , a left C -como dule M is C / A - coﬂa t if and only if Coto r C i ( N , M ) = 0 for any A - ﬂat righ t C - como dule N and all i < 0, since the cobar resolution of the como dule N is a complex of A -ﬂat righ t C -como dules, exact except in degree 0 and split ov er A . The rest is ob vious.  Remark. A m uc h more general construction of the double-sided deriv ed functor Cotor C ∗ ( N , M ) deﬁned for arbitrary C - como dules M and N will b e giv en, in the as- sumptions of 1.1 .3, in Section 2. Using this construction, one can pro v e somewhat stronger results. In particular, Cotor C i ( M , N ) = 0 for an y C / A -coﬂat left C -como dule M , an y right C - como dule N , and all i < 0, since the k -mo dules Cotor C i ( M , N ) can b e computed using a left resolution of N consisting of A -ﬂat righ t C -como dules (see 2.8). Therefore, any A - ﬂat C / A -coﬂat C -como dule is coﬂat. It follows tha t the construction of Lemma 1.1.3 assigns to an y C / A -coﬂat C - como dule a surjectiv e map on to it from a coﬂat C -como dule with a C / A -coﬂat kernel. 1.2.3. No w let C b e an arbitr a ry coring. Let us call a left C -como dule M quasic oﬂat if the functor of cotensor pro duct with M is right exact on the catego ry of rig h t C -como dules, i. e., this functor preserv es cok ernels. An y coinduced C - como dule is quasicoﬂat. An y quasicoﬂat C - como dule is C / A -coﬂat. Prop osition. L et N b e a right C -c omo dule, K b e a left C -c omo dule endo w e d with a right action of a k -algebr a B by c omo dule endomorphisms, and M b e a left B -mo dule. 39 Then ther e i s a natur al k -mo dule map ( N  C K ) ⊗ B M − → N  C ( K ⊗ B M ) , which is an isomorphism, at le ast, in the fol lowing c ases: (a) M is a ﬂat left B -m o dule; (b) N is a quasic o ﬂat right C -c omo dule; (c) C is a ﬂat right A -mo dule, N is a ﬂat right A -m o dule, K is a C / A -c oﬂat left C -c omo dule, K is a ﬂat right B -mo dule, an d the ring B has a ﬁnite we ak homolo g ic al d imension; (d) K as a le ft C -c omo dule with a right B -mo dule structur e is c oinduc e d fr om an A - B -bimo dule. Besides, in the c ase (c) the c otensor pr o duct N  C K is a ﬂat right B -m o dule. Pr o of . The map ( N  C K ) ⊗ B M − → N ⊗ A K ⊗ B M obtained b y taking the tensor pro duct of t he map N  C K − → N ⊗ A K with the B -module M has equal comp ositions with t w o maps N ⊗ A K ⊗ B M ⇒ N ⊗ A C ⊗ A K ⊗ B M , hence there is a natura l ma p ( N  C K ) ⊗ B M − → N  C ( K ⊗ B M ). The case (a) is obv ious. In the case (b), it suﬃces to presen t M as the cok ernel of a map of ﬂat B -mo dules. T o pro v e ( c) and ( d) , consider the cobar complex (3) N  C K − − → N ⊗ A K − − → N ⊗ A C ⊗ A K − − → N ⊗ A C ⊗ A C ⊗ A K − − → · · · In the case (c) this complex is exact, since it is the cot ensor pro duct of a C / A -coﬂat C -como dule K with an A -split exact complex of A - ﬂat C -como dules N − → N ⊗ A C − → N ⊗ A C ⊗ A C − → · · · Since all the terms of the complex (3), except p ossibly the leftmost one, ar e ﬂat righ t B -mo dules and the w eak homological dimension of the ring B is ﬁnite, the leftmost term K  C M is also a ﬂat B - mo dule and the tensor pro duct of this complex with the left B - mo dule M is exact. In the case ( d), the complex (3) is exact and split as a complex of right B -mo dules.  1.2.4. Let C b e a coring ov er a k -a lg ebra A and D b e a coring o v er a k -alg ebra B . A C - D -bic o m o dule K is an A - B - bimo dule in t he category of k -mo dules endo w ed with a left C -como dule and a righ t D -como dule structures suc h that the righ t D -coaction map K − → K ⊗ B D is a morphism of left C -como dules and the left C - coaction map K − → C ⊗ A K is a morphism of right B -mo dules, or equiv alen tly , the righ t D -coaction map is a morphism of left A -mo dules and the left C -coa ction map is a morphism of righ t D - como dules. Equiv alen tly , a C - D -bicomo dule is a k -mo dule endo w ed with an A - B -bimo dule structure and an A - B - bimo dule map of bic o action K − → C ⊗ A K ⊗ B D satisfying the coasso ciativity and counity equations. W e will denote the category of C - D -bicomo dules by C – como d – D . Assume that C is a ﬂat righ t A -mo dule and D is a ﬂat left B -mo dule. Then the category of C - D -bicomo dules is ab elian and the fo rgetful functor C – como d – D − → k – mo d is exact. Let E b e a coring ov er a k - algebra F . Let N b e a C - E -bicomo dule and M b e a E - D - bicomo dule. Then the cotensor pro duct N  E M can b e endo w ed 40 with a C - D - bicomo dule structure a s the k ernel o f a pair of bicomo dule morphisms N ⊗ F M ⇒ N ⊗ F E ⊗ F M . More generally , let C , D , and E b e arbitrary corings. Assume that the functor of tensor pro duct with C ov er A and with D ov er B preserv es the k ernel of the pair of maps N ⊗ F M ⇒ N ⊗ F E ⊗ F M , tha t is the natural map C ⊗ A ( N  E M ) ⊗ B D − → ( C ⊗ A N )  E ( M ⊗ B D ) is an isomorphism. Then o ne can deﬁne a bicoaction map N  E M − → C ⊗ A ( N  E M ) ⊗ B D taking the cotensor pro duct o v er E of the left C -coaction map N − → C ⊗ A N and the rig h t D -coaction map M − → M ⊗ B D . One can easily see that this bicoaction is counital and coasso ciative, a t least, if the natural maps C ⊗ A C ⊗ A ( N  E M ) − → ( C ⊗ A C ⊗ A N )  E M and ( N  E M ) ⊗ B D ⊗ B D − → N  E ( M ⊗ B D ⊗ B D ) are also isomorphisms. In particular, if C is a ﬂat righ t A -mo dule and either D is a ﬂat left B -mo dule, or N is a quasicoﬂat right E -como dule, or N is a ﬂat right F -mo dule, E is a ﬂa t right F -mo dule, M is an E /F -coﬂat left E -como dule, M is a ﬂat rig h t B - mo dule, and B has a ﬁnite w eak homological dimension, o r M as a left E - como dule with a rig ht B -mo dule structure is coinduced f rom a n F - B - bimo dule, t hen the cotensor pro duct N  E M has a natural C - D -bicomo dule structure. 1.2.5. Let C b e a coring ov er a k -algebra A and D b e a coring ov er a k -algebra B . Prop osition. L et N b e a righ t C -c omo d ule, K b e a C - D -bic omo dule, and M b e a le ft D -c omo dule. Then the iter ate d c otenso r pr o d ucts ( N  C K )  D M and N  C ( K  D M ) ar e natur al ly isomorphic, a t le ast, in the fol lowing c ases: (a) C is a ﬂat right A -mo dule, N is a ﬂat right A -mo dule, D is a ﬂat left B -mo dule, and M is a ﬂat left B -mo dule; (b) C is a ﬂat right A -mo dule an d N is a c oﬂat right C -c omo dule; (c) C is a ﬂat right A -mo dule, N is a ﬂat right A -m o dule, K is a C / A -c oﬂat left C -c omo dule, K is a ﬂat right B -mo dule, an d the ring B has a ﬁnite we ak homolo g ic al d imension; (d) C is a ﬂat right A -mo dule, N is a ﬂat right A -mo d ule, and K as a left C -c o- mo dule with a right B -mo dule structur e is c oinduc e d f r om an A - B -bimo dule; (e) M is a quasic oﬂat left C -c omo dule and K as a left C -c omo dule w ith a right B -mo d ule structur e is c oinduc e d fr om an A - B -bimo dule; (f ) K as a left C -c omo dule with a right B -mo dule structur e is c oinduc e d fr om an A - B -bimo dule and K as a right D -c omo dule with a left A -mo dule s tructur e is c oinduc e d fr om an A - B -bimo dule. Mor e pr e cisely, i n al l c ases in this list the natur a l maps fr om b oth iter ate d c otenzo r pr o ducts to the k -mo dule N ⊗ A K ⊗ B M ar e inje ctive, their im ages c oincide and ar e e qual to the interse c tion of two subm o dules ( N ⊗ A K )  D M and N  C ( K ⊗ B M ) in the k -mo dule N ⊗ A K ⊗ B M . 41 Pr o of . One can easily see that whenev er b oth maps ( N  C K ) ⊗ B M − → N  C ( K ⊗ B M ) and ( N  C K ) ⊗ B D ⊗ B M − → N  C ( K ⊗ B D ⊗ B M ) are isomorphisms, the natural map ( N  C K )  D M − → N ⊗ A K ⊗ B M is injectiv e and its image coincides with the desired in tersection of tw o submo dules in N ⊗ A K ⊗ B M . Thus it remains to apply Prop osition 1.2 .3.  When a sso ciativit y of cotensor pro duct of f o ur or more (bi)como dules is an issue, it b ecomes imp orta n t to kno w tha t the p en tagonal diagrams of asso ciativit y isomor- phisms are comm utativ e. Since eac h of the ﬁv e itera t ed cotensor pro ducts of four factors of the form N  C K  E L  D M is endo w ed with a natural map in to the tensor pro duct N ⊗ A K ⊗ F L ⊗ B M and the asso ciativit y isomorphisms are, presumably , compatible with these maps, it suﬃces to c hec k that at least one of these ﬁv e maps is injectiv e in o rder to sho w that the p entagonal diagram commutes . In particular, if the ab o v e Prop o sition prov ides all t he ﬁv e asso ciativity isomorphisms constituting the p entagonal diagra m and either M is a ﬂat left B -mo dule, or N is a ﬂat right A -mo dule, or b oth K and L as left (right) como dules with righ t (left) mo dule struc- tures are coinduced from bimo dules, then the p en tagona l diagram is commutativ e. W e will say that a m ultiple cotensor pro duct of sev eral bicomo dules N  C · · ·  D M is asso ciative if for any w a y of putting paren theses in this pro duct all the in termedi- ate cotensor pro ducts can b e endo w ed with bicomo dule structures via the construc- tion of 1.2.4, all p ossible asso ciativit y isomorphisms b et w een intermediate cotensor pro ducts exist in the sense of the la st assertion of Prop osition and preserv e bicomo d- ule structures, and a ll the p en tagonal diag rams comm ute. This deﬁnition a llo ws to consider asso ciativit y of cotensor pro ducts a s a pr o p erty rat her than an a dditional structure. In particular, asso ciativit y isomorphisms and bicomo dule structures on as- so ciativ e m ultiple cotensor pro ducts are preserv ed b y the morphisms b et w een them induced b y any bicomo dule morphisms of the factors. 1.3. Semialgebras and semimo dules. 1.3.1. Assume tha t the coring C ov er A is a ﬂat rig h t A -mo dule. It follo ws from Prop o sition 1.2.5(b) that the category of C - C -bicomo dules whic h are coﬂa t rig h t C -como dules is an asso ciativ e tensor category with a unit ob ject C , the category of left C -como dules is a left mo dule catego r y o v er it, and the category of coﬂat r igh t C - como dules is a right mo dule catego ry ov er this tensor category . F urthermore, it follows fro m Prop osition 1.2.5(c) tha t whenev er the ring A has a ﬁnite we ak homolo gical dimension, the C - C -bicomo dules that are ﬂat right A -mo dules and C / A -coﬂat left C -como dules also for m a tensor category , left C -como dules form a left mo dule category o v er it, and A -ﬂat right C -como dules form a righ t mo dule category ov er this tensor catego ry . F inally , it follo ws from Prop osition 1.2.5(a) that whenev er the ring A is absolutely ﬂat, the categories of left and right C -como dules are left and righ t mo dule categories ov er the tensor category of C - C -bicomo dules. In 42 eac h case, the cotensor pro duct op eration provides a pairing b etw een these left and righ t mo dule catego ries compatible with their mo dule category structures and ta king v alues in the category o f k -mo dules. A semialgeb r a o v er C is a ring ob ject with unit in one of the tens or categories of C - C - bicomo dules o f the kind described ab ov e. In other w o r ds, a semialgebra S o v er C is a C - C -bicomo dule satisfying appropriat e (co)ﬂatness conditions guarante eing asso ciativit y of cotensor pro ducts S  C · · ·  C S of a n y num b er of copies of S and endo w ed with tw o bicomo dule morphisms o f semimultiplic ation S  C S − → S a nd semiunit C − → S satisfying the asso ciativity and unit y equations. Namely , tw o comp ositions S  C S  C S ⇒ S  C S − → S of the morphisms S  C S  C S ⇒ S  C S induced b y the semim ultiplicatio n mor phism with the semim ultiplicatio n mor phism S  C S − → S should coincide with eac h other and tw o comp ositions S ⇒ S  C S − → S of the morphisms S ⇒ S  C S induced b y the semiunit morphism with the semim ultiplication morphism should coincide with t he iden tit y morphism of S . A left semimo dule o v er S is a mo dule ob ject in one of the left mo dule cate- gories of C -como dules of the ab ov e kind ov er t he ring ob ject S in the corresp ond- ing tensor catego r y of C - C -bicomo dules. In other w ords, a left S -semimodule M is a left C -como dule endo wed with a left C -como dule morphism of left semia ction S  C M − → M satisfying the asso ciativity and unit y equations. Namely , t w o com- p ositions S  C S  C M ⇒ S  C M − → M o f the morphisms S  C S  C M ⇒ S  C M induced by the semim ultiplication and the semiaction morphisms with the semiac- tion morphism S  C M − → M should coincide with each other and the comp osition M − → S  C M − → M of the morphism M − → S  C M induc ed b y the semi- unit morphism with the semiaction morphism should coincide with the iden tity mor- phism of M . F or this deﬁnition to mak e sense, (co)ﬂatness conditions imp osed on S and/or M m ust guaran tee asso ciativit y of multiple cot ensor pro ducts of the form S  C · · ·  C S  C M . Right semimo dules ov er S a re deﬁned in the a na logous w a y . If L is a left C -como dule for whic h the multiple cotensor pro ducts S  C · · ·  C S  C L are asso ciative , then there is a natural left S -semimo dule structure on the cotensor pro duct S  C L . The left semimo dule S  C L is called the left S -semimo dule induc e d from a C - como dule L . According to Lemma 1.1.2, the k -mo dule o f semimo dule homomorphisms f rom t he induced S - semimo dule to an a rbitrary S -semimodule is described b y the f o rm ula Hom S ( S  C L , M ) ≃ Hom C ( L , M ). W e will denote the category of left S -semimo dules by S – simo d and the category of right S -semimo dules b y simod – S . This notatio n presumes that one can sp eak of (left or right) S - semimo dules with no ﬂa tness conditions imp osed on them. If S is a coﬂat right C -como dule, the category of left semimo dules o v er S is a b elian and the forgetful functor S – simo d − → C – como d is exact. Assume t ha t either S is a coﬂat rig h t C - como dule, or S is a ﬂat righ t A -mo dule and a C / A -coﬂat left C - como dule a nd A has a ﬁnite w eak homological dimension, 43 or A is absolutely ﬂa t . Then b oth inﬁnite direct sums and inﬁnite pro ducts exist in the category of left S -semimo dules, and b oth are preserv ed by the forgetful functor S – simo d − → C – como d , ev en though only inﬁnite direct sums are preserv ed b y the full f orgetful functor S – simo d − → A – mo d . If S is a ﬂa t righ t A -mo dule and a C / A -coﬂat left C -como dule and A has a ﬁnite w eak homological dimension, then the category of A -ﬂat righ t S -semimo dules is exact. Of course, if S is a coﬂat right C -como dule, then the category of A -ﬂa t left S -semimo d- ules is exact. In b oth cases there are exact categories of C - coﬂat right S -semimo dules and C / A -coﬂat left S -semimo dules. If A is absolutely ﬂat, there are exact categories of C - coﬂat left a nd right S - semimo dules. Inﬁnite direct sums exist in all of these exact categories, and the forg etful functors preserv e them. 1.3.2. Assume that the coring C is a ﬂat left and right A - mo dule, the semialgebra S is a ﬂa t left A - mo dule and a coﬂat right C -como dule, and the ring A has a ﬁnite w eak homological dimension. Lemma. T her e exists a (not always additive) functor assigning to an y left S -semi- mo dule a surje ctive map onto it fr om an A -ﬂat left S -semimo dule. Pr o of . Let P ( M ) − → M denote the functorial surjectiv e morphism on to a C -como dule M from an A -ﬂat C -como dule P ( M ) constructed in Lemma 1.1.3. Then for an y left S -semimo dule M the comp o sition of maps S  C P ( M ) − → S  C M − → M pro vides the desired surjectiv e morphism of S -semimo dules. According to the last assertion of Prop o sition 1.2.3 (with the left and righ t sides switc hed), the A - mo dule P ( M ) = S  C P ( M ) is ﬂa t .  Remark. In the ab ov e assumptions, the same construction provides also a (not alw a ys additive) functor assigning t o an y C / A - coﬂat righ t S -semimo dule a surjectiv e map o n to it from a semiﬂat righ t S -semimo dule (see 1.4.2) with a C / A -coﬂat k ernel. This follows from Lemma 1.2.2 and Remark 1.2.2, since the cotensor pro duct with S o v er C preserv es the k ernel of the morphism P ( N ) − → N and the kernel o f the map N  C S − → N is isomorphic to a direct summand of N  C S as a righ t C - como dule. 1.3.3. Assume that the coring C is a ﬂat righ t A -mo dule, t he semialgebra S is a C / A -coﬂat left C -como dule and a coﬂat righ t C -como dule, and t he ring A has a ﬁnite w eak homological dimension. Lemma. The r e exists an exact functor assigning to any A -ﬂat right S -semimo d- ule an inje ctive morp h i sm f r om it into a c o ﬂat right S -semimo dule with an A -ﬂat quotient semimo d ule. Besi d es, ther e exists an exact functor assigning to any left S -semimo dule an inje ctive morphism fr o m it into a C / A -c oﬂat lef t S -semimo dule. Pr o of . F or any A -ﬂat right C - como dule N , set G ( N ) = N ⊗ A C . Then the coaction map N − → G ( N ) is a n injectiv e morphism of C -como dules, the como dule G ( N ) is 44 coﬂat, a nd the quotient como dule G ( N ) / N is A -ﬂa t. No w let N b e an A -ﬂat righ t S -semimodule. The semiaction map N  C S − → N is a surjectiv e morphism of A -ﬂat S -semimodules; let K ( N ) denote its kerne l. The map N  C S − → G ( N )  C S is an injectiv e mo r phism of A -ﬂat S - semimo dules with an A -ﬂat quotien t semimo dule ( G ( N ) / N )  C S . L et Q ( N ) b e the cok ernel of the comp osition K ( N ) − → N  C S − → G ( N )  C S . Then the comp osition o f maps N  C S − → G ( N )  C S − → Q ( N ) fa cto r izes through the surjection N  C S − → N , so there is a natural injectiv e morphism of S -semimodules N − → Q ( N ). The quotient semimodule Q ( N ) / N is isomorphic to ( G ( N ) / N )  C S , hence b oth Q ( N ) / N a nd Q ( N ) are ﬂat A - mo dules. Notice that the semimo dule morphism N − → Q ( N ) can b e lif t ed to a como dule morphism N − → G ( N )  C S . Indeed, the map N − → Q ( N ) can b e presen ted as the comp osition N − → N  C S − → G ( N )  C S − → Q ( N ), where the map N − → N  C S is induced b y the semiunit morphism C − → S of the semialgebra S . Iterating this construction, w e obtain an inductiv e system of C -como dule mor- phisms N − → G ( N )  C S − → Q ( N ) − → G ( Q ( N ))  C S − → Q ( Q ( N )) − → · · · , where the maps N − → Q ( N ) − → Q ( Q ( N )) − → · · · are injectiv e morphisms of S -semimo d- ules with A -ﬂat cok ernels, while the C - como dules G ( N )  C S , G ( Q ( N ))  C S , . . . a re coﬂat. D enote b y J ( N ) the inductiv e limit of this system; then N − → J ( N ) is an in- jectiv e morphism of S -semimo dules with an A - ﬂa t cok ernel a nd the C -como dule J ( N ) is coﬂat (since the functor of cotensor pro duct preserv es ﬁltered inductiv e limits). A functorial injection M − → J ( M ) of any left S -semimo dule M in to a C / A -coﬂat left S -semimo dule J ( M ) is pro vided b y the same construction (with the left and righ t sides switc hed). The only c hanges are that A - mo dules a re no longer ﬂat, for an y left C -como dule M the C -como dule G ( M ) = C ⊗ A M is C / A -coﬂat, and therefore the S -semimodule S  C G ( M ) is C / A -coﬂa t . Both functors J are exact, since t he k ernels of surjectiv e maps, the cok ernels of injectiv e maps, and the ﬁltered inductiv e limits preserv e exact triples.  1.4. Semitensor pro duct. 1.4.1. Assume that the coring C is a ﬂat righ t A -mo dule, t he semialgebra S is a ﬂat right A - mo dule and a C / A -coﬂat left C - como dule, and the ring A has a ﬁnite w eak homological dimension. Let M b e a left S - semimo dule and N b e an A -ﬂat righ t S - semimo dule. The semitensor pr o duct N ♦ S M is a k -mo dule deﬁned as the cok ernel of the pair of maps N  C S  C M ⇒ N  C M one of whic h is induced b y the S -semiaction in N and another by the S -semiaction in M . Eve n under the strongest of our ( co)ﬂatness conditions on C and S , the ﬂatness of either N or M is still needed to guaran tee that the triple cotensor pro duct N  C S  C M is asso ciativ e. F or an y A -ﬂat rig ht S -semimo dule N and a n y left C -como dule L there is a natural isomorphism N ♦ S ( S  C L ) ≃ N  C L . Analogously , fo r a n y A -ﬂat right C -como dule R 45 and an y left S -semimo dule M there is a natural isomorphism ( R  C S ) ♦ S M ≃ R  C M . These assertions follow from Lemma 1.2.1 . 1.4.2. If the coring C is a ﬂat righ t A - mo dule and the semialgebra S is a coﬂa t right C -como dule, one can deﬁne the semitensor pro duct of a C -coﬂat rig ht S -semimo dule and an arbitrary left S -semimodule. In these assumptions, a C -coﬂat right S -semi- mo dule N is called semiﬂat if the functor of semitensor pro duct with N is exact on the ab elian catego ry of left S -semimo dules. The S -semimo dule induced from a coﬂat C -como dule is semiﬂat. If C is a ﬂat righ t A -mo dule, S is a coﬂat righ t C -como dule and C / A - coﬂat left C -como dule, and the ring A has a ﬁnite w eak homological dimension, one can de- ﬁne se miﬂat S -semimo dules as A -ﬂat rig h t S -semimo dules suc h that the functors of semitensor pro duct with them are exact. Then one can pro v e t hat a n y semiﬂat S -semimodule is a coﬂat C -como dule. When t he ring A is absolutely ﬂat, the semitensor pro duct of arbitrar y tw o S -semi- mo dules is deﬁned without an y conditions on the cor ing C and the semialgebra S . 1.4.3. Let S b e a semialgebra ov er a coring C o v er a k -algebra A and T b e a semi- algebra o v er a coring D o v er a k -algebra B . Let K denote a C - D -bicomo dule. One can sp eak ab out S - T -bisemimo dule structures on K if the ( co)ﬂatness conditions im- p osed on S , T , and K guaran tee associativity of m ultiple cotensor pro ducts of t he form S  C · · ·  C S  C K  D T  D · · ·  D T . Assuming that this is so, K is called a n S - T -bisemimo dule if it is endow ed with a left S - semimo dule and a right T -semimo d- ule structures suc h that the right T - semiaction map K  D T − → K is a morphism of left S -semimo dules and the left S -semiaction map S  C K − → K is a mo r phism of righ t D -como dules, or equiv alen tly , the righ t T -semiaction map is a mor phism of left C -como dules and t he left S -semiaction map is a morphism of right T - semimo dules. Equiv alen tly , the C - D - bicomo dule K is called an S - T -bisemimo dule if it is endo w ed with a C - D -bicomo dule morphism of bisemiaction S  C K  D T − → K satisfying the asso ciativit y and unit y equations. In particular, one can sp eak ab out S - T -bisemimo dules K without imp osing a ny (co)ﬂatness conditions on K if C is a ﬂat rig h t A -mo dule and either S is a coﬂat righ t C -como dule, or S is a ﬂat r ig h t A -mo dule and a C / A -coﬂat left C - como dule and A has a ﬁnite weak homological dimension, while D is a ﬂat left B -mo dule and either T is a coﬂat left D -como dule, or T is a ﬂat left B -mo dule and a D / B -coﬂat righ t D - como dule and B has a ﬁnite w eak homological dimens ion. W e will denote the category of S - T -bisemimodules by S – simo d – T . Besides, o ne can consider B -ﬂa t S - T -bisemimo dules if C is a ﬂat rig h t A - mo dule and S is a coﬂat righ t C -como dule, while D is a ﬂa t right B - mo dule, T is a ﬂat right B -mo dule and a D /B -coﬂat righ t D -como dule, a nd B has a ﬁnite w eak homolo gical dimension; and o ne can consider 46 D -coﬂat S - T -bisemim o dules if C is a ﬂat righ t A -mo dule and S is a coﬂat righ t C -como dule, while D is a ﬂat rig ht B -mo dule and T is a coﬂat rig h t D -como dule. 1.4.4. Let R b e a semialgebra o v er a coring E ov er a k -a lgebra F . Let N b e an S - R -bisemimo dule and M b e an R - T - bisemimo dule. W e would lik e to deﬁne a n S - T -bisemimo dule structure on the semitensor pro duct N ♦ R M . Assume that multiple cotensor pro ducts of the form S  C · · ·  C S  C N  E R  E M  D T  D · · ·  D T are asso ciative. T hen, in particular, the semitensor pro ducts ( S  n  C N ) ♦ R ( M  D T  m ) can b e deﬁned. Assume in addition that multiple cotensor pro ducts of the form S  C · · ·  C S  C N  E M  D T  D · · ·  D T are asso ciative. Then the semitens or pro ducts ( S  n  C N ) ♦ R ( M  D T  m ) hav e natural C - D -bicomo d- ule structures as cok ernels of C - D -bicomo dule morphisms. Assume that multiple cotensor pro ducts of the for m S  C · · ·  C S  C ( N ♦ R M )  D T  D · · ·  D T are also asso ciativ e. Finally , assume that the semitensor pro duct with S  n o v er C and with T  m o v er D preserv es the cok ernel o f t he pair of morphisms N  E R  E M ⇒ N  E M for n + m = 2, that is the bicomo dule morphisms ( S  n  C N ) ♦ R ( M  D T  m ) − → S  n  C ( N ♦ R M )  D T  m are isomorphisms. Then one can deﬁne an asso ciativ e and unital bisemiaction morphism S  C ( N ♦ R M )  D T − → N ♦ R M taking the semitensor pro duct ov er R of the morphism o f S -semiaction in N and the morphism of T -semiaction in M . F or example, if C is a ﬂat right A -mo dule, S is a coﬂat righ t C - como dule, D is a ﬂat left B -mo dule, T is a coﬂat righ t D -como dule, E is a ﬂat righ t F - mo dule, R is a ﬂat righ t F -mo dule and a E /F - coﬂa t left E -como dule, and F ha s a ﬁnite w eak homological dimension, then the semitensor pro duct of any F - ﬂa t S - R -bisemi- mo dule N and any R - T -bisemimo dule M has a natural S - T -bisemimodule structure. Since the cat ego ry of S - T - bisemimo dules is ab elian in this case, the bisemimo dule N ♦ R M can b e simply deﬁned as the cok ernel o f the pair of bisemimo dule morphisms N  E R  E M ⇒ N  E M . Prop osition. L et N b e a right S -semimo dule, K b e an S - T -bisemimo dule, and M b e a left T -semimo dule. Then the iter ate d semitensor pr o ducts ( N ♦ S K ) ♦ T M and N ♦ S ( K ♦ T M ) ar e wel l-deﬁn e d and natur al ly is o morphic, at le ast, in the fol l o wing c ases: (a) C i s a ﬂat right A -mo dule, S is a c oﬂat right C -c omo dule, N is a c oﬂat rig h t C -c omo dule, D is a ﬂat left B -mo dule, T is a c oﬂat left D -c omo dule, an d M is a c oﬂat left D -c omo dule; (b) C is a ﬂat right A -mo dule, S is a c oﬂat right C -c omo dule, N is a semiﬂat right S -semimo dule, and either • D is a ﬂat right B -mo dule, T is a c oﬂat rig h t D -c omo dule, and K is a c oﬂat right D -c omo dule, or 47 • D is a ﬂat right B -m o dule, T is a ﬂat right B -mo dule and a D / B -c oﬂat left D -c omo d ule, the ring B ha s a ﬁnite we ak homolo g ic al di m ension, and K is a ﬂat rig ht B -mo dule, or • D is a ﬂat left B -m o dule, T is a ﬂat le f t B -mo dule and a D /B -c oﬂat right D -c omo dule, the ring B has a ﬁni te we ak homolo gic al dimen sion, and M is a ﬂat left B -mo dule, or • the ring B is absolutely ﬂat; (c) C is a ﬂat right A -mo dule, S is a c oﬂat right C -c omo dule, N is a c oﬂat right C -c omo dule, a n d ei ther • D is a ﬂat right B -mo dule, T is a c oﬂat right D - c omo dule, and K a s a left S -semimo dule with a righ t D -c omo dule structur e i s induc e d fr om a D -c oﬂat C - D -bic om o dule, or • D is a ﬂat right B -m o dule, T is a ﬂat right B -mo dule and a D / B -c oﬂat left D -c omo d ule, the ring B ha s a ﬁnite we ak homolo g ic al di m ension, and K as a left S -s e m imo dule with a rig h t D -c omo dule structur e is i nduc e d fr om a B -ﬂat C - D -bic omo dule, or • D is a ﬂat left B -m o dule, T is a ﬂat le f t B -mo dule and a D /B -c oﬂat right D -c omo dule, the ring B has a ﬁni te we ak homolo gic al dimen sion, K as a left S -s e m imo dule with a rig h t D -c omo dule structur e is i nduc e d fr om a C - D -bic omo dule, and M is a ﬂat left B -m o dule, or • the ring B is absolutely ﬂat and K as a left S -semimo dule with a right D -c omo dule structur e is induc e d fr om a B -ﬂat C - D -bic om o dule. Mor e pr e cisely, in al l c ases in this list the natur al maps into b oth iter ate d semitensor pr o ducts fr om the k -mo dule ( N  C K )  D M ≃ N  C ( K  D M ) a r e surje ctive, their kernels c oincide and ar e e qual to the sum of the kernels of two maps fr om this mo d ule onto its quotient mo dules ( N  C K ) ♦ T M a n d N ♦ S ( K  D M ) . Pr o of . It follows from Prop osition 1.2.5 that all multiple cotensor pro ducts o f the form N  C S  C · · ·  C S  C K  D T  D · · ·  D T  D M are asso ciativ e. Multiple cotensor pro ducts N  C S  C · · ·  C S  C ( K ♦ T M ) and ( N ♦ S K )  D T  D · · ·  D T  D M are also a sso ciativ e b y the same Prop osition (here one has to notice that the semitensor pro duct N ♦ S K is a coﬂat rig ht D -como dule whenev er K is a coﬂat righ t D -como dule and N is a semiﬂat righ t S -semimo dule). The map N  C K  D M − → ( N ♦ S K )  D M factorizes through the surjection N  C K  D M − → N ♦ S ( K  D M ), hence there is a na t ur a l map N ♦ S ( K  D M ) − → ( N ♦ S K )  D M . One can easily see that whenev er this map and the analogous maps f o r T , T  D T , and T  D M in place of M are isomorphisms, the iterated semitensor pro duct ( N ♦ S K ) ♦ T M is deﬁned, the natural map N  C K  D M − → ( N ♦ S K ) ♦ T M is surjectiv e, and its k ernel is equal to the desired sum of t w o kerne ls of maps fro m N  C K  D M on to its quotien t modules. Thus it remains to prov e that the map N ♦ S ( K  D M ) − → 48 ( N ♦ S K )  D M is an isomorphism, i. e., the exact sequence of right D - como dules N  C S  C K − → N  C K − → N ♦ S K − → 0 remains exact after taking the cotensor pro duct with M ov er D . This is obvious if M is a quasicoﬂat D -como dule. If N is a semiﬂat S - semimo dule, it suﬃces to pr esen t M a s a k ernel of a mo r phism of (quasi)coﬂat D -como dules. F inally , if K as a left S -semimo dule with a rig h t D -como dule structure is induced from a C - D -bicomo dule, then our exact sequence of righ t D -como dules splits.  49 2. Derived Functor SemiTor 2.1. Co deriv ed categories. A complex C • o v er an exact category A is called exact if it is comp osed of exact triples Z i → C i → Z i +1 in A . A complex o v er A is called acyclic if it is homotop y equiv alen t to an exact complex (or equiv alen tly , if it is a direct summand of an exact complex). Acyclic complexes form a thick sub catego ry Acycl ( A ) of the ho motop y category Hot ( A ) of complexes ov er A . All a cyclic complexes o v er A are exact if and only if A con tains images o f idemp otent endomorphisms [3 7]. The quotient category D ( A ) = Hot ( A ) / Acycl ( A ) is called the deriv ed catego ry of A . Let A b e an exact category where all inﬁnite direct sums exist and the functors of inﬁnite direct sum are exact. By the total complex of an exact triple ′ K • → K • → ′′ K • of complexes ov er A w e mean the total complex of t he corresp onding bicomplex with three r o ws. A complex C • o v er A is called c o acyclic if it b elongs to the minimal triangulated sub category Acycl co ( A ) of the homotopy category Hot ( A ) containing all the tota l complexes of exact triples o f complexes o v er A and closed under inﬁnite direct sums. Any coacyclic complex is acyclic. Acyclic complexes are not alw a ys coacyclic ( see 0.2.2). It follow s from the next Lemma that an y acyclic complex b ounded from b elow is coacyclic. Lemma. L e t 0 → M 0 , • → M 1 , • → · · · b e an ex a ct se quenc e, b ounde d fr om b elow, of arbitr ary c omplex e s over A . Then the total c omplex T • of the bic omplex M • , • c onstructe d by taking inﬁnite dir e ct sums along the diago n als is c o acyclic. Pr o of . An exact sequence of complexes 0 → M 0 , • → M 1 , • → · · · can b e presen ted as the inductiv e limit of ﬁnite exact sequences of complexes 0 → M 0 , • → · · · → M n, • → Z n +1 , • → 0. The total complex T • n of t he latter ﬁnite exact sequence is homotop y equiv alent to a complex obtained from to t a l complexes of the exact triples Z n, • → M n, • → Z n +1 , • using the op erations of shift and cone. Hence the complexes T • n are coacyclic. The complex T • is their inductiv e limit; moreov er, the inductive system of T • n is obtained b y applying the functor of tota l complex to a lo cally stabilizing inductiv e system of bicomplexes. Therefore, the construction of homotopy inductiv e limit prov ides an exact triple o f complexes L n T • n − → L n T • n − → T • . Since the total complex o f this exact triple is coa cyclic and the direct sum of coacyclic complexes is coacyclic, the complex T • is coacyclic. (In fact, this exact tr iple of complexes is split in ev ery degree, so its to t a l complex is ev en contractible.)  The category of coacyclic complexes Acycl co ( A ) is a thic k sub category of the ho- motop y category Hot ( A ), since it is a tria ng ulated subcatego r y with inﬁnite direct sums [37, 38 ]. The c o derive d c ate gory D co ( A ) of an exact cat ego ry A is deﬁned as the quotien t category Hot ( A ) / Acycl co ( A ). Remark. If a n exact category A has a ﬁnite homolog ical dimension, then the mini- mal triangulated sub category of the homotopy category Hot ( A ) containing the total 50 complexes o f exact triples of complexes o v er A coincides with the sub category of acyclic complexes. Indeed, let C • b e a n exact complex ov er A and n b e a num b er greater tha n the homological dimension of A . Let Z i b e the ob jects of cycles of the complex C • . Then for any in teger j the Y oneda extension class represen ted b y the extension Z 2 j n → C 2 j n → · · · → C 2 j n + n − 1 → Z 2 j n + n is trivial, and therefore, this extension can b e connected with the split extension b y a pair of extension morphisms ( Z 2 j n → C 2 j n → · · · → C 2 j n + n − 1 → Z 2 j n ) − → ( Z 2 j n → ′ C 2 j n → · · · → ′ C 2 j n + n − 1 → Z 2 j n + n ) ← − ( Z 2 j n → Z 2 j n → 0 → · · · → 0 → Z 2 j n + n → Z 2 j n + n ). Let ′ C • b e the complex obtained by replacing a ll the ev en segmen ts C 2 j n → · · · → C 2 j n + n − 1 of t he complex C • with the segmen ts ′ C 2 j n → · · · → ′ C 2 j n + n − 1 while lea ving the o dd segmen ts C 2 j n + n → · · · → C 2( j +1) n − 1 in place, and let ′′ C • b e the complex ob- tained b y replacing the same ev en segmen ts of the complex C • with the segmen ts Z 2 j n → 0 → · · · → 0 → Z 2 j n + n while leav ing the o dd segmen ts in place. Then the complex ′′ C • and the cones of b oth morphisms C • − → ′ C • and ′′ C • − → ′ C • are homotop y equiv alent to complexes obtained from total complexes of exact triples of complexes with zero diﬀeren tials using the op eration of cone rep eatedly . 2.2. Coﬂat complexes. Let C b e a coring ov er a k -algebra A . The cotensor pro duct N •  C M • of a complex of righ t C -como dules N • and a complex of left C -como dules M • is deﬁned a s the total complex of the bicomplex N i  C M j , constructed b y taking inﬁnite direct sums alo ng the diagonals. Assume that C is a ﬂat right A - mo dule. Then the category of left C - como dules is a n ab elian category with exact functors o f inﬁnite direct sums, so the co deriv ed category D co ( C – como d ) is deﬁned. When sp eaking ab out c o acyclic c omple x es of C - como dules, w e will a lw a ys mean coacyclic complexe s with resp ect to the abelian category of C -como dules, unless another exact category of C -como dules is explicitly men tioned. A complex of right C -como dules N • is called c oﬂat if the complex N •  C M • is acyclic whenev er a complex of left C -como dules M • is coacyclic. Lemma. Any c omplex of c oﬂat C -c omo dules is c oﬂat. Pr o of . Let N • b e a comple x of coﬂat C -como dules. Since the functor of cotensor pro duct with N • preserv es shifts, cones, and inﬁnite dir ect sums, it suﬃces to sho w the complex N •  C M • is acyclic whenev er M • is the t o tal complex of an exact triple of complexes of left C -como dules ′ K • → K • → ′′ K • . In this case, the triple of complexes N •  C ′ K • − → N •  C K • − → N •  C ′′ K • is also exact, b ecause N • is a complex of coﬂat C -como dules, and the complex N •  C M • is the total complex of this exact t r iple.  If the r ing A has a ﬁnite we ak homological dimension, then an y coﬂat complex of C -como dules is a ﬂa t complex of A -mo dules in the sense of 0 .1 .1. (Indeed, if V • is a complex of right A -mo dules suc h that the tensor pro duct of V • with an y coacyclic complex o f left A -mo dules is acyclic, t hen the tensor pro duct of V • with any acyclic 51 complex U • of left A -mo dules is also acyclic, since one can construct a morphism in to U • from an acyclic complex of ﬂa t A -mo dules with a coacyclic cone.) The complex of C -como dules V • ⊗ A C coinduced from a ﬂat complex of A -mo dules V • is coﬂat. Remark. The co deriv ed category D co ( C – como d ) can b e only t hough t of as the “r igh t” v ersion of exotic un b ounded deriv ed category of C -como dules (e. g., for the purp o ses of deﬁning the deriv ed functors Cotor C and Co ext C , constructing the equiv alence of deriv ed categories of C -como dules and C - contramo dules, etc.) when the ring A has a ﬁnite (we ak or left) homological dimension. Indeed, what is needed is a deﬁnition of “relativ e co derive d category” o f C -como dules suc h that for C = A it would coincide with the deriv ed category of A -mo dules, while when C is a coalgebra o v er a ﬁeld it w ould b e t he co deriv ed category of C -como dules deﬁned ab ov e. (The same applies to the semideriv ed category D si ( S – simo d ) of S -semimo dules—it only app ears to b e the “righ t” deﬁnition when the ring A has a ﬁnite homological dimension.) 2.3. Semideriv ed categories. Let S b e a semialgebra o ve r a coring C . Assume that C is a ﬂat right A -mo dule and S is a coﬂat right C -como dule, so that the category o f left S -semimo dules is ab elian. The semideriv e d c ate gory of left S - semi- mo dules D si ( S – simo d ) is deﬁned as the quotient category of the homotop y category Hot ( S – simo d ) by the thic k sub category Acycl co - C ( S – simo d ) of complexes of S -semi- mo dules that are c o acyclic as c omp lexes of C -c omo dules . Remark. There is no claim that the semideriv ed category e x ists in the sense tha t morphisms b etw een a given pair o f ob jects form a set rather than a class. Rather, w e think of our lo calizations of categories as of “v ery lar g e” catego ries with classes of morphisms instead of sets. W e will explain in 5.5 and 6.5 ho w to compute the mo dules of homomorphisms in semideriv ed categories in terms of resolutions; then it will follow that the semideriv ed category do es exist, under certain assumptions. 2.4. Semiﬂat complexes. Let S b e a semialgebra. The semitensor pro duct N • ♦ S M • of a complex of right S -semimo dules N • and a complex o f left S -semimo dules M • is deﬁned as the to t a l complex of the bicomplex N i ♦ S M j , constructed by taking inﬁnite direct sums along the diagonals. Of course, appropria te (co)ﬂatness conditions m ust b e imp osed on S , N • , and M • for this deﬁnition to make sense. Assume t ha t the cor ing C is a ﬂat righ t A -mo dule, the semialgebra S is a coﬂat righ t C -como dule and a C / A -coﬂat left C - como dule, and the ring A has a ﬁnite w eak homological dimension. A complex of A - ﬂat rig h t S -semimo dules N • is called semiﬂat if the complex N • ♦ S M • is acyclic whenev er a complex o f left S -semimo d- ules M • is C -coacyclic. Any semiﬂat complex of S -semimo dules is a coﬂat complex of C -como dules. The complex o f S -semimo dules R •  C S induced from a coﬂat complex of A -ﬂat C -como dules R • is semiﬂat. If it is only kno wn that C is a ﬂat right A -mo dule and S is a coﬂat rig h t C -como dule, one can deﬁne semiﬂat complexes of C -coﬂat righ t S -semimo dules. Then the complex 52 of S -semimo dules induced from a complex of coﬂat C -como dules is semiﬂat; it is also a complex of semiﬂat semimo dules. Notice that not every c omplex of s emiﬂat semimo dules is semiﬂat (see 0.1.1). In particular, it follows from Theorem 2.6 and Lemma 2.7 b elow that (in the assumptions of 2.6) a C -coacyclic complex of A - ﬂat right S -semimo dules N • is semiﬂat if and only if its semitensor pro duct with any complex of left S - semimo dules M • (or just with an y left S -semimo dule M ) is acyclic. Th us a C -coacyclic complex of semiﬂat S -semi- mo dules is semiﬂat if and only if all of its semimo dules of co cycles are semiﬂat. On t he other hand, an y complex of semiﬂat semimo dules b ounded fr o m a b o v e is semiﬂat. Moreov er, if · · · → N − 1 , • → N 0 , • → 0 is a complex, b ounded from ab ov e, of semiﬂat complexes of S -semimo dules, then the tota l complex E • of the bicomplex N • , • constructed by taking inﬁnite direct sums along the diagonals is semiﬂat. Indeed, the category of semiﬂat complexes is closed under shifts, cones, and inﬁnite direct sums, so one can apply the followin g L emma. Lemma. L et · · · → N − 1 , • → N 0 , • → 0 b e a c omp l e x , b o unde d fr om ab ove, of arbitr ary c omplex e s o v er an additive c ate gory A w h er e inﬁnite dir e ct sums exist. Then the total c omplex E • of the bic om plex N • , • up to the ho m otopy e quivalenc e c an b e ob taine d fr om the c omplexe s N − i, • using the op er ations of shift, c one, a n d in ﬁ nite dir e ct sum. Pr o of . Let E • n b e the total complex o f the ﬁnite complex of complexes 0 → N − n, • → · · · → N 0 , • → 0. Then the complex E • is the inductive limit of the complexes E • n , and in a ddition, the em b eddings of complexes E • n − → E • n +1 split in ev ery degree. Th us the triple of complexes L n E • n − → L n E • n − → E • is split exact in ev ery degree and the complex E • is homoto py equiv alent t o the cone of the morphism L n E • n − → L n E • n (the homotopy inductiv e limit of the complexes E • n ).  2.5. Main theorem for como dules. Assume t ha t the coring C is a ﬂat left and righ t A -mo dule and the ring A has a ﬁnite weak ho mo lo gical dimension. Theorem. The functor m apping the quotient c ate gory of the homotopy c ate gory of c omplex e s o f c oﬂat C -c o mo dules ( c oﬂat c omplexes of C -c omo dules) by its interse ction with the thick sub c ate gory of c o acyclic c omplexes of C -c omo dules into the c o derive d c ate go ry of C -c om o dules is an e quivalenc e of triangulate d c ate gories. Pr o of . W e will show that any complex of C - como dules K • can b e connected with a complex of coﬂat C -como dules in a functorial w ay b y a c hain o f t w o morphisms K • ← − R 2 ( K • ) − → R 2 L 1 ( K • ) with coacyclic cones. Moreo v er, if the complex K • is a complex o f coﬂat C -como dules (coﬂat complex of C -como dules), then the intermediate complex R 2 ( K • ) in this chain is also a complex of coﬂat C -como dules (coﬂat complex of C -como dules). Then we will apply the follow ing Lemma. Lemma. L et C b e a c ate gory and F b e its f ul l sub c ate gory. L et S b e a class of morphisms in C c ontaining the thir d morp hism of any triple of morphi s m s s , t , and 53 st when it c ontains two of them. Supp ose that for any obje ct X in C ther e is a chai n of morphisms X ← F 1 ( X ) → · · · ← F n − 1 ( X ) → F n ( X ) b elongin g to S and functorial ly dep endi n g on X s uch that the ob j e ct F n ( X ) b elongs to F f o r an y X ∈ C an d al l the obje cts F i ( X ) b elong to F for a ny X ∈ F . Then the functor F [( S ∩ F ) − 1 ] − → C [ S − 1 ] induc e d by the emb e dding F − → C is an e quivalenc e of c ate gories. Pr o of . It is ob vious that the functor b etw een the lo calized categories is surjectiv e o n the isomorphism classes of ob jects; let us sho w tha t it is bijectiv e on morphisms. It fol- lo ws from t he condition on the class S that the functor s F i preserv e it . Let U and V be t w o ob jects o f F and φ : U − → V b e a morphism b et w een them in the category C [ S − 1 ]. Applying the functor F n : C − → F , we obtain a mo r phism F n ( φ ) : F n ( U ) − → F n ( V ) in the category F [( S ∩ F ) − 1 ]. The square diag ram of morphisms in the cat ego ry C [ S − 1 ] formed by the morphism φ , the isomorphism b et w een U and F n ( U ), the mor- phism F n ( φ ), and the isomorphism b etw een V and F n ( V ) is comm utative , since it is comp osed from comm utativ e squares of morphisms in the category C . Since the other three morphisms in this commu tative square lift to F [( S ∩ F ) − 1 ], the morphism φ b elongs to the image of the functor F [( S ∩ F ) − 1 ] − → C [ S − 1 ]. Now suppose that t w o morphisms φ and ψ : U − → V in the catego ry F [( S ∩ F ) − 1 ] map to the same morphism in C [ S − 1 ]. Applying the f unctor F n , w e see t hat the morphisms F n ( φ ) and F n ( ψ ) are equal in F [( S ∩ F ) − 1 ]. So w e ha v e tw o comm utativ e squares in the category F [( S ∩ F ) − 1 ] with the same v ertices U , V , F n ( U ), and F n ( V ), the same morphism F n ( U ) − → F n ( V ), the same isomor phisms U ≃ F ( U ) a nd V ≃ F ( V ), and t w o morphisms φ and ψ : U − → V . It follows that the lat t er t w o morphisms are equal.  Let K • b e a comple x of C -como dules. Let P ( M ) − → M denote the functorial surjectiv e morphism on to an arbitrary C -como dule M from an A -ﬂat C -como dule P ( M ) constructed in Lemma 1.1 .3. The functor P is not alw ay s additiv e, but as an y functor from an additiv e categor y to an ab elian o ne it is the direct sum of a constan t functor M 7− → P (0 ) and a functor P + ( M ) = k er( P ( M ) → P (0)) = coker( P (0) → P ( M )) sending zero ob j ects to zero ob jects a nd zero mo r phisms to zero morphisms. F or any C -como dule M , the como dule P + ( M ) is A -ﬂat and the morphism P + ( M ) − → M is surjectiv e. Set P 0 ( K • ) = P + ( K • ), P 1 ( K • ) = P + (k er( P 0 ( K • ) → K • )), etc. F or d large enough, the ke rnel Z ( K • ) o f the morphism P d − 1 ( K • ) − → P d − 2 ( K • ) will b e a complex of A -ﬂat C -como dules. Let L 1 ( K • ) b e the total complex o f the bicomplex Z ( K • ) − − → P d − 1 ( K • ) − − → · · · − − → P 1 ( K • ) − − → P 0 ( K • ) . Then L 1 ( K • ) is a complex of A -ﬂat C -como dules and the cone o f the morphism L 1 ( K • ) − → K • is the total complex of a ﬁnite exact sequence of complexes of C -como dules, and therefore, a coacyclic complex. 54 No w let L • b e a complex of A -ﬂat left C -como dules. Consider the cobar construc- tion C ⊗ A L • − − → C ⊗ A C ⊗ A L • − − → C ⊗ A C ⊗ A C ⊗ A L • − − → · · · Let R 2 ( L • ) b e the total complex of this bicomplex, constructed b y taking inﬁnite direct sums along the diagonals. Then R 2 ( L • ) is a complex of coﬂat C -como dules. The functor R 2 can b e extended to arbitrary complexes of C -como dules; for an y complex K • , the cone of the morphism K • − → R 2 ( K • ) is coacyclic by Lemma 2 .1. Finally , if K • is a coﬂat complex of C -como dules, then R 2 ( K • ) is also a coﬂat complex of C - como dules, since the cotensor product of R 2 ( K • ) with a complex of righ t C -como dules N • coincides with the cotensor pro duct of K • with t he t o tal cobar complex R 2 ( N • ), a nd the latter is coa cyclic whenev er N • is coacyclic. W e hav e constructed the chain of morphisms K • ← − R 2 ( K • ) − → R 2 L 1 ( K • ) with the desired prop erties. The only remaining pro blem is tha t the functor L 1 is not addi- tiv e and therefore not deﬁned on the homotopy category of complexes o f C -como dules, but only on the (a b elian) category of complexes and their morphisms. So w e ha v e to apply Lemma to the catego ry C of complexes of C - como dules, the full sub category F of complexes of coﬂat C -como dules (coﬂat complexes of C -como dules) in it, and the class S of morphisms with coacyclic cones. The corresp onding lo calizations will coincide with the desired quotien t catego ries of homotop y categories due to the following g eneral fact [25, I I I.4.2-3]. F or any DG-category DG where shifts and cones exist the lo calization of the category of closed morphisms in DG with resp ect to the class of homotop y equiv alences coincides with the homoto p y category of DG (i. e., closed mo r phisms homoto pic in DG b ecome equal after inv erting homotop y equiv alences). In particular, this is true f o r any category of complexes ov er an additive category that is closed under shifts and cones.  Remark. Another pro of of Theorem ( f or complexes of coﬂat como dules or coﬂat complexes of A -ﬂat como dules) can b e fo und in 2.6. After Theorem has b een prov en, it turns out that the functors L 1 and R 2 can b e also applied in the rev erse order: for an y complex of C -como dules L • , the complex R 2 ( L • ) is a complex of C / A -coﬂat C -como dules, and for a ny complex of C / A - coﬂat C -como dules K • , the complex L 1 ( K • ) is a complex of coﬂat C - como dules (b y Remark 1 .2.2, whic h dep ends o n Theorem). 2.6. Main theorem for semimo dules. Assume that the coring C is a ﬂa t left and righ t A -mo dule, the semialgebra S is a coﬂat left and right C -como dule, and the ring A has a ﬁnite w eak homolog ical dimension. Theorem. The functor m apping the quotient c ate gory of the homotopy c ate gory of semiﬂat c omplexes of A - ﬂat ( C -c oﬂat, semiﬂat) S -semimo dules by its interse ction with the thick sub c ate gory of C -c o ac yclic c omple x es of S -semimo dules i n to the s e mi- derive d c ate gory of S -semim o dules is an e quivalenc e of triangulate d c ate gories. 55 Pr o of . W e will sho w that in the c hain o f functors mapping the quotient catego r y of (the homoto py category of ) semiﬂat complexes of C - coﬂat (semiﬂat) S -semimo dules b y C -coacyclic semiﬂat complexes o f C -coﬂat S -semimo dules in to the quotien t cate- gory of complex es of C -coﬂa t S -semimo dules by C -coacyclic complexes of C -coﬂat S -semimodules into the quotient category of complexes of A -ﬂat S -semimo dules b y C -coa cyclic complexes o f A -ﬂat S -semim o dules in to the semideriv ed category of S -semimodules all the three functors are equiv alences of categories. Analo g ously , in the c hain of functors mapping the quotien t catego r y of (the homot o p y category of ) semiﬂat complexes of A -ﬂat S -semimodules b y C -coacyclic semiﬂat complexes of A -ﬂat S -semimodules in to the quotien t category of C -coﬂat complexes of A -ﬂat S -semimodules b y C -coacyclic C -coﬂat complexes of A -ﬂat S -semimodules in to the quotien t category of complexes o f A -ﬂat S -semimo dules by C -coacyclic complexes of A -ﬂat S -semimo dules in to the semide riv ed category of S -semimo dules all the three functors are equiv alences o f categories. In or der to prov e this, w e will construct for an y complex of S - semimo dules K • a morphism L 1 ( K • ) − → K • in to K • from a complex of A -ﬂat S -semimo dules L 1 ( K • ), for any complex of A -ﬂat S -semimo dules L • a morphism L • − → R 2 ( L • ) from L • in to a complex of C -coﬂa t S -semimo dules R 2 ( L • ), and for any C -coﬂat complex of A -ﬂat S -semimodules ( complex o f C -coﬂat S -semimo dules) M • a morphism L 3 ( M • ) − → M • in to M • from a semiﬂat complex of A -ﬂat (semiﬂat) S -semimo dules L 3 ( M • ) suc h that in eac h case the cone of this morphism will b e a C -coacyclic complex o f S -semimo d- ules. Then w e will apply t he follow ing Lemma. Lemma. L et H b e a c ate gory and F b e its ful l s ub c ate g ory. L et S b e a l o c alizi n g (i. e., satisfying the O r e c onditions) class of morphisms in H . Assume that for any obje ct X of H ther e exists an obje ct U of F to gether with a morphism U − → X b elonging to S (or for any obje ct X of H ther e exists an obje ct V of F to gether with a morphism X − → V b elonging to S ) . Th e n the functor F [( S ∩ F ) − 1 ] − → H [ S − 1 ] in duc e d by the emb e dding F − → H i s an e quivalenc e of c ate gories. Pr o of . It is ob vious that the functor b etw een the lo calized categories is surjectiv e o n the isomorphism classes of ob jects; let us show that it is bijectiv e o n morphisms. An y morphism in the category H [ S − 1 ] b et w een t w o ob jec ts U and V from F can b e represen ted b y a fraction U ← X → V , where X is an ob ject of H and the morphism X → U b elongs to S . By our assumption, there is an ob ject W fro m F to gether with a morphism W → X from S . Then the fractions U ← X → V and U ← W → V represen t the same morphism in H [ S − 1 ], while the second fraction represen ts also a certain morphism in F [( S ∩ F ) − 1 ]. F urthermore, any t w o morphisms from an ob ject U to an ob jec t V in the cat ego ry F [( S ∩ F ) − 1 ] can b e represen ted b y t w o fractions of the form U ← U ′ ⇒ V , with the same morphism U → U ′ from S ∩ F and t w o diﬀeren t morphisms U ′ ⇒ V (since the class of morphisms S ∩ F in the category F satisﬁes the 56 righ t Ore conditions). If the ima g es of these morphisms in the category H [ S − 1 ] a re equal, then there is a morphism X → U ′ from S with an ob ject X from H suc h tha t t w o comp ositions X → U ′ ⇒ V coincide. Aga in there is an o b ject W from F together with a morphism W → X belonging to S . Since the t w o comp ositions W → U ′ ⇒ V coincide in F , the morphisms represen ted b y the t w o fractions U ← U ′ ⇒ V are equal in F [( S ∩ F ) − 1 ].  Let K • b e a complex of S - semimo dules. Let P ( M ) − → M denote the f unctoria l surjectiv e morphism onto an a r bitrary S -semimo dule M from an A -ﬂat S - semimo dule P ( M ) constructed in Lemma 1.3.2. As explained in the pro of of Theorem 2.5, the functor P is the direct sum of a constan t functor M 7− → P (0) and a functor P + send- ing zero morphisms to zero morphisms. F or any S -semimodule M , the semimo dule P + ( M ) is A -ﬂat and the morphism P + ( M ) − → M is surjectiv e. Set P 0 ( K • ) = P + ( K • ), P 1 ( K • ) = P + (k er( P 0 ( K • ) → K • )), etc. F o r d large enough, the k ernel Z ( K • ) of the morphism P d − 1 ( K • ) − → P d − 2 ( K • ) will b e a complex of A -ﬂat S -semimo dules. Let L 1 ( K • ) b e the total complex of the bicomplex Z ( K • ) − − → P d − 1 ( K • ) − − → · · · − − → P 1 ( K • ) − − → P 0 ( K • ) . Then L 1 ( K • ) is a complex of A -ﬂat S -semimo dules and the cone of the morphism L 1 ( K • ) − → K • is the to tal complex of a ﬁnite exact sequence of complexes of S -semi- mo dules, and therefore, a C -coacyclic complex (and ev en a n S -coacyclic complex). No w let L • b e a complex of A -ﬂat S - semimo dules. Let M − → J ( M ) denote the functorial injectiv e morphism from an arbitrary A -ﬂat S -semimo dule M in to a C -coﬂat S - semimo dule J ( M ) with an A -ﬂat cokerne l J ( M ) / M constructed in Lemma 1.3.3. Set J 0 ( L • ) = J ( L • ), J 1 ( L • ) = J (cok er( L • → J 0 ( L • ))), etc. Let R 2 ( L • ) b e the total complex o f the bicomplex J 0 ( L • ) − − → J 1 ( L • ) − − → J 2 ( L • ) − − → · · · , constructed b y t a king inﬁnite direct sums along the diagonals. Then R 2 ( L • ) is a complex of C -coﬂat S -semimodules and the cone of the morphism L • − → R 2 ( L • ) is a C -coacyclic (and ev en S -coacyclic) complex b y Lemma 2.1. Finally , let M • b e a C -coﬂat complex o f A -ﬂat left S -semimo dules. Then the complex S  C M • is a semiﬂat complex of A -ﬂa t S -semimo dules. Moreov er, if M • is a complex of C -coﬂa t S -semimo dule s, then S  C M • is a semiﬂat complex of semiﬂat S -semimodules. Consider the bar construction · · · − − → S  C S  C S  C M • − − → S  C S  C M • − − → S  C M • . Let L 3 ( M • ) b e the total complex of this bicomplex, constructed b y taking inﬁnite direct sums alo ng the diagonals. Then the complex L 3 ( M • ) is semiﬂat b y 2.4 and the cone of the morphism L 3 ( M • ) − → M • is not o nly C -coacyclic, but ev en C -contractible (the con tracting homot o p y b eing induced b y t he semiunit morphism C − → S ) .  57 Remark. It is clear t ha t the constructions of complexes R 2 ( L • ) and L 3 ( M • ) can b e applied to arbitrary complexes of S -semimo dules, with no (co)ﬂatness conditions im- p osed on them. F or example, an alternative w a y of provin g Theorem is to show that the functors mapping the quotien t categor y of semiﬂat complexes of C -coﬂat (semi- ﬂat) S -semimo dules by C -coacyclic semiﬂat complexes into the quotient categor y of complexes of C / A -coﬂat S -semimo dules b y C -coacyclic complexes in to the semide- riv ed category of S -semimo dules are b oth equiv alences of cat ego ries. Indeed, for an y complex of S -semimo dules L • the complex R 2 ( L • ) is a complex of C / A -coﬂat S -semi- mo dules b y Lemma 1 .3.3 and for any complex of C / A - coﬂa t S -semimo dules K • the complex L 1 ( K • ) is a complex of C -coﬂat S -semimo dules by Remark 1.3.2 (hence the complex L 3 L 1 ( K • ) is a semiﬂat complex o f semiﬂat S - semimo dules). Y et anot her use- ful approac h to prov ing Theorem w as presen ted in 2 .5: an y complex of S -semimo dules K • can b e connected with a semiﬂat complex of semiﬂat S -semimo dules in a f unctorial w a y by a chain of three mo r phisms K • ← − L 3 ( K • ) − → L 3 R 2 ( K • ) ← − L 3 R 2 L 1 ( K • ) with C -coacyclic cones, a nd when K • is a semiﬂat complex of ( A -ﬂat, C -coﬂat , o r semiﬂat) S -semimo dules, a ll the complexes in this c hain a re also semiﬂat complexes of ( A -ﬂat, C -coﬂat, or semiﬂat) S -semimo dules. Question. Is the quotien t category of C -coﬂat complexes of S -semimo dules by the thic k subcategory of C -coacyclic C - coﬂat complex es equiv alen t to the se mideriv ed category of S -semimo dules? 2.7. Derived functor SemiT or. The fo llo wing Lemma prov ides a general approach to double-sided deriv ed functors of (partially deﬁned) functors of t w o arguments . Lemma. L et H 1 and H 2 b e two c ate gories, H b e a (not ne c essarily ful l) sub c ate gory in H 1 × H 2 , and S 1 and S 2 b e lo c alizing clas ses of mo rp h isms in H 1 and H 2 . L et K b e a c ate gory and Θ : H − → K b e a functor. L et F 1 and F 2 b e sub c ate gories in H 1 and H 2 . Assume that b oth functors F i [( S i ∩ F i ) − 1 ] − → H i [ S − 1 i ] ind uc e d by the emb e ddings F i − → H i ar e e quivalenc es of c ate gories and the sub c ate gory H c ontains b oth sub c at- e gories F 1 × H 2 and H 1 × F 2 . F urthermor e, assume that the mo rphisms Θ( U, t ) and Θ( s, V ) ar e isomorphis m s in the c ate gory K for any obje cts U ∈ F 1 , V ∈ F 2 and any morphisms s ∈ S 1 , t ∈ S 2 . Then the r estrictions of the functor Θ to the sub c ate gories F 1 × H 2 and H 1 × F 2 factorize thr ough their lo c aliz a tions by their interse ctions with S 1 × S 2 , so on e c an deﬁne derive d functors D 1 Θ , D 2 Θ : H 1 [ S − 1 1 ] × H 2 [ S − 1 2 ] − → K by r estricting the functor Θ to these sub c ate gories. Mor e over, the deriv e d functors D 1 Θ and D 2 Θ ar e natur al ly isom orphic to e ach other and ther efor e do not dep end o n the choic e of sub c a te gories F 1 and F 2 , pr ovide d that b oth sub c ate gorie s exist. Pr o of . Let us show that for any morphism s ∈ S 1 ∩ F 1 and a n y ob ject X ∈ H 2 the morphism Θ( s, X ) is an isomorphism in K . By assumptions of Lemma, the image of X in H 2 [ S − 1 2 ] is isomorphic to the imag e of a certain ob ject V ∈ F 2 . First supp o se 58 that there exists a fraction X ← Y → V of morphisms from S 2 connecting X and V . Then b oth morphisms of morphisms Θ( s, Y ) − → Θ( s , X ) and Θ( s, Y ) − → Θ( s, V ) are isomorphisms of morphisms, since t he source a nd the target o f s b elong t o F 1 . No w the mor phism Θ( s, X ) is an isomorphism, b ecause the morphism Θ( s, V ) is an isomorphism. In the general case, there exist a fraction X ← Y → V connecting X and V a nd t w o morphisms Y ′ → Y and V → V ′ suc h that the mor phism Y → X and tw o comp ositions Y ′ → Y → V and Y → V → V ′ b elong to S 2 . Then the comp ositions of mor phisms of morphisms Θ( s, Y ′ ) − → Θ( s , Y ) − → Θ( s, V ) and Θ( s, Y ) − → Θ( s, V ) − → Θ( s, V ′ ) are isomorphisms of morphisms, so the morphism of morphisms Θ( s, Y ) − → Θ( s, V ) is b oth left and righ t in v ertible, and therefore, is an isomorphism of morphisms. Since the morphism of morphisms Θ( s, Y ) − → Θ( s, X ) is also an isomorphism of mor phisms and the mo r phism Θ( s, V ) is an isomorphism, one can conclude that the mor phism Θ( s, X ) is also an isomorphism. Th us the deriv ed functor D 1 Θ is deﬁned; it remains to construct an isomorphism b et w een D 1 Θ and D 2 Θ. But the comp ositions of the functors D 1 Θ and D 2 Θ with the functor F 1 [( S 1 ∩ F 1 ) − 1 ] × F 2 [( S 2 ∩ F 2 ) − 1 ] − → H 1 [ S − 1 1 ] × H 2 [ S − 1 2 ] coincide by deﬁnition, and t he latter functor is an equiv alence of categories.  Assume that the coring C is a ﬂat left and righ t A - mo dule, the semialgebra S is a coﬂat left and righ t C -como dule, a nd the ring A has a ﬁnite w eak homological dimension. The double-sided deriv ed functor SemiT or S on the Carthesian pro duct of the semi- deriv ed categories of right and left S -semimo dules is deﬁne d as follows. Consider the partially deﬁned functor of semitensor pro duct of complexes of S -semimodules ♦ S : Hot ( simo d – S ) × Hot ( S – simo d ) 99 K Hot ( k – mo d ). This functor is deﬁned on the full sub catego ry of the Carthesian pro duct of homotopy categories that consists of pairs of complexes ( N • , M • ) suc h that either N • or M • is a complex of A -ﬂa t S - semi- mo dules. Compose it with the functor of lo calization Hot ( k – mo d ) − → D ( k – mo d ) and restrict to the Carthesian pro duct of the homotopy category of semiﬂat com- plexes of A -ﬂat right S -semimodules and the homotop y category o f complexes of left S -semimodules. By the deﬁnition, the functor so obtained factorizes through the semideriv ed cat e- gory of left S -semimodules in the second a rgumen t, a nd it follows from Theorem 2.6 and the ab ov e Lemma that it factorizes through the quotien t category of the homo- top y category o f semiﬂat complexes of A -ﬂat righ t S -semimo dules by its interse ction with the thic k sub catego r y of C - coacyclic complexes in t he ﬁrst argumen t. Explicitly , let N • b e a C - coacyclic semiﬂat complex of A - ﬂat right S -semimo dules and M • b e a complex of left S -semimo dules. Using the constructions f r o m the pro of of Theorem 2.6, connect M • with a semiﬂat complex of A -ﬂat left S -semimodules L • b y a c hain of morphisms with C -coacyclic cones. Then the complexes N • ♦ S M • 59 and N • ♦ S L • are connected by a c hain of quasi-isomorphisms, and since the complex N • ♦ S L • is acyclic, the complex N • ♦ S M • is acyclic, to o. Th us w e hav e constructed the double-sided derive d functor SemiT or S : D si ( simo d – S ) × D si ( S – simo d ) − − → D ( k – mo d ) . According to Lemma, the same derived functor can b e obtained b y restricting the functor of semitensor pro duct to the Carthesian pro duct of the homoto py category of complexes of left S -semimo dules and the homot op y category of semiﬂat complexes of A - ﬂat rig h t S -semimo dules, or indeed, to the Carthesian pro duct of the homotopy categories of semiﬂat complexes of A -ﬂat righ t and left S -semimo dules. One can a lso use semiﬂat complexes of C -coﬂat S -semimo dules or semiﬂat complexes o f semiﬂat S -semimodules instead of semiﬂat complexes of A -ﬂat S -semimo dules. In particular, when the coring C is a ﬂat left and righ t A - mo dule and the ring A has a ﬁnite we ak homolog ical dimension, one deﬁnes the do uble-sided deriv ed functor Cotor C : D co ( como d – C ) × D co ( C – como d ) − − → D ( k – mo d ) b y comp osing t he functor o f cotensor pro duct  C : Hot ( como d – C ) × Hot ( C – como d ) − → Hot ( k – mo d ) with the functor of lo calization Hot ( k – mo d ) − → D ( k – mo d ) a nd r e- stricting it to the Carthesian pro duct of the homotopy category of complexes of coﬂat rig h t C - como dules and the homotopy category of arbitrary complexes of left C -como dules. The same deriv ed functor is obtained b y restricting the functor of cotensor pro duct to the Carthesian pro duct of the homotop y category of a rbitrary complexes of righ t C -como dules and the homotopy categor y of complexes o f coﬂat left C -como dules, o r indeed, to the Carthesian pro duct of the homotopy categories of coﬂat righ t C -como dules and coﬂat left C -como dules. One can a lso use coﬂat com- plexes of C -como dules or coﬂat complexes of A -ﬂat C -como dules instead o f complexes of coﬂat C -como dules. Remark. One can deﬁne a vers ion of deriv ed functor Cotor without making a n y ho- mological dimension a ssumptions b y considering pro- ob jects in the spirit o f [23, 24]. Let k – mo d ω denote the category of pro-ob jects ov er the categor y k – mo d that can b e represen ted b y coun table ﬁltered pr o jectiv e systems o f k - mo dules; this is an ab elian tensor category with exact functors of coun table ﬁltered pro jectiv e limits and a righ t exact functor of tensor pro duct comm uting with countable ﬁltered pro jec tiv e limits. Let A b e a ring ob ject in k – mo d ω ; then o ne can consider righ t and left A -mo dule ob jects and A - A -bimo dule ob jects in k – mo d ω , which we will simply call right and left A -mo dules and A - A -bimo dules. F urt hermore, let C b e a coring ob ject in t he tensor category of A - A -bimo dules; we will consider C -como dule ob jects in the categories of righ t and left A -mo dules and call them righ t and left C -como dules. Deﬁne the functor of cotensor pro duct ov er C taking v alues in the category k – mo d ω in the usual 60 w a y and extend it to the Carthesian pro duct o f the homotop y categories of com- plexes o f right and left C - como dules b y taking inﬁnite pro ducts along the dia g onals in the bicomplex of cotensor pro ducts. The categories of right and left A -mo dules are ab elian. Assume that C is a ﬂat left a nd righ t A - mo dule; then t he categories of righ t and left C -como dules are also ab elian. Deﬁne the semideriv ed categories o f righ t and left C -como dules as the quotien t categor ies of the homoto p y categories by the thic k sub cat ego ries o f A -contraacyclic complexes (the con traacyclic complexes b eing deﬁned in terms of countable pro ducts). The n one can use Lemma to deﬁne the double-sided derived functor ProCotor C of cotensor pro duct on the Carthesian pro duct of the semideriv ed categories of right and left C -como dules in terms of coﬂat complexes of C -como dules. In order to obtain for any complex of C -como dules M • a coﬂat complex of C -como dules connected with M • b y a functorial chain of tw o mor- phisms with A -contraacyclic cones one needs to construct a surjectiv e morphism onto an y C -como dule M from an A -ﬂat C -como dule F ( M ). This construction is dual t o that of Lemma 1.3.3 and uses the surjectiv e map on to any A - mo dule M from an A -ﬂa t A -mo dule G ( M ) = A ⊗ ω k M ′ , where M ′ is a pro- k - mo dule represen ted b y a coun table ﬁltered pro jec tiv e sys tem of ﬂat k - mo dules mapping onto the pro- k - mo dule M and ⊗ ω k denotes t he functor of tensor pro duct in k – mo d ω . The A -ﬂat C -como dule F ( M ) is obtained as the pro jectiv e limit in k – mo d ω of the pro jectiv e system of C -como dules M ← − Q ( M ) ← − Q ( Q ( M )) ← − · · · G iv en a complex of A -ﬂat C -como dules M • , a coﬂat complex of C -como dules endo wed with a morphism fro m the complex M • with an A -contractible cone is obta ined as the total complex of the cobar complex of M • , constructed b y ta king inﬁnite pro ducts along the diagona ls. One can also consider the category of arbitrary pro- k - mo dules in place o f k – mo d ω . Notice tha t for a conv en tional coalgebra C ov er a ﬁeld A = k and complexes o f C - como dules N • and M • in the category of k -v ector spaces that are b oth b o unded from ab ov e o r from b elo w the o b ject of the derived category o f k -ve ctor spaces obtained by applying the deriv ed functor of pro jectiv e limit to the ob ject ProCotor C ( N • , M • ) of the deriv ed category D ( k – vect ω ) coincides with Cotor C ,I ( N • , M • ) ( see 0.2.10). 2.8. R elatively semiﬂat complexes. W e k eep the a ssumptions and nota tion of 2.5, 2.6, and 2.7. One can compute the deriv ed functor Coto r C using resolutions of a diﬀerent kind. Namely , the cotensor pro duct N •  C M • of a complex of A -ﬂa t right C - como dules N • and a complex of C / A -coﬂat C -como dules M • represen ts an ob ject natura lly isomorphic to Cotor C ( M • , N • ) in the derive d cat ego ry of k -mo dules. Indeed, the complex R 2 ( N • ) is a complex of coﬂat C -como dules and the cone of the morphism N • − → R 2 ( N • ) is coacyclic with r esp ect to the exact category of A -ﬂat right C -como dules, hence the morphism N •  C M • − → R 2 ( N • )  C M • is a quasi-isomorphism. One can pro v e that the cotensor pro duct of a complex coa cyclic with resp ect to the 61 exact catego ry of A -ﬂat C -como dules and a complex of C / A -coﬂat C -como dules is acyclic in the w ay completely analogous to the pro of of Lemma 2 .2. One can also compute the deriv ed functor SemiT or S using resolutions of diﬀeren t kinds. Namely , a complex of left S -semimo dules is called sem iﬂat r elative to A if its semitensor pro duct with any complex of A -ﬂat right S -semimo dules that as a complex of C -como dules is coacyclic with resp ect to exact category of A -ﬂat right C -como dules is acyclic (cf. Theorem 7.2.2(a)). F or example, the complex of S -semimo dules induced from a complex of C / A - coﬂat C -como dules is semiﬂat relative to A , hence the com- plex L 3 R 2 ( K • ) is semiﬂat relativ e to A for any complex of left S - semimo dules K • . The semitensor pro duct N • ♦ S M • of a complex of A -ﬂat rig h t S -semimo dules N • and a complex of left S -semimo dules M • semiﬂat relativ e to A represen ts an ob ject naturally isomorphic to SemiT or S ( N • , M • ) in the derived category of k - mo dules. In- deed, L 3 R 2 ( N • ) is a semiﬂat complex of right S -semimo dules connected with N • b y a c hain of morphisms N • − → R 2 ( N • ) ← − L 3 R 2 ( N • ) whose cones are coa cyclic with resp ect to the exact categor y of A -ﬂat C -como dules and con tractible ov er C , resp ec- tiv ely . Hence there is a c hain of t w o quasi-isomorphisms connecting N • ♦ S M • with L 3 R 2 ( N • ) ♦ S M • . Analogously , a complex of left S -semimo dules is called sem iﬂat r elative to C if it s semitensor pro duct with an y C -contractible complex of C -coﬂat right S -semimo dules is acyclic. F or example, the complex of S -semimo dules induced from an y complex of C -como dules is semiﬂat relativ e to C , hence the complex L 3 ( K • ) is semiﬂat relativ e to C for any complex of left S -semimo dules K • . The semitensor pro duct N • ♦ S M • of a complex of C -coﬂat righ t S -semimodules N • and a complex of left S -semimo dules M • semiﬂat relativ e to C represen ts an ob ject naturally isomorphic to SemiT or S ( N • , M • ) in the deriv ed category of k -mo dules. Indeed, L 3 ( N • ) is a semiﬂat complex of righ t S -semimodules and the cone o f the morphism L 3 ( N • ) − → N • is a C -contr a ctible complex of C -coﬂat right S -semimo dules. It follow s that the semitensor pr o duct of a complex of left S -semimo dules semiﬂat relativ e to C with a C -coacyclic complex of C -coﬂat righ t S -semimo dules is acyclic. A t last, a complex of A -ﬂa t right S -semimo dules is called se miﬂat r elative to C r el- ative to A ( S / C / A - semiﬂat) if its semitensor pro duct with any C - contr actible complex of C / A -coﬂat left S -semimo dules is acyclic. F or example, the complex of S -semimo d- ules induced from a complex o f A -ﬂat C - como dules is S / C / A -semiﬂat, hence the com- plex L 3 L 1 ( K • ) is S / C / A -semiﬂat for any complex of righ t S - semimo dules K • . The semitensor pro duct N • ♦ S M • of an S / C / A -semiﬂat complex of A -ﬂat righ t S -semi- mo dules N • and a complex of C / A - coﬂat left S -semimo dules M • represen ts an ob ject naturally isomorphic to SemiT or S ( N • , M • ) in the derived category of k - mo dules. In- deed, L 3 ( M • ) is a complex of left S -semimo dules semiﬂat r elativ e to A and the cone of the morphism L 3 ( M • ) − → M • is a C -contractible complex of C / A -coﬂat righ t S -semimodules. It follows t ha t the semitensor pro duct of an S / C / A -semiﬂat complex 62 of A -ﬂat r igh t S -semimo dules with a C -coacyclic complex of C / A -coﬂat left S - semi- mo dules is acyclic. The functors mapping the quotien t catego ries of the homotop y categories o f com- plexes of S -semimo dules semiﬂat relative to A , complexes of S -semimo dules semiﬂat relativ e to C , and S / C / A -semiﬂat complexes of A -ﬂat S -semimo dules b y their in ter- sections with the thic k sub cat ego ry of C -coacyclic complexes in to the semideriv ed category o f S -semimo dules are equiv alences of t riangulated categories. The same applies to complexes of A -ﬂat, C -coﬂat, or C / A - coﬂat S -semimo dules. These results follo w easily from either of Lemmas 2 .5 or 2.6 . So one can deﬁne the deriv ed func- tor SemiT or S b y restricting the functor of semitensor pro duct to these categories of complexes of S -semimo dules as explained ab ov e. Remark. Assuming that C is a ﬂat righ t A -mo dule, S is a coﬂat righ t and a C / A -coﬂat left C -como dule, and A has a ﬁnite w eak homological dimension, o ne can deﬁne the double-sided deriv ed functor SemiT or S on the Carthesian pro duct of the semideriv ed catego r y of A -ﬂat right S - semimo dules and t he semideriv ed category of left S -semimo dules. The former is deﬁned as the quotien t catego ry of the homo- top y category of complexes of A -ﬂat right S -semimo dules by the thic k sub category of complexes that as complexes of C -como dules are coacyclic with resp ect to the exact category of A -ﬂat rig ht C -como dules. The deriv ed f unctor is constructed by restrict- ing the f unctor of semitensor pro duct to the Carthesian pro duct of the homotop y category o f complexes of A -ﬂat right S -semimo dules and the homo t op y catego ry of complexes of left S -semimo dules semiﬂat relativ e to A , or the Carthesian pro duct of the homotopy category o f semiﬂat complexes of A -ﬂat righ t S -semimo dules and the homotop y category o f complexes of left S -semimo dules. Assuming that C is a ﬂat left and right A -mo dule, S is a ﬂat left A -mo dule and a coﬂat righ t C - como dule, and A has a ﬁnite w eak homological dimension, one can deﬁne the left deriv ed func- tor SemiT or S on the Carthesian pro duct of the semideriv ed category of C / A -coﬂat righ t S - semimo dules and the semideriv ed catego ry of left S -semimo dules. The for- mer is deﬁned as the quotien t category of t he homotop y category of complexes o f C / A -ﬂat right S -semimo dules by the thick sub category of complexes that as com- plexes of C -como dules are coacyclic with r esp ect to the exact category of C / A -coﬂat righ t C -como dules (cf. Remark 7.2 .2 ). The deriv ed functor is constructed by restrict- ing the f unctor of semitensor pro duct to the Carthesian pro duct of the homotop y category of complexes of C / A -coﬂat right S - semimo dules and the homotopy category of S / C / A -semiﬂat complexes of A -ﬂat left S - semimo dules, or the Carthesian pro duct of the homot o p y categor y of semiﬂat complexes of C -coﬂat right S -semimo dules and the homotop y category of complexes of left S -semimo dules. Both of t hese deﬁnitions of deriv ed functors are pa r t icular cases of Lemma 2.7. 63 2.9. R emarks on derived semitensor pro duct of bisemimo dules. W e w ould lik e to deﬁne the double-sided deriv ed functor of semitensor pro duct o f bisemimo dules and in suc h a w ay that deriv ed semitensor pro ducts of sev eral fa ctors w ould b e asso ciativ e. It app ears that there are t w o approache s to this problem, ev en in the case of modules ov er ring s. First supp ose that w e only wish to ha v e asso ciative deriv ed semitensor pro ducts o f three factors. Let S b e a semialgebra o v er a coring C and T b e a semialgebra o v er a coring D , b oth satisfying the conditions o f 2 .6 . The semideriv ed categor y of S - T - bisemimo dules D si ( S – simo d – T ) is deﬁned as the quotien t category of the homotopy category Hot ( S – simo d – T ) by the thic k sub cate- gory o f complexes of bisemimo dules that as complexes of C - D -bicomo dules a re co- acyclic with r esp ect t o the ab elian category of C - D -bicomo dules. W e would lik e to deﬁne deriv ed functors of semitensor pro duct ♦ D S : D si ( simo d – S ) × D si ( S – simo d – T ) − − → D si ( simo d – T ) ♦ D T : D si ( S – simo d – T ) × D si ( T – simo d ) − − → D si ( S – simo d ) and prov e the asso ciativity isomorphism SemiT or T ( N • ♦ D S K • , M • ) ≃ SemiT or S ( N • , K • ♦ D T M • ) . Let us call a complex of C -coﬂat righ t S -semimo dules quite semiﬂat if it b elongs to the minimal triangula t ed sub category of the homotop y cat ego ry of S -semimo dules con taining the complexes induced fr o m complexes of coﬂat right C -como dules and closed under inﬁnite direct sums. One can show (see Remark 7.2.2 and the pro of of Theorem 8.2.2) that the quotient category of the catego r y of quite semiﬂat complexes of C -coﬂat S -semimo dules by its minimal tria ngulated sub category containing the complexes of S -semimo dules induced from complexes of C - como dules coacyclic with resp ect to the exact category of C -coﬂat C -como dules and closed under inﬁnite direct sums is equiv alen t to the semideriv ed category of S -semimo dules. In ot her w ords, an y C -coacyclic quite semiﬂat complex of C -coﬂa t S -semimo dules can b e obtained from the complexes of S -semimo dules induced from the total complexes o f exact triples of complexes of coﬂat C -como dules using the op erations of cone and inﬁnite direct sum. It fo llows (b y Lemmas 2.2 and 1.2.2) that the r estriction of the functor of semi- tensor pro duct Hot ( simo d – S ) × Hot ( S – simo d – T ) 99K D si ( simo d – T ) to the Carthesian pro duct of the homot op y category of quite semiﬂat complexes of C -coﬂat r igh t S -semi- mo dules and the homoto py category of complexes of S - T - bisemimo dules factor izes through the Carthesian pro duct of semideriv ed categories of right S -semimo dules and S - T -bisemimo dules. So the desired deriv ed functors are deﬁned; a nd the asso ciativity isomorphism follo ws from Prop osition 1 .4.4. Not ice that this deﬁnition of a double- sided deriv ed functor is n o t a particular case of the construction of Lemma 2.7. 64 Question. Can o ne use arbitrary semiﬂat complexes of C - coﬂat S -semimo dules or, at least, semiﬂat complexes o f semiﬂat S - semimo dules instead of quite semiﬂat com- plexes in this construction? In other w ords, a ssume that N • is a C - coacyclic semiﬂat complex of semiﬂat right S -semimodules and K is an S - T -bisemimo dule. Is t he complex N • ♦ S K necess arily D -coacyclic? (Cf. 4 .9.) No w supp ose that w e w ant to ha v e deriv ed semitensor pro ducts of an y num b er of factors. Let S b e a semialgebra ov er a coring C ov er a k -a lgebra A , T b e a semialgebra o v er a coring D ov er a k -algebra B , and R b e a semialgebra o v er a coring E ov er a k -alg ebra F , all three satisfying the conditions of 2.6. W e w ould lik e to deﬁne the deriv ed functor of semitensor pro duct ♦ D R : D si ( S – simo d – R ) × D si ( R – simo d – T ) − − → D si ( S – simo d – T ) . This can b e done, assuming tha t the k -alg ebras A , B , and F are ﬂat k -mo dules. Let us call a complex o f F -ﬂat S - R -bisemimo dules str ongly R -semiﬂat if its semi- tensor pro duct ov er R with any E - D -coacyclic complex of R - T -bisemimo dules is a C - D -coacyclic complex of S - T -bisemim o dules for an y semialgebra T . Using bimo dule v ersions of the constructions of L emmas 1.3.2 and 1.3.3, one can pro v e that the quo- tien t catego ry of the homotop y category of strongly R -semiﬂat complexes o f F -ﬂat S - R -bisemimo dules by it s in tersection with the thic k sub category of C - E -coacyclic bi- semimo dules is equiv alen t to the semideriv ed category of S - R -bisemimo dules, and the analogous result holds f or the homotopy category of strongly S -semiﬂat and strongly R -semiﬂat complexes of A -ﬂat and F -ﬂat S - R -bisemimo dules. One just uses the func- tor G ( M ) = L m ∈ M A ⊗ k F in the construction o f Lemma 1.1.3, considers the bicoac- tion and bisemiaction mor phisms in place o f t he coaction and semiaction morphisms, etc. (As w e only w an t our A - F -bimo dules t o b e A -ﬂat a nd F -ﬂat, no assumption ab out the homological dimension of A ⊗ k F is needed.) So Lemma 2.7 is applica- ble to the functor of semitensor pro duct Hot ( S – simo d – R ) × Hot ( R – simo d – T ) 99K D si ( S – simo d – T ) and we obtain the desired double-sided deriv ed f unctor. There is a n asso ciativit y isomorphism ( N • ♦ D S K • ) ♦ D T M • ≃ N • ♦ D S ( K • ♦ D T M • ). In the case of deriv ed cotensor pro duct of bicomo dules, one do es not need to in tro duce quite coﬂat or stro ng ly coﬂat complexes . It suﬃces to consider complexes of C -coﬂat C -como dules or complexes of ( C -coﬂat and) E -coﬂat C - E - bicomo dules. One can deﬁne double-sided derive d functors  D C : D co ( como d – C ) × D co ( C – como d – D ) − − → D co ( como d – D )  D D : D co ( C – como d – D ) × D co ( D – como d ) − − → D co ( C – como d ) and prov e the asso ciativity isomorphism Cotor D ( N •  D C K • , M • ) ≃ Coto r C ( N • , K •  D D M • ) 65 b y replacing the complex of righ t C -como dules N • with a complex of coﬂat righ t C -como dules and the complex of left D -como dules M • b y a complex of coﬂat left D -como dules represen ting the same ob ject in the co deriv ed category of como dules. The deriv ed functors  D C and  D D are w ell-deﬁned, since an y coacyclic complex of coﬂat como dules is coacyclic with resp ect to the exact category of coﬂat como dules (see 7.2.2). If the k -mo dules A and F are ﬂat, one can pro v e that the quotient category of the homotopy category of E -coﬂat C - E -bicomo dules b y its in tersection with the thick sub category o f coacyclic complexes of C - E -bicomo dules is equiv alent to the co deriv ed category o f bicomo dules, and the same applies to the ho mo t o p y category of C -coﬂat and E -coﬂat C - E - bicomo dules. Then one can apply Lemma 2.7 in order to deﬁne the double-sided deriv ed functor  D E : D co ( C – como d – E ) × D co ( E – como d – D ) − − → D co ( C – como d – D ) and t here is an asso ciativit y isomorphism ( N •  D C K • )  D D M • ≃ N •  D C ( K •  D D M • ). 66 3. Semicontramodules and Semihomomorphisms Throughout Sections 3–11, k ∨ is an injective cogenerator of the category of k -mo dules. One can alw ay s t ak e k ∨ = Hom Z ( k , Q / Z ). 3.1. Contramodules. F or tw o k -algebras A and B , we will denote b y A – mo d – B the category of k -mo dules with an A - B -bimo dule structure. 3.1.1. The iden tit y Hom A ( K ⊗ A M , P ) ≃ Hom A ( M , Hom A ( K , P ) for left A -mo dules M , P and an A - A -bimo dule K means that the category opp o site to the category of left A -mo dules is a r igh t mo dule category o v er the tensor category of A - A -bimo dules with the functor of righ t a ction ( N , P op ) 7− → Hom( N , P ) op . Therefore, one can consider mo dule ob jects in this mo dule catego r y o v er ring ob jects in A – mo d – A an como dule ob jects in this mo dule category ov er coring ob jects in A – mo d – A . Clearly , a ring ob j ect B in A – mo d – A is just a k -a lgebra endo w ed with a k - algebra morphism A − → B . A B -mo dule in A – mo d op is an A -mo dule P endo w ed with a map P − → Hom A ( B , P ); so one can easily see that B -mo dules in A – mo d op are just (ob jects of the cat ego ry opp osite to the catego ry of ) usual left B -mo dules. Let C b e a coring o v er A . The category o f lef t c ontr amo dules ov er C is the op- p osite category to the category of como dule ob jects in the right mo dule cat ego ry A – mo d op o v er the cor ing ob ject C in the tensor category A – mo d – A . In other w ords, a left C -contramo dule P is a left A -mo dule endo w ed with a left c ontr aaction ma p Hom A ( C , P ) − → P , whic h should b e a morphism of left A -mo dules satisfying the follo wing c ontr aas s o ciativity and c ounity equations. F ir st, t w o maps f rom the mo d- ule Hom A ( C ⊗ A C , P ) = Hom A ( C , Hom A ( C , P ) to the mo dule Hom A ( C , P ), one of whic h is induced b y the com ultiplication map of C and the other b y the con traaction map, should ha v e equal comp o sitions with the contraaction map Hom A ( C , P ) − → P , and second, the comp osition P = Hom A ( A, P ) − → Hom A ( C , P ) − → P o f the map induced b y the counit map of C with the con traaction ma p should b e equal to the iden tit y map of P . A ri g ht c ontr amo dule R ov er C is a right A -mo dule endow ed with a right c ontr aaction map Hom A op ( C , R ) − → R , whic h should b e a map of right A -mo dules satisfying t he analogous equations. 3.1.2. The standard example of a C -contramo dule: for an y righ t C -como dule N endo w ed with a left a ction o f a k -algebra B by C -como dule endomorphisms and an y left B - mo dule V , the left A -mo dule Hom B ( N , V ) has a natural left C -contra mo dule structure. The left C - contramo dule Hom A ( C , V ) is called the C -contramo dule induc e d from a left A -mo dule V . According to Lemma 1.1.2, t he k -mo dule of con tramo dule homomorphisms fr o m the induced C -contramo dule to an arbitrar y C -contramo dule is described b y the form ula Hom C (Hom A ( C , V ) , P ) ≃ Hom A ( V , P ). W e will denote the category of left C -contramo dules by C – contra a nd the category of righ t C -contra mo dules b y contra – C . The category of left C - contramo dules is a b elian 67 whenev er C is a pro jectiv e left A -mo dule. Moreov er, the left A -mo dule C is pro jectiv e if and only if the category C – contra is ab elian and the forgetful functor C – contra − → A – mo d is exact. This can b e pro v en by the same adjoin t f unctor ar g umen t as the analogous result for C - como dules. F or an y coring C , there are tw o natural exact categories of left con tramo dules: the exact category of A -injectiv e C -contramo dules and the exact category of arbit r a ry C -contramo dules with A - split exact triples. Besides , an y morphism of C -contramo d- ules has a k ernel and the forgetful functor C – contra − → A – mo d preserv es k ernels. When a morphism of C -contramo dules has the pr o p ert y that its cok ernel in t he category of A -mo dules is preserv ed b y the functors of homomorphisms from C and C ⊗ A C o v er A , this cok ernel has a natural C -contramo dule structure, which mak es it the cok ernel of that morphism in the category of C -contramo dules. Inﬁnite pro ducts alwa ys exist in t he category o f C -contramo dules and the forgetful functor C – contra − → A – mo d preserv es them. The induction functor A – mo d − → C – contra pres erv es b oth inﬁnite direct sums and inﬁnite pro ducts. T o construct direct sums of C -contramo dules, one can presen t them as cok ernels of mor phisms of induced con tramo dules, a nd all cokerne ls exist in the category of C - contr a mo dules [4 ], so the category of C -contra mo dules has inﬁnite direct sums. Question. If C is a ﬂa t right A -mo dule, then subcomo dules o f ﬁnite direct sums of copies of C constitute a set of generators of the categor y of left C - como dules [16 ]. Do es the category of C - contr a mo dules ha v e a set of cogenerators? 3.1.3. Assume that the coring C is a pro jec tiv e left and a ﬂat right A -mo dule and the ring A has a ﬁnite left homological dimension (ho mological dimension o f the category of left A -mo dules). Lemma. (a) Ther e exists a (not always additive) functor a ssigning to any le f t C -c omo dule a s urje ctive map onto it fr om an A -pr oje c tive C -c omo d ule. Mor e o v e r, the kernel of this map is an iter ate d extension of c oinduc e d C -c omo dules. (b) Ther e ex i s ts a ( n ot always additive) functor assigni n g to any lef t C -c ontr a- mo dule an inje ctive map fr om it into an A -inje ctive C -c o ntr amo dule. Mor e over, the c okernel of this map is an iter ate d extension of induc e d C -c ontr amo dules. Pr o of . The pro of o f part (a) is completely ana logous to the pro of of Lemma 1.1.3 and part (b) is pro v en in the f ollo wing w ay . Let P − → G ( P ) b e an injectiv e map from an A -mo dule P in t o an injectiv e A -mo dule G ( P ) functorially dep ending on P . F or example, one can tak e G ( P ) to b e the direct pro duct of copies of the A -mo dule Hom A ( A, k ∨ ) n um b ered b y all k -mo dule homomorphisms P − → k ∨ . Let P b e a left C - contramo dule. Consider the con traa ctio n map Hom A ( C , P ) − → P ; it is a surjectiv e morphism o f C -contra mo dules; let K ( P ) denote its k ernel. Let Q ( P ) b e the cok ernel of the comp osition K ( P ) − → Hom A ( C , P ) − → Hom A ( C , G ( P )). Then the 68 comp osition of maps Hom A ( C , P ) − → Hom A ( C , G ( P )) − → Q ( P ) fa ctorizes through the surjection Hom A ( C , P ) − → P , so there is a natural injectiv e morphism of C - con- tramo dules P − → Q ( P ). Let us show that the injective dimension di A Q ( P ) of the A -mo dule Q ( P ) is smaller than that of P . Indeed, the A -mo dule Hom A ( C , G ( P )) is injectiv e, hence di A Q ( P ) = di A K ( P ) − 1 6 di A Hom A ( C , P ) − 1 6 di A ( P ) − 1, b ecause the A -mo dule K ( P ) is a direct summand of the A -mo dule Hom A ( C , P ) and an injectiv e resolution of the A -mo dule Hom A ( C , P ) can b e constructed by applying the functor Hom A ( C , − ) to an injectiv e resolution of P . Notice that t he coke rnel of the map P − → Q ( P ) is an induced C -contramo dule Hom A ( C , G ( P ) / P ). It remains to iterate the functor P 7− → Q ( P ) suﬃcien tly many times.  3.2. Cohomomorphisms. 3.2.1. The k -mo dule o f c ohomom orphisms Cohom C ( M , P ) from a left C -como dule M to a left C - contramo dule P is deﬁned as the cok ernel of the pair of maps Hom A ( C ⊗ A M , P ) = Hom A ( M , Hom A ( C , P )) ⇒ Hom A ( M , P ) one o f whic h is induced b y the C - coaction in M and the o t her b y the C -con traaction in P . The functor of co- homomorphisms is neither left no r right exact in general; it is right exact if the ring A is semisimple. F or any left A - mo dule U a nd a ny left C -contramo dule P there is a nat- ural isomorphism Cohom C ( C ⊗ A U, P ) ≃ Hom A ( U, P ), and for any left C -como dule M and an y left A -mo dule V there is a na t ur a l isomorphism Cohom C ( M , Hom A ( C , V )) ≃ Hom C ( M , V ). These assertions fo llo w fro m Lemma 1.2.1. Explicitly , the ﬁrst iso- morphism can b e obtained b y applying the f unctor Hom A ( U, − ) to the split exact sequence of A -mo dules Hom A ( C ⊗ A C , P ) − → Hom A ( C , P ) − → P and the second one can b e o bta ined by applying the functor Hom A ( − , V ) to the split exact sequence of A -mo dules M − → C ⊗ A M − → C ⊗ A C ⊗ A M . 3.2.2. Assuming that C is a pro jectiv e left A -mo dule, a left como dule M ov er C is called c o pr oje ctive if the functor o f cohomomorphisms from M is exact on the category o f left C - contramo dules. It is easy to see that any copro jective C - como dule is a pro jective A - mo dule. The C - como dule coinduced from a pro jectiv e A -mo dule is copro jectiv e. Assuming that C is a ﬂat righ t A -mo dule, a left con tramo dule P o v er C is called c o inje ctive if the functor of cohomomorphisms in to P is exact o n t he category of left C -como dules. An y coinjectiv e C -contramo dule is an injectiv e A -mo dule. The C -contramo dule induced fro m an injective A -mo dule is coinjective . A left como dule M ov er C is called c opr oje ctive r elative to A ( C / A - copro jectiv e) if the functor of cohomomorphisms from M maps exact triples of A -inj ective C -contra- mo dules to exact triples. A left contramodule P ov er C is called c oinje ctive r ela tive to A ( C / A -coinjectiv e) if the functor of cohomomorphisms in to P maps exact triples of A -projective C -como dules to exact triples. Any coinduced C -como dule is C / A -co- projectiv e and any induced C - contramo dule is C / A -coinjectiv e. 69 F or any right C -como dule N and any left C -como dule M there is a natural isomor- phism Hom k ( N  C M , k ∨ ) ≃ Cohom C ( M , Hom k ( N , k ∨ )). Therefore, an y copro jec- tiv e C -como dule M is coﬂat and any C / A -copro jectiv e C -como dule M is C / A -coﬂa t. Besides, a righ t C -como dule N is coﬂat if and only if the left C - contramo dule Hom k ( N , k ∨ ) is coinjectiv e; if a righ t C -como dule N is C / A -coﬂat, then the left C -con- tramo dule Hom k ( N , k ∨ ) is C / A -coinjectiv e ( a nd the con v erse can b e deduced from Lemma 3.1.3(a) and the pro of of Lemma b elow in the assumptions of 3.1.3). It app ears t hat the notion of a relative ly copro jective left C -como dule is useful when C is a ﬂat right A -mo dule, and the notion of a relatively coinjectiv e left C -con- tramo dule is useful when C is a pro jectiv e left A -mo dule. Lemma. ( a) Assume that C is a ﬂat right A -mo dule. Then the class of C / A -c opr oje c- tive left C -c omo dules is close d under extensi o ns and c ok ernels of in je ctive m o rphisms. The f unc tor of c o homomorphism s in to an A -inje ctive l e ft C -c ontr amo dule maps exact triples o f C / A -c opr oje ctive left C -c omo dules to exact triples. (b) Assume that C is a pr oje ctive left A -mo dule. T h en the class of C / A -c oinje c tive left C -c ontr am o dules is c l o se d und e r extensions an d kernels of surje ctive mo rp hisms. The functor of c ohomomorphis m s fr om an A -pr oje ctive left C -c omo dule maps exact triples o f C / A -c oinje ctive left C -c ontr amo dules to exact triples. Pr o of . P art (a): these results fo llo w from the standard prop erties of the left deriv ed functor o f the righ t exact functor of cohomomorphisms on the Cart hesian pro duct of the ab elian category of left C -como dules and the exact category of A -injectiv e left C -contramo dules. One can deﬁne the k -mo dules Co ext i C ( M , P ), i = 0, − 1, . . . a s the homolog y of the bar complex · · · − → Hom A ( C ⊗ A C ⊗ A M , P ) − → Hom A ( C ⊗ A M , P ) − → Hom A ( M , P ) for any left C -como dule M a nd an y A -inj ective left C - contra- mo dule P . Then Co ext 0 C ( M , P ) ≃ Cohom C ( M , P ) and there are long exact sequenc es of Co ext ∗ C asso ciated with exact triples of como dules and con tramo dules. Now a left C -como dule M is C / A -coprojective if and only if Co ext i C ( M , P ) = 0 for an y A -injec- tiv e left C -contramo dule P and all i < 0. Indeed, the “if ” assertion follow s from the homological exact sequence, and “ o nly if ” holds since the bar complex is isomorphic to the complex of cohomomorphisms from the C -como dule M into the bar resolution · · · − → Hom A ( C , Hom A ( C , P )) − → Hom A ( C , P ) o f the C -contramo dule P , whic h is a complex of A -injectiv e C -contramo dules, exact except in degree 0 and split o v er A . The pro of o f part (b) is completely a nalogous; it uses the left deriv ed functor of the functor o f cohomomorphisms o n the Carthesian pro duct of the exact catego ry of A -projectiv e left C -como dules and the ab elian category of left C -contramo dules.  Remark. It f ollo ws fr o m Lemma 5 .2 that an y extension of an A - projectiv e C -como d- ule by a copro jectiv e C -como dule splits, and any extens ion o f a coinjectiv e C -con- tramo dule b y an A -injectiv e C -contra mo dule splits. The analogues of the results 70 of Remark 1.2.2 also hold for (r elat ively ) copro jectiv e como dules and coinjectiv e con tramo dules in the assumptions of 3.1.3; see the pro of o f Lemma 5.3.2 fo r details. Question. Are all relativ ely coﬂat C -como dules relativ ely copro jectiv e? Are all A -projectiv e coﬂat C -como dules copro jectiv e? 3.2.3. Let C b e an ar bitrary coring. Let us call a left C -como dule M quasic opr oje ctive if the f unctor o f cohomomorphisms f rom M is left exact on the catego ry o f left C -contramo dules, i. e., this functor preserv es kernels . Any coinduced C - como dule is quasicopro jectiv e. An y quasicopro jectiv e como dule is quasicoﬂat. Let us call a left C -contramo dule P quasic o i n je ctive if the functor of cohomomorphisms in to P is left exact on the catego r y of left C -como dules, i. e., this functor maps cokerne ls to k ernels. An y induced C -contramo dule is quasicoinjectiv e. (Cf. Lemma 5.2.) Prop osition 1. L et M b e a left C -c omo dule, K b e a right C -c omo d ule en d owe d with a left action of a k -algebr a B by c omo dule endomorphisms, and P b e a left B -mo d ule. Then ther e is a natur al k -mo dule map Cohom C ( M , Hom B ( K , P )) − → Hom B ( K  C M , P ) , which is an isomorph i sm, at le a st, in the fol lowing c ases: (a) P is an inje ctive left B -mo dule; (b) M is a quasic opr oje ctive left C -c omo dule; (c) C is a pr oje ctive left A -mo dule, M is a pr oje ctive left A -mo dule, K is a C / A -c oﬂat right C -c o m o dule, K is a pr oje ctive left B -mo dule, and the ring B has a ﬁn ite left homolo gic al dimen sion; (d) K as a rig h t C -c omo dule with a left B -mo dule structur e is c oinduc e d fr om a B - A -bimo dule. Besides, in the c ase (c) the left B -m o dule K  C M is p r oje ctive. Pr o of . The map Hom B ( K ⊗ A M , P ) − → Hom B ( K  C M , P ) annihilates the diﬀer- ence of tw o maps Hom B ( K ⊗ A C ⊗ A M , P ) ⇒ Hom B ( K ⊗ A M , P ) and this pair of maps can b e iden tiﬁed with the pair of maps Hom A ( C ⊗ A M , Ho m B ( K , P )) ⇒ Hom A ( M , Hom B ( K , P )) whose cok ernel is, by the deﬁnition, the cohomomorphism mo dule Cohom C ( M , Hom B ( K , P )). Hence there is a natural map Cohom C ( M , Hom B ( K , P )) − → Hom B ( K  C M , P ). The case (a ) is obvious . In the case (b), it suﬃces to presen t P as the kerne l of a map of injectiv e B -mo dules. The rest o f the pro of is completely analogous to the pro of o f Prop osition 1.2.3 (with ﬂat mo dules replaced b y pro j ective ones and the left and rig h t sides switc hed).  Prop osition 2. L et P b e a left C -c ontr amo dule, K b e a left C -c om o dule endowe d with a right action of a k -a lgebr a B by c om o dule en d omorphisms, and M b e a left B -mo dule. Then ther e is a natur al k -mo dule map Cohom C ( K ⊗ B M , P ) − → Hom B ( M , Cohom C ( K , P )) , wh ich is an isomorphism, at le ast, in the fol lowing c ases : (a) M is a p r oje ctive left B -mo dule; 71 (b) P is a q uasi c oinje ctive left C -c ontr amo dule; (c) C is a ﬂat right A -mo dule, P is a n inje ctive left A -mo dule, K i s a C / A -c o- pr oje ctive le f t C -c om o dule, K is a ﬂat right B -m o dule, a n d the ring B has a ﬁnite left ho molo gic al dimension; (d) K as a le ft C -c omo dule with a right B -mo dule structur e is c oinduc e d fr om an A - B -bimo dule. Besides, in the c ase (c) the left B -m o dule Cohom C ( K , P ) is inje ctive. Pr o of . The map Hom B ( M , Hom A ( K , P )) − → Hom B ( M , Cohom C ( K , P )) a nnihilates the diﬀerence of tw o maps Hom B ( M , Hom A ( C ⊗ A K , P )) ⇒ Hom B ( M , Hom A ( K , P )) and t his pair of maps can b e iden tiﬁed with the pair o f maps Hom A ( C ⊗ A K ⊗ B M , P ) ⇒ Hom A ( K ⊗ B M , P ) whose cok ernel is, b y the deﬁnition, the cohomomorphism mo dule Cohom C ( K ⊗ B M , P ). Hence there is a natural ma p Cohom C ( K ⊗ B M , P ) − → Hom B ( M , Cohom C ( K , P )). The case (a) is ob vious. In the case (b), it suﬃces to presen t M a s the cokerne l o f a map of pro jectiv e B - mo dules. T o prov e (c) and (d), consider the bar complex (4) · · · − − → Hom A ( C ⊗ A C ⊗ A K , P ) − − → Hom A ( C ⊗ A K , P ) − − → Hom A ( K , P ) − − → Cohom C ( K , P ) . In the case (c) this complex is exact, since it is the complex of cohomomorphisms from a C / A - coprojectiv e C -como dule K in to an A -split exact complex of A -injec- tiv e C -contr amo dules · · · − → Hom A ( C ⊗ A C , P ) − → Hom A ( C , P ) − → P . Since a ll the terms of the complex (4), except p ossibly t he righ tmost one, are injectiv e left B -mo dules and the left homological dimension of the ring B is ﬁnite, the righ tmost term Cohom C ( K , P ) is also an injectiv e B -mo dule, t he complex of left B -mo dules (4) is con tractible, and the complex of B -mo dule homomorphisms from the left B -mo dule M in to (4 ) is exact. In the case (d), the complex (4) is also a split exact complex of left B -mo dules.  3.2.4. Let C b e a coring ov er a k -alg ebra A and D b e a coring o v er a k -algebra B . Assume t ha t D is a pro jectiv e left B -module. Let K b e a C - D -bicomo dule and P b e a left C -contramo dule. Then the mo dule of cohomo mo r phisms Cohom C ( K , P ) is endo w ed with a left D -contramo dule structure as the cok ernel of a pair of contra- mo dule morphisms Hom A ( C ⊗ A K , P ) ⇒ Hom A ( K , P ). More generally , let C and D b e arbitra ry corings. Assume that the functor of homomorphisms from D o v er B preserv es the cok ernel of the pair of maps Hom A ( C ⊗ A K , P ) ⇒ Hom A ( K , P ), that is the natural map Cohom C ( K ⊗ B D , P ) − → Hom B ( D , Cohom C ( K , P )) is an isomorphism. Then one can deﬁne a left contraaction map Hom B ( D , Cohom C ( K , P )) − → Cohom C ( K , P ) taking the cohomomo r phisms o v er C from the right D -coaction map K − → K ⊗ B D in to the contramo dule P . This con traaction is counital and con traasso ciativ e, at least, if the natural map 72 Cohom C ( K ⊗ B D ⊗ B D , P ) − → Hom B ( D ⊗ B D , Cohom C ( K , P )) is a lso an iso- morphism. In particular, if one of the conditions of Prop osition 3.2.3 .2 is satisﬁed (fo r M = D ), then the left B - mo dule Cohom C ( K , P ) has a nat ura l D -contramo dule structure. 3.2.5. Let C b e a coring ov er a k -algebra A and D b e a coring ov er a k -algebra B . Prop osition. L et M b e a left D -c omo dule, K b e a C - D -b i c omo d ule, and P b e a left C -c ontr a mo dule. Th en the iter ate d c ohom omorphism mo dules Cohom C ( K  D M , P ) and Cohom D ( M , Cohom C ( K , P )) ar e natur al ly iso morphic, at le ast, in the fol lowing c ases: (a) D is a pr oje c tive left B -mo dule, M is a pr oje ctive left B -mo dule, C is a ﬂat right, and P is a n inje ctive left A -mo dule; (b) D i s a pr oje ctive left B -mo d ule and M is a c opr oje ctive lef t D -c omo dule; (c) C is a ﬂat right A -mo dule and P is a c oinje ctive left C -c ontr amo dule; (d) D is a pr oje ctive left B -mo dule, M is a pr oje ctive left B -mo dule, K is a D /B -c oﬂat right D -c omo dule, K is a pr oj e ctive left A -mo dule, and the ring A has a ﬁnite left homo lo gic al dim ension; (e) C is a ﬂat right A -mo dule, P is a n inje ctive left A -mo dule, K i s a C / A -c o- pr oje ctive le f t C -c om o dule, K is a ﬂat right B -m o dule, a n d the ring B has a ﬁnite left ho molo gic al dimension; (f ) D is a pr oje ctive left B -mo dule, M is a pr oje ctive left B -mo dule, and K as a right D -c omo dule wi th a left A -mo dule structur e is c oinduc e d fr om an A - B -bimo dule; (g) C is a ﬂat right A -mo dule, P is an inje ctive left A -mo dule, and K as a left C -c omo dule w i th a right B -mo dule structur e is c oinduc e d fr om an A - B -bimo dule; (h) M is a quasic opr oje ctive left D -c omo dule and K as a left C -c o m o dule with a right B -mo d ule structur e is c oinduc e d fr om an A - B - b imo dule; (i) P is a quasic oinje ctive left C -c ontr amo dule a nd K as a right D -c omo dule with a left A -mo dule structur e is c oind uc e d fr om an A - B -bimo dule; (j) K as a left C -c omo dule with a right B -m o dule structur e i s c oinduc e d fr om an A - B -bimo dule and K as a right D -c omo dule with a left A -mo dule s tructur e is c oinduc e d fr om an A - B -bimo dule. Mor e pr e cisely, in al l c ase s in this list the natur al map s fr om the k -mo dule Hom A ( K ⊗ B M , P ) = Hom B ( M , Hom A ( K , P )) into b oth iter ate d c ohomomo rp hism mo dules under c onsider ation ar e surje ctive, their kern e l s c oincide and ar e e qual to the sum of the k e rn els of two map s fr om this mo dule onto its quotient m o dules Cohom C ( K ⊗ B M , P ) an d Cohom D ( M , Hom A ( K , P )) . 73 Pr o of . One can easily see that whenev er b oth maps Cohom D ( M , Hom A ( K , P )) − → Hom A ( K  D M , P ) and Cohom D ( M , Hom A ( K , Hom A ( C , P ))) − → Hom A ( K  D M , Hom A ( C , P )) are isomorphisms, the natural map Hom A ( K ⊗ B M , P ) − → Cohom C ( K  D M , P ) is surjectiv e and its k ernel coincides with the desired sum of t w o kerne ls of maps f r om Hom A ( K ⊗ B M , P ) onto it s quotien t mo dules. Analo- gously , whenev er b oth maps Cohom C ( K ⊗ B M , P ) − → Hom B ( M , Cohom C ( K , P )) and Cohom C ( K ⊗ B D ⊗ B M , P ) − → Hom B ( D ⊗ B M , Cohom C ( K , P )) are isomorphisms, the natural map Hom B ( M , Hom A ( K , P )) − → Cohom D ( M , Cohom C ( K , P )) is surjectiv e and it k ernel coincides with the desired sum of tw o ke rnels in Hom B ( M , Hom A ( K , P )). Th us it remains to apply Prop ositions 3.2.3.1 a nd 3.2 .3.2.  Comm utativit y of p entagonal diagrams of asso ciativit y isomorphisms b et w een it- erated cohomomorphism mo dules can be established in the w ay ana logous to the case of iterat ed cotensor pro ducts. Namely , eac h of the ﬁve iterated cohomo- morphism mo dules Cohom C (( K  E L )  D M , P ), Cohom C ( K  E ( L  D M ) , P ), Cohom E ( L  D M , Cohom C ( K , P )), Cohom D ( M , Cohom E ( L , Cohom C ( K , P ))), and Cohom D ( M , Cohom C ( K  E L , P )) is endo w ed with a nat ura l map in to it fr om t he homomorphism mo dule Hom A ( K ⊗ F L ⊗ B M , P ), and since the a sso ciativit y isomor- phisms are, presumably , compatible with these maps, it suﬃces to c hec k that at least one of these ﬁv e maps is surjectiv e in order to sho w that the p en tagonal dia g ram comm utes. In particular, if the ab o v e Prop osition together with Prop osition 1.2.5 pro vide all the ﬁv e isomorphisms constituting the p en tagonal diagra m and either M is a pro jec tiv e left B -mo dule, or P is an injectiv e left A -mo dule, or b oth K and L as left (righ t) como dules with righ t (left) mo dule structures are coinduced f r om bimo dules, then the p en tagonal diag ram is comm utativ e. W e will say that multiple cohomomorphisms b et w een sev eral bicomo dules and a con tramo dule Cohom C ( K  E · · ·  D M , P ) a re asso ciative if the multiple cotensor pro duct K  E · · ·  D M is asso ciativ e and for any p ossible wa y of r epresen ting this m ultiple cohomomorphism mo dule in terms of iterated cot ensor pro duct and cohomo- morphism op erations all the interme diate cohomomorphism mo dules can b e endo w ed with contramodule structures via the construction of 3.2.4 , all p ossible asso ciativit y isomorphisms b et w een iterated cohomomorphism mo dules exist in the sense of the last assertion of Prop osition a nd preserv e con tramo dule structures, and all the p en- tagonal diagrams comm ute. Asso ciativity isomorphisms and con tramo dule structures on asso ciativ e multiple cohomomorphisms are preserv ed b y the morphisms b et w een them induced b y a n y bicomo dule and contramo dule morphisms o f the factors. 3.3. Semicon tramo dules. 3.3.1. Dep ending on the (co)ﬂatness, (co)pro jectivity , and/or (co)injectivit y con- ditions imp o sed, there are sev eral w ays to ma ke the category opp osite to a cate- gory of left C -contramo dules into a rig h t mo dule catego ry ov er a tensor category o f 74 C - C -bicomo dules with resp ect to the functor Cohom C . Moreo v er, a category of left C -como dules t ypically can b e ma de into a left mo dule catego r y o v er the same tensor category , so that the functor Cohom C w ould pro vide a lso a pairing b et w een these left and r ig h t mo dule catego ries taking v alues in the category k – mo d op . It follo ws from Prop osition 3.2.5(b) that whenev er C is a pro jectiv e left A -mo dule, the category opp osite to t he category of left C -contramo dules is a r ig h t mo dule category ov er the tens or cat ego ry of C - C - bicomo dules that are copro jec tiv e left C -como dules; the category of copro jective left C -como dules is a left mo dule cate- gory ov er this tensor category . If follo ws f rom Prop osition 3.2 .5(c) that whenev er C is a ﬂat righ t A -mo dule, the category opp osite to the category of coinjectiv e left C -contramo dules is a righ t mo dule cat ego ry o v er the t ensor category of C - C - bico- mo dules that are coﬂat right C - como dules; the category of left C -como dules is a left mo dule catego ry ov er this tensor category . It follows from Prop osition 3.2.5( d) that whenev er C is a pro jectiv e left A -mo dule a nd the ring A has a ﬁnite left homolog- ical dimension, the catego ry opp osite to the category of left C -contra mo dules is a righ t mo dule catego r y o v er the tensor category of C - C -bicomo dules that are pro jec- tiv e left A -mo dules a nd C / A - coﬂa t righ t C -como dules; t he category of A -pro jec tiv e left C - como dules is a left mo dule category o v er this tensor category . It follo ws fro m Prop osition 3 .2 .5(e) that whenev er C is a ﬂat righ t A -mo dule and the ring A has a ﬁnite left homo lo gical dimension, the category opp osite to the category of A -injectiv e left C -contramo dules is a rig h t mo dule category ov er the tensor category of C - C - bi- como dules that are ﬂat righ t A -mo dules and C / A - coprojectiv e left C -como dules; the category of left C - como dules is a left mo dule category o v er this tensor category . Fi- nally , it follows from Prop osition 3 .2.5(a) t hat whenev er the ring A is semisimple, the category opp o site to the catego ry of left C -contramo dules is a right mo dule category o v er the t ensor category o f C - C -bicomo dules; the category of left C -como dules is a left mo dule category ov er this tensor category . In eac h case, there is a pairing be- t w een t hese left and righ t mo dule catego ries compatible with their mo dule category structures and taking v alues in the category opp osite t o the category of k -mo dules. A left semic ontr amo dule ov er a semialgebra S is an ob ject of the category o pp osite to the categor y of mo dule ob jects in one of the righ t mo dule categories of the a b o v e kind (opp osite to a category o f left C - contramo dules) ov er the ring o b ject S in the corresp onding tensor category of C - C -bicomo dules. In ot her w ords, a left S -semicon- tramo dule P is a left C -contramo dule endow ed with a left C -contr a mo dule morphism of left semic ontr aaction P − → Cohom C ( S , P ) satisfying the asso ciativit y and unity equations. Namely , tw o comp ositions P − → Cohom C ( S , P ) ⇒ Cohom C ( S  C S , P ) of the se micon traaction morphism P − → Cohom C ( S , P ) w ith the morphisms Cohom C ( S , P ) ⇒ Cohom C ( S  C S , P ) = Cohom C ( S , Cohom C ( S , P )) induced b y the semim ultiplication morphism of S and the semicon t raaction morphism should coincide with eac h other and the comp osition P − → Cohom C ( S , P ) − → P of the 75 semicon tra action mor phism with t he morphism induced b y t he semiunit morphism of S should coincide with the iden tity morphism of P . F or this deﬁnition to mak e sense, (co)ﬂatness, (co)pro jectivity , and/or (co)injectivit y conditions imp osed on S and/or P mus t guarantee asso ciativity of m ultiple coho mo mor phism mo dules of the form Cohom C ( S  C · · ·  C S , P ). R ight semic o n tr amo d ules ov er S are deﬁned in t he analogous w a y . If Q is a left C -contramo dule for whic h m ultiple coho mo mo r phisms Cohom C ( S  C · · ·  C S , Q ) are asso ciativ e, then there is a natural left S -semicontramo dule structure on the cohomomorphism mo dule Cohom C ( S , Q ). The semicon tramo dule Cohom C ( S , Q ) is called the S -semicontramo dule c oinduc e d fro m a C - contramo dule Q . According to Lemma 1.1.2, the k -mo dule of semicon tramo dule homomorphisms fro m an arbitrary S -semicontramo dule in to the coinduced S - semicontramo dule is described b y the formula Hom S ( P , Cohom C ( S , Q )) ≃ Hom C ( P , Q ). W e will denote t he category of left S -semicontramo dules by S – sicntr and the cat- egory of right S -semicontramo dules b y sicntr – S . This nota tion presumes that one can sp eak o f (left or right) S -semicontramo dules with no (co)injectivit y conditions imp osed on them. If C is a pro jectiv e left A -mo dule and S is a copro jectiv e left C -como dule, then the category of left semicon tramo dules o v er S is ab elian and the forgetful functor S – sicntr − → C – contra is exact. If C is a pro jectiv e left A - mo dule and either S is a copro jectiv e left C -como dule, or S is a pro jectiv e left A -mo dule and a C / A -coﬂat right C -como dule and A has a ﬁnite left homological dimension, or A is semisimple, then b oth inﬁnite direct sums and inﬁnite pro ducts exist in the categor y of left S -semicontramo dules and b oth are preserv ed by t he forg etful functor S – sicntr − → C – contra , ev en t ho ugh only inﬁnite pro ducts are preserv ed b y the full forg etful functor S – sicntr − → A – mo d . If C is a ﬂat righ t A -mo dule, S is a ﬂat righ t A -mo dule a nd a C / A -coprojectiv e left C -como dule, a nd A has a ﬁnite left homological dimension, then the category o f A -injectiv e left S -semicontramo dules is exact. If C is a pro jectiv e left A -mo dule, S is a pro jectiv e left A -mo dule and a C / A -coﬂa t right C -como dule, and A has a ﬁnite left homological dimension, then the category of C / A -coinjectiv e left S -semicontramo d- ules is exact. If C is a ﬂat right A - mo dule and S is a coﬂat right C -como dule, then the categor y of C -coinjectiv e left S -semicontramo dules is exact. If A is semisimple, the category of C -coinjectiv e S -semimo dules is exact. Inﬁnite pro ducts exist in all of these exact categories, and the fo rgetful functors preserv e them. Question. When C is a ﬂat right A -mo dule and S is a coﬂat righ t C -como dule, a righ t adjoint f unctor to t he forgetf ul functor S – simo d − → C – como d exists according to t he abstract adjo in t f unctor existence theorem [36]. Indeed, the forgetful functor preserv es colimits and the category o f left S -semimo dules has a set of generators (since the category of left C -como dules do es; see Question 3 .1.2). Do es a left adjoint 76 functor to the forgetful f unctor S – sicntr − → C – contra exist? Can one describ e these functors more explicitly? 3.3.2. Assume that the coring C is a pro j ective left and a ﬂat rig ht A -mo dule and the ring A has a ﬁnite left homo lo gical dimension. Lemma. (a) If the semia lgebr a S is a c oﬂat righ t C -c o m o dule and a pr oje ctive left A -mo dule, then ther e exists a (not always additive) functor ass i g ning to any left S -semimo dule a surje ctive map onto it fr om an A -pr o je ctive S -semimo dule. (b) I f the semialgebr a S is a c opr oje ctive le ft C -c omo dule and a ﬂat right A -mo d ule, then ther e exists a (not a l w ays additive) functor assig n ing to any left S -semic ontr a- mo dule an inje ctive map fr o m it into an A -inje ctive S -semi c ontr amo dule. Pr o of . The pro of o f part (a) is completely ana logous to the pro of of Lemma 1.3.2 (with the last assertion of Prop osition 3 .2.3.1 used as needed); and part (b) is prov en in the follo wing w a y . Let P − → I ( P ) denote the functorial injectiv e morphism from a C -contramo dule P in to an A -injectiv e C -contramo dule I ( P ) constructed in Lemma 3.1 .3. Then for any S -semicontramo dule P t he comp osition of maps P − → Cohom C ( S , P ) − → Cohom C ( S , I ( P )) pro vides the desired injectiv e morphism of S -semicontramo dules. According to the last assertion of Prop osition 3.2 .3.2, the A -mo dule I ( P ) = Cohom C ( S , I ( P )) is injectiv e.  Remark. The analogues of the result o f Remark 1.3.2 hold for C / A -coprojec- tiv e/semiprojectiv e S - semimo dules and C / A -coinjectiv e/semiinjectiv e S -semicontra- mo dules; see the pro of of Lemma 9.2.1 for details. 3.3.3. Let S b e a semialgebra o v er a coring C ov er a k -alg ebra A . Lemma. (a) Assume that C is a pr o je ctive left A -m o dule, S is a c opr oje ctive le f t C -c omo dule and a C / A -c oﬂat rig ht C -c omo dule, a n d the ring A has a ﬁn ite left ho- molo gic al dimen s ion. Then ther e exist • an exa ct functor assignin g to any A -pr o je ctive left S -semim o dule an A -split inje ctive morphism fr om it into a C -c opr oje ctive S -semimo dule, and • an exact functor assign i n g to any le f t S -s emic ontr amo dule a surje ctive mor- phism onto it fr om a C / A -c oinje ctive S -semic ontr amo dule. (b) Assume that C i s a ﬂat righ t A -mo dule, S is a c oﬂat right C -c omo dule and a C / A -c opr oje ctive left C -c omo dule, and the ring A ha s a ﬁnite left homolo gic al dimen- sion. The n ther e exist • an exact functor assigning to any A -inje ctive left S -semic ontr amo dule a n A -split surje ctive mo rp h ism onto it fr om a C -c o i n je ctive S -semic ontr amo d ule, and • an exact f unc tor assigning to any left S -semimo dule an inje ctive morphism fr om it into a C / A -c opr oje ctive S -semimo dule. 77 (c) When b oth the assumptions of (a) and (b) ar e satisﬁe d, the two functors acting in c a te gories of semi m o dules (c an b e made to) agr e e a nd the two functors acting in c ate go ries o f sem ic ontr amo dules (c an b e made to) agr e e. Pr o of . The pro of of the ﬁrst assertion of part (a) and the second assertion of part (b) is based o n the construction completely ana logous to that of the pro of of Lemma 1.3 .3, with (co)ﬂat (co)mo dules r eplaced by ( co) pr o jectiv e ones, and the left and righ t sides switc hes as needed. The only diﬀerence is that the inductiv e limit of a sequence of copro jectiv e como dules do es no t hav e to b e copro jectiv e, b ecause ev en the inductiv e limit of a sequence o f pro jectiv e mo dules do es not hav e to b e pro jec tiv e. This obstacle is dealt with in the fo llo wing w ay . Sublemma A. Assume that C is a p r oje ctive left A -mo dule. L et U 1 − → U 2 − → U 3 − → U 4 − → · · · b e an in ductive system of le ft C -c omo dules, wher e the c omo dules U 2 i ar e c opr oje ctive, while the morph i s ms of c omo d ules U 2 i − 1 − → U 2 i +1 ar e in j e ctive and split over A . Then the inductive limi t lim − → U j is a c opr oje ctive C - c omo dule. Pr o of . Let us ﬁrst sho w that for a ny C -contra mo dule P there is an isomorphism Cohom C (lim − → U j , P ) = lim ← − Cohom C ( U j , P ). Denote by G • j the bar complex · · · − − → Hom A ( C ⊗ A C ⊗ A U j , P ) − − → Hom A ( C ⊗ A U j , P ) − − → Hom A ( U j , P ); w e will denote the terms of this complex b y upp er indices, so that G n j = 0 for n > 0 and H 0 ( G • j ) = Cohom C ( U j , P ). Clearly , w e hav e H 0 (lim ← − G • j ) = Cohom C (lim − → U j , P ). Since the como dules U 2 i are copro jec tiv e, H n ( G • 2 i ) = 0 fo r n 6 = 0, as the complex G • 2 i can b e obtained b y applying the functor Cohom C ( U 2 i , − ) to the complex of C -contra- mo dules · · · − → Hom A ( C ⊗ A C , P ) − → Hom A ( C , P ), which is exact except at degree 0. Since the maps of A -mo dules U 2 i − 1 − → U 2 i +1 are split injectiv e, the morphisms of complexes G • 2 i +1 − → G • 2 i − 1 are surjectiv e. Therefore, lim ← − 1 G • j = lim ← − 1 G • 2 i − 1 = 0 , hence there is a “univ ersal co eﬃcien ts” sequence [47] 0 − − → lim ← − 1 H n − 1 ( G • j ) − − → H n (lim ← − G • j ) − − → lim ← − H n ( G • j ) − − → 0 . In particular, for n = 0 we obtain the desired isomorphism H 0 (lim ← − G • j ) = lim ← − H 0 ( G • j ), b ecause lim ← − 1 H − 1 ( G • j ) = lim ← − 1 H − 1 ( G • 2 i ) = 0. No w for any exact triple of C - contramo dules P ′ → P → P ′′ w e ha v e an exact t riple of pro jectiv e syste ms Cohom C ( U 2 i , P ′ ) − → Cohom C ( U 2 i , P ) − → Cohom C ( U 2 i , P ′′ ) and lim ← − 1 Cohom C ( U 2 i , P ′ ) = lim ← − 1 Cohom C ( U 2 i − 1 , P ′ ) = 0, hence the triple remains exact after passing to the pro jec tiv e limit.  Sublemma B. Assume that C is a ﬂat right A -mo d ule. L e t U 1 − → U 2 − → U 3 − → U 4 − → · · · b e an inductive system of left C - c omo d ules , w her e the c omo dules U 2 i ar e C / A -c opr oje ctive, while the morphisms of c om o dules U 2 i − 1 − → U 2 i +1 ar e inj e ctive. Then the inductive lim it lim − → U j is a C / A -c opr oje ctive C -c omo dule. 78 Pr o of . Analogous to the pro o f of Sublemma A, the only c hanges b eing that P , P ′ , P ′′ are no w A -injectiv e C -contramo dules and the complex · · · − → Ho m A ( C ⊗ A C , P ) − → Hom A ( C , P ) − → P is an A -split exact sequence of A -injectiv e C -contramo dules.  Pro of of the ﬁrst ass ertion of part (b): fo r an y A -injectiv e C -contramo dule P , set G ( P ) = Hom A ( C , P ). Then the con traaction map G ( P ) − → P is a surjec- tiv e morphism of C -contramo dules, the con tr a mo dule G ( P ) is coinjectiv e, and the k ernel of this morphism is A -injectiv e. No w let P b e a n A - injectiv e left S -semicon- tramo dule. The semicon traaction map P − → Cohom C ( S , P ) is an injectiv e mor- phism o f A -injectiv e S -semic ontramo dules; let K ( P ) denote it s cok ernel. The map Cohom C ( S , G ( P )) − → Cohom C ( S , P )) is a surjectiv e mo r phism of S -semicontra- mo dules with a n A -injectiv e kernel Cohom C ( S , ke r ( G ( P ) → P )). Let Q ( P ) b e the k ernel of the comp osition Cohom C ( S , G ( P )) − → Cohom C ( S , P ) − → K ( P ) . Then the comp o sition of maps Q ( P ) − → Cohom C ( S , G ( P )) − → Cohom C ( S , P ) f a ctorizes through the injection P − → Cohom C ( S , P ), so there is a na tural surjectiv e morphism of S -semicontramo dules Q ( P ) − → P . The kerne l o f the map Q ( P ) − → P is iso- morphic to the k ernel of the map Cohom C ( S , G ( P )) − → Cohom C ( S , P ), hence b oth k er( Q ( P ) → P ) and Q ( P ) are injectiv e A -mo dules. Notice that the semicon tramo dule morphism Q ( P ) − → P can b e extended to a con tramo dule morphism Cohom C ( S , G ( P )) − → P . Indeed, the map Q ( P ) − → P can b e presen ted as the comp osition Q ( P ) − → Cohom C ( S , G ( P )) − → Cohom C ( S , P ) − → P , where t he map Cohom C ( S , P ) − → P is induced b y the semiunit morphism C − → S o f the semialgebra S . Iterating t his construction, w e obtain a pro jectiv e system of C -contramo dule morphisms P ← − Cohom C ( S , G ( P )) ← − Q ( P ) ← − Cohom C ( S , G ( Q ( P ))) ← − Q ( Q ( P )) ← − · · · , where the maps P ← − Q ( P ) ← − Q ( Q ( P )) ← − · · · a re A -split surjectiv e morphisms of A - injectiv e S -contramo dules, while the C -contra- mo dules Cohom C ( S , G ( P )), Cohom C ( S , G ( Q ( P ))), . . . ar e coinjectiv e. Denote by F ( P ) the pro jec tiv e limit of this system; then F ( P ) − → P is a n A -split surjectiv e morphism of S -semicontramo dules, while coinjectivit y o f the C -contr a mo dule F ( P ) follo ws from the next Sublemma. Sublemma C. Assume that C is a ﬂat right A -mo dule. L et U 1 ← − U 2 ← − U 3 ← − U 4 ← − · · · b e a pr oje ctive system of left C -c ontr amo d ules , wher e the c on tr amo d ules U 2 i ar e c oinje ctive, while the m orphisms of c ontr amo dules U 2 i +1 − → U 2 i − 1 ar e surje c- tive and split over A . Then the p r oje ctive limit lim ← − U j is a c oin je ctive C -c ontr amo d ule. Pr o of . Completely analog o us to the pro of of Sublemma A. One considers the pro- jectiv e system of bar- complexes · · · − → Hom A ( C ⊗ A C ⊗ A M , U j ) − → Hom A ( C ⊗ A M , U j ) − → Hom A ( M , U j ), etc.  79 The pro of o f the second assertion of part (a) is based on the same construction; the only changes a r e that A -mo dules are no longer injectiv e, for a ny left C -contramo d- ule P the C -contramo dule G ( P ) = Hom A ( C , P ) is C / A -coinjectiv e, a nd therefore the S -semicontramo dule Cohom C ( S , G ( P )) is C / A -coinjective . The pro jectiv e limit F ( P ) is C / A -coinjectiv e according to the follo wing Sublemma. Sublemma D. Assume that C is a pr oje ctive left A -mo dule. L et U 1 ← − U 2 ← − U 3 ← − U 4 ← − · · · b e a pr oje ctive system of left C -c ontr a mo dules, wher e the c o n tr a- mo dules U 2 i ar e C / A -c oinje ctive, while the morphisms of c ontr amo dules U 2 i +1 − → U 2 i − 1 ar e surje ctive. Then the pr oje ctive lim i t lim ← − U j is a C / A -c oinje ctive C -c ontr a- mo dule.  Both functors F are exact, since the cok ernels of injectiv e maps, the k ernels of surjectiv e maps, a nd the pro jectiv e limits of Mittag- Leﬄer sequences o f k -mo dules preserv e exact triples. P art (c) is clear from the constructions.  3.4. Semihomomorphisms. 3.4.1. Assume that the coring C is a pro jectiv e left A -mo dule, the semialgebra S is a pro jectiv e left A -mo dule and a C / A -coﬂat right A - mo dule, and the r ing A has a ﬁnite left homological dimension. Let M b e an A -projectiv e left S -semimo dule and P b e a left S -semicontramo dule. The k -mo dule o f semihomomorp h isms SemiHom S ( M , P ) is deﬁned as the k ernel of the pair o f maps Cohom C ( M , P ) ⇒ Cohom C ( S  C M , P ) = Cohom C ( M , Cohom C ( S , P )) one of whic h is induced b y the S -semiaction in M and the other b y the S -semicon traaction in P . F or an y A -proj ective left C - como dule L and an y left S -semicontramo dule P there is a natural isomorphism SemiHom S ( S  C L , P ) ≃ Cohom C ( L , P ). Analogously , for any A -projectiv e left S -semimo dule M and a n y left C - contramo dule Q there is a natural isomorphism SemiHom S ( M , Cohom C ( S , Q )) ≃ Cohom C ( M , Q ). These assertions follow from Lemma 1.2.1 . 3.4.2. Assume that the coring C is a ﬂat righ t A -mo dule, t he semialgebra S is a ﬂat righ t A -mo dule and a C / A -coprojective left A -mo dule, and the ring A has a ﬁnite left homological dimension. Let M b e a left S -semimo dule a nd P b e a n A - in- jectiv e left S -semicontramo dule. As ab o v e, the k -mo dule of semihomomorphisms SemiHom S ( M , P ) is deﬁned as the k ernel of the pair of maps Cohom C ( M , P ) ⇒ Cohom C ( S  C M , P ) = Cohom C ( M , Cohom C ( S , P )) one of which is induced by the S -semiaction in M and the other by the S -semicon traaction in P . F or an y left C - como dule L and an y A -injectiv e left S -semicontramo dule P there is a natural isomorphism SemiHom S ( S  C L , P ) ≃ Cohom C ( L , P ). Analogously , for any left S -semimo dule M and an y A -injectiv e left C -contramo dule Q there is a natural isomorphism SemiHom S ( M , Cohom C ( S , Q )) ≃ Cohom C ( M , Q ). 80 Notice that ev en under the strongest o f our assumptions on A , C and S , the A -pro jectivit y of M or t he A -injectivit y of P is still needed to guarantee that the triple cohomomorphisms Cohom C ( S  C M , P ) are asso ciative . 3.4.3. If the coring C is a pro jectiv e left A -mo dule and the semialgebra S is a co- pro jectiv e left C -como dule, one can deﬁne the mo dule of semihomomorphisms from a C -coproj ective left S -se mimo dule in to an arbitrary left S -semicontramo dule. In these assumptions, a C -copro jectiv e left S -semimo dule M is called semipr oje ctive if the functor of semihomomorphisms from M is exact on the a b elian category of left S -semicontramo dules. The S - semimo dule induced from a copro jective C -como dule is semipro jectiv e. An y semipro jective S -semimo dule is semiﬂat. If the coring C is a ﬂat righ t A -mo dule and the semialgebra S is a coﬂat righ t C -como dule, one can deﬁne t he mo dule of semihomomorphisms from an a r bitrary left S -semimodule in to a C -coinjectiv e left S -semicontramo dule. In these assumptions, a C -coinjectiv e left S -semicontramo dule P is called semiinje c tive if the functor o f semi- homomorphisms in to P is exact o n the ab elian category of left S -semimo dules. The S -semicontramo dule coinduced fr o m a coinjectiv e C -contramo dule is semiinjectiv e. When the ring A is semisimple, the mo dule of semihomomorphisms from an ar- bitrary S -semimodule in to an arbitrar y S - semicontramo dule is deﬁned without an y conditions on the coring C a nd the semialgebra S . 3.4.4. Let S b e a semialgebra ov er a coring C o v er a k -algebra A and T b e a semi- algebra ov er a coring D ov er a k -alg ebra B . Let K b e an S - T - bisemimo dule and P b e a left S -semicontramo dule. W e w ould lik e t o deﬁne a left T -semic ontramo dule structure on the mo dule o f semihomomorphisms SemiHom S ( K , P ). Assume that m ultiple cohomomo r phisms of the form Cohom C ( S  C K  D T  D · · ·  D T , P ) are asso ciative. Then, in particular, the k -mo dules of semihomo- morphisms SemiHom S ( K  D T  D · · ·  D T , P ) can b e deﬁned. Assume in ad- dition tha t m ultiple cohomomorphisms of the f orm Cohom C ( K  D T  D · · ·  D T , P ) are asso ciativ e. T hen the semihomomorphism mo dules SemiHom S ( K  D T  D · · ·  D T , P ) hav e natural left D -contramo dule structures a s k ernels of D -contramo dule morphisms. Assume that m ultiple cohomomorphisms of the form Cohom D ( T  D · · ·  D T , SemiHom S ( K , P )) are also asso ciative. F inally , assume that the cohomomorphisms from T  m preserv e the ke rnel of the pair of morphisms Cohom C ( K , P ) ⇒ Cohom C ( S  C K , P ) for m = 1 and 2 , tha t is the con tramo d- ule morphisms Cohom D ( T  m , SemiHom S ( K , P )) − − → SemiHom S ( K  D T  m , P ) are isomorphisms. Then one can deﬁne an asso ciat ive and unital semicon traaction morphism SemiHom S ( K , P ) − → Cohom D ( T , SemiHom S ( K , P )) ta king the semiho- momorphisms ov er S from the right T -semiaction morphism K  D T − → K into the semicon tra mo dule P . 81 F or example, if D is a pro jectiv e left B -mo dule, T is a copro jectiv e left D -como dule, A has a ﬁnite left homo lo gical dimension, and either C is a pro jectiv e left A - mo dule, S is a pro jectiv e left A -mo dule and a C / A - coﬂat r igh t C -como dule, and K is a pro- jectiv e left A - mo dule, or C is a ﬂat r ig h t A -mo dule, S is a ﬂat right A -mo dule and a C / A -coprojectiv e left C -como dule, and P is an injectiv e left A -mo dule, then the mo dule of semihomomorphisms SemiHom S ( K , P ) has a natural left T -semicontra- mo dule structure. Since the catego r y of left T -semicontramo dules is ab elian in this case, the T -semicontramo dule SemiHom S ( K , P ) can b e simply deﬁned as t he k ernel of the pa ir of semicon tramo dule morphisms Cohom C ( K , P ) ⇒ Cohom C ( S  C K , P ). Prop osition. L et M b e a left T -semimo dule, K b e an S - T -bisemim o dule, and P b e a left S -semic ontr amo dule. T h e n the iter ate d sem ihomomorphis m mo dules SemiHom S ( K ♦ T M , P ) a n d SemiHom T ( M , SemiHom S ( K , P )) ar e wel l-d eﬁne d an d natur al ly isomorph ic, at le a s t, in the fol low ing c ases: (a) D is a pr oje ctive left B -mo dule, T is a c op r oje ctive left D -c o mo dule, M is a c opr oje ctive left D -c omo dule, C is a ﬂat right A -mo dule, S is a c o ﬂat right C -c omo dule, a n d P is a c o i n je ctive left C -c ontr amo dule; (b) D is a pr oje ctive left B -mo dule, T is a c o pr oje ctive left D -c omo dule, M is a semipr oje ctive left T -semimo dule, and either • C is a p r oje ctive left A -mo dule, S is a c opr oje ctive le f t C -c om o dule, a n d K is a c opr oje ctive left C -c omo dule, or • C is a pr oje c tive left A -mo dule, S is a pr oje ctive left A -mo dule a nd a C / A -c oﬂat right C -c omo dule, the ring A has a ﬁnite left hom olo gic al di- mension, a nd K is a pr oje ctive left A -mo dule, or • C is a ﬂat right A -m o dule, S is a ﬂat right A -m o dule and a C / A -c opr o- je ctive left C -c omo dule, and P is an inje ctive left A -mo dule, o r • the ring A is semisim ple; (c) C is a ﬂat right A -mo dule, S is a c oﬂat right C -c omo dule, P i s a semiin je ctive left S -semic ontr amo dule, and either • D is a ﬂat right B -mo dule, T is a c oﬂat rig h t D -c omo dule, and K is a c oﬂat right D -c omo dule, or • D is a ﬂat right B -mo dule, T is a ﬂat right B -m o dule and a D /B -c opr o- je ctive left D -c omo dule, the ring B has a ﬁnite left homo lo gic al dimens ion, and K is a ﬂat right B -mo dule, or • D is a pr oje ctive left B -mo dule, T is a pr oje ctive left B -mo dule and a D /B -c oﬂat ri g ht D -c omo dule, the rin g B has a ﬁnite left homolo gic al dimension, and M is a pr oje ctive left B -mo dule, o r • the ring B is semisimp le; (d) D is a pr oje ctive left B -mo dule, T is a c o pr oje ctive left D -c omo dule, M is a c opr oje ctive left D -c omo dule, and either 82 • C is a p r oje ctive left A -mo dule, S is a c opr oje ctive le f t C -c om o dule, a n d K as a right T -sem imo dule with a left C -c om o dule structur e is induc e d fr om a C -c opr oje ctive C - D -bi c omo d ule, or • C is a pr oje c tive left A -mo dule, S is a pr oje ctive left A -mo dule a nd a C / A -c oﬂat right C -c omo dule, the ring A has a ﬁnite left hom olo gic al di- mension, and K as a rig ht T -semim o dule with a left C -c omo dule structur e is in duc e d f r om an A -pr oje ctive C - D -bic omo dule, or • C is a ﬂat right A -mo dule, S is a ﬂat right A -mo dule and a C / A -c opr oje c- tive left C -c omo d ule, the ring A has a ﬁnite left homolo gic al d imension, K as a right T -sem imo dule with a left C -c om o dule structur e is induc e d fr om a C - D -bic omo dule, and P i s an in j e ctive left A -mo dule, or • the ring A is semisimple and K as a right T -se mimo dule with a left C -c omo dule structur e is in d uc e d fr o m a C - D -bic omo dule; (e) C is a ﬂat right A -mo dule, S is a c oﬂat rig h t C -c omo dule, P i s a c oin j e ctive left C -c o ntr amo dule, and either • D is a ﬂat right B -mo dule, T is a c oﬂat ri g ht D - c omo d ule, and K as a left S -semimo dule with a righ t D -c omo dule structur e i s induc e d fr om a D -c oﬂat C - D -bic om o dule, or • D is a ﬂat right B -mo dule, T is a ﬂat right B -m o dule and a D /B -c opr o- je ctive left D - c omo dule, the ring B has a ﬁnite left homolo gic al dimen- sion, an d K as a left S -semimo dule w ith a right D -c omo dule structur e is induc e d fr om a B -ﬂat C - D -bic om o dule, o r • D is a pr oje ctive left B -mo dule, T is a pr oje ctive left B -mo dule and a D /B -c oﬂat ri g ht D -c omo dule, the rin g B has a ﬁnite left homolo gic al dimension, K as a left S -semimo dule with a righ t D -c omo dule s tructur e is in duc e d fr om a C - D -bic omo dule, and M is a pr oje ctive left B -mo dule, or • the ring B is se m isimple and K as a le ft S -semimo dule with a right D -c omo dule structur e is induc e d fr om a C - D -bic omo dule. Mor e pr e cisely, in al l c ases in this list the natur al maps fr om b oth iter ate d semiho- momorphism m o dules under c onsi d er ation into the iter ate d c oho momorphism mo dule Cohom C ( K  D M , P ) ≃ Cohom D ( M , Cohom C ( K , P )) ar e inje ctive, their im ages c o- incide and ar e e qual to the interse ction o f two submo dules SemiHom S ( K  D M , P ) and SemiHom T ( M , Cohom C ( K , P )) in this k -mo dule. Pr o of . Analogous to the pro of of Prop osition 1.4.4 (see also the pro o f of Prop osi- tion 3.2.5).  83 4. Derived Functor SemiExt 4.1. Contraderiv ed categories. Let A b e an exact category in whic h all inﬁnite pro ducts exist and the functors o f inﬁnite pro duct are exact. A complex C • o v er A is called c ontr aacyclic if it b elongs to the minimal triangulated sub category Acycl ctr ( A ) of the homot o p y category Hot ( A ) con taining all the total complexes of exact triples ′ K • → K • → ′′ K • of complex es ov er A and closed under inﬁnite pro ducts. Any con traacyclic complex is acyclic. It follo ws from the next Lemma that any acyclic complex b ounded from ab o v e is contraacyclic. Lemma. L et · · · → P − 1 , • → P 0 , • → 0 b e an exact se quenc e, b ound e d fr om ab ove, of arbitr ary c om p lexes o v er A . Th en the total c omp lex T • of the bic omple x P • , • c onstructe d by taking inﬁnite pr o ducts along the d iagonals is c ontr aac yclic. Pr o of . See the pro of of Lemma 2.1 .  The cat ego ry of con traacyclic complexes Acycl ctr ( A ) is a thic k sub category o f the homotop y category Hot ( A ), since it is a triangulated sub category with inﬁnite pro d- ucts. The c on tr aderive d c ate gory D ctr ( A ) of an exact category A is deﬁned as the quotien t category Hot ( A ) / Acycl ctr ( A ). Remark. One can ch ec k tha t fo r an y exact category A and any thic k sub category T in Hot ( A ) contained in the t hick sub category of acyclic complexes, containing all b ounded acyclic complexe s, and con t a ining with eve ry exact complex its sub- complexes and quotien t complexes of canonical ﬁltratio n, the g roups of homomor- phisms Hom Hot ( A ) / T ( X , Y [ i ]) b et w een complexes with a single nonzero term coin- cide with the Y oneda extension groups Ext i A ( X , Y ). Moreov er, the natura l functors Hot + / − /b ( A ) / ( T ∩ Hot + / − /b ( A )) − → Hot ( A ) / T b et w een the “ T -deriv ed categories” with v arious b ounding conditions are all fully faithful. In particular, these a sser- tions hold if T ⊂ Hot ( A ) consists of acyclic complexes and contains either a ll a cyclic complexes b ounded fro m ab o v e o r all a cyclic complexes b ounded from b elo w. 4.2. Copro jectiv e and coinjectiv e complexes. Let C b e a coring o v er a k -alge- bra A . The complex of cohomomorphisms Cohom C ( M • , P • ) f rom a complex of left C -como dules M • in to a complex of left C -contra mo dules P • is deﬁned as the total complex of the bicomplex Cohom C ( M i , P j ), constructed b y taking inﬁnite pro ducts along the diagonals. If C is a pro jectiv e left A -mo dule, the category o f left C -contramo dules is an ab elian category with exact functors of inﬁnite pro ducts, so the con traderiv ed cat- egory D ctr ( C – contra ) is deﬁned. When speaking ab out c ontr aacyclic c omplex e s of C -contramo dules, w e will alwa ys mean con t r a acyclic complexes with resp ect to the ab elian category of C -contramo dules, unless another exact category of C - contramo d- ules is explicitly men tioned. 84 Assuming that C is a pro jectiv e left A - mo dule, a complex of left C -como dules M • is called c opr oje ctive if the complex Cohom C ( M • , P • ) is acyclic whenev er a com- plex of left C -contramo dules P • is contraacyclic . Assuming that C is a ﬂat right A -mo dule, a complex of left C -contramo dules P • is called c oinje ctive if the complex Cohom C ( M • , P • ) is acyclic whenev er a complex of left C -como dules M • is coacyclic. Lemma. (a) Any c omplex of c opr oje ctive C -c omo dules is c opr oj e ctive. (b) A ny c om plex of c oinje ctive C -c on tr amo d ules is c o i n je ctive. Pr o of . Argue as in the pro of of L emma 2.2, using the fact that the functor of co- homomorphisms of complexes maps inﬁnite direct sums in the ﬁrst argumen t into inﬁnite pro ducts a nd preserv es inﬁnite pro ducts in t he second argumen t.  If the ring A has a ﬁnite left homological dimension, t hen an y copro jectiv e complex of left C -como dules is a pro jectiv e complex of A - mo dules in the sense of 0.1 .2 and an y coinjectiv e complex of left C -contramo dules is an injectiv e complex of A -mo dules. The complex of C -como dules C ⊗ A U • coinduced from a pro jectiv e complex of A - mo dules U • is copro jectiv e and the complex of C - contramo dules Hom A ( C , V • ) induced from an injectiv e complex o f A -mo dules is coinjectiv e. 4.3. Semideriv ed categories. Let S b e a semialgebra o ve r a coring C . Assume that C is a pro jectiv e left A -mo dule and the sem ialgebra S is a copro jectiv e left C -como dule, so that the catego ry of left S - semicontramo dules is ab elian. The s e m i- derive d c ate gory of left S - semicontramo dules D si ( S – sicntr ) is deﬁned as the quo- tien t catego r y of the homotopy category Hot ( S – sicntr ) b y the thic k sub category Acycl ctr - C ( S – sicntr ) of comple xes of S -semicontramo dules that are c ontr aacyclic as c omplex e s of C -c ontr amo dules . 4.4. Semipro jectiv e and semiinjectiv e complexes. Let S b e a semialgebra. The complex o f semihomomorphisms SemiHom S ( M • , P • ) f rom a complex of left S - semi- mo dules M • to a complex of left S -semicontramo dules P • is deﬁned as the t o tal complex of the bicomplex SemiHom S ( M i , P j ), constructed by taking inﬁnite pro d- ucts along the diagona ls. Of course, appropriate conditions m ust b e imp o sed on S , M • , and P • for this deﬁnition to make sense. Assume that the coring C is a pro jectiv e left A -mo dule and a ﬂat righ t A -mo dule, the semialgebra S is a copro jectiv e left S -semimo dule and a coﬂat righ t S -semimo d- ule, and the ring A has a ﬁnite left homological dimension. A complex of A -projectiv e left S -semimo dules M • is called semipr o j e ctive if the complex SemiHom S ( M • , P • ) is a cyclic whenev er a complex of left S -semicontramo d- ules P • is C -contraa cyclic. An y semipro jectiv e complex of S -semimo dules is a copro- jectiv e complex of C -como dules. The complex of S -semimo dules S  C L • induced f r o m a copro jectiv e complex of A -ﬂat C -como dules is semipro j ectiv e. An y semipro jectiv e complex of S -semimo dules is semiﬂat. Analogously , a complex of A -injectiv e left 85 S -semicontramo dules P • is called se miinje ctive if t he complex SemiHom S ( M • , P • ) is acyclic whenev er a complex of left S -semimo dules M • is C -coacyclic. An y semiin- jectiv e complex o f S -semicontramo dules is a coinjectiv e complex of C -contramo dules. The complex of S -semicontramo dules Cohom C ( S , Q • ) coinduced from a coinjectiv e complex of A -injectiv e C - contramo dules is semiinjectiv e. Notice that not ev ery complex of semipro jectiv e semimo dules is semipro jectiv e and not ev ery complex of semiinjectiv e semicon tramo dules is semiinjectiv e. On the other ha nd, any complex o f semipro j ective semimo dules b ounded from ab ov e is semi- pro jectiv e. Moreov er, if · · · → M − 1 , • → M 0 , • → 0 is a complex, b ounded from ab ov e, of semipro jectiv e complexes of S -semimo dules, then the total complex E • of the bicomplex M • , • constructed b y taking inﬁnite direct sums along the diagonals is semipro jectiv e. Indeed, the category of semipro jectiv e complexes is closed under shifts, cones, and inﬁnite direct sums, so o ne can apply Lemma 2.4. Analog ously , an y complex of semiinje ctiv e semicon tr a mo dules b ounded from b elo w is semiinjec- tiv e. Moreo v er, if 0 → P 0 , • → P 1 , • → · · · is a complex, b o unded from b elo w, of semiinjectiv e complexes of S - semicontramo dules, then the total complex E • of the bicomplex P • , • constructed b y taking inﬁnite pro ducts along the diagonals is semiin- jectiv e. Indeed, the category of semiinjectiv e complexes is closed under shifts, cones, and inﬁnite pro ducts, so one can apply the followin g L emma. Lemma. L et 0 → P 0 , • → P 1 , • → · · · b e a c o m plex, b ounde d fr om b elow, of arbitr ary c omplex e s o ver a n additive c ate gory A wher e in ﬁnite p r o ducts exi s t. Then the total c omplex E • of the bic omp lex P • , • up to the homo topy e quivale n c e c an b e obtaine d fr om the c omple xes P i, • using the op er ations of shift, c one, a n d in ﬁ nite pr o duct. Pr o of . See the pro of of Lemma 2.4 .  4.5. Main theorem for como dules and con tramo dules. Assume that the coring C is a pro jectiv e left and a ﬂat right A -mo dule and the ring A has a ﬁnite left homological dimension. Theorem. (a) T he functor mapp i n g the quotient c ate gory of the homotopy c ate- gory of c om plexes of c opr oje ctive left C -c omo dules (c op r oje ctive c omplexes of left C -c omo dules) by its interse ction with the thick sub c ate gory of c o acyclic c omplexes of C -c omo dules into the c o derive d c ate gory of left C - c omo d ules is an e quivalenc e of triangulate d c ate g o ri e s. (b) The functor mapping the quotient c ate gory of the hom otopy c ate gory of c om- plexes of c oinje ctive left C -c ontr amo dules (c oinje ctive c omplexes of left C -c ontr a- mo dules) by its interse ction with the thick sub c ate gory of c ontr aacyclic c o mplexes of C -c ontr a mo dules into the c ontr aderive d c ate gory of left C -c ontr amo dules is an e q uiv- alenc e of triangulate d c a te gories. 86 Pr o of . The pro of of part (a) is completely analogous to the pro of of Theorem 2.5. It is based on the same constructions of resolutions L 1 and R 2 , a nd uses the result of Lemma 3.1.3(a) instead of Lemma 1.1 .3. T o pro v e part (b), we will sho w that any complex of left C -contramo dules K • can b e connected with a complex of coinjectiv e C -contramo dules in a functorial w a y b y a chain of t w o morphisms K • − → L 2 ( K • ) ← − L 2 R 1 ( K • ) with contraacyc lic cones. Moreo v er, if the complex K • is a complex of coinjectiv e C -contramo dules ( coinj ective complex of C -contramo dules), t hen the in termediate complex L 2 ( K • ) is also a complex of coinjectiv e C - contramo dules (coinjectiv e complex of C -contramo dules). Then w e will apply Lemma 2.5 in the wa y explained in the end of the pro of of Theorem 2.5. Let K • b e a complex of left C -contramo dules. Let P − → I ( P ) denote the func- torial injectiv e morphism f rom an arbitrary left C -contramo dule P in to an A -in- jectiv e C - contramo dule I ( P ) constructed in Lemma 3.1.3 (b). The functor I is the direct sum of a constan t functor P 7− → I (0 ) and a functor I + sending zero mor- phisms to zero morphisms. F or any C -contramo dule P , t he con tramo dule I + ( P ) is A -injectiv e and the morphism P − → I + ( P ) is injective . Set I 0 ( K • ) = I + ( K • ), I 1 ( K • ) = I + (cok er( K • → I 0 ( K • ))), etc. F or d large enough, the cok ernel Z ( K • ) of t he morphism I d − 2 ( K • ) − → I d − 1 ( K • ) will b e a complex o f A - injectiv e C -contramo dules. Let R 1 ( K • ) b e the total complex o f the bicomplex I 0 ( K • ) − − → I 1 ( K • ) − − → · · · − − → I d − 1 ( K • ) − − → Z ( K • ) . Then R 1 ( K • ) is a complex of A - injectiv e C -contra mo dules and the cone of t he mor- phism K • − → R 1 ( K • ) is the tot a l complex of a ﬁnite exact sequence of complexes of C -contramo dules, and therefore, a con traacyclic complex. No w let R • b e a complex of A -injective left C -contramo dules. Consider the bar construction · · · − − → Hom A ( C , Hom A ( C , R • )) − − → Hom A ( C , R • ) . Let L 2 ( R • ) b e the total complex of this bicomplex, constructed b y taking inﬁnite pro ducts alo ng the diagonals. Then L 2 ( R • ) is a complex of coinjectiv e C -contramo d- ules. The functor L 2 can b e extended to arbitrary complexes of C -contra mo dules; for an y complex K • , the cone of the morphism L 2 ( K • ) − → K • is con t raacyclic b y Lemma 4.1. Finally , if K • is a coinjectiv e complex of C -contramo dules, then L 2 ( K • ) is also a coinjectiv e complex of C -contramo dules, since the complex of cohomomorphisms from a complex of left C -como dules M • in to L 2 ( K • ) coincides with the complex of cohomomorphisms in to K • from t he total cobar complex R 2 ( M • ), and the lat t er is coacyclic whenev er M • is coacyclic.  Remark. Another pro of o f Theorem (for complexes of copro jectiv e como dules a nd complexes of coinjectiv e contramo dules) can b e deduced from the results of Section 5. 87 In addition, it will fo llow that an y coacyclic complex of copro jective left C - como dules is contractible and an y con traacyclic complex of coinjectiv e left C -contramo dules is con tractible (see Remark 5.5 ) . 4.6. Main theorem for semimo dules and semicon tramo dules. Assume that the coring C is a pro jectiv e left a nd a ﬂat rig h t A -mo dule, t he semialgebra S is a copro jectiv e left and a coﬂat righ t C -como dule, and the ring A has a ﬁnite left homological dimension. Theorem. (a) The functor ma p ping the quotient c ate gory of the h o m otopy c a te gory of semipr oje ctive c omplexes of A -pr oje ctive ( C -c o pr oje ctive, semipr oje ctive) left S -semi- mo dules by its interse ction with the thick sub c ate gory of C -c o acyclic c omplexes of S -semimo dules into the semiderive d c ate gory of left S -semim o dules is an e quivale n c e of triang ulate d c ate g ories. (b) The functor mapping the quotient c ate gory of the homotopy c ate g ory of semiin- je ctive c omplex e s of A -inje ctive ( C -c oinje c tive, semiinje ctive) left S -semic ontr amo d- ules by its in terse ction with the thick sub c ate g ory of C -c ontr aacyclic c omplexes of S -semic ontr amo dules into the semideriv e d c ate gory of left S -semic ontr amo dules is an e quivalenc e of triangulate d c ate gories . Pr o of . There are tw o approaches : one can ar gue as in 2.5 o r as in 2.6. Either w a y , the pro of is based on the constructions of intermediate resolutions L i and R j . F or part (a ) , it is the same constructions that w ere presen ted in the pro of of Theorem 2.6. One just has to use the results of Lemmas 3.3.2(a) and 3.3 .3(a) instead of Lemmas 1.3.2 and 1 .3.3. Let us introduce the analogous constructions for part (b). Let K • b e a complex of left S -semicontramo dules. Let P − → I ( P ) denote the functorial injectiv e morphism from an arbitra r y left S -semicontramo dule P in to an A -injectiv e S -semicontramo dule I ( P ) constructed in Lemma 3.3.2(b). The functor I is the direct sum of a constan t functor P 7− → I (0) and a f unctor I + sending zero morphisms to zero morphisms. F or an y S -semicontramo dule P , the semicon- tramo dule I + ( P ) is A -injectiv e and the morphism P − → I + ( P ) is injective . Set I 0 ( K • ) = I + ( K • ), I 1 ( K • ) = I + (cok er( K • → I 0 ( K • ))), etc. F o r d lar ge enough, the cok ernel Z ( K • ) of the mo r phism I d − 2 ( K • ) − → I d − 1 ( K • ) will b e a complex of A -injectiv e S - semicontramo dules. Let R 1 ( K • ) b e the total complex of the bicomplex I 0 ( K • ) − − → I 1 ( K • ) − − → · · · − − → I d − 1 ( K • ) − − → Z ( K • ) . Then R 1 ( K • ) is a complex of A -injective S -semicontramo dules and the cone of the morphism K • − → R 1 ( K • ) is t he total complex of a ﬁnite exact sequen ce of com- plexes of S -semicontramo dules, and therefore, a C - contraacyclic complex (and ev en an S -contraacyclic complex). 88 No w let R • b e a complex of A -injectiv e left S - semicontramo dules. Let F ( P ) − → P denote the functoria l surjectiv e morphism on to an arbitrary A -injectiv e S -semi- contramo dule P from a C -coinjectiv e S -semicontramo dule F ( P ) with an A -inj ective k ernel k er( F ( P ) → P ) constructed in Lemma 3.3.3(b). Set F 0 ( R • ) = F ( R • ), F 1 ( R • ) = F (ke r ( F 0 ( R • ) → R • )), etc. Let L 2 ( R • ) b e the total complex of the bicomplex · · · − − → F 2 ( R • ) − − → F 1 ( R • ) − − → F 0 ( R • ) , constructed b y ta king inﬁnite pro ducts along the diagonals. Then L 2 ( R • ) is a com- plex of C -coinjectiv e S -semicontramo dules. Since the surjection F ( P ) − → P can b e deﬁned for ar bitr ary left S - semicontramo dules, the functor L 2 can b e extended to arbitrary complexes o f S -semicontramo dules. F or a n y complex K • , the cone of the morphism L 2 ( K • ) − → K • is a C -contra acyclic complex (and ev en an S -contraacyclic complex) b y Lemma 4.1. Finally , let P • b e a C -coinjectiv e complex of A -injectiv e left S -semicontramo d- ules. Then the complex Cohom C ( S , P • ) is a semiinjectiv e complex of A -injective left S -semicontramo dules. Moreov er, if P • is a complex of C -coinjectiv e S -semicontra- mo dules, then Cohom C ( S , P • ) is a semiinjectiv e complex of semiinjectiv e S -semicon- tramo dules. Consider the cobar construction Cohom C ( S , P • ) − − → Cohom C ( S , Cohom C ( S , P • )) − − → · · · Let R 3 ( P • ) b e the to t a l complex of this bicomplex, constructed b y taking inﬁnite pro ducts along the diagonals. Then complex R 3 ( P • ) is semiinjectiv e b y Lemma 4.4. The functor R 3 can b e extended to arbitrary complexes of S -semicontramodules; for an y complex K • , the cone of the morphism K • − → R 3 ( K • ) is not only C -con- traacyclic, but eve n C - contractible (the contracting homoto py b eing induced by the semiunit morphism C − → S .) It follo ws that the natural functors b et w een the quotien t categories of the ho- motop y categories of semiinjectiv e complexes o f semiinjectiv e S -semicontramo dules, semiinjectiv e complexe s of C -coinjectiv e S -semicontramo dules, complexes of C - coin- jectiv e S -semicontramo dules, semiinjectiv e complexes of A -injectiv e S -semicontra- mo dules, C -coinj ective complexes of A - injectiv e S -semicontramo dules, complexe s o f A -injectiv e S -semicontramo dules b y their inters ections with the thic k sub category of C -contraacycliccomplexes and the semideriv ed category of left S -semicontramo d- ules are all equiv alences of triangulated catego r ies. Moreo v er, an y complex of left S -semicontramo dules K • can b e connected with a se miinjectiv e complex of semi- injectiv e S -semicontramo dules in a functorial w ay by a chain of three morphisms K • − → R 3 ( K • ) ← − R 3 L 2 ( K • ) − → R 3 L 2 R 1 ( K • ) with C -contraa cyclic cones , and when K • is a semiinjectiv e complex of ( A -injectiv e, C -coinjectiv e, or semiinjectiv e) S -semicontramo dules, all complexes in this c hain ar e also semiinjectiv e complexes of ( A -injectiv e, C -coinjectiv e, or semiinjectiv e) S -semicontramo dules.  89 Remark. One can show using the metho ds dev elop ed in Section 6 that an y C -coacyclic semipro jectiv e complex of C -copro jectiv e left S -semimo dules is con tract- ible, and analog ously , an y C - contraacyclic semiinjectiv e complex of C - coinj ective left S -semicontramo dules is contractible (see Remark 6.4). 4.7. Derived functor SemiExt. Assume that the coring C is a pro jective left and a ﬂat righ t A -mo dule, the semialgebra S is a copro jectiv e left and a coﬂat right C -como dule, and the ring A has a ﬁnite left homological dimension. The double-sided deriv ed functor SemiExt S : D si ( S – simo d ) × D si ( S – sicntr ) − − → D ( k – mo d ) is deﬁned as follows . Consider the partially deﬁned functor of semihomomorphisms of complexes SemiHom S : Hot ( S – simo d ) op × Hot ( S – sicntr ) 99K Hot ( k – mo d ). This functor is deﬁned on the full sub category of the Carthesian pro duct of homotopy categories that consists of pairs o f complexes ( M • , P • ) suc h that either M • is a complex of A -projectiv e S -semimo dules, or P • is a complex of A -injectiv e S -semicontramo d- ules. Comp ose it with the functor of lo calization Hot ( k – mo d ) − → D ( k – mo d ) and restrict either to the Carthesian pro duct of the homotop y category of semipro jectiv e complexes of A -proj ective S -semimo dules and the homotopy category of S - semicon- tramo dules, or to the Carthesian pro duct of the homotopy categor y of S -semimo dules and t he homotopy category of semiinjectiv e complexes of A -inj ective S -semicontra- mo dules. By Theorem 4.6 and Lemma 2.7, b oth functors so obta ined factor ize through the Carthesian pro duct of semideriv ed categories of left semimo dules and left semicon- tramo dules and t he deriv ed functors so deﬁned are naturally isomorphic. The same deriv ed functor is obtained b y restricting the functor of semihomomorphisms to the Carthesian pro duct of the homotop y categories of semipro jectiv e complexes of A - pr o - jectiv e S -semimo dules a nd semiinjectiv e complexes of A -injectiv e S -semicontramo d- ules. One can also use semipro jectiv e complexes of C -coproj ectiv e S -semimo dules or semiinjectiv e complexes of C -coinj ective S -semicontramo dules, etc. In particular, when the coring C is a pro jectiv e left and a ﬂat right A -mo dule and the ring A has a ﬁnite left homological dimension, one deﬁnes the double-sided deriv ed functor Co ext C : D co ( C – como d ) op × D ctr ( C – contra ) − − → D ( k – mo d ) b y comp osing the functor of cohomo mo r phisms Cohom C : Hot ( C – como d ) op × Hot ( C – contra ) − → Hot ( k – mo d ) with the functor of lo calization Hot ( k – mo d ) − → D ( k – mo d ) and restricting it to either the Carthesian pro duct of the ho mo t o p y category of complexes o f copro jectiv e C -como dules and the homot o p y category of arbitrary complexes of C -contramo dules, or the Carthesian pro duct of t he homotopy category of arbitra r y complexes of C - como dules and the homotopy category of 90 complexes of coinjectiv e C -contra mo dules. The same deriv ed functor is obtained b y restricting the functor of cohomomorphisms to the Carthesian pro duct of the homotop y categories of copro j ectiv e C -como dules and coinjectiv e C -contramo dules. One can a lso use copro jec tiv e complexes of C -como dules or coinjectiv e complexes of C -contramo dules. Question. Assuming only that C is a ﬂat left and righ t A -mo dule, one can deﬁne the double-sided derive d functor Cotor C on the Carthesian pro duct of co deriv ed cat- egories of the exact categories of right and left C -como dules of ﬂat dimension o v er A not exceeding d , for any giv en d , usin g Lemma 2.7 and the corresp onding vers ion of Lemma 1.1.3. Analogously , assuming that C is a pro jectiv e left and a ﬂat righ t A -mo dule, one can deﬁne the double-sided deriv ed functor Co ext C on the Cart he- sian pro duct of the co deriv ed category of left C -como dules of pro jectiv e dimension o v er A not exceeding d and the con traderiv ed category of C -contramo dules of in- jectiv e dimension o v er A not exceeding d . One can ev en do with the homological dimension assumption on only one of the argumen ts of Cotor C and Co ext C , using t he corresp onding v ersions of the results o f Theorem 7.2.2. Can one deﬁne, at least, a de- riv ed functor SemiT or S for complexes of A - ﬂa t S -semimo dules and a deriv ed functor SemiExt S for complexes of A -projectiv e S -semimo dules and A -injectiv e S -semicon- tramo dules without the homological dimension assumptions o n A ? The only problem one encoun ters attempting to do so comes from the homological dimension conditions in Prop ositions 1.2.3(c) and 3.2.3.1-2(c) and consequen tly in Lemmas 1.3.3 and 3.3.3; when S satisﬁes the conditions of Prop osition 1.2.5(f ) there is no problem. Remark. In the wa y completely ana lo gous to Remark 2.7, without an y homological dimension assumptions one can deﬁne the do uble-sided deriv ed functor IndCoext C for complexes of left C -como dules in k – mo d ω and complexes o f left C -contr a mo dules in the category k – mo d ω of ind-ob jects o v er k – mo d represen ta ble b y coun table ﬁltered inductiv e systems o f k -mo dules. Here the category opp osite to k – mo d ω is considered as a mo dule category o v er the tensor category k – mo d ω and C is a coring ov er a ring A in k – mo d ω . Appropriate coﬂatness and “contrapro jectivity ” conditions hav e to b e imp osed on C . The coun t a bilit y assumption can b e dropp ed. 4.8. R elatively semipro jective and semiinjectiv e complexes. W e k eep the as- sumptions and notat io n o f 4 .5, 4.6, and 4.7. One can compute the derive d functor Co ext C using resolutions of other kinds. Namely , the complex o f cohomomorphisms Cohom C ( M • , P • ) from a complex of C / A -coprojective left C -como dules M • in to a complex o f A -injectiv e left C -contra- mo dules P • represen ts an ob ject naturally isomorphic to Co ext C ( M • , P • ) in the de- riv ed category of k -mo dules. Indeed, the complex L 2 ( P • ) is a complex of coinjectiv e C -contramo dules and the cone of t he morphism L 2 ( P • ) − → P • is con traacyclic with resp ect to the exact category o f A - injectiv e C -contramo dules, hence the morphism 91 Cohom C ( M • , L 2 ( P • )) − → Cohom C ( M • , P • ) is an isomorphism. Analogo usly , the complex of cohomomorphisms Cohom C ( M • , P • ) fr o m a complex of A -projectiv e left C -como dules M • in to a complex of C / A - coinjective left C -contramo dules P • repre- sen t s an ob ject naturally isomorphic t o Co ext C ( M • , P • ) in the deriv ed category of k -mo dules. One can also compute the deriv ed functor SemiExt C using resolutions of other kinds. Namely , a complex o f left S -semimo dules is called sem ipr oje ctive r e lative to A if the complex of semihomomorphisms fr om it in to any complex of A -injec- tiv e left S -semicontramo dules that as a complex of C - contramo dules is con traacyclic with resp ect to the exact category of A -injectiv e C -contramo dules is acyclic (cf. The- orem 7.2.2(c)). The complex of se mihomomorphisms SemiHom C ( M • , P • ) from a complex of left S - semimo dules M • semipro jectiv e relativ e to A in to a complex of A -injectiv e left S -semicontramo dules P • represen ts an ob ject naturally isomorphic to SemiExt S ( M • , P • ) in the deriv ed category of k -mo dules. Indeed, R 3 L 2 ( P • ) is a semiinjectiv e complex o f S -semicontramo dules connected with P • b y a chain of mor- phisms P • ← − L 2 ( P • ) − → R 3 L 2 ( P • ) whose cones are con traacyclic with resp ect to the exact category of A -injectiv e C -contramo dules and contractible o v er C , respec- tiv ely . Analogously , a complex of left S -semicontramo dules is called semiinje ctive r elative to A if the complex o f semihomomorphisms into it fr o m a n y complex of A -pro- jectiv e left S -semimo dules that a s a complex of C -como dules is coa cyclic with resp ect to t he exact catego r y of A -projectiv e C -como dules is acyclic (cf. Theorem 7.2.2(b)). The complex of semihomomorphisms SemiHom C ( M • , P • ) from a complex of A - pro- jectiv e left S -semimo dules to a complex of left S -semicontramo dules semiinjectiv e relativ e to A represen ts an ob ject na turally isomorphic to SemiExt S ( M • , P • ) in the deriv ed catego ry of k -mo dules. F or example, the complex of S -semimo dules induced from a complex of C / A -coprojectiv e left C -como dules is semipro jectiv e relative to A and the complex of S -semicontramo dules coinduced from a complex of C / A - coinjec- tiv e left C -contramo dules is semiinjectiv e relative to A . A complex of left S -semimo dules is called sem ipr oje ctive r elative to C if the com- plex of semihomomorphisms fro m it in t o an y C -contractible complex of C -coin- jectiv e left S -semicontramo dules is acyclic. The complex of semihomomorphisms SemiHom C ( M • , P • ) from a complex of left S -semimo dules M • semipro jectiv e relativ e to C in to a complex of C -coinjectiv e left S -semicontramo dules P • represen ts a n ob ject naturally isomorphic to SemiExt S ( M • , P • ) in the deriv ed category of k -mo dules. In- deed, R 3 ( P • ) is a semiinjectiv e complex of S -semicontramo dules and the cone of the morphism P • − → R 3 ( P • ) is a C -contractible complex o f C -coinjectiv e S - semicontra- mo dules. Analogously , a complex of left S - semicontramo dules is called sem i i n je ctive r elative to C if the complex of semihomomorphisms in to it f r om a n y C -contractible complex of C - coprojectiv e left S -semimo dules is acyclic. The complex o f semihomo- morphisms SemiHom C ( M • , P • ) f r om a complex of C -coprojectiv e left S -semimo dules 92 M • in to a complex of left S -semicontramo dules P • semiinjectiv e relativ e to C rep- resen ts an ob ject natura lly isomorphic to SemiExt S ( M • , P • ) in the derived category of k -mo dules. It follows that the complex of semihomomorphisms from a complex of left S -semimo dules semipro jectiv e relativ e to C in to a C -contraacyclic complex of C -coinjectiv e left S -semicontramo dules is acyclic, and the complex of semihomomor- phisms into a complex of left S -semicontramo dules semiinjectiv e relative to C fro m a C -coacyclic complex of C -copro jectiv e left S -semimo dules is a cyclic. F or example, the complex of S -semimo dules induced from a complex of left C - como dules is semi- pro jectiv e relative t o C and the complex of S -semicontramo dules coinduced f r o m a complex of left C -contra mo dules is semiinjectiv e relativ e to C . A t last, a complex of A -projectiv e left S -semimo dules is called semipr oj e ctive r ela- tive to C r elative to A ( S / C / A -semiprojectiv e) if the complex of semihomomorphisms from it in to any C - contractible complex of C / A -coinjectiv e left S -semicontramo d- ules is acyclic. The complex o f semihomomorphisms SemiHom C ( M • , P • ) fro m an S / C / A -semiprojectiv e complex of A -projectiv e left S -semimo dules M • in to a complex of C / A -coinjectiv e left S -semicontramo dules P • represen ts an ob ject naturally iso- morphic to SemiExt S ( M • , P • ) in the derive d category of k -mo dules. Indeed, R 3 ( P • ) is a complex of left S -semicontramo dules semiinjectiv e relative to A and the cone of the morphism P • − → R 3 ( P • ) is a C -contractible complex of C / A - coinjectiv e S -semi- contramo dules. Analogously , a complex of A -injectiv e left S -semicontramo dules is called semiinje ctive r ela tive to C r elative to A ( S / C / A -semiinjectiv e) if the complex of semihomomorphisms in to it from an y C -contractible complex of C / A -coprojectiv e left S -semimodules is acyclic. The complex o f semihomomorphisms SemiHom C ( M • , P • ) from a complex of C / A - coprojectiv e left S -semimo dules M • in to an S / C / A -semi- injectiv e complex of A -injectiv e left S -semicontramo dules P • represen ts a n ob ject naturally isomorphic t o SemiExt S ( M • , P • ) in the deriv ed category of k -mo dules. It follo ws tha t t he complex of semihomomorphisms f r o m an S / C / A -semiprojectiv e com- plex of A -projectiv e left S -semimo dules in to a C -contraacyclic complex o f C / A -coin- jectiv e left S -semicontramo dules is acyclic, and the complex of semihomomorphisms in to an S / C / A -semiinjectiv e complex of A -injectiv e left S -semicontramo dules from a C -coacyclic complex of C / A - coprojectiv e left S -semimo dules is acyclic. F or ex- ample, the complex of S -semimo dules induced from a complex of A -pro jectiv e left C -como dules is S / C / A -semiprojectiv e and t he complex of S -semicontramo dules coin- duced from a complex of A -injectiv e left C -contramo dules is S / C / A -semiinjectiv e. The functors mapping the quotien t catego ries of the homotop y categories o f com- plexes of S -semimodules semipro j ective relative to A , complexes of S -semimo dules semipro jectiv e relativ e to C , and S / C / A -semiprojectiv e complexes o f S -semimo dules b y their in tersections with the thic k subcat ego ry of C -coacyclic complexes in to the semideriv ed category of left S -semimo dules are equiv alences of triangulated cate- gories. Analogously , the functors mapping the quotien t catego r ies o f the homotop y 93 categories of complexes of S -semicontramo dules semiinjectiv e relativ e to A , com- plexes of S -semicontramo dules semiinjectiv e r elat ive to C , and S / C / A -semiinjectiv e complexes of S -semicontramo dules b y their interse ctions with the thic k subcatego r y of C -contraacyclic complexes into the semideriv ed category of left S - semicontramo d- ules ar e equiv alences of triangulated categories. The same applies to complexes of A -projectiv e, C -coprojectiv e, and C / A - coprojectiv e S -semimo dules and complexes of A -injectiv e, C -coinjectiv e, and C / A -coinjectiv e S -semicontramodules. These results follo w easily fro m either of Lemmas 2.5 o r 2.6. So one can deﬁne t he deriv ed functor SemiExt S b y restricting the functor of se mihomomorphisms to these categories of complexes as explained ab o v e. Remark. One can deﬁne the double-sided or right derive d functor SemiExt S in the assumptions analogous to tho se of Remark 2.8 in the completely analogous w a ys. 4.9. R emarks on deriv ed semihomomorphis ms from bisemimo dules. Let S b e a semialgebra o v er a coring C and T b e a semialgebra o v er a coring D , b oth satisfying the conditions of 4.6 . One can deﬁne the double-sided deriv ed functor D SemiHom S : D si ( S – simo d – T ) op × D si ( S – sicntr ) − − → D si ( T – sicntr ) b y restricting the functor o f semihomomorphisms SemiHom S : D si ( S – simo d – T ) op × D si ( S – sicntr ) 99K D si ( T – sicntr ) to the Carthesian pro duct of the homotop y category of complexes of S - T - bisemimodules and the homotopy category o f semiinjectiv e com- plexes of C -coinjectiv e left S -semicontramo dules (using the result of Remark 6 .4). There is an asso ciativit y isomorphism SemiExt S ( K • ♦ D T M • , P • ) ≃ SemiExt T ( M • , D SemiHom S ( K • , P • )) . Let R b e a semialgebra ov er a coring E satisfying the conditions of 4.6. If the k -alg ebra A is a ﬂat k -mo dule and the k - algebras B a nd F are pro jective k -mo dules, then the deriv ed functor D SemiHom can b e deﬁned using Lemma 2.7 in terms of str ongly S -semipr oje ctive complexes of A -projectiv e S - T - bisemimo dules and semiin- jectiv e complexes of C -coinjectiv e left S - semicontramo dules (or str ong l y sem iinje ctive complexes of A -injectiv e left S -semicontramo dules). Here a complex of A - projectiv e S - T -bisemimo dules K • is called strongly S -semiprojectiv e if for any C -contra a cyclic complex of left S -semicontramo dules P • the complex of left T -semicontramodules SemiHom S ( K • , P • ) is D -contraacyclic; strongly semiinjectiv e complexes are deﬁned in the analogo us w ay . In this case, there is an asso ciativity isomorphism D SemiHom S ( K • ♦ D T M • , P • ) ≃ D SemiHom T ( M • , D SemiHom S ( K • , P • )) for any complex of T - R -bisemim o dules M • , a n y complex of S - T -bisemimo dules K • , and a n y complex o f left S -semicontramo dules P • . 94 In particular, without an y conditions on the k -mo dule A for any complex of rig h t S -semimodules N • and a ny complex o f left S -semimo dules M • there is a natural isomorphism Hom k (SemiT or S ( N • , M • ) , k ∨ ) ≃ SemiExt S ( M • , Hom k ( N • , k ∨ )). 95 5. Comodule-Contramodule Corresp ondence 5.1. Contratensor pro duct and como dule/con t ramo dule homomorphisms. Let C b e a coring ov er a k -alg ebra A . 5.1.1. The c ontr atenso r pr o duct N ⊙ C P of a right C -como dule N and a left C -contramo dule P is a k -mo dule deﬁne d as the cok ernel of the pair of maps N ⊗ A Hom A ( C , P ) ⇒ N ⊗ A P one of whic h is induced b y the C -contraaction in P , while the other is the comp osition of the map induced b y the C -coaction in N and the map induced by the ev aluatio n map C ⊗ A Hom A ( C , P ) − → P . The contratensor pro duct op eration is dual to homomorphisms in the category of con tra mo dules: for an y right C -como dule N with a left action of a k -algebra B b y C -como dule endomorphisms, an y left C -contramo dule P , and any left B - mo dule U there is a natural isomorphism Hom B ( N ⊙ C P , U ) ≃ Hom C ( P , Hom B ( N , U )). Indeed, bo t h k -mo dules are isomorphic to the ke rnel o f the same pa ir o f maps Hom A ( P , Hom B ( N , U )) ⇒ Hom A (Hom A ( C , P ) , Hom B ( N , U )). T aking B = k , one can conclude t hat for an y right C -como dule N and an y left A -mo dule V there is a natural isomorphism N ⊙ C Hom A ( C , V ) ≃ N ⊗ A V . When C is a pro j ectiv e left A - mo dule, t he functor of contratensor pro duct o v er C is righ t exact in b oth its argumen ts. 5.1.2. Let D b e a coring ov er a k -algebra B . F or an y C - D -bicomo dule K and an y left C -como dule M , the k - mo dule Hom C ( K , M ) has a nat ura l left D - contr a mo dule structure as the k ernel of a pa ir of D - contramo dule morphisms Hom A ( K , M ) ⇒ Hom A ( K , M ⊗ B D ). Analogo usly , for an y D - C -bicomo dule K and any left C -con- tramo dule P , the k -mo dule K ⊙ C P has a natural left D -como dule structure a s the cok ernel of a pair of D -como dule morphisms K ⊗ A Hom A ( C , P ) ⇒ K ⊗ A P . F or any left D -como dule M , an y D - C -bicomo dule K , and any left C -contramo dule P there is a natura l isomorphism Hom D ( K ⊙ C P , M ) ≃ Hom C ( P , Hom D ( K , M )). Indeed, a B -mo dule map K ⊗ A P − → M factorizes through K ⊙ C M if and only if the corresp onding A - mo dule map P − → Hom B ( K , M ) is a C -contramo dule morphism, and a B -mo dule map K ⊗ A P − → M is a D -como dule morphism if and only if the corresp onding A -mo dule map P − → Hom B ( K , M ) factorizes through Hom D ( K , M ). In particular, there is a pair of adjoint f unctors Ψ C : C – como d − → C – contra a nd Φ C : C – contra − → C – como d b etw een the categories of left C - como dules and left C -con- tramo dules deﬁned by the rules Ψ C ( M ) = Hom C ( C , M ) and Φ C ( P ) = C ⊙ C P . 5.1.3. A left C - como dule M is called quite in je ctive r elative to A ( quite C / A -injective ) if the functor of C - como dule homomorphisms in to M maps A -split exact triples of left C -como dules t o exact triples. It is easy t o see that a C - como dule is quite C / A -in- jectiv e if and only if it is a direct summand of a coinduced C - como dule. Analogously , a left C -contramo dule P is called quite pr oje ctive r elative to A (quite C / A - pro jectiv e) 96 if the functor of C -contramo dule homomorphisms fr om P maps A -split exact triples of left C -contramo dules to exact triples. A C -contramo dule is quite C / A -projectiv e if and o nly if it is a direct summand of an induced C -contr amo dule. The r estrictions o f the functors Ψ C and Φ C on the sub categories o f quite C / A -in- jectiv e left C - como dules and quite C / A -pro jectiv e left C -contramo dules are m utually in v erse equiv alences b etw een these sub categories. Indeed, one ha s Hom C ( C , C ⊗ A V ) = Hom A ( C , V ) a nd C ⊙ Hom A ( C , V ) = C ⊗ A V . 5.1.4. A left C -como dule M is called inje ctive r ela tive t o A ( C / A -injectiv e) if the functor of homomo r phisms in to M maps exact t r iples of A - projectiv e left C -como dules to exact triples. A left C -contramo dule P is called pr oje ctive r elative to A ( C / A - pro jectiv e) if the functor of homomorphisms from P maps exact triples of A -injectiv e left C - contramo dules to exact triples. (Cf. Lemma 5.3 .2 .) Remark. What w e call quite relatively injectiv e como dules are usually called rela- tiv ely injectiv e como dules [16]. W e chose this non traditional terminology for coher- ence with our deﬁnitions of relativ e coﬂat ness, etc., and also b ecause what w e call relativ ely injectiv e como dules is a more imp ortan t notion f r o m o ur p oin t of view. Question. One can compute mo dules Ext in the exact category of left C -como dules with A -split exact triples in terms of the cobar resolution. When C is a pro jectiv e left A -mo dule, this resolution can b e also used t o compute mo dules Ext in the exact category of A -projectiv e left C -como dules, whic h therefore turn out to b e the same. Ho w can one compute mo dules Ext in the exact category of A -projectiv e C -como dules without making any pro jectivity assumptions on C ? 5.1.5. When C is a ﬂat righ t A -mo dule, the coinduction functor A – mo d − → C – como d preserv es injectiv e ob jects. It fo llo ws easily that an y left C - como dule is a sub como dule of an injectiv e C -como dule; a C -como dule is injectiv e if and only if it is a direct summand of a C -como dule coinduced fr o m a n injectiv e A - mo dule. Analogously , when C is a pro jectiv e left A - mo dule, the induction functor A – mo d − → C – contra pre- serv es pro jectiv e ob jects. Hence an y left C - contramo dule is a quotien t contramo dule of a pro jectiv e C -contramo dule; a C -contra mo dule is pro jectiv e if and only if it is a direct summand of a C -contramo dule induced from a pro jectiv e A - mo dule. 5.1.6. When C is a ﬂat left A -mo dule, a left C - contramo dule P is called c o ntr a- ﬂat if the functor of contratens or pr o duct with P is exact on the category of right C -como dules. The C - contramo dule induced fro m a ﬂat A - mo dule is contraﬂat. An y pro jectiv e C - contr a mo dule is contraﬂat. A left C - contramo dule P is called quite C / A -c ontr aﬂat if the functor of con tra- tensor pro duct with P maps those exact triples of rig h t C -como dules whic h as exact triples of A -mo dules remain exact after the tensor pro duct with an y left A -mo dule to exact t r iples. An y quite C / A -proj ective C - contramo dule is quite C / A -contr a ﬂat. 97 A left C -contramo dule P is called C / A -c ontr aﬂat if the functor of con tratensor pro d- uct with P maps exact triples of A -ﬂat righ t C - como dules to exact triples. Using the dualization functor Hom k ( − , k ∨ ), one can easily c hec k tha t any C / A -projectiv e C -como dule is C / A -contraﬂat. 5.2. A sso ciativit y isomorphisms. Let C b e a coring ov er a k -algebra A and D b e a coring ov er a k - algebra B . The f o llo wing three Prop ositions will b e mostly applied to the case of K = D = C in the sequel. Prop osition 1. L et N b e a right D -c omo dule, K b e a D - C -bi c omo d ule, an d P b e a left C -c ontr am o dule. Then ther e is a natur al map ( N  D K ) ⊙ C P − → N  D ( K ⊙ C P ) whenever the c otensor pr o duct N  D K is en d o we d with a righ t C -c om o dule structur e such that the map N  D K − → N ⊗ B K is a C -c omo d ule morp h ism. T h is natur al map is an iso morphism, at le ast, in the fol lowing c ases: (a) C is a ﬂat left A -mo dule and P is a c ontr aﬂat left C -c ontr amo dule, (b) P is a quite C / A -c ontr aﬂat l e f t C -c o ntr amo dule and K as a left D -c omo dule with a rig h t A -mo dule structur e is c o nduc e d fr om a B - A -bimo dule; (c) P is a C / A -c ontr aﬂat left C -c ontr am o dule, D is a ﬂat righ t B -mo dule, N is a ﬂat right B -mo dule, and K as a left D -c omo dule with a right A -mo d ule structur e is c o induc e d fr om an A -ﬂat B - A - b i m o dule; (d) P is a C / A -c on tr aﬂat left C -c ontr a mo dule, D is a ﬂat righ t B -m o dule, N is a ﬂat right B -mo dule, K is a ﬂat right A -m o dule, K is a D /B -c oﬂat left D -c omo dule, and the ring A has a ﬁnite we a k homolo gic al dimension; (e) N is a quasic oﬂat right D -c omo dule. Pr o of . The ma p ( N  D K ) ⊙ C P − → N ⊗ B K ⊙ C P ha s equal comp ositions with t w o maps N ⊗ B K ⊙ C P ⇒ N ⊗ B D ⊗ B K ⊙ C P , so there is a natural map ( N  D K ) ⊙ C P − → N  D ( K ⊙ C P ). Besides, the comp osition ( N  D K ) ⊗ A P − → N  D ( K ⊗ A P ) − → N  D ( K ⊙ C P ) annihilates t he diﬀerence b et w een tw o ma ps ( N  D K ) ⊗ A Hom A ( C , P ) ⇒ ( N  D K ) ⊗ A P , whic h leads to the same nat ura l map ( N  D K ) ⊙ C P − → N  D ( K ⊙ C P ). T o prov e cases (a-d), one show s that the sequence 0 − → N  D K − → N ⊗ B K − → N ⊗ B D ⊗ B K remains exact after ta king the contratensor pro duct with P . Indeed, the case (a ) is obvious, in the cases (b- c) this exact sequence of right A -mo dules splits, and in the cases (c-d) this sequenc e of right A -mo dules is exact with resp ect to the exact category of ﬂat A -mo dules (see the pro of of Prop osition 1.2.3). T o pro v e (e), one notices that the sequence K ⊗ A Hom A ( C , P ) − → K ⊗ A P − → K ⊙ C P − → 0 remains exact after taking the cotensor pro duct with N and uses Prop osition 1.2 .3 (b).  Prop osition 2. L et L b e a left D -c o mo dule, K b e a C - D -bic omo dule, and M b e a left C -c omo dule. T h en ther e is a natur al map Cohom D ( L , Hom C ( K , M )) − → Hom C ( K  D L , M ) w henever the c o tensor pr o duct K  D L is endowe d with a left C -c omo dule 98 structur e such that the map K  D L − → K ⊗ B L is a C -c omo dule morphism . This natur al map is an isomorphism, at le ast, in the fol lowing c ase s : (a) C is a ﬂat right A -mo dule and M is a n inje ctive lef t C -c omo dule; (b) M is a quite C / A -inje ctive l e f t C -c omo dule and K as a right D -c omo dule with a left A -mo dule structur e is c oind uc e d fr om an A - B -bimo dule; (c) M is a C / A -inje ctive left C -c om o dule, D is a pr oje ctive left B -mo dule, L is a pr oje ctive left B -mo dule, and K as a right D -c omo dule with a left A -mo dule structur e is c o induc e d fr om an A -pr oje ctive A - B -bimo dule; (d) M is a C / A -inje ctive left C -c o m o dule, D is a pr oje c tive left B -mo dule, L is a pr oje ctive le f t B -mo dule, K is a pr oje ctive left A -mo dule, K is a D /B -c oﬂat right D -c omo dule, and the ring A has a ﬁnite le f t hom olo gic al dimensi o n; (e) L is a quasic op r oje ctive left D -c omo dule. Pr o of . Analogous to the pro o f of Prop osition 1 and Prop osition 3 b elo w (see also the pro of of Prop osition 3 .2.3.1). In particular, to prov e (e) one no t ices that the sequenc e 0 − → Ho m C ( K , M ) − → Hom A ( K , M ) − → Hom A ( K , C ⊗ A M ) remains exact af ter taking the cohomomorphisms f r o m L .  Prop osition 3. L et P b e a left C -c ontr a mo dule, K b e a D - C -b ic omo dule, and Q b e a left D -c ontr amo dule. Then ther e is a natur al map Cohom D ( K ⊙ C P , Q ) − → Hom C ( P , Cohom D ( K , Q )) whe n ever the c oho momorphism mo dule Cohom D ( K , Q ) is endowe d with a left C -c ontr amo dule s tructur e such that the map Hom B ( K , Q ) − → Cohom D ( K , Q ) is a C -c ontr amo dule morphism. This natur al map is an isomo rphism, at le ast, in the fol lowing c ases: (a) C is a pr oje ctive le ft A -mo dule and P is a pr oje ctive left C -c ontr amo dule; (b) P is a quite C / A -pr oje ctive le ft C -c ontr amo dule and K as a left D -c omo dule with a rig h t A -mo dule structur e is c o induc e d fr om a B - A -bimo dule, (c) P is a C / A -pr oje ctive left C -c ontr amo dule, D is a ﬂat right B -mo dule, Q is an inje ctive le ft B -mo dule, and K as a left D -c omo dule with a right A -mo dule structur e is c o induc e d fr om an A -ﬂat B - A - b i m o dule; (d) P is a C / A -pr oje c tive left C -c ontr amo dule, D is a ﬂat right B -mo dule, Q is an inje ctive le ft B -mo dule, K is a ﬂat right A -mo dule, K is a D /B -pr oje ctive left D -c omo dule, and the ring A has a ﬁnite lef t hom olo gic al dimens i o n; (e) Q is a quasic oinje ctive left D -c o ntr amo dule. Pr o of . The map Hom C ( P , Hom B ( K , Q )) − → Hom C ( P , Cohom D ( K , Q )) a nnihilates the diﬀerence of tw o maps Hom C ( P , Hom B ( D ⊗ B K , Q )) − → Hom C ( P , Hom B ( K , Q )) and t his pair of maps can b e identiﬁed with the pair of maps Hom B ( D ⊗ B K ⊙ C P , Q ) ⇒ Hom B ( K ⊙ C P , Q ) whose cok ernel is, b y the deﬁnition, the cohomomorphism mo dule Cohom D ( K ⊙ C P , Q ). Hence there is a natura l ma p Cohom D ( K ⊙ C P , Q ) − → Hom C ( P , Cohom D ( K , Q )). Bes ides, the composition Cohom D ( K ⊙ C P , Q ) − → 99 Cohom D ( K ⊗ A P , Q ) − → Hom A ( P , Cohom D ( K , Q )) has equal comp ositions with t w o maps Hom A ( P , Cohom D ( K , Q )) ⇒ Hom A (Hom A ( C , P ) , Cohom D ( K , Q )), whic h leads to the same natura l map Cohom D ( K ⊙ C P , Q ) − → Hom C ( P , Cohom D ( K , Q )). T o prov e cases (a-d), one sho ws that the sequence Hom B ( D ⊗ B K , Q ) − → Hom B ( K , Q ) − → Cohom D ( K , Q ) − → 0 remains exact after applying the functor Hom C ( P , − ). Indeed, the case (a) is ob vious, in the cases (b-d) this sequence of left A -mo dules splits, and in the cases (c-d) it is also an exact sequence of injectiv e A -mo dules (see the pro of of Prop osition 3 .2.3.2). T o prov e (e), one notices that the sequence K ⊗ A Hom A ( C , P ) − → K ⊗ A P − → K ⊙ C P − → 0 remains exact after taking the cohomomorphisms into Q and uses Prop osition 3.2.3.2 (b).  In the case of K = D = C , the natural maps deﬁned in Prop ositions 2–3 ha v e the follo wing prop erty of compat ibility with the adj o in t functors Ψ C and Φ C : for an y left C -como dule M and any left C -contramo dule P the maps Cohom C (Φ C ( P ) , Ψ C ( M )) − → Hom C (Φ C ( P ) , M ) and Cohom C (Φ C ( P ) , Ψ C ( M )) − → Hom C ( P , Ψ C ( M )) form a comm u- tativ e diagram with the isomorphism Hom C (Φ C ( P ) , M ) ≃ Hom C ( P , Ψ C ( M )). The following imp or t a n t Lemma is deduced as a corolla ry of Prop ositions 2– 3 . Lemma. (a) A C -c omo dule is quasic opr o j e ctive if and only if it is quite C / A -inje c- tive. I f C is a pr oje ctive left A -mo dule, then a left C -c omo dule is c opr oje c tive if and only if it is a di r e ct summand of a c omo dule c oinduc e d fr om a p r oje ctive A -mo dule. (b) A C -c ontr amo dule is quasic o i n je ctive if an d on ly if it is quite C / A -pr oje ctive. If C is a ﬂat right A -mo dule, then a left C -c ontr amo dule is c oin je ctive if and only if it is a dir e ct summand of a c ontr a mo dule induc e d fr om an inje ctive A -mo d ule. Pr o of . P art (a): let M be a quasicopro jectiv e left C -como dule. Denote b y l the coaction map M − → C ⊗ A M . It is an A -split injective morphism of quasicopro jectiv e C -como dules. According to Prop osition 2(e), we hav e an isomorphism of morphisms Hom C ( l , M ) ≃ Cohom C ( l , Hom C ( C , M )). But the map Cohom C ( l , P ) is surjectiv e for an y left C - contr a mo dule P . Therefore, the map Ho m C ( l , M ) is a lso surjectiv e, hence the morphism l splits and t he como dule M is quite C / A -injectiv e. Now supp ose that M is copro jectiv e; then w e already kno w that M is quite C / A -injectiv e. Set P = Ψ C ( M ). It follows from Prop o sition 3(b) that there is an isomorphism of functors Hom C ( P , − ) ≃ Cohom C ( M , − ) on the category of left C -contramo dules. Therefore, the C - contramo dule P is pro jectiv e, hence it is a direct summand of a C -contra mo dule induced fro m a pro jectiv e A -mo dule a nd M is a direct summand o f the C -como dule coinduced fr o m the same pro jec tiv e A -mo dule. The pro of of pa r t (b) is completely analogous; it uses Prop ositions 3(e) and 2(b).  Question. Are there any analogues of the results of Lemma for (quasi)coﬂat como d- ules a nd (quite relatively ) con traﬂat contramodules? 100 5.3. R elatively injectiv e como dules and relativ ely pro jectiv e contram o d- ules. Assume that C is a pro jectiv e left A -mo dule. F or an y righ t C -como dule N and any left C -contramo dule P denote by Ctrtor C i ( N , P ) the sequenc e of left deriv ed functors in the second argumen t of the righ t exact functor of con tratensor pro duct N ⊙ C P . By t he deﬁnition, the k -mo dules Ctrtor C i ( N , P ) are computed using a left pro jectiv e resolution of the C -contramo dule P . Since pro jectiv e con tr amo dules ar e con traﬂat, the functor Ctrtor C ∗ ( N , P ) assigns long exact sequences to exact triples in either of its argumen ts. Question. Can one compute the deriv ed functor Ctrto r using con traﬂat resolutions of the second argument? In o t her w ords, is it true that Ctrtor C i ( N , P ) = 0 for any righ t C -como dule N , any con traﬂat left C -contramo dule P , and all i > 0? Also, is it true that Ctrtor C i ( N , P ) = 0 for any A - ﬂat right C -como dule N , any (quite) C / A -con- traﬂat left C -contramo dule P , a nd all i > 0? A related question: is Ctrtor C > 0 ( N , P ) an eﬀaceable functor of its ﬁrst argumen t? No w a ssume that C is a pro jectiv e left and a ﬂat righ t A - mo dule and the ring A has a ﬁnite left homological dimension. Lemma 1. (a) A left C -c omo dule M is C / A -inje ctive if and only if for any A -p r o- je ctive left C -c omo dule L the k -mo dules Ext i C ( L , M ) of Y one da extensions in the ab elian c ate gory of left C -c o m o dules vanish for al l i > 0 . In p articular, the functor of C -c omo dule homomorphisms fr om an A -pr oje ctive left C -c omo dule L m a ps ex- act triples of C / A -inje ctive left C -c om o dules to exa c t triples. Besides, the class of C / A -inje ctive left C -c om o dules is close d under extensions and c okernels of inje ctive morphisms. (b) A left C -c ontr amo dule P is C / A -pr oje c tive if a n d only if for any A -inje c- tive left C -c ontr amo dule Q the k -mo dules Ext C ,i ( P , Q ) of Y one d a extensions in the ab elian c ate gory of le ft C -c on tr amo d ules v anish for al l i > 0 . In p articular, the functor of C -c ontr amo dule homomorphism s in to an A -inje ctive left C -c on tr amo dule Q maps exact triples of C / A -pr oje ctive left C -c ontr amo dules to exact triples. B e sides, the class of C / A -pr o je ctive left C -c o ntr amo dules is close d under extensions and kernels of surje ctive morphisms. (c) F or any C / A -pr oje ctive left C -c ontr amo dule P a nd an y A -ﬂat ri g ht C -c om o dule N the k -mo dules Ctrtor C i ( N , P ) van ish f o r al l i > 0 . In p articular, the functor o f c ontr a tens o r pr o duct w i th an A -ﬂat right C -c omo dule maps exact triples of C / A -p r o- je ctive left C -c ontr amo d ules to exac t triples. Pr o of . P art (a): the “if ” part of the ﬁrst assertion is ob vious; let us pro v e the “only if ” part . An arbitrary elemen t of Ext i C ( L , M ) can b e represe n ted b y a morphism of degree i from an exact complex · · · → L i → · · · → L 0 → L → 0 to the como dule M . According to Lemma 3.1.3(a), any left C -como dule is a surjectiv e 101 image of an A -pro jectiv e C - como dule. Therefore, one can assume tha t the comod- ules L i are A - pro jectiv e. Now if L is also A - projectiv e, then our exact complex of C -como dules is comp o sed o f exact triples of A -projective C -como dules, so if M is C / A -injectiv e, then the complex of ho momorphisms in t o M from t his complex of C -como dules is acyclic. The remaining t w o assertions follow from the ﬁrst one. The pro of of pa rt (b) is comple tely analo gous. T o pro v e (c), notice the isomorphism Hom k (Ctrtor C i ( N , P ) , k ∨ ) ≃ Ext C ,i ( P , Hom k ( N , k ∨ )).  Remark. Analogues of the t hird a ssertion of Lemma 1 (a) and the third assertion of Lemma 1(b) are not true for quite relative ly injectiv e como dules a nd quite relativ ely pro jectiv e contramodules (see Remark 7.4.3; cf. R emark 9.1). Theorem. F or any C / A -inje ctive left C -c omo d ule M the left C -c on tr amo dule Ψ C ( M ) is C / A -pr oje ctive and for any C / A -pr oje ctive left C - c ontr amo dule P the left C -c omo d- ule Φ C ( M ) is C / A -inj e ctive. The r estrictions of the functors Ψ C and Φ C to the ful l sub c a te gories of C / A -inje ctive C -c omo dules and C / A -pr oje ctive C -c ontr a m o dules ar e mutual ly inverse e quivalenc es b etwe en these sub c ate gories. Pr o of . Let us ﬁrst sho w that the injectiv e dimension of a C / A -injectiv e left C -co- mo dule M in the ab elian catego ry of C -como dules do es not exceed the left homo- logical dimension d of the ring A . Indeed, it follows from Lemma 3.1.3(a) that an y left C - como dule L has a ﬁnite resolution 0 → L d → · · · → L 0 → L → 0 with A -projectiv e C -como dules L j ; and since Ext i ( L j , M ) = 0 for all j and all i > 0 , the complex Ho m C ( L • , M ) computes Ext ∗ C ( L , M ). So the C -como dule M has a ﬁ- nite injectiv e resolution, and consequen t ly it has a ﬁnite resolution 0 → M → K 0 → · · · → K d → 0 consisting o f quite C / A -injectiv e C -como dules K j . Accord- ing to Lemma 1( a ), this exact sequence is comp osed of exact triples o f C / A -in- jectiv e C -como dules, whic h the functor Ψ C maps to exact triples; so the sequenc e 0 − → Ψ C ( M ) − → Ψ C ( K 0 ) − → · · · − → Ψ C ( K d ) − → 0 is also exact. Since the C -con- tramo dules Ψ C ( K j ) are quite C / A -projectiv e, it follo ws from Lemma 1(b) that the C -contramo dule Ψ C ( M ) is C / A -projectiv e and the latter exact seque nce is comp osed of exact triples of C / A -projectiv e C -contramo dules. Thus it follows fro m Lemma 1(c) that the sequence 0 − → Φ C Ψ C ( M ) − → Φ C Ψ C ( K 0 ) − → · · · − → Φ C Ψ C ( K d ) − → 0 is also exact. No w since the adjunction maps Φ C Ψ C ( K j ) − → K j are isomorphisms, the adjunction map Φ C Ψ C ( M ) − → M is a lso an isomorphism. The remaining assertions are prov en in the completely analogous w a y .  Lemma 2. (a) I n the ab ov e assumptions, a left C -c omo dule is C / A -c opr oje ctive if and only if it is C / A -inje ctive. (b) In the ab ove assumptions, a le ft C -c ontr amo dule is C / A -c oi nje ctive if and only if it is C / A -pr oje ctive. 102 Pr o of . P art (a) in the “if ” direction: it follows from Prop o sition 5.2.3(c) that when- ev er a left C -contra mo dule P is C / A -projectiv e, the C -como dule Φ C ( P ) is C / A -co- projectiv e. Now if a left C -como dule M is C / A -injectiv e, then the C -contramo dule P = Ψ C ( M ) is C / A - pro jectiv e and M = Φ C ( P ) b y the a b o v e Theorem. Part (a) in the “only if ” direction: in view of Lemma 1(a), the construction of L emmas 1.1.3 and 3 .1 .3(a) represen ts an y left C -como dule M as the quotient como dule o f a n A - pr o - jectiv e C -como dule P ( M ) b y a C / A -inj ective C -como dule. W e will show that whenev er M is a C / A - coprojectiv e C - como dule, P ( M ) is a copro jectiv e C -como dule; then if will follo w that M is a C / A -injectiv e C -como dule b y Lemma 5.2(a) and Lemma 1(a). Indeed, an extension of C / A - copro jectiv e left C -como dules is C / A -coprojectiv e b y Lemma 3.2.2(a); let us chec k that an A -projectiv e C / A -coprojectiv e C -como dule is copro jectiv e. F or any left C -como dule M and any left C -contramo dule P denote by Co ext i C ( M , P ) the cohomology of the ob j ect Coext C ( M , P ) of the deriv ed category D ( k – mo d ) that w as constructed in 4.7. This deﬁnition a g rees with the deﬁnition o f Cotor ∗ C ( M , P ) for a n A - projectiv e C -como dule M or an A - injectiv e C -contramo dule P giv en in the pro o f o f Lemma 3 .2.2. The functor Cotor ∗ C ( M , P ) assigns long ex- act sequences to exact triples in either o f its argumen ts. F or an y A - pro jectiv e left C -como dule M and a ny left C -contr a mo dule P one ha s Co ext i C ( M , P ) = 0 for all i > 0 and Co ext 0 C ( M , P ) ≃ Cohom C ( M , P ). Therefore, an A - projectiv e left C -como dule M is copro j ective if and only if Cohom i C ( M , P ) = 0 for an y left C -contra mo dule P and all i 6 = 0. F or an y C / A -copro j ectiv e left C -como dule M and an y left C -como dule P one has Co ext i C ( M , P ) = 0 for all i < 0, since one can compute Co ext C ( M , P ) using a ﬁnite A -injective right resolution of P by the result of 4.8. Th us an A -pro jectiv e C / A -coprojective left C -como dule is copro jectiv e. The pro of of part ( b) is completely analogous; it uses Prop osition 5.2.2(c) and Lemma 3 .1.3(b).  Question. It follows f rom Prop osition 1(c) that if C is a ﬂat righ t A -mo dule, then whenev er a left C -contramo dule P is C / A -contraﬂat the C -como dule Φ C ( P ) is C / A -coﬂat. Do es the con v erse hold? 5.4. Como dule-con t r amo dule corresp ondence. Assume that the coring C is a pro jectiv e left and a ﬂat r igh t A -mo dule and the ring A has a ﬁnite left homological dimension. The categories of C / A -injectiv e left C -como dules and C / A -projectiv e left C -contra- mo dules ha v e nat ural exact category structures as full sub categories, closed under extensions, of the ab elian categories of left C -como dules and left C -contramo dules. Theorem. (a) The functor mappin g the quotient c ate gory of the homotopy c ate gory of c omplex es of C / A -inje ctive left C - c omo d ules b y its minimal triagulate d sub c ate- gory c ontaining the tot al c omp lexes of e xact triples of c om plexes of C / A -inje ctive C -c omo dules into the c o derive d c ate go ry of left C -c om o dules is an e quivalen c e of tri- angulate d c ate gories. 103 (b) The functor mapping the quotient c ate gory of the hom otopy c ate gory of c om- plexes of C / A -pr oje ctive left C -c ontr amo dules by its minimal trian g ulate d sub c ate gory c ontaining the total c omplexes of C / A -pr oje ctive C -c ontr am o dules i n to the c ontr ade- rive d c ate gory of left C -c ontr am o dules is an e quivalenc e of triangulate d c ate gories. Pr o of . P art (a): let M • b e a complex of left C -como dules. Then the t o tal complex of the cobar bicomplex C ⊗ A M • − → C ⊗ A C ⊗ A M • − → · · · is a complex of (quite) C / A -in- jectiv e C -como dules, the complex M • maps in to this total complex, and the cone of this map is coacyclic. Hence it follo ws from Lemma 2.6 that the co deriv ed category of left C - como dules is equiv alen t to the quotien t catego r y of the ho mo t o p y category of complexes of C / A - inj ective C - como dules b y its in tersection with the thick sub category of coacyclic complexes o f C -como dules. It remains to show that this in tersection of sub categories coincides with the minimal triangulated subcategor y con taining the total complexes of exact triples of complexes of C / A -injective C -como dules. Lemma. (a) F or any exa c t c ate gory A wher e inﬁnite dir e ct sums exist and pr e s erve exact triples, the c omplex o f ho m omorphisms fr om a c o acycli c c om plex over A into a c omplex of inje ctive obje cts with r esp e c t to A i s acyclic. (b) F or any exact c ate g o ry A wher e i n ﬁnite pr o ducts exist and pr eserve exact triples, the c o m plex of homomo rphisms fr om a c omple x of pr oje ctive obje cts with r esp e ct to A into a c ontr aacyclic c om plex over A is acyclic . Pr o of . Analogous to the pro ofs of Lemmas 2.2 and 4.2. Part (a) : let M • b e a com- plex of injectiv e ob jects with resp ect to A . Since the functor of homomorphisms in to M • maps distinguished triangles in the homotop y category to distinguished triangles and inﬁnite direct sums to inﬁnite pro ducts, it suﬃces to c heck that the complex Hom A ( L • , M • ) is acyclic whenev er L • is t he tota l complex of an exact triple ′ K • → K • → ′′ K • of complexes ov er A . But t he complex Hom A ( L • , M • ) is the to- tal complex of an exact triple of complexes of ab elian gr o ups Ho m A ( ′′ K • , M • ) − → Hom A ( K • , M • ) − → Hom A ( ′ K • , M • ) in this case. The pro of o f part (b) is dual.  W e will sho w that (i) the minimal tr iangulated sub category con taining the total complexes of exact triples of complexes of C / A -injectiv e C -como dules and (ii) the homotop y category of complexes of injectiv e C - como dules form a semiorthogonal de- comp osition of the homot op y cat ego ry o f complexes of C / A -injectiv e left C - como dules. This means, in addition to the sub category (i) b eing left orthogonal to the sub cate- gory (ii), that for an y complex K • of C / A -injectiv e C -como dules there exists a (unique and functorial) distinguished tria ng le L • → K • → M • → L • [1] in the homotop y cat- egory of C -como dules, where L • b elongs to the sub category (i) a nd M • b elongs to the sub category ( ii) . It will fo llow that the sub cat ego ry (i) is the maximal sub category of the homotop y category of complexes of C / A -injectiv e C -como dules left orthogonal to 104 the sub category (ii), hence the sub catego ry (i) contains the in tersection of the homo- top y category of complexes of C / A - inj ectiv e C -como dules with t he thic k sub category of coacyclic complexes of C -como dules. Indeed, let K • b e a complex of C / A -injectiv e left C -como dules. Cho ose for eve ry n an injection j n of the C - como dule K n in to an injectiv e C -como dule J n . Consider the complex E • = E ( K • ) whose t erms are the C -como dules E n = J n ⊕ J n +1 and the diﬀeren tial d n E : E n − → E n +1 maps J n +1 in to itself by t he iden tit y map and v anishes in the restriction to J n and in the pro jec tion to J n +2 . There is a natura l injectiv e morphism of complexes K • − → E • formed b y the C -como dule maps K n − → E n whose comp onen ts are j n : K n − → J n and j n +1 d n K : K n − → J n +1 . Set 0 E • = E ( K • ), 1 E • = E ( 0 E • / K • ), etc. As it w as sho wn in the pro o f of Theorem 5.3 , the injectiv e dimension of a C / A -injectiv e left C -como dule do es not exceed the left homo lo gical dimension d of the ring A . Therefore, the complex Z • = cok er( d − 2 E • → d − 1 E • ) is a complex of injectiv e C -como dules. Now it is clear that the total complex M • of the bicomplex 0 E • − → 1 E • − → · · · − → d − 1 E • − → Z • is a complex of injectiv e C -como dules and the cone L • of the morphism K • − → M • b elongs to the minimal triangulated sub category containing the tota l complexes of exact triples of complexes of C / A -injectiv e C -como dules by Lemma 5.3.1(a) . P art (a) is prov en; the pro of of par t (b) is completely analogo us and uses Lemma 5.3.1(b).  Remark. Let A b e an exact category where inﬁnite direct sums exist and preserv e exact triples, ev ery ob jec t admits a n admissible monomor phism in to an ob jec t in- jectiv e relative to A , and the class of suc h injectiv e ob jects is closed under inﬁnite direct sums. Then t he thick sub category of coacyclic complexes with resp ect to A and the triangulat ed sub category of complexes of injectiv e ob jects fo r m a semiorthogo nal decomp osition of the homotop y category Hot ( A ), so the co deriv ed category D co ( A ) is equiv alent t o the homotopy category of complexes of injectiv es in A . Indeed, or- thogonality is already prov en in Lemma, so it remains to construct a morphism from an y complex C • o v er A in to a complex of injectiv es M • with a coacyclic cone. T o do so, one pro ceeds a s in t he pro of of Theorem, constructing a morphism from C • in to a complex of injectiv es 0 E • that is an admissible monomorphism in eve ry de- gree, taking the quotien t complex, constructing an analogous morphism from it in to a complex of injectiv es 1 E • , etc. Finally , one constructs the to tal complex M • of the bicomplex • E • b y taking inﬁnite direct sums alo ng the diagonals; then M • is a complex of injectiv es a nd the cone o f the morphism C • − → M • is coacyclic b y Lemma 2.1. Conseq uen tly , the homotopy category of acyclic complexes of injectiv es in A is equiv alen t to the quotien t category Acycl ( A ) / Acycl co ( A ) and to the kerne l of the lo calization functor D co ( A ) − → D ( A ) (cf. [34]). When A has a ﬁnite homolog ical dimension, the condition that the class of injectiv es is closed under inﬁnite direct sums is not needed in this argumen t. This is a somewhat t r ivial situatio n, though; 105 see Remark 2.1. Moreov er, let A b e an exact category where inﬁnite direct sums exist and preserv e exact triples and ev ery ob ject admits an admissible monomorphism into an injectiv e. Let F ⊂ A b e a class of ob jects closed under cok ernels of admissible monomorphisms, con taining t he injectiv es, and consisting of o b jects of ﬁnite injectiv e dimension. Then ev ery complex o v er F that is coacyclic a s a complex ov er A b elongs to the minimal triangulated sub category of Hot ( A ) containing the t o tal complexes of exact t r iples of complexes o v er F . When there is a class F ⊂ A closed under inﬁnite direct sums, consisting of o b jects of ﬁnite injectiv e dimension, and suc h that eve ry ob ject of A admits an admissible monomorphism into an ob ject o f F , the co deriv ed category D co ( A ) is equiv alen t to the homo t o p y category of complexes of injectiv es in A (cf. [29 ], where in the dual situation the role of the class F is pla y ed b y ﬂa t mo d- ules). T o show this, one has to rep eat t wice the ab o v e construction of a resolution • E • , ta king inﬁnite direct sums along the diagonals for the ﬁrst time and ﬁnite directs sums along the diago nals of the canonical truncation for the second time. When A is the ab elian categor y of C -como dules, one can ta k e F to b e the class of C / A -injectiv e C -como dules or quite C / A - injectiv e C - como dules. The relat ed results for como dules and contramo dules are obta ined in Theorem 5.5 and Remark 5.5. Corollary . The r estrictions of the functors Ψ C and Φ C (applie d to c omp lexes term- wise) to the homotopy c ate gory of c omplexes of C / A -inje ctive C -c omo dules and the homotopy c ate g ory of c omp l e x es of C / A -pr oje ctive C -c ontr am o dules deﬁne mutual ly inverse e quivalen c es R Ψ C and L Φ C b etwe en the c o derive d c ate gory of lef t C -c omo dules and the c ontr aderiv e d c ate gory of left C -c ontr amo dules. Pr o of . By Theorem 5.3, the functors Ψ C and Φ C induce m utually in v erse equiv alences b et w een the homotop y categories of C / A -injectiv e left C -como dules and C / A -projec- tiv e left C -contr a mo dules. According to Lemma 5.3.1( a) and (c), the total complexes of exact triples of complexes of C / A - injectiv e C -como dules corresp ond to the to t a l complexes of exact triples of complexes of C / A -projectiv e C -contramo dules under this equiv alence. So it remains to apply the ab ov e Theorem.  Question. Can one obtain a ve rsion of the deriv ed como dule-con tramo dule corre- sp ondence (a n equiv alence b et w een appro priately deﬁned exotic deriv ed categories of left C -como dules and left C -contramo dules) not dep ending o n an y assumptions ab out the homological dimension of the ring A ? It is not diﬃcult to see that one can w eak en the assumption that A has a ﬁnite left homological dimens ion to the assumption that A is left Gorenstein, i. e., the classes of left A -mo dules o f ﬁnite pro jectiv e and injectiv e dimensions coincide. In this case, the co deriv ed category of left C - como dules and t he con tr a deriv ed category of left C - contramo dules are natu- rally equiv alent whenev er the coring C is a pro jectiv e left and a ﬂat righ t A -mo dule. Indeed, arg uing as in Theorem 5.5 b elo w, one can sho w tha t the co deriv ed category 106 of left C -como dules is equiv alen t t o the quotien t category of the homoto p y cate- gory of complexes of C -como dules coinduced f rom left A -mo dules of ﬁnite pro jec tiv e (injectiv e) dimension b y its minimal tria ng ulated sub category containing the total complexes o f exact triples of complexes o f C -como dules that at ev ery term are exact triples of C -como dules coinduced from exact triples of A -mo dules of ﬁnite pro jective (injectiv e) dimension. Analogously , the con traderiv ed category of left C -contra mo d- ules is equiv alen t to the quotien t category of the homotopy categor y of complexes of C -contramo dules induced fro m left A - mo dules of ﬁnite pro jectiv e (injectiv e) di- mension by its minimal triangula t ed sub category con taining the total complexes of exact triples of complexes of C -contramo dules that at ev ery term a r e exact tr iples of C -contramo dules induced from exact triples of A - mo dules of ﬁnite pro jec tiv e (injec- tiv e) dimension. The k ey step is to notice that the class of left A -mo dules of ﬁnite pro jectiv e (injectiv e) dimension is closed under inﬁnite direct sums and pro ducts. 5.5. Derived functor Ctrt or. The fo llo wing analog ue of Theorem 5.4 holds under sligh tly w eak er conditions. Theorem. (a) Assume that the c ori n g C is a ﬂat right A -m o dule and the ring A has a ﬁnite left hom olo gic al dimens ion. Then the functor mapping the ho m otopy c ate gory of c omplex e s of inj e ctive left C -c om o dules into the c o derive d c ate go ry of left C -c om o dules is a n e quivalenc e of triangulate d c ate gories. I n addition, the functor mapping the quotient c ate gory of the homotopy c a te gory of c omplexes of quite C / A -inje ctive left C -c omo dules by the minim a l triangulate d sub c ate gory c ontaining the total c omplexes of exact triples of c om p lexes of c oinduc e d C -c omo dules that at every term ar e exac t triples of C -c omo dules c oinduc e d fr om exact triples of A -mo dules into the c o derive d c ate go ry of left C -c omo dules is an e quivalenc e of triangulate d c ate gorie s. (b) Assume that the c oring C is a pr oje ctive lef t A -mo dule and the ring A has a ﬁn i te left homolo gic al dimension. The n the functor mapp ing the homotopy c ate- gory of c omplexes of pr oje ctive left C -c o n tr amo d ules into the c ontr a d erive d c ate gory of left C -c ontr amo dules is a n e quivalenc e of triangulate d c ate gories. I n addition, the func tor mapp ing the quotient c ate gory of the homotopy c ate gory of c omplexes o f quite C / A -pr oje ctive left C -c ontr am o dules by the min imal triangulate d sub c ate gory c ontaining the total c omplexes of ex a ct triples of c o m plexes of induc e d C -c ontr amo d- ules that at every term ar e exact triples of C -c ontr amo dules induc e d fr om e x act triples of A -mo dules into the c ontr ad e ri v e d c ate gory of left C -c ontr a mo dules is an e quival e nc e of triang ulate d c ate g ories. Pr o of . P art (a): when C is also a pro jectiv e left A -mo dule, the ﬁrst assertion f ollo ws from the pro o f of Theorem 5 .4. T o pro v e b oth assertions in the general case, w e will sho w that (i) the minimal tr ia ngulated sub category containing the total complexes of exact triples of complexes of coinduced C -como dules that at ev ery term are exact 107 triples of C -como dules coinduced from exact triples of A - mo dules a nd (ii) t he homo- top y category of complexes of injective C -como dules fo rm a semiorthogonal decomp o - sition of t he homotopy category o f complexes of quite C / A -injective left C -como dules. Then we will argue as in the pro of of Theorem 5.4. An y complex of quite C / A -injectiv e C -como dules is homotopy equiv alen t to a com- plex of coinduced C -como dules. Let K • b e a complex o f coinduced left C -como dules; then K n ≃ C ⊗ A V n for certain A -mo dules V n . Let V i − → I i b e injectiv e maps of the A -mo dules V n in to injectiv e A -mo dules I n ; set J n = C ⊗ A I n . Then J n are injectiv e C -como dules endow ed with injectiv e C -como dule morphisms K n − → J n . As in the pro of of Theorem 5.4, we construct the complex of injectiv e C -como dules E • with E n = J n ⊕ J n +1 and the inj ective morphism of complexes K • − → E • . Let us show that there exists an auto morphism of the C -como dule E n suc h that its comp osition with the injection K n − → E n is the injection whose comp onen ts are j n : K n − → J n and the zero map K n − → J n +1 . Since the como dule J n +1 is injectiv e, the comp onent K n − → J n +1 of the morphism K n − → E n can b e extended from the como dule K n to como dule J n con taining it. Denote the morphism so obtained b y h n : J n − → J n +1 ; then the automorphism of the como dule E n whose comp onents are − h n , the iden tit y automorphisms of J n and J n +1 , and zero has the desired prop ert y . Now it is clear that the triple K • − → E • − → E • / K • is an exact triple of complexe s of coinduced C -como dules whic h at ev ery term is an exact triple of C -como dules coinduced from an exact triple o f A -mo dules. Moreo v er, E n / K n ≃ C ⊗ A W n , where the injectiv e dimen- sion di A W n is equal to di A V n − 1. So w e can iterate this (nonfunctorial) construction, setting 0 E • = E ( K • ) = E • , 1 E • = E ( 0 E • / K • ), etc., and Z • = cok er( d − 2 E • → d − 1 E • ). Then the total complex M • of the bicomplex 0 E • − → 1 E • − → · · · − → d − 1 E • − → Z • is a complex of injectiv e C -como dules and the cone L • of the morphism K • − → M • b elongs t o the minimal tria ngulated sub catego r y con taining the t o tal complexes of exact triples of complexes of coinduced C - como dules that at ev ery t erm is an exact triple of C -como dules coinduced f rom an exact triple of A - mo dules.  Remark. The ab o v e Theorem provides an alternative w a y o f proving Corollary 5.4. Besides, it follows f rom 5.1.3, Lemma 5.2, a nd the ab o v e Theorem that in the as- sumptions of 5.4 the functor mapping the homotopy category of complexes of co- pro jectiv e left C - como dules in to the co derive d category of left C -como dules and the functor mapping the homotopy category of complexes of coinjectiv e left C -contra- mo dules in to t he con traderiv ed category of left C -contra mo dules are equiv alences of triangulated categories. This is a stro ng er result than Theorem 4 .5. The con tratensor pro duct N • ⊙ C P • of a complex of right C -como dules N • and a complex of left C -contramo dules P • is deﬁned as the total complex of the bicomplex N i ⊙ C P j , constructed by taking inﬁnite direct sums along the diagonals. Assume that the coring C is a pro jective left A -mo dule and the ring A has a ﬁnite left homolo g ical 108 dimension. One can prov e in the w ay completely analogous t o the pro of of Lemma 2.2 that the con tr a tensor pro duct of a coacyclic complex of righ t C -como dules and a complex of con traﬂat (and in particular, pro jectiv e) left C -contramo dules is acyclic. The left deriv ed functor of contratensor pro duct Ctrtor C : D co ( como d – C ) × D ctr ( C – contra ) − − → D ( k – mo d ) is deﬁned by restricting the functor o f con tratensor pro duct to the Carthesian pro d- uct of the homotop y category of righ t C -como dules and the homotop y category of complexes of pro jectiv e left C -contramo dules. The same deriv ed functor can b e obtained b y restricting the functor of contratensor pro duct to the Cart hesian pro duct of the ho motop y catego r y of complexes of A -ﬂat righ t C -como dules and the homotopy category of complexes of quite C / A -projec- tiv e left C -contramo dules. Indeed, it follows from part (b) of Theorem that the con tratensor pro duct o f a complex of A -ﬂat right C -como dules and a con traacyclic complex of quite C / A - projectiv e left C - contramo dules is acyclic. No w if N • is a complex of A -ﬂat righ t C -como dules, P • is a complex of quite C / A -projectiv e left C -contramo dules, and ′ P • − → P • is a mo r phism from a complex of pro jec tiv e C -con- tramo dules ′ P • in to P • with a contraacyc lic cone, then the map N • ⊙ C P • − → N • ⊙ C ′ P • is a quasi-isomorphism. In particular, if the complex N • is coacyclic, then the complex N • ⊙ C P • is acyc lic, since the complex N • ⊙ C ′ P • is. When C is also a ﬂat righ t A -mo dule, one can use complexes of C / A -projectiv e C -contra- mo dules instead of complexes of quite C / A -pro jectiv e C -contramo dules, b ecause the con tratensor pro duct o f a complex of A -ﬂat right C -como dules and a con traacyclic complex of C / A -projectiv e left C - contramo dules is a cyclic b y Theorem 5.4(b) and Lemma 5.3.1(c). Notice that this deﬁnition of the deriv ed functor Ctrto r C is not a particular case of Lemma 2.7 (instead, it is a particular case of Lemma 6.5.2 b elo w). Analogously , assume that the coring C is a ﬂat right A -mo dule and the ring A has a ﬁnite left homological dimension. According to L emma 5.4, the complex of ho mo - morphisms fro m a coacyclic complex o f left C -como dules in to a complex o f injectiv e left C -como dules is acyclic. There fore, the natural map Hom Hot ( C – comod ) ( L • , M • ) − → Hom D co ( C – como d ) ( L • , M • ) is an isomorphism whene v er M • is a complex of injective C -como dules. So the functor o f homomorphisms in the co derived category of left C -como dules can b e lifted to a functor Ext C : D co ( C – como d ) op × D co ( C – como d ) − − → D ( k – mo d ) , whic h is deﬁned by restricting t he functor of homomorphisms of complexes of C -como dules to the Carthesian pro duct o f the homoto p y category of left C -como dules and t he homotop y category of complexes of injectiv e left C -como dules. 109 The same functor Ext C can b e obtained by restricting the functor of homomor- phisms to the Carthesian pro duct of the ho motop y category of complexes of A -pro- jectiv e left C -como dules and the homotop y category of complexes of quite C / A -in- jectiv e left C -como dules. Indeed, it follows from part (a) of Theorem that the com- plex of homomorphisms from a complex of A -projectiv e left C -como dules in to a co- acyclic complex of quite C / A -injectiv e left C -como dules is acyclic. No w if L • is a complex of A -projective left C - como dules, M • is a complex of quite C / A -injectiv e left C -como dules, and M • − → ′ M • is a morphism from M • in to a complex o f in- jectiv e C -como dules ′ M • with a coacyclic cone, then the map Hom C ( L • , M • ) − → Hom C ( L • , ′ M • ) is a quasi-isomorphism. When C is also a pro jective left A -mo dule, one can use complexes of C / A -injectiv e C -como dules instead of complexes of quite C / A -injectiv e C -como dules. Finally , assume that the cor ing C is a pro jectiv e left A -mo dule and t he ring A ha s a ﬁnite left homological dimension. By Lemma 5.4 , the natural map Hom Hot ( C – contra ) ( P • , Q • ) − → Hom D ctr ( C – contra ) ( P • , Q • ) is a n isomorphism whenev er P • is a complex of pro jectiv e C -contramo dules. So the functor of homomorphisms in the con traderiv ed category o f left C -contramo dules can b e lifted to a functor Ext C : D ctr ( C – contra ) op × D ctr ( C – contra ) − − → D ( k – mo d ) , whic h is deﬁned b y restricting t he functor of homomorphisms of complexes of C -con- tramo dules to the Carthesian pro duct of the homotop y category of complexes of pro jectiv e left C -contr a mo dules a nd the homotopy category of left C -contramo dules. The same functor Ext C can b e obtained by restricting the functor of homomor- phisms to the Carthesian pro duct of the homotopy category of complexes of quite C / A -proj ective left C - contramo dules and the homotop y category of complexes o f A -in- jectiv e left C -contramo dules. When C is also a ﬂat righ t A -mo dule, one can use complexes of C / A -pr o jectiv e C -contramo dules instead of complexes o f quite C / A -pro- jectiv e C -contramo dules. 5.6. Co ext and Ext, Cot or and Ctrtor . Assume that the coring C is a pro jectiv e left and a ﬂat right A -mo dule and the ring A has a ﬁnite left homological dimension. Corollary . (a) Ther e ar e natur al isom orphisms of functors Co ext C ( M • , P • ) ≃ Ext C ( M • , L Φ C ( P • )) ≃ Ext C ( R Ψ C ( M • ) , P • ) on the C arthesian pr o duct of the c ate gory opp osite to the c o d erive d c ate gory of left C -c om o dules and the c ontr aderive d c ate gory of le f t C -c ontr amo dules. (b) Ther e is a natur al isom o rphism of functors Cotor C ( N • , M • ) ≃ Ctrtor C ( N • , R Ψ C ( M • )) on the Carthesian pr o duct of the c o derive d c ate gory of right C -c o mo dules and the c o derive d c ate gory of left C - c omo dules. Pr o of . Clearly , it suﬃces to construct natural isomorphisms Co ext C ( L • , R Ψ C ( M • )) ≃ Ext C ( L • , M • ), Co ext C ( L Φ C ( P • ) , Q • ) ≃ Ext C ( P • , Q • ), and Cotor C ( N • , L Φ C ( P • )) ≃ 110 Ctrtor C ( N • , P • ). In the ﬁrst case, represen t the image of M • in D co ( C – como d ) by a complex of injectiv e C -como dules, notice that the functor Ψ C maps injectiv e como d- ules t o coinjectiv e con tramo dules, and use Prop osition 5.2.2(a). Alternative ly , r ep- resen t the image of M • in D co ( C – como d ) b y a complex of C / A -injectiv e C -como dules and the image of L • in D co ( C – como d ) by a complex of copro jectiv e C -como dules, and use Prop osition 5.2.2(e); or represen t the image of M • in D co ( C – como d ) by a complex of C / A -injectiv e C -como dules and the image of L • in D co ( C – como d ) b y a complex of A -projectiv e C - como dules, and use Prop osition 5.2.2(c), Lemma 5.3.2(b), and the result of 4.8. In the second case, represen t the image of P • in D ctr ( C – contra ) by a complex o f pro jective C -contramo dules, no t ice that the f unctor Φ C maps pro jec tiv e con tramo dules to copro jectiv e como dules, a nd use Prop osition 5.2.3(a). Alterna- tiv ely , represen t the image of P • in D ctr ( C – contra ) b y a complex of C / A - projectiv e C -contramo dules and the imag e of Q • in D ctr ( C – contra ) b y a complex of coinjec- tiv e C -contra mo dules, and use Prop osition 5.2.3(e); or represen t the imag e of P • in D ctr ( C – contra ) by a complex of C / A -projectiv e C -contramo dules and the image of Q • in D ctr ( C – contra ) by a complex of A - injectiv e C -contramo dules, and use Prop osi- tion 5.2.3(c), Lemma 5 .3.2(a), and the result of 4.8. In the third case, represen t the image of P • in D ctr ( C – contra ) by a complex o f pro jectiv e C -contramo dules, notice that the functor Φ C maps pro jectiv e con tramo dules to copro jectiv e como dules, and use Prop osition 5.2 .1 (a). Alternativ ely , represen t the image o f P • in D ctr ( C – contra ) by a complex of C / A -projectiv e C -contramo dules and the image of N • in D co ( como d – C ) b y a complex of coﬂat C - como dules, and use Prop osition 5.2.1(e); or represen t the image of P • in D ctr ( C – contra ) b y a complex of C / A -projectiv e C -contr a mo dules and the image of N • in D co ( como d – C ) by a complex of A -ﬂat C -como dules, a nd use Prop o- sition 5.2.1(c), Lemma 5.3.2( a), and the result of 2.8. Finally , to show that the three pairwise isomorphisms betw een the functors Co ext C ( M • , P • ), Ext C ( M • , L Φ C ( P • )), and Ext C ( R Ψ C ( M • ) , P • ) f orm a comm utativ e diagram, one can represen t the image of M • in D co ( C – como d ) b y a complex of copro- jectiv e C -como dules a nd the image o f P • in D ctr ( C – contra ) b y a complex of coinjective C -contramo dules (having in mind Lemma 5.2), and use a result of 5.2.  111 6. Semimodule-Se micontramodule Correspondence 6.1. Contratensor pro duct and semimo dule/semicon tramo dule homomor- phisms. Let S b e a semialgebra ov er a coring C . 6.1.1. W e would lik e to deﬁne the op eration of contratensor pro duct of a righ t S -semimodule and a left S -semicontramo dule. Dep ending on the (co)ﬂatness and/or (co)pro jectivit y conditions on C and S , one can sp eak of S -semimo dules a nd S -semi- contramo dules with v arious (co)ﬂatness and (co)injectivity conditions imp osed on them. In particular, when C is a pro jectiv e left A -mo dule and either S is a copro- jectiv e left C -como dule, o r S is a pro jectiv e left A -mo dule and a C / A - coﬂa t rig ht C -como dule and A ha s a ﬁnite left homological dimension, o r A is semisimple, one can consider righ t S -semimo dules and left S -semicontramo dules with no (co)ﬂatness or (co)injectivit y conditions impo sed. When C is a ﬂat rig ht A -mo dule, S is a ﬂat righ t A -mo dule a nd a C / A -projective left C -como dule, and A ha s a ﬁnite left ho- mological dimension, one can conside r A - ﬂat righ t S -semimo dules and A -injectiv e left S -semicontramo dules. When C is a ﬂat righ t A - mo dule and S is a coﬂat righ t C -como dule, one can consider C - coﬂat righ t S -semimo dules and C -coinjectiv e left S -semicontramo dules. The c ontr atensor pr o duct N ⊚ S P of a right S -semimo dule N and a left S -semi- contramo dule P is a k -mo dule deﬁned as the coke rnel of the following pair of maps ( N  C S ) ⊙ C P ⇒ N ⊙ C P . The ﬁrst map is induce d b y the righ t S -semiaction morphism N  C S − → N . The second map is the comp osition o f the map induced b y the left S - semicon t r aaction morphism P − → Cohom C ( S , P ) and the na tural “ev aluation” map η S : ( N  C S ) ⊙ C Cohom C ( S , P ) − → N ⊙ C P . The latter is deﬁned in the follo wing generalit y . Let C b e a coring ov er a k - algebra A and D b e a coring ov er a k -algebra B . Let K b e a C - D -bicomo dule, N b e a right C -como dule, and P b e a left C -contramo dule. Supp ose that the cotensor pro duct N  C K is endow ed with a right D -como dule structure via the construction o f 1.2.4 and the cohomomorphism mo dule Cohom C ( K , P ) is endo w ed with a left D -contra- mo dule structure via the construction of 3.2.4. Then the comp osition of maps ( N  C K ) ⊗ B Hom A ( K , P ) − → N ⊗ A K ⊗ B Hom A ( K , P ) − → N ⊗ A P − → N ⊙ C P factor izes through the surjection ( N  C K ) ⊗ B Hom A ( K , P ) − → ( N  C K ) ⊙ D Cohom C ( K , P ), so there is a natural map η K : ( N  C K ) ⊙ D Cohom C ( K , P ) − → N ⊙ C P . Indeed, the kernel of this surjection is equal to the sum o f t he diﬀerence of tw o maps ( N  C K ) ⊗ B Hom A ( K ⊗ B D , P ) ⇒ ( N  C K ) ⊗ B Hom A ( K , P ) and the diﬀerence of t w o maps ( N  C K ) ⊗ B Hom A ( C ⊗ A K , P ) ⇒ ( N  C K ) ⊗ B Hom A ( K , P ). The diﬀerence of the ﬁrst pair of maps v anishes already in the comp osition with the map ( N  C K ) ⊗ B Hom A ( K , P ) − → N ⊗ A P , while the second pair of maps can b e presen ted as the comp osition of the map ( N  C K ) ⊗ B Hom A ( C ⊗ A K , P ) − → N ⊗ A Hom A ( C , P ) and the pair o f maps N ⊗ A Hom A ( C , P ) ⇒ N ⊗ A P whose cokerne l is, by the deﬁnition, 112 N ⊙ C P . The “ev a luation” map η K is dual to the map Hom k ( η K , k ∨ ) = Cohom C ( K , − ) : Hom C ( P , Hom k ( N , k ∨ )) − − → Hom D (Cohom C ( K , P ) , Cohom C ( K , Hom k ( N , k ∨ ))) . 6.1.2. The op eratio n of contratensor pro duct ov er S is dua l to homomorphisms in the category of left S -semicontramo dules: for an y r igh t S -semimo dule N and any left S -semicontramo dule P there is a natural isomorphism Hom k ( N ⊚ S P , k ∨ ) ≃ Hom S ( P , Hom k ( N , k ∨ )). Indeed, b oth k -mo dules are isomorphic to the ke rnel of the same pair of maps Hom C ( P , Hom k ( N , k ∨ )) ⇒ Hom C ( P , Cohom C ( S , Hom k ( N , k ∨ ))). It follows that for an y righ t C -como dule R for whic h the induced right S -semimo dule R  C S is deﬁned and an y left S -semicontramo dule P the compo sition of the map ( R  C S ) ⊙ C P − → ( R  C S ) ⊙ C Cohom C ( S , P ) induced b y the S - semicon t r a action in P with the “ev alua t ion” map ( R  C S ) ⊙ C Cohom C ( S , P ) − → R ⊙ C P induces a natural isomorphism ( R  C S ) ⊚ S P ≃ R ⊙ C P . When C is a pro jective left A -mo dule and S is a copro jectiv e left C -como dule, t he functor of con tratensor pro duct ov er S is rig h t exact in b oth its argumen ts. 6.1.3. Let S b e a semialgebra ov er a coring C and T b e a semialgebra ov er a cor ing D . W e w ould lik e to deﬁne a T -semimo dule structure on the contratensor pro duct of a T - S -bisemimo dule and an S -semicontramo dule, and an S -semicontramo dule struc- ture on semimo dule homomorphisms from a T - S -bisemimodule to a T -semimo dule. Let N b e a r ig h t D -como dule, K b e D - C -bicomo dule with a right S -semimo dule structure suc h that the m ultiple cotensor products N  D K  C S  C · · ·  C S are asso ciativ e and the semiaction map K  C S − → K is a left D -como dule morphism, and P b e a left S -semicontramo dule. Then the contratensor pro duct K ⊚ S P has a natural left D -como dule structure as the coke rnel of a pair of D -como dule morphisms ( K  C S ) ⊙ C P ⇒ K ⊙ C P . The comp o sition of maps ( N  D K ) ⊙ C P − → N  D ( K ⊙ C P ) − → N  D ( K ⊚ S P ) factorizes through the surjection ( N  D K ) ⊙ C P − → ( N  D K ) ⊚ S P , so t here is a natural map ( N  D K ) ⊚ S P − → N  D ( K ⊚ S P ). Indeed, the comp osition of the pair of maps ( N  D K  C S ) ⊙ C P ⇒ ( N  D K ) ⊙ C P whose cokerne l is, by the deﬁnition, ( N  D K ) ⊚ S P , with the map ( N  D K ) ⊙ C P − → N  D ( K ⊙ C P ) is equal to the comp osition of the map ( N  D K  C S ) ⊙ C P − → N  D (( K  C S ) ⊙ C P ) with the pair o f ma ps N  D (( K  C S ) ⊙ C P ) ⇒ N  D ( K ⊙ C P ). No w let K b e a T - S -bisemimodule and P b e a left S -semicontramo dule. Assume that the m ultiple cotensor pro ducts T  D · · ·  D T  D ( K ⊚ S P ) are asso ciativ e and the D - como dule morphisms ( T  m  D K ) ⊚ S P − → T  m  D ( K ⊚ S P ) are isomorphisms fo r m 6 2. Then one can deﬁne an asso ciativ e and unital semiaction morphism T  D ( K ⊚ S P ) − → K ⊚ S P taking the con tratensor pro duct ov er S of the semiaction morphism T  D K − → K with the semicon tramo dule P . 113 Analogously , let L b e a left C - como dule, K b e a D - C - bicomo dule with a left T -semi- mo dule structure suc h that the multiple cotensor pro ducts T  D · · ·  D T  D K  C L are asso ciativ e a nd the semiaction map T  D K − → K is a right C -como dule morphism, and M b e a left T -semimo dule. Then the mo dule of homomorphisms Hom T ( K , M ) has a natural left C -contramo dule structure as the k ernel of a pair of C -contramo d- ule morphisms Hom D ( K , M ) ⇒ Hom D ( T  D K , M ). The comp osition of maps Cohom C ( L , Hom T ( K , M )) − → Cohom C ( L , Hom D ( K , M )) − → Hom D ( K  C L , M ) factorizes t hr o ugh the injection Hom T ( K  C L , M ) − → Hom D ( K  C L , M ), so there is a natural map Cohom C ( L , Hom T ( K , M )) − → Hom T ( K  C L , M ). No w let K b e a T - S -bisemimo dule and M b e a left T -semimo dule. Assume that the mu ltiple cohomomor phisms Cohom C ( S  C · · ·  C S , Hom T ( K , M )) are asso ciative and the C - contramo dule morphisms Cohom C ( S  n , Hom T ( K , M )) − → Hom T ( K  C S  n , M ) are isomorphisms for n 6 2. Then o ne can deﬁne a n asso ciative and unital semicon tra action morphism Hom T ( K , M ) − → Cohom C ( S , Hom T ( K , M )) taking the T -semimo dule homomorphisms fro m the semiaction morphism K  C S − → K into the semimo dule M . 6.1.4. Let M b e a left T -sem imo dule, K b e a T - S -bisemim o dule, and P b e a left S -semicontramo dule. Assume that a left T -semimodule structure on K ⊚ S P and a left S -semicontramo dule structure on Hom T ( K , M ) are deﬁned via t he constuctions of 6 .1.3. Then there is a natural adj unction isomorphism Hom T ( K ⊚ S P , M ) ≃ Hom S ( P , Hom T ( K , M )). Indeed, the mo dule Hom T ( K ⊚ S P , M ) is the k ernel of the pair of maps Hom D ( K ⊚ S P , M ) ⇒ Hom D ( T  D K ⊚ S P , M ) a nd there is an injection Hom D ( T  D K ⊚ S P , M ) − → Hom D (( T  D K ) ⊙ C P , M ). The mo dule Hom D ( K ⊚ S P , M ) is the ke rnel of the pair of maps Hom D ( K ⊙ C P , M ) ⇒ Hom D (( K  C S ) ⊙ C P , M ). There is a pair of natural maps Hom D ( K ⊙ C P , M ) ⇒ Hom D (( T  D K ) ⊙ C P , M ) (o ne of whic h go es through Ho m D ( T  D ( K ⊙ C P ) , M )) extending the pair of maps Hom D ( K ⊚ S P , M ) ⇒ Hom D ( T  D K ⊚ S P , M ). Therefore, the module Hom T ( K ⊚ S P , M ) is isomorphic to the intersec tion of the k er- nels of t w o pairs of maps Hom D ( K ⊙ C P , M ) ⇒ Hom D (( K  C S ) ⊙ C P , M ) and Hom D ( K ⊙ C P , M ) ⇒ Hom D (( T  D K ) ⊙ C P , M ). Analogously , t he mo dule Hom S ( P , Hom T ( K , M )) is em b edded in to Hom C ( P , Hom D ( K , M )) by the comp osition of maps Hom S ( P , Hom T ( K , M )) − → Ho m C ( P , Hom T ( K , M )) − → Hom C ( P , Hom D ( K , M )) and its image coincide s with the intersec tion of the k er- nels of t w o pair s of maps Hom C ( P , Hom D ( K , M )) ⇒ Hom C ( P , Hom D ( T  D K , M )) and Hom C ( P , Hom D ( K , M )) ⇒ Hom C ( P , Hom D ( K  C S , M )). These a re the same t w o pairs of maps. 114 In order to obtain adjoin t functors and equiv alences b etw een speciﬁc categories of left semimo dules and left semicon tra mo dules, w e will ha v e to pro v e asso ciativit y isomorphisms needed for the constructions of 6.1.3 to w ork. 6.2. A sso ciativit y isomorphisms. Let S b e a semialgebra o ver a coring C o v er a k -alg ebra A and T b e a semialgebra o v er a coring D ov er a k -algebra B . The follo wing three Prop ositions will b e mostly applied to the cases of K = T = S or T = D = C , K = S in the sequel. Prop osition 1. L et N b e a right T -semimo dule, K b e a T - S -bisemi m o dule, and P b e a le f t S -semi c ontr amo dule. Then ther e is a natur al map ( N ♦ T K ) ⊚ S P − → N ♦ S ( K ⊚ S P ) whenever b o th mo dules ar e de ﬁ ne d via the c onstructions of 1.4.4 and 6.1.3. This map is an isomorphism, at le ast, in the fol lowing c ase s : (a) D is a ﬂat left B -mo dule, C is a pr oje ctive l e ft A -mo dule, P is a c ontr aﬂat left C -c o ntr amo dule, and either • T is a c oﬂat left D -c omo dule, S is a c opr oje ctive left C -c o m o dule, and K as a rig h t S -se m imo dule with a left D -c omo dule structur e is induc e d fr om a D - c oﬂat D - C - b ic omo dule, or • T is a ﬂat left B -mo dule and a D /B -c oﬂat right D -c o mo dule, S is a pr oje ctive left A -mo dule and a C / A -c oﬂat right C -c omo dule, the ring A (r esp., B ) has a ﬁnite left (r esp., we ak) h o molo gic al dimension, K as a right S -semimo dule with a lef t D -c omo dule structur e i s induc e d fr om a B -ﬂat and C / A -c oﬂat D - C -bic om o dule, and K as a left T -sem i m o dule with a right C -c omo dule structur e is i n duc e d fr om a B -ﬂat D - C -bic omo d- ule, or • the ring A is semisimple, the ring B is absolutely ﬂat, K as a right S -semimo dule with a left D -c omo dule structur e is induc e d fr om a D - C -bi- c omo dule, and K as a left T -semimo dule with a right C -c omo dule struc- tur e is induc e d fr om a D - C -bic omo dule; (b) N is a ﬂat right B -mo dule, D is a ﬂat rig h t B -mo dule, T is a ﬂat right B -mo d ule and a D /B -c oﬂat left D -c o mo dule, C is a ﬂat right A -mo dule, S is a ﬂat right A -m o dule and a C / A -c opr oje ctive l e ft C -c omo dule, the ring A (r esp., B ) has a ﬁnite left (r esp., we ak) homol o gic al dim e nsion, K as a right S -sem i m o dule with a left D -c omo dule structur e is induc e d fr om an A -ﬂat and D /B -c oﬂat D - C -bic omo d ule, K as a left T -semimo dule w i th a right C -c omo dule structur e is induc e d fr o m an A -ﬂat a nd D /B -c o ﬂat D - C -bic o mo d- ule, and P is an A -inje ctive and C / A -c ontr aﬂat left C -c ontr amo dule; (c) N is a ﬂat right B -mo dule, D is a ﬂat right B -mo dule, T is a ﬂat right B -mo d ule and a D /B -c oﬂat left D -c omo dule, the ring B has a ﬁnite we ak homolo g ic al dimension, K as a right S -semimo dule with a left D -c omo dule 115 structur e is induc e d fr om an A -ﬂat D - C -bic omo dule, C is a pr o j e ctive left A -mo dule, P is a C / A -c ontr aﬂat left C -c o n tr amo d ule, and either • S is a c opr oje ctive left C -c omo dule and the rin g A has a ﬁnite we ak ho- molo gic al dimen s ion, or • S is a pr o je ctive left A -mo dule and a C / A -c oﬂat right C -c omo dule, the ring A ha s a ﬁnite left homolo gic al dimension, and K as a left T -sem imo dule with a rig h t C -c om o dule structur e is induc e d fr om a D - C -bic omo dule; (d) D is a ﬂat right B -mo dule, T i s a c oﬂat righ t D -c om o dule, N is a c oﬂat right D -c omo dule, and either • C is a pr o j e ctive left A -mo dule and S i s a c opr o je ctive left C -c omo dule, or • C is a pr oje c tive left A -mo dule, S is a pr oje ctive left A -mo dule a nd a C / A -c oﬂat rig h t C -c omo dule, the right A has a ﬁni te lef t homolo gic al di- mension, and K as a le ft T -semim o dule with a right C -c omo dule structur e is in duc e d f r om a D - C -bic om o dule, or • C is a ﬂat right A -mo dule, S is a ﬂat right A -mo dule and a C / A -c opr oje c- tive left C -c omo d ule, the ring A has a ﬁnite left homolo gic al d imension, K as a left T -sem i m o dule with a right C -c omo dule structur e is induc e d fr om an A -ﬂat D - C -bic omo dule, and P is an inje c tive left A -mo dule, or • C is a ﬂat right A -mo d ule, S is a c oﬂat right C -c om o dule, K as a left T -s e mimo dule with a right C -c omo dule s tructur e is induc e d f r om a C -c oﬂat D - C - bic omo dule, and P i s a c oinje ctive left C -c ontr amo dule. Pr o of . If N ′′′ → N ′′ → N ′ → 0 is a sequence of r igh t S -semimo dule morphisms whic h is exact in the category of A -mo dules and remains exact after ta king the cotensor pro duct with S ov er C , then for any left S -semicontramo dule P there is an exact sequence N ′′′ ⊚ S P − → N ′′ ⊚ S P − → N ′ ⊚ S P − → 0. Hence whenev er a righ t S - semi- mo dule structure on N ♦ T K is deﬁned via the construction of 1.4.4, the k -mo dule ( N ♦ T K ) ⊚ S P is the cokerne l of the pair of maps ( N  D T  D K ) ⊚ S P ⇒ ( N  D K ) ⊚ S P . By the deﬁnition, the semitensor pro duct N ♦ T ( K ⊚ S P ) is the cok ernel of the pair o f maps N  D T  D ( K ⊚ S P ) ⇒ N  D ( K ⊚ S P ). There are natural maps ( N  D K ) ⊚ S P − → N  D ( K ⊚ S P ) and ( N  D T  D K ) ⊚ S P − → N  D T  D ( K ⊚ S P ) constructed in 6.1.3. Whenev er the left T -semimo dule structure on K ⊚ S P is deﬁned via the construction of 6.1 .3, the corresp onding (t w o) square diagrams comm ute. So there is a natural map ( N ♦ T K ) ⊚ S P − → N ♦ T ( K ⊚ S P ), whic h is a n isomorphism pro vided that the map ( N  D K ) ⊚ S P − → N  D ( K ⊚ S P ) and the analog o us map for N  D T in place of N are isomorphisms; and the left T -semimo dule structure on K ⊚ S P is deﬁned pro vided that the analogous map for T in place of N is an isomorphism. It is straightforw a rd to chec k that in eac h case (a-d) a right S - semi- mo dule structure on N ♦ T K is deﬁned via the construction of 1.4.4 (that is where the conditions tha t K as a left T -semimodule with a righ t C -como dule structure is 116 induced from a D - C - bimo dule are used). It is also easy to ve rify the (co)ﬂatness conditions on K ⊚ S P that are needed to guara n tee that the se mitensor product N ♦ T ( K ⊚ S P ) is deﬁned in the case (a). Th us it remains to sho w that the map ( N  D K ) ⊚ S P − → N  D ( K ⊚ S P ) is an isomorphism. In the case (d), the map ( N  D K ) ⊙ C P − → N  D ( K ⊙ C P ) and the analo gous map for K  C S in pla ce of K a r e isomorphisms by Prop osition 5.2.1 (e) and the mo dule N  D ( K ⊚ S P ) is the cok ernel of the pair of maps N  D (( K  C S ) ⊙ C P ) ⇒ N  D ( K ⊙ C P ), so it is clear from the construction of the map ( N  D K ) ⊚ S P − → N  D ( K ⊚ S P ) that it is an isomorphism. In the cases (a-c), o ne has K ≃ K  C S and the m ultiple cotensor pro ducts N  D K  C S  C · · ·  C S are asso ciativ e. So the ma p ( N  D K ) ⊚ S P − → N  D ( K ⊚ S P ) is naturally isomorphic to the map ( N  D K ) ⊙ C P − → N  D ( K ⊙ C P ). The latter is an isomorphism b y Prop osition 5.2.1(a) in the case (a ) a nd b y Prop osition 5.2.1(d) in the cases (b- c).  Prop osition 2. L et L b e a left C -semimo d ule, K b e a T - S -bisemimo dule, and M b e a lef t T -semimo dule. Then ther e is a na tur al m ap SemiHom S ( L , Hom T ( K , M )) − → Hom T ( K ♦ S L , M ) whenever b oth mo dules a r e deﬁne d via the c o n structions of 1.4.4 and 6.1.3. This map is an isomorphism, at le ast, in the fol lowing c ase s : (a) C is a ﬂat right A -mo d ule, D is a ﬂat right B -mo dule, M is a n inje ctive left D -c omo dule, and either • S is a c oﬂat right C -c omo dule, T is a c oﬂat right D -c omo dule, and K as a lef t T -semimo dule with a right C -c om o dule structur e is induc e d fr om a C -c oﬂat D - C - bic omo dule, or • S is a ﬂat right A -mo dule and a C / A -c op r oje ctive left C -c omo dule, T is a ﬂat right B -m o dule and a D / B -c oﬂat left D - c omo d ule, the ring A (r esp., B ) has a ﬁnite left (r esp., we ak) h o molo gic al dimension, K as a left T -sem imo dule with a right C -c omo dule structur e is in d uc e d fr om an A -ﬂat and D / B -c oﬂat D - C -bic o m o dule, and K as a right S -se m imo dule with a left D -c o m o dule structur e is in duc e d fr om an A -ﬂat D - C -bi c omo d- ule, or • the ring A is semisimple , the rin g B is absolutely ﬂat, K as a left T -se m i- mo dule with a right C -c o mo dule structur e is in d uc e d fr om a D - C -bic om o d- ule, and K as a right S -semimo dule with a left D -c omo dule structur e is induc e d fr om a D - C -bic omo dule; (b) L i s a pr oje ctive left A -mo dule, C is a pr oje ctive left A -m o dule, S is a pr oje ctive left A -mo dule and a C / A -c oﬂat right C -c omo dule, D is a ﬂat le f t B -mo dule, T is a ﬂat left B - m o dule and a D /B -c oﬂat right D -c o mo dule, the rings A and B have ﬁn i te left homolo gic al dimensi ons, K as a lef t T -semimo dule with a righ t C -c omo d ule structur e is i n duc e d fr om a B -pr oje ctive an d C / A -c oﬂat D - C -bic o m o dule, K as a right S -semimo dule with a left D -c omo dule structur e 117 is induc e d fr om a B -ﬂat and C / A -c oﬂat D - C -bic omo dule, a n d M is a B - ﬂat and D /B - inje ctive left D -c omo dule; (c) L is a pr oje ctive left A -mo dule, C is a pr oje ctive left A -m o dule, S is a p r oje ctive left A -m o dule and a C / A -c oﬂat right C -c omo dule, the rings A and B have ﬁnite left h o molo gic al dim e nsions, K as a left T -semimo dule with a righ t C -c om o dule structur e is induc e d fr om a B -pr oje ctive D - C -bic omo dule, D is a ﬂat right B -mo d ule, M is a D /B -inje ctive left D -c omo dule, a nd e ither • T is a c oﬂat ri g ht D -c omo dule, or • T is a ﬂat right B -mo dule and a D /B -c oﬂat left D -c omo dule, and K as a right S -semim o dule with a left D -c omo dule s tructur e is induc e d fr om a D - C -bic o m o dule; (d) C is a pr oje ctive left A -mo dule, S is a c opr oje ctive left C -c omo dule, L is a c opr oje ctive left C -c omo dule, and either • D is a ﬂat right B -mo dule an d T is a c oﬂat right D -c omo d ule, or • D is a ﬂat right B -m o dule, T is a ﬂat right B -mo dule and a D / B -c oﬂat left D -c o mo dule, and K as a right S -semimo d ule with a lef t D -c o m o dule structur e is induc e d fr om a D - C -bic omo dule, or • D is a ﬂat left B -m o dule, T is a ﬂat le f t B -mo dule and a D /B -c oﬂat right D -c omo dule, K a s a right S -semimo dule with a left D -c omo dule structur e is induc e d fr om a B -ﬂat D - C -bic om o dule, and M is a ﬂat left B -mo d ule, or • D is a ﬂat left B -mo dule, T is a c o ﬂat left D -c omo dule, K as a right S -semimo dule with a left D -c omo dule structur e is in duc e d fr om a D -c oﬂat D - C -bic o m o dule, a nd M is a c oﬂat left D -c omo dule. Pr o of . An y sequence L ′′ → L ′′ → L ′ → 0 of T - semimo dule morphisms which is exact in the category of B - mo dules and remains exact after taking the coten- sor pro duct with T ov er D is exact in the category of T -semimo dules, i. e., for any T - semimo dule M there is a n exact seque nce 0 − → Ho m T ( L ′ , M ) − → Hom T ( L ′′ , M ) − → Hom T ( L ′′′ , M ). Hence whenev er a left T -semimodule structure is deﬁned on K ♦ S L via the construction o f 1.4.4, t he k -mo dule Hom T ( K ♦ S L , M ) is the k ernel of the pair of maps Hom T ( K  C L , M ) ⇒ Hom T ( K  C S  C L , M ). By the deﬁnition, the k - mo dule SemiHom S ( L , Hom T ( K , M )) is the k ernel of the pair of maps Cohom C ( L , Hom T ( K , M )) ⇒ Cohom C ( S  C L , Hom T ( K , M )) = Cohom C ( L , Cohom C ( S , Hom T ( K , M ))). There are natura l maps Cohom C ( L , Hom T ( K , M )) − → Hom T ( K  C L , M ) and Cohom C ( S  C L , Hom T ( K , M )) − → Hom T ( K  C S  C L , M ) constructed in 6.1.3. Whenev er the left S -semicon- tramo dule structure on Hom T ( K , M ) is deﬁned via the construction of 6.1 .3, the corresp onding (t w o) square diagra ms comm ute. So there is a nat ur a l map SemiHom S ( L , Hom T ( K , M )) − → Hom T ( K ♦ S L , M ), whic h is an isomorphism pro vided that the map Cohom C ( L , Hom T ( K , M )) − → Hom T ( K  C L , M ) and the 118 analogous map for S  C L in place of L a re isomorphisms; a nd the left S - semicontra- mo dule structure on Hom T ( K , M ) is deﬁned provide d that the analogous map for S in place of L is an isomorphism. It is straigh tforward to c hec k that in eac h case (a-d) a left T -semimo dule structure on K ♦ S L is deﬁned via the construction of 1.4.4. It is a lso easy to very fy (using Proposition 5.2.2 (a)) the (co)inj ectivit y conditions on Hom T ( K , M ) that are needed to gua r a n tee that the semihomomorphism mo dule SemiHom S ( L , Hom T ( K , M )) is deﬁned in the case (a). Th us it remains to sho w that the map Cohom C ( L , Hom T ( K , M )) − → Hom T ( K  C L , M ) is an isomorphism. In the case (d), the map Cohom C ( L , Hom D ( K , M )) − → Hom D ( K  C L , M ) and the analogous map for T  D K in place of K are isomorphisms b y Prop osi- tion 5.2.2(e) and t he mo dule Cohom C ( L , Hom T ( K , M )) is the kerne l of the pair o f maps Cohom C ( L , Hom T ( K , M )) ⇒ Cohom C ( L , Hom T ( T  D K , M )), so it is clear from the construction of the map Cohom C ( L , Hom T ( K , M )) − → Hom T ( K  C L , M ) that it is an isomorphism. In the cases (a-c), one has K = T  D K and m ul- tiple cotensor pro ducts T  D · · ·  D T  D K  C L are a sso ciativ e. So the map Cohom C ( L , Hom T ( K , M )) − → Hom T ( K  C L , M ) is naturally isomorphic to the map Cohom C ( L , Hom D ( K , M )) − → Hom D ( K  C L , M ). The latter is an isomor- phism b y Prop o sition 5.2.2(a) in the case (a) and b y 5.2.2(d) in the cases (b- c).  Prop osition 3. L et P b e a left S -semic ontr amo dule, K b e a T - S -bise mimo dule, and Q b e a left T -sem ic ontr amo dule. Then ther e is a natur al map SemiHom T ( K ⊚ S P , Q ) − → Hom S ( P , SemiHom T ( K , Q )) whenever b o th mo dules ar e deﬁn e d via the c onstructions of 3.4.4 and 6.1.3. This ma p is an isomorphism, at le ast, in the fol- lowing c ases: (a) D i s a pr oje ctive left B -mo dule, C is a pr o je ctive left A -mo dule, P is a pr o- je ctive left C -c ontr amo d ule, an d either • T is a c opr oje c tive left D -c omo dule, S is a c opr oje ctive left C -c omo dule, and K as a right S -semimo dule with a lef t D -c omo dule structur e is i n - duc e d fr o m a D -c opr oje ctive D - C -bic om o dule, or • T is a pr oje ctive left B -mo dule and a D /B -c oﬂat right D -c o m o dule, S is a pr oje ctive le ft A -mo dule and a C / A -c oﬂat right C -c omo dule, the rings A and B have ﬁnite left homolo gic al dimensio n s, K as a right S -semimo d- ule with a left D -c omo dule structur e is induc e d fr om a B -pr oje ctive and C / A -c oﬂat D - C -bic om o dule, and K as a left T -semimo dule with a right C -c omo dule structur e is in d uc e d fr o m a B -pr oje ctive D - C -bic om o dule, o r • the rings A an d B ar e semisim ple, K as a right S -semim o dule with a left D -c om o dule structur e is ind uc e d fr om a D - C -bic om o dule, and K as a lef t T -semimo dule with a right C -c om o dule structur e is induc e d fr om a D - C -bic o m o dule; 119 (b) D is a ﬂat righ t B -mo dule, T is a ﬂat righ t B - m o dule and a D /B -c oﬂat left D -c omo dule, Q is an inje ctive left B -mo dule, C is a ﬂat right A -m o dule, S is a ﬂat right A -mo dule and a C / A -c opr oje ctive left C -c omo dule, the rings A and B have ﬁni te left hom o lo gic al dim ensions, K as a right S -semimo dule with a lef t D - c omo d ule structur e is induc e d fr om an A -ﬂat and D / B -c opr oje ctive D - C -bic o m o dule, K as a le ft T -semimo dule w ith a righ t C -c om o dule structur e is induc e d fr om an A -ﬂat and D / B -c opr oje ctive D - C -bic omo dule, and P is a c oinje ctive le f t C -c ontr amo dule; (c) D is a ﬂat right B -mo dule, T is a ﬂat right B -mo dule and a D / B -c oﬂat left D -c omo dule, Q is an inje ctive left B -mo dule, the rings A and B have ﬁnite left homolo g ic al d imensions, K as a righ t S -semimo dule with a left D -c o m o dule structur e is induc e d fr om an A -ﬂat D - C -bic omo dule, C is a pr o j e ctive left A -mo dule, P is a C / A -pr o je ctive left C -c ontr amo dule, and either • S is a c opr oje ctive left C -c omo dule, or • S is a pr oje ctive left A -mo dule an d a C / A -c oﬂat right C -c o m o dule, a nd K as a left T -sem i m o dule with a right C -c omo dule structur e is induc e d fr om a D - C -bic omo dule; (d) D is a ﬂat right B -mo d ule, T is a c oﬂat right D -c omo dule, Q is a c oinje ctive left D -c omo dule, a n d one of the c onditions of the list of Pr op osition 1(d) holds. Pr o of . Let Q b e a left D -contr amo dule, K b e a D - C -bicomo dule with a rig ht S -semi- mo dule structure suc h that m ultiple cohomomor phisms Cohom D ( K  C S  C · · ·  C S , Q ) are asso ciative and the semiaction map K  C S − → K is a left D - como dule morphism, and P b e a left S -semicontramo dule. Then there is a nat ural left S - semicontramo dule structure on the mo dule Cohom D ( K , Q ). The comp osition of maps Cohom D ( K ⊚ S P , Q ) − → Cohom D ( K ⊙ C P , Q ) − → Hom C ( P , Cohom D ( K , Q )) factorizes thro ugh the injection Hom S ( P , Cohom D ( K , Q )) − → Hom C ( P , Cohom D ( K , Q )), so there is a natural map Cohom D ( K ⊚ S P , Q ) − → Hom S ( P , Cohom D ( K , Q )). The rest of the pro of is analo g ous to the pro ofs of Prop ositions 1 a nd 2.  Assume that C is a pro jectiv e left A -mo dule, S is a copro jec tiv e left C -como dule, D is a ﬂat righ t B -mo dule, and T is a coﬂat righ t D -como dule. Then it follows from 6 .1.4 together with Prop ositions 1(d) and 2(d) tha t fo r any left T -semimo dule P , and T - S - bisemimo dule K , and an y left S -semicontramo dule P there is a na tural isomorphism Hom T ( K ⊚ S P , M ) ≃ Hom S ( P , Hom T ( K , M )). In particular, when C is a pro jectiv e left and a ﬂat righ t A - mo dule and S is a copro jectiv e left and a coﬂat righ t C -como dule, there is a pa ir of adjoin t functors Ψ S : S – simo d − → S – sicntr and Φ S : S – sicntr − → S – simo d compatible with the func- tors Ψ C : C – como d − → C – contra and Φ C : C – contra − → C – como d . In o ther words, the S -semimodule Ψ S ( M ) as a C -como dule is naturally isomorphic to Ψ C ( M ) and the S -semicontramo dule Φ S ( P ) a s a C - contramo dule is natura lly isomorphic to Φ C ( P ). 120 Assume that C is a pro jectiv e left A -mo dule and either S is a copro jective left C -como dule, or S is a pro jectiv e left A -mo dule a nd a C / A -coﬂat righ t C -como dule and A has a ﬁnite left homological dimension. Then it follo ws from Prop o sitions 1(a) and 2 (b,d) that the categories of C - copro jectiv e left S -semimo dules and C -projectiv e left S -semicontramo dules are naturally equiv alen t. Assume t ha t C is a ﬂat righ t A - mo dule and either S is a coﬂat righ t C - como dule, or S is a ﬂat righ t A -mo dule and a C / A -coproj ective left C -como dule and A has a ﬁnite left homological dimension. Then if fo llo ws from Prop ositions 1(b,d) and 2 (a) that the categories of C -injectiv e left S -semimo dules and C -coinjectiv e left S -semi- contramo dules are natura lly equiv alen t. Assume that C is a pro jectiv e left A -mo dule a nd a ﬂat righ t A -mo dule, A has a ﬁnite left homological dimension, and either S is a copro jectiv e left C -como dule and a ﬂat right A -mo dule, o r S is a pro jectiv e left A -mo dule and a coﬂa t righ t C -como dule. Then it follows f rom Prop o sitions 1(c,d) and 2(c,d) tha t t he categories of C / A -injec- tiv e left S -semimodules and C / A -projectiv e left S -semicontramo dules are naturally equiv alen t. Finally , assume that the ring A is semisimple . Then it follow s from Prop osi- tions 1(a) and 2(a ) that the categories of C -injectiv e left S - semimo dules and C -pro- jectiv e left S -semicontramo dules are naturally equiv alen t. In eac h of these cases, the na t ura l maps deﬁned in Prop ositions 2–3 in the case of K = T = S hav e the following prop erty of compatibility with the adjo in t f unc- tors b etw een categories of S -semimo dules and S -semicontramo dules. F or any left S -semimodule M and an y left S -semicontramo dule P suc h that the S -semimo dule Φ S ( P ) = S ⊚ S P , the S -semicontramo dule Ψ S ( M ) = Hom S ( S , M ), and the k - mo dule of semihomomorphisms SemiHom S (Φ S ( P ) , Ψ S ( M )) are deﬁned via the construc- tions of 6.1.3 and 3.4.4, the maps SemiHom S (Φ S ( P ) , Ψ S ( M )) − → Hom S (Φ S ( P ) , M ) and SemiHom S (Φ S ( P ) , Ψ S ( M )) − → Hom S ( P , Ψ S ( M )) for m a comm utativ e diagram with the adjunction isomorphism Hom S (Φ S ( P ) , M ) ≃ Hom S ( P , Ψ S ( M )). 6.3. Semimo dule-semicon tramo dule corresp ondence. Assume that the coring C is a pro jec tiv e left a nd a ﬂa t r igh t A -mo dule, the semialgebra S is a copro jectiv e left and a coﬂat right C - como dule, and the ring A has a ﬁnite left ho mo lo gical dimension. Theorem. (a) The functor mapping the quotient c ate gory of c omplexes of C / A -inje c- tive l e f t S -semimo dules by the thic k sub c ate g o ry of C -c o acyclic c omplexes of C / A -in- je ctive S -sem imo dules into the semideriv e d c ate gory of left S -semimo dules is an e q uiv- alenc e of triangulate d c a te gories. (b) T he functor m apping the quotient c ate gory of c omplex e s o f C / A -pr o j e ctive left S -semic ontr amo dules b y the thick sub c ate gory of C -c ontr aa c ycli c c ompl e xes of C / A -pr oje ctive S -semic ontr amo dules into the s e miderive d c ate go ry of left S -semic on - tr amo d ules is an e quivalenc e of triangulate d c ate gories. 121 Pr o of . P art (b) follows from Lemma 5.3.2(b) and Lemma 2.6 applied to the construc- tion of the mor phism of complexes L 2 ( P • ) − → P • from the pro o f o f Theorem 4.6(b). As a n alternativ e to using Lemma 5.3.2 , one can sho w that L 2 ( P • ) is a complex of C / A -proj ective S -semicontramo dules in the following w ay . Use Lemma 3.3 .2 (b) to construct a ﬁnite righ t A -injectiv e resolution o f ev ery term of the complex of left S -semicontramo dules P • , then apply the functor L 2 , whic h maps exact triples of complexes to exact triples, and use Lemmas 3 .3 .3(c), 5.2(b), and 5.3.1(b). The pro of of part (a) is completely a na logous.  Remark. The analogue of Theorem for complexes of quite C / A -injectiv e S -semi- mo dules and quite C / A -projectiv e S -semicontramo dules is true. Moreo v er, for an y complex of left S -semimo dules M • there exists a morphism from M • in to a com- plex of C -injectiv e S -semimo dules with a C -coacyclic cone, and for any complex of left S - semicontramo dules P • there exists a morphism in to P • from a complex of C -projectiv e S - semicontramo dules with a C - contraacyclic cone. Indeed, consider the complex of C / A -injectiv e S -semimodules Φ S L 2 ( P • ) and apply to it the construc- tion of the morphism of complexes L 1 ( K • ) − → K • from the pro of of Theorems 2.6 and 4.6(a). F o r an y complex of C / A -injectiv e S - semimo dules K • , t he complex L 1 ( K • ) is a complex o f copro jectiv e S -semimo dules b y Remark 3.2.2 and Lemma 5.3.2(a) (or simply b ecause the class of C / A -injectiv e left C -como dules is closed under extensions and any A -projectiv e C / A - injectiv e left C -como dule is copro jectiv e, whic h is easy to c hec k). So the complex of C -coprojectiv e S -semimo dules L 1 (Φ S L 2 ( P • )) maps in to Φ S L 2 ( P • ) with a C -coacyclic cone, hence the complex of C -projectiv e S - semicontra- mo dules Ψ S L 1 (Φ S L 2 ( P • )) maps in to L 2 P • and P • with C -contraacyclic cones. Corollary . The r estrictions of the functors Ψ S and Φ S (applie d to c omp lexes term- wise) to the homotopy c ate gory of c omplexes of C / A -inje ctive S -s emimo d ules and C / A -pr oje ctive S -semic ontr amo d ules deﬁne mutual ly inverse e quivalenc es R Ψ S and L Φ S b etwe en the semiderive d c ate gory of left S -semimo dules a n d the semideriv e d c ate go ry of left S -semic ontr am o dules. Pr o of . By Corollary 5.4, the restrictions of f unctors Ψ S and Φ S induce m utually in- v erse equiv alences b etw een the quotien t category of the homotopy category of C / A -in- jectiv e S -semimo dules b y its in tersection with the thic k sub category of C -coacyclic complexes a nd the quotien t category of the homotop y catego ry of C / A -pro j ectiv e S -semicontramo dules b y its inters ection with t he thic k sub catego ry of C -contraacyclic complexes. Th us it remains to t a k e in accoun t the ab ov e Theorem.  6.4. B ir elativ ely con traﬂat, pro jectiv e, and injective complexes. W e k eep the assumptions of 6.3. A complex of left S -semimo dules M • is called pr oje ctive r elative to C r elative to A ( S / C / A -projectiv e) if the complex o f homomorphisms ov er S from M • in to 122 an y C -coacyclic complex of C / A -injectiv e S -semimodules is acyclic. A complex of left S -semicontramo dules P • is called in je ctive r elative to C r elative to A ( S / C / A -in- jectiv e) if the complex of homomorphisms ov er S into P • from an y C -contra acyclic complex of C / A - projectiv e S -semicontramo dules is acyclic. The con tratensor pro duct N • ⊚ S P • of a complex N • of right S -semimo dules a nd a complex P • of left S -semicontramodules is deﬁned as the tot al complex of the bicomplex N i ⊚ S P j , constructed b y taking inﬁnite direct sums a long the diagonals. A complex of righ t S -semimo dules N • is called c ontr aﬂat r elative to C r elative to A ( S / C / A -contraﬂat ) if the contratensor pro duct o v er S of the complex N • an y an y C -contraacyclic complex of C / A -projectiv e left S -semicontramo dules is acyclic. It follows from Theorem 5 .4 and Lemma 5.3.1 that the complex of left S -semimo d- ules induced from a complex of A -projectiv e C -como dules is S / C / A - projectiv e, the complex of left S - semicontramo dules coinduced from a complex of A -injectiv e C -con- tramo dules is S / C / A -injectiv e, a nd the complex of rig h t S -semimo dules induced from a complex of A -ﬂat C -como dules is S / C / A -contr a ﬂat. Lemma. (a) Any S / C / A -semiﬂat c omplex o f A -ﬂat right S -semimo dules (i n the sense of 2.8 ) is S / C / A -c ontr aﬂat. (b) A c omplex of A -pr oje ctive left S -semim o dules is S / C / A -p r oje ctive if and only if it is S / C / A -sem ipr oje ctive (in the sense of 4.8 ). (c) A c omple x of A -in j e ctive left S -semic ontr amo d ules is S / C / A -inje ctive i f a n d only if it is S / C / A -semii n je ctive (in the sense of 4.8). Pr o of . The functors Ψ S and Φ S deﬁne an equiv a lence b etw een the category of C -coacyclic complexes of C / A -injectiv e left S -semimo dules and the catego ry of C -con- traacyclic complexes of C / A -pro jectiv e left S -semicontramo dules. Therefore, par t (a) follo ws from Propo sition 6 .2.1(c) (applied to K = T = S ) and Lemma 5.3.2(a), part (b) follow s from Prop osition 6.2.2(c) and Lemma 5.3.2(b), and part (c) follows from Prop osition 6.2 .3 (c) a nd Lemma 5.3.2(a) .  In view of the relev an t results of 4.8, it is also clear that a complex of A -pro- jectiv e left S -semimodules is S / C / A - projectiv e if the complex of S -semimo dule ho- momorphisms f r om it into an y C -contractible complex of C / A -injectiv e S -semimo d- ules is a cyclic. Analogously , a complex of A -injectiv e left S -semicontramo dules is S / C / A -injectiv e if the complex of S -semicontramo dule homomorphisms in to it from an y C -contractible complex o f C / A -proj ective S -semicontramo dules is acyclic. Question. One can sho w using t he construction of the morphism of complexes of left S -semimodules L • − → R 2 ( L • ) and Lemma 1.2 .2 that an y S / C / A -contraﬂat complex of (appropriately deﬁned) S / C / A - semiﬂat righ t S - semimo dules is S / C / A -semiﬂat. One can a lso sho w using the functor SemiT or S that an y A -ﬂat S / C / A -contraﬂat righ t S -semimo dule (deﬁned in terms of exact triples of C / A -pro jectiv e or C / A -con- traﬂat left S -semicontramo dules) is S / C / A -semiﬂat; the con v erse is clear (cf. 9.2). 123 Are all S / C / A -contraﬂat (in either deﬁnition) righ t S -semim o dules A -ﬂat? Are all S / C / A -contraﬂat complexes of A -ﬂat righ t S -semimo dules S / C / A -semiﬂat? The functor mapping the quotient category o f S / C / A -contra ﬂa t complexes of right S -semimodules b y its in tersection with the thick sub catego ry of C -coacyclic complexes in to the semideriv ed category of righ t S -semimo dules is an equiv alence o f triangulated categories, since the complex L 3 L 1 ( K • ) is S / C / A -contraﬂa t for an y complex of right S -semimodules K • . The a nalogous results for S / C / A -projectiv e complexes of left S -semimodules and S / C / A -injectiv e complexes of left S -semicontramo dules follow from the corresp onding results of 4.8. Remark. It follo ws from the a b ov e Lemma and Lemma 5.2 that an y C -coacyclic semipro jectiv e complex of C -coprojective left S -semimo dules is contractible . Indeed, suc h a comple x is sim ultaneously an S / C / A -pr o jectiv e complex and a C - coacyclic complex of C / A -injectiv e S -semimodules. Analogo usly , an y C -contra a cyclic semiin- jectiv e complex of C -coinjectiv e left S -semicontramo dules is con tractible. Hence the homotop y cat ego ry of semipro jectiv e complexes of C -coprojective S -semimo dules is equiv alen t to the semideriv ed catego r y of left S -semimodules and the homotopy cat- egory of semiinjectiv e complexes of C -coinjectiv e S -semicontramo dules is equiv alent to the semideriv ed catego ry of left S -semicontramo dules. F urthermore, it follo ws that the homotopy category of semipro jectiv e complexes of C -coprojective S -semi- mo dules is the minimal triangulated sub catego r y con taining the complexes of left S -semimodules induced from complexes of C -coprojectiv e C -como dules and closed under inﬁnite direct sums. Analogously , the homotop y category o f semiinjectiv e com- plexes of C - coinjectiv e S -semicontramo dules is the minimal triang ula ted sub category con taining the complexes of left S - semicontramo dules coinduced from complexes of C -coinjectiv e C -contra mo dules and closed under inﬁnite pro ducts. (Cf. 2.9.) 6.5. Derived functor CtrT or. The follo wing Lemmas provide a g eneral approach to one-sided derived functors of any num b er of argumen ts. They are essen tially due to P . Deligne [1 8]. Lemma 1. L et H b e a c ate gory and S b e a lo c alizing class o f morphis m s in H . L et P and J b e ful l s ub c ate gories of H such that either (a) the map Hom H ( Q, j ) i s bije ctive for any obj e ct Q ∈ P and any morphism j ∈ S ∩ J , and fo r any obje ct Y ∈ H ther e i s an o bje ct J ∈ J to gether with a morphism Y − → J b elonging to S , o r (b) the map Hom H ( q , J ) is bije ctive for any morphism q ∈ S ∩ P and any obje ct J ∈ J , and for an y obje c t X ∈ H ther e is an obje ct Q ∈ P to gether with a morphism Q − → X b elonging to S . Then for any obje cts P ∈ P and I ∈ J the natur al ma p Hom H ( P , I ) − → Hom H [ S − 1 ] ( P , I ) is bije ctive. 124 Pr o of . P art (b): any elemen t of Hom H [ S − 1 ] ( P , I ) can b e represen ted b y a fraction of morphisms P ← − X − → I in H , where the morphism X − → P b elongs to S . Cho ose an ob ject Q ∈ P to g ether with a morphism Q − → X b elonging to S . Then the comp o- sition Q − → X − → P belong s to S ∩ P , hence the map Hom H ( P , I ) − → Hom H ( Q, I ) is bijectiv e and there exists a morphism P − → I that forms a commutativ e triangle with the mor phisms Q − → X − → P and Q − → X − → I . Obv iously , this morphism P − → I represen ts the same morphism in H [ S − 1 ] that the fraction P ← − X − → I . No w supp ose that there are tw o morphisms P ⇒ I in H whose images in H [ S − 1 ] coin- cide. Then there exists a morphism X − → P b elonging to H whic h has equal comp o si- tions with the morphisms P ⇒ I . Cho ose an ob ject Q ∈ P tog ether with a morphism Q − → X b elonging to H again. The comp osition Q − → X − → P has equal comp o- sitions with the morphisms P ⇒ I , and since the map Hom H ( P , I ) − → Hom H ( Q, I ) is bijectiv e, our tw o morphisms P ⇒ I are equal. Proof of par t (a ) is dual.  Lemma 2. L e t H i , i = 1 , . . . , n b e sever al c ate gories, S i b e lo c alizing cla s ses of morphisms in H i , and F 1 b e ful l sub c ate gories of H i . Assume that for any obje ct X in H i ther e is an obje ct U i n F i to gether with a morphism U − → X fr om S i . L et K b e a c a te gory and Θ : H 1 × · · · × H n − → K b e a functor such that the morphism Θ( U 1 , . . . , U i − 1 , t, U i +1 , . . . , U n ) is an isomorphi s m for an y obje c ts U j ∈ F j and any morphism t ∈ S i ∩ F i . Th e n the left derive d functor L Θ : H 1 [ S − 1 1 ] × · · · × H n [ S − 1 n ] − → K obtaine d by r estricting Θ to F 1 × · · · × F n is a univers a l ﬁnal obje ct in the c ate gory of al l functors Ξ : H 1 × · · · × H n − → K factorizable thr ough H 1 [ S − 1 1 ] × · · · × H n [ S − 1 n ] and endowe d with a morphism of f unctors Ξ − → Θ . Pr o of . It suﬃces to consider a single catego ry H = H 1 × · · · × H n with the class of morphisms S = S 1 × · · · × S n , the full sub category F = F 1 × · · · × F n , a nd the functor of one argumen t Θ : H − → K . The functor F [( S ∩ F ) − 1 ] − → H [ S − 1 ] is an equiv alence of categories by Lemma 2.6 , so the derive d functor L Θ can b e deﬁned. F or any ob ject X ∈ H , c ho ose an ob ject U X ∈ F together with a morphism U X − → X from S ; then we hav e t he induced morphism L Θ( X ) = Θ( U X ) − → Θ( X ). F or an y morphism X − → Y in H there exists an o b ject V in F together with a morphism V − → U X b elonging to S and a mor phism V − → U Y in H forming a comm utativ e diagram with the morphisms U X − → X − → Y and V X − → Y . So we ha v e constructed a morphism of functors L Θ − → Θ. Now if a functor Ξ : H − → K fa cto r izable through H [ S − 1 ] is endo w ed with a morphism of functors Ξ − → Θ, then the desired morphism of functors Ξ − → L Θ can b e obtained b y restricting the morphism of f unctors Ξ − → Θ to the sub category F ⊂ H .  Notice the diﬀerence b etw een the construction of a double-sided derive d functor of t w o argumen ts in Lemma 2.7 and the construction of a left derive d functor of any n um b er of argumen ts in Lemma 2. While in the former construction only one of t he t w o a rgumen ts is resolve d, and the conditio ns imp o sed on the resolutions guar a n tee 125 that the tw o deriv ed functors obtained in this wa y coincide, in the latter construction al l of the argumen ts are resolv ed at once and it would not suﬃce to resolv e o nly some of them. In other w ords, the construction of Lemma 2.7 only works to deﬁne b alanc e d double-sided deriv ed functors, while construction of Lemma 2 allows to deﬁne nonb alanc e d one-sided deriv ed functors. Assume that the semialgebra S satisﬁes the conditions of 6 .3 . According to Lemma 1(a) and (the pro of of ) Theorem 6.3(a) , the natural map Hom Hot ( S – si mod ) ( L • , M • ) − → Hom D si ( S – simo d ) ( L • , M • ) is an isomorphism whenev er L • is a complex of S / C / A -projectiv e S -semimo dules and M • is a complex of C / A -injec- tiv e S -semimo dules. So t he functor of homomorphisms in the semideriv ed categor y of left S -semimo dules can b e lifted to a functor Ext S : D si ( S – simo d ) op × D si ( S – simo d ) − − → D ( k – mo d ) , whic h is deﬁned b y restricting t he functor of homomorphisms of complexes o f left S -semimodules to the Carthesian pro duct of the homotop y category of S / C / A - projec- tiv e complexes of S -semimo dules and the homoto py category of complexes of C / A -in- jectiv e S -semimo dules. By Lemma 2, this construction o f the righ t deriv ed functor Ext S do es not dep end o n the c hoice o f sub categories of adjusted complexes. Analogously , a ccording to Lemma 1(b) and (the pro of of ) Theorem 6.3(b), the natural map Hom Hot ( S – si cntr ) ( P • , Q • ) − → Hom D si ( S – sicntr ) ( P • , Q • ) is an isomorphism whenev er P • is a complex of C / A - injectiv e S -semicontramo dules and Q • is a complex of S / C / A -injectiv e S - semicontramo dules. So the functor of homomorphisms in the semideriv ed category of left S -semicontramo dules can b e lifted to a functor Ext S : D si ( S – sicntr ) op × D si ( S – sicntr ) − − → D ( k – mo d ) , whic h is deﬁned by restricting t he functor of homomorphisms of complexes of left S -semicontramo dules to the Carthesian pro duct of the homotop y category o f complexese of C / A -projectiv e S -semicontramo dules and the homotopy category of S / C / A -injectiv e complexes of S -semicontramo dules. Finally , the left deriv ed functor of contratensor pro duct CtrT or S : D si ( simo d – S ) × D si ( S – sicntr ) − − → D ( k – mo d ) is deﬁned b y restricting the functor of con tratensor pro duct ov er S to the Carthesian pro duct of the homotop y category o f S / C / A - contraﬂat complexes of righ t S -semi- mo dules and the homotopy category of complexes of C / A -projectiv e left S -semicon- tramo dules. By the deﬁnition, this restriction factorizes thro ugh t he semideriv ed category of left S -semicontramo dules in the second a rgumen t; let us sho w that it also factorizes throug h the semideriv ed category of right S -semimo dules in the ﬁrst argumen t. The complex of left S -semicontramo dules Hom k ( N • , k ∨ ) is S / C / A -injec- tiv e whenev er a complex of right S - semimo dules N • is S / C / A -contraﬂat; and the complex Hom k ( N • , k ∨ ) is C -contraacyclic whenev er the complex N • is C -coacyclic. 126 Hence if N • is a C -coacyclic S / C / A -contraﬂat complex of right S -semimo dules and P • is a complex of C / A - pr o jectiv e left S - semicontramo dules, then the complex Hom S ( P • , Hom k ( N • , k ∨ )) is a cyclic, so the complex N • ⊚ S P • is also acyclic. By Lemma 2, this construction of the left deriv ed functor CtrT or S do es not dep end on the c hoice of sub categories of adjusted complexes. Notice that the constructions of deriv ed functors R Ψ S and L Φ S in Corollar y 6.3 are a lso particular cases o f Lemma 2. Remark. T o deﬁne/compute the comp osition m ultiplication Ext S ( L • , M • ) ⊗ L k Ext S ( K • , L • ) − → Ext S ( K • , M • ) it suﬃces to represen t the imag es of K • , L • , and M • in the semideriv ed cat ego ry of left S -semimodules by semipro jectiv e complexes of C -coprojectiv e S -semimodules. The same applies to the functor Ext S and semiinjec- tiv e complexes of C -coinjectiv e S -semicontramo dules. Besides, one can compute the functors Ext S , Ext S , and CtrT or S using resolutions o f other kinds. In particular, one can use complexes of C -injectiv e S -semimo dules a nd complexes o f C -proj ectiv e S - semi- contramo dules (see R emark 6.3 ) together with (appropriately deﬁned) S / C -pro j ectiv e complexes of left S -semimo dules, S / C - inj ective complexes of left S - semicontramo d- ules, a nd S / C -contraﬂat complexes of right S -semimo dules. One can a lso compute the functor Ext S in terms of injectiv e complexes of S -semimo dules (deﬁned as complexes righ t ortho gonal to C -coacyclic complexes in Hot ( S – simo d )) and the functor Ext S in terms of pro jectiv e complexes of S -semicontramo dules. The se can b e obtained b y applying the functor Φ S to semiinjectiv e complexes of C -coinjectiv e S -semicontramo d- ules and the functor Ψ S to semipro jectiv e complexes of C -coprojectiv e S -semimo dules, and using Prop ositions 6.2.2(a) a nd 6 .2.3(a). Injectiv e complexes of S -semimo d- ules can b e also constructed using the functor right adjoin t to the forgetful functor S – simo d − → C – como d (see Question 3.3.1) and inﬁnite pro ducts of complexes o f S -semimodules; this approac h w orks a ssuming o nly that C is a ﬂat rig ht A - mo dule, S is a coﬂat right C -como dule, a nd A has a ﬁnite left homolog ical dimension. 6.6. SemiExt and Ext , SemiT or and Ct r T or. W e kee p the assumptions of 6.3. Corollary . (a) Ther e ar e natur al isomorphisms of functors SemiExt S ( M • , P • ) ≃ Ext S ( M • , L Φ S ( P • )) ≃ Ext S ( R Ψ S ( M • ) , P • ) on the Carthesian pr o duct of the c ate- gory opp osite to the semideriv e d c ate gory of left S -semimo dules and the semiderive d c ate go ry of left S -semic ontr am o dules. (b) Ther e is a natur al isomorphis m of functors SemiT or S ( N • , M • ) ≃ CtrT or S ( N • , R Ψ S ( M • )) on the Carthesian pr o duct of the semiderive d c ate gory of rig h t S -semimo dules and the sem iderive d c ate go ry of left S -semimo dules. Pr o of . It suﬃce s to construct natural isomorphisms SemiEx t S ( L • , R Ψ S ( M • )) ≃ Ext S ( L • , M • ), SemiExt S ( L Φ S ( P • ) , Q • ) ≃ Ext S ( P • , Q • ), and SemiT or S ( N • , L Φ S ( P • )) ≃ CtrT or S ( N • , P • ). In the ﬁrst case, represen t the image of M • in 127 D si ( S – simo d ) by a complex of C / A -injectiv e S -semimodules a nd the ima g e of L • in D si ( S – simo d ) by a semipro j ective complex of C -copro j ectiv e S -semimo dules, and use Prop osition 6.2.2(d) and Lemma 6 .4(b). Alternativ ely , repres en t the image of M • in D si ( S – simo d ) by a complex of C / A -injectiv e S -semimo dules and the ima g e of L • in D si ( S – simo d ) b y an S / C / A - semiprojectiv e complex of A -pro jectiv e S - semimo d- ules (see 4.8), and use Propo sition 6.2 .2(c), Lemma 6.4(b), a nd Lemma 5.3.2(b). In the second case, represe n t the image of P • in D si ( S – sicntr ) b y a complex of C / A -proj ective S -semicontramo dules and the image of Q • in D si ( S – sicntr ) by a semiinjectiv e complex of C -coinjectiv e S -semimo dules, and use Prop osition 6.2.3(d) and Lemma 6.4(c). Alternativ ely , represen t the image of P • in D si ( S – sicntr ) b y a complex of C / A - projectiv e S - semicontramo dules and the image of Q • in D si ( S – sicntr ) b y an S / C / A - semiinjectiv e complex of A -injectiv e S -semicontramo dules (see 4.8), and use Prop osition 6.2.3(c), Lemma 6.4(c), and Lemma 5 .3 .2(a). In t he third case, represen t the imag e of P • in D si ( S – sicntr ) by a complex o f C / A - projectiv e S -semi- contramo dules and the image o f N • in D si ( simo d – S ) b y a semiﬂat complex of C -coﬂat S -semimodules, and use Prop osition 6.2 .1(d) and Lemma 6.4(a) . Alternativ ely , represen t the imag e of P • in D si ( S – sicntr ) by a complex o f C / A - projectiv e S -semi- contramo dules and the image of N • in D si ( simo d – S ) b y an S / C / A -semiﬂat complex of A -ﬂat S -semimo dules (see 2.8) , and use Prop osition 6.2.1(c), Lemma 6.4(a), a nd Lemma 5.3.2(a). Finally , to show that the three pairwise isomorphisms betw een the functors SemiExt S ( M • , P • ), Ext S ( M • , L Φ S ( P • )), and Ext S ( R Ψ S ( M • ) , P • ) form a commu- tativ e dia g ram, one can represen t the image of M • in D si ( S – simo d ) b y a semipro jec- tiv e complex o f C -coproj ectiv e S -semimo dules a nd the image of P • in D si ( S – sicntr ) b y a semiin jectiv e complex of C -coinjectiv e S -semicontramo dules (ha ving in mind Lemmas 6.4 and 5.2), and use a r esult o f 6.2.  128 7. Functoriality in the Coring 7.1. Compatible morphisms. Let C b e a coring ov er a k -algebra A and D b e a coring o v er a k -algebra B . 7.1.1. W e will sa y that a map C − → D is compatible with a k - algebra morphism A − → B if the biaction ma ps A ⊗ k C ⊗ k A − → C and B ⊗ k D ⊗ k B − → D form a comm utativ e diagram with the maps C − → D and A ⊗ k C ⊗ k A − → B ⊗ k D ⊗ k B (in other words, the map C − → D is an A - A -bimo dule mor phism) a nd the comultipli- cation maps C − → C ⊗ A C and D − → D ⊗ B D , a s w ell as t he counit maps C − → A and D − → B , form comm utative diagra ms with the maps A − → B , C − → D , and C ⊗ A C − → D ⊗ B D . Let C − → D b e a map of coring s compatible with a k -algebra map A − → B . Let M b e a left como dule o v er C a nd N b e a left como dule ov er B . W e will sa y that a map M − → N is compatible with the maps A − → B and C − → D if the action maps A ⊗ k M − → M and B ⊗ k N − → N form a comm utativ e dia gram with the maps M − → N and A ⊗ k M − → B ⊗ k N (that is the map M − → N is an A -mo dule morphism) and the coaction maps M − → C ⊗ A M and N − → D ⊗ B N form a commutativ e diag r am with the maps M − → N and C ⊗ A M − → D ⊗ B M . Analogously , let P b e a left contramo dule ov er C and Q b e a left con tramo dule o v er D . W e will say that a map Q − → P is compatible with the maps A − → B and C − → D if the action maps P − → Hom k ( A, P ) and Q − → Hom k ( B , Q ) form a comm utativ e diagra m with the maps Q − → P and Hom k ( B , Q ) − → Hom k ( A, P ) (that is the map Q − → P is an A -mo dule mor phism) and the contraaction maps Hom A ( C , P ) − → P and Hom B ( D , Q ) − → Q form a comm utativ e diagram with the maps Q − → P and Hom B ( D , Q ) − → Hom A ( C , P ). Let M ′ − → N ′ b e a map fro m a right C - como dule M ′ to a right D - como dule N ′ compatible with the maps A − → B and C − → D , and M ′′ − → N ′′ b e a map from a left C -como dule M ′′ to a left D - como dule N ′′ compatible with the maps A − → B and C − → D . Then there is a natural map M ′  C M ′′ − → N ′  D N ′′ . Analogously , let M − → N b e a map from a left C -como dule M to a left D -como dule N compatible with the maps A − → B and C − → D , and Q − → P b e a map from a left D - con- tramo dule Q to a left C -contramo dule P compatible with the ma ps A − → B and C − → D . Then there is a natural map Cohom D ( N , Q ) − → Cohom C ( M , P ). 7.1.2. Let C − → D b e a map of corings compat ible with a k -algebra map A − → B . Then t here is a functor fro m the catego r y of left C -como dules t o t he category of left D -como dules assigning to a C -como dule M the D -como dule B M = B ⊗ A M with the D -coaction map deﬁned as the comp osition B ⊗ A M − → B ⊗ A C ⊗ A M − → D ⊗ A M = D ⊗ B ( B ⊗ A M ) of the map induced b y the C -coaction in M and the map induced b y the map C − → D and the left B -action in D . The functor M 7− → M B from 129 the category of right C - como dules to the category of right D -como dules is deﬁned in the analogous w a y . F urthermore, there is a functor from the category of left C -con- tramo dules to the category of left D -contramo dules assigning to a C -contramo dule P the D -contramo dule B P = Hom A ( B , P ) with the con traa ctio n map deﬁned as the comp osition Hom B ( D , Hom A ( B , P )) = Hom A ( D , P ) − → Hom A ( C ⊗ A B , P ) = Hom A ( B , Hom A ( C , P )) − → Hom A ( B , P ) of the map induced b y the map C − → D and t he righ t B -a ction in D with the map induced by the C - con traaction in P . If C is a ﬂat righ t A -mo dule, then the f unctor M 7− → B M has a righ t adjoint functor assigning to a left D -como dule N the left D - como dule C N = C B  D N , where C B = C ⊗ A B is a C - D -bicomo dule with the right D - como dule structure pro- vided by the ab ov e construction. These functors are adjo int since b oth k - mo dules Hom D ( B M , N ) and Hom C ( M , C N ) are isomorphic to the k - mo dule of all maps of co- mo dules M − → N compatible with the maps A − → B and C − → D . Without any assumptions on the cor ing C , the functor N 7− → C N is deﬁned on the f ull sub category of left D -como dules suc h that the cotensor pro duct C B  D N can b e endo w ed with a left C -como dule structure via the construction of 1.2.4 ; this includes, in particular, quasicoﬂat D -como dules. Analogo usly , if C is a ﬂa t left A -mo dule, then the functor M 7− → M B has a righ t adjoin t functor assigning to a righ t D -como dule N the right C -como dule N C = N  D B C , where B C = B ⊗ A C is a D - C -bicomo dule with the left D -como dule structure provided by the ab ov e construction. F urthermore, if C is a pro jective left A -mo dule, then the functor P 7− → B P has a left a djoin t functor assigning to a left D -contramo dule Q the left C -contra - mo dule C Q = Cohom D ( B C , Q ). These functors are adjoint since bo t h k -mo dules Hom D ( Q , B P ) and Hom C ( C Q , P ) are isomorphic to the k -mo dule of a ll maps o f con- tramo dules Q − → P compatible with the maps A − → B and C − → D . Without an y assumptions on the coring C , the functor Q 7− → C Q is deﬁned on the full sub category of left D -contramo dules suc h that the cohomomorphism mo dule Cohom D ( B C , Q ) can b e endow ed with a left C -contramo dule structure via the construction of 3.2.4; t his includes, in particular, quasicoinjectiv e D -contra mo dules. If C is a pro jectiv e left A -mo dule, then for an y righ t C -como dule M and a ny left D -contramo dule Q there is a natural isomorphism M B ⊙ D Q ≃ M ⊙ C C Q . Indeed, b oth k -mo dules are isomorphic to t he cok ernel of the pair of maps M ⊗ A Hom B ( D , Q ) ⇒ M ⊗ A Q , one of whic h is induced by the D -contraaction in Q and t he other is the comp osition of the map induced b y t he C -coa ction in M a nd the map induced b y the ev aluation map C B ⊗ B Hom B ( D , Q ) − → Q . This is obv ious for M B ⊙ D Q , and in order to show this for M ⊙ C C Q it suﬃces to represen t C Q as the cok ernel of the pair of C -contramo dule mo r phisms Hom B ( B C , Hom B ( D , Q )) ⇒ Hom B ( B C , Q ). Without an y a ssumptions on the coring C , there is a nat ural isomorphism M B ⊙ D Q ≃ M ⊙ C C Q for an y right C -como dule M and any left D -contramo dule Q for whic h the C -contra- mo dule C Q = Cohom D ( B C , Q ) is deﬁned via the construction of 3.2.4. 130 7.1.3. Let C − → D b e a map of corings compat ible with a k -algebra map A − → B . Prop osition. ( a) F or any left C -c omo dule M and any right D -c omo dule N for which the rig h t C -c omo dule N C is deﬁne d ther e is a natur al map N C  C M − → N  D B M , which is a n isomorp hism, at le ast, when C an d M ar e ﬂat lef t A -mo dules or N is a quasic oﬂat right D -c omo dule. (b) F or any left C -c ontr amo dule P and any left D -c omo dule N for whi c h the left C -c omo dule C N is deﬁne d ther e is a n a tur al m ap Cohom D ( N , B P ) − → Cohom C ( C N , P ) , whic h is an isom o rp hism, a t le ast, when either C is a ﬂat right A -mo dule and P is an inje ctive left A -mo dule, or N is a quasic o p r oje ctive left D -c omo dule. (c) F or a ny left C -c omo dule M and any left D -c o ntr amo dule Q for which the left C -c ontr amo d ule C Q i s deﬁne d ther e is a natur al map Cohom D ( B M , Q ) − → Cohom C ( M , C Q ) , whi c h is an i s omorphism, at le ast, when C and M ar e pr o je ctive left A -mo dules or Q is a quasic oinje ctive left D -c ontr amo dule. Pr o of . P art (a): for any left C -como dule M and an y rig h t D -como dule N there are maps of como dules M − → B M and N C − → N compatible with the maps A − → B and C − → D . So there is the induced map N C  C M − → N  D B M . On the other hand, for any left C -como dule M there is a natural isomorphism of left D - como dules B M ≃ B C  C M , hence N C  C M = ( N  D B C )  C M and N  D B M = N  D ( B C  C M ). Let us c heck that the maps N C  C M − → N  D B M , N C  C M − → N ⊗ B B C ⊗ A M and N  D B M − → N ⊗ B B C ⊗ A M form a comm utativ e diagram. Indeed, the map ( N  D B C ) ⊗ A M − → N ⊗ B B C ⊗ A M is equal to the comp osition of the ma p ( N  D B C ) ⊗ A M − → ( N  D B C ) ⊗ A C ⊗ A M induced b y t he C -coaction in N  D B C with the map ( N  D B C ) ⊗ A C ⊗ A M − → N ⊗ B B C ⊗ A M induced by the maps N  D B C − → N a nd C − → B C ; while the comp osition of maps ( N  D B C ) ⊗ A M − → N ⊗ B ( B C  C M ) − → N ⊗ B B C ⊗ A M is equal to the comp osition o f the map ( N  D B C ) ⊗ A M − → ( N  D B C ) ⊗ A C ⊗ A M induced by the C -coa ctio n in M with the same map ( N  D B C ) ⊗ A C ⊗ A M − → N ⊗ B B C ⊗ A M . It remains to apply Prop osition 1.2.5 (d) and (e) with the left and righ t sides switc hed. The pro ofs of parts (b) and (c) are completely analogous; the pro of o f (b) uses Prop o sition 3 .2 .5(g,h) and t he pro of of (c) uses Prop osition 3.2.5(f,i).  7.1.4. Let C − → D b e a map of corings compat ible with a k -algebra map A − → B . Assume that C is a pro jectiv e left and a ﬂat rig h t A -mo dule. Then fo r an y left D -contramo dule Q there is a natural morphism of C -como dules Φ C ( C Q ) − → C (Φ D Q ), whic h is a n isomorphism, at least, when D is a ﬂat right B -mo dule a nd Q is a D /B -contraﬂat left D -contramo dule. Indeed, Φ C ( C Q ) = C ⊙ C C Q ≃ C B ⊙ D Q as a left C -como dule and C (Φ D Q ) = C B  D ( D ⊙ D Q ), so it remains to apply Pro p osition 5.2.1( c). Analogously , f or any left D -como dule N there is a natural morphism of C -contramo dules C (Ψ D N ) − → Ψ C ( C N ), whic h is an 131 isomorphism, at least, when D is a pro jectiv e left B -mo dule and N is a D /B -in- jectiv e left D -como dule. Inde ed, Ψ C ( C N ) = Hom C ( C , C N ) ≃ Hom D ( B C , N ) as a left C -contramo dule and C (Ψ D N ) = Cohom D ( B C , Hom D ( D , N ) ), so it remains to apply Prop osition 5.2.2(c). Without a n y assumptions on t he corings C and D , there is a nat ura l isomor phism Φ C ( C Q ) ≃ C (Φ D Q ) for an y quite D /B -projective D - contramo dule Q and a natural isomorphism C (Ψ D N ) ≃ Ψ C ( C N ) for any quite D / B -injectiv e D - como dule N . The natural morphisms Φ C ( C Q ) − → C (Φ D Q ) and C (Ψ D N ) − → Ψ C ( C N ) ha v e t he follo wing compatibilit y prop erty . F or an y left D -como dule N and left D -contramo d- ule Q for whic h the C -como dule C N and t he C - contramo dule C Q are deﬁned via the constructions of 1 .2.4 and 3.2.4, fo r any pair of morphisms Φ D Q − → N and Q − → Ψ D N corresp onding to each o ther under the adjunction of functors Ψ D and Φ D , the comp ositions Φ C ( C Q ) − → C (Φ D Q ) − → C N and C Q − → C (Ψ D N ) − → Ψ C ( C N ) corresp ond to eac h other under the adjunction of functors Ψ C and Φ C . 7.2. P rop erties of the pull-bac k and push-forw ard functors. 7.2.1. Let C − → D b e a map of corings compat ible with a k -algebra map A − → B . Theorem. (a) Assume that C is a ﬂat ri g ht A -mo d ule. Then the functor N 7− → C N maps D /B -c oﬂat ( D /B -c opr oje ctive) left D -c omo d ules to C / A -c oﬂat ( C / A -c o- pr oje ctive) left C -c om o dules. Assume additional ly that D is a ﬂat right B -mo dule. Then the sam e functor a pplie d to c omple x es maps c o acyclic c o m plexes of D /B -c oﬂat D -c omo dules to c o acyclic c omplex e s of C - c omo dules. (b) Assume that C is a pr oje ctive left A -m o dule. Then the functor Q 7− → C Q maps D /B -c oinje ctive left D -c ontr amo dules to C / A -c oinje ctive left C -c ontr amo d ules. Assume additional ly that D is a pr oje ctive left B -mo dule. Then the sa m e functor applie d to c omple x es maps c o ntr aacyclic c o m plexes of D / B -c oin j e ctive D -c ontr amo d- ules to c ontr a acyclic c omplexes of C -c ontr amo dules. Pr o of . P art (a): the ﬁrst assertion follo ws fr om parts (a ) (with the left and r ig h t sides switc hed) and (b) of Prop osition 7.1.3 . T o prov e the second assertion, denote b y K • the cobar resolution C B ⊗ B D − → C B ⊗ B D ⊗ B D − → · · · of the right D -como dule C B . Then K • is a complex of D -coﬂat C - D -bicomo dules a nd the cone of the morphism C B − → K • is coacyclic with resp ect to the exact cat ego ry of B - ﬂat C - D -bicomo d- ules. Th us if N • is a coacyclic complex of left D -como dules, then the complex of left C -como dules K •  D N • is coacyclic and if N • is a complex o f D /B - coﬂat left D -como dules, then the cone of the morphism C B  D N • − → K •  D N • is coacyclic. The pr o of of part (b) is completely analogous.  132 7.2.2. It is obvious that the functor M 7− → B M maps complexes of A -ﬂat C -como dules to complexes of B -ﬂat D -como dules. It will follow from the next The- orem tha t it maps coacyclic complexes of A -ﬂa t C -como dules t o coacyclic complexes of D -como dules. Theorem. (a) Assume that the c oring C is a ﬂat lef t and right A -mo dule and the ring A ha s a ﬁnite w e ak hom olo gic al dimension. T h en any c omplex of A -ﬂat C -c o m o dules that is c o acyclic as a c ompl e x of C -c omo d ules is c o acyclic with r esp e ct to the e xact c ate go ry of A -ﬂat C -c omo d ules . (b) Assume that the c oring C is a pr oje ctive left and a ﬂat right A -mo dule and the ring A has a ﬁnite l e ft hom olo gic al dimension. Then any c omplex of A -p r oje ctive left C -c omo dules that is c o acyclic as a c omp l e x o f C -c omo dules i s c o ac ycli c with r esp e ct to the exact c ate gory of A -pr o j e ctive left C -c omo dules. (c) I n the assumptions of p art (b), any c omplex of A -in je ctive left C -c ontr amo d ules that is c ontr aacyclic as a c omp l e x of C -c o n tr amo d ules is c ontr aacyclic with r esp e ct to the exact c ate gory of A -inje ctive le f t C -c ontr amo dules. Pr o of . The pro o f is not diﬃcult when k is a ﬁeld, as in this case the functors of Lemmas 1 .1.3 and 3.1.3 can b e made a dditiv e and exact. Then it follows that for any coacyclic complex of C -como dules M • the complex L 1 ( M • ) is coacyclic with respect to the exact category o f A -ﬂat C - como dules, while it is clear that f o r any complex of A -ﬂat C -como dules M • the cone of the morphism L 1 ( M • ) − → M • is coacyclic with resp ect to the exact category of A - ﬂa t C -como dules. Besides, parts (b) and (c) can b e deriv ed from the result of Remark 5.5 using the cobar and bar constructions fo r C -como dules and C -contramo dules. Finally , part (a) can b e deduced from part (b) using Lemma 3.1.3(a), but this arg umen t requires stronger assumptions on C and A . Here is a direct pro of of part (a). Let us call a complex of C -como dules m - ﬂat if its terms considered as A -mo dules ha v e w eak homolo g ical dimensions not exceed- ing m , and let us call an m -ﬂat complex of C -como dules m -coacyclic if it is coacyclic with resp ect to the exact categor y of C - como dules whose w eak homological dimen- sion ov er A do es not exceed m . W e will sho w that for an y m -coa cyclic complex of C -como dules M • there ex ists an ( m − 1)-coacyclic complex of C -como dules L • together with a surjectiv e morphism of complexes L • − → M • whose k ernel K • is also ( m − 1)- coacyclic. It will fo llo w that any ( m − 1)-ﬂat m -coacyclic complex of C -como dules M is ( m − 1)-coacyclic, since the total complex of the exact triple K • → L • → M • is ( m − 1 ) - coacyclic, as is the cone of the morphism K • − → L • . By induction w e will deduce that an y 0-ﬂat d - coa cyclic complex of C - como dules is 0-coacyclic, where d denotes the w eak homolog ical dimension of t he ring A ; that is a reform ulation of the a ssertion (a). Let M • b e the tota l complex of an exact triple o f m -ﬂat complexes of C -como dules ′ M • → ′′ M • → ′′′ M • . Let us c ho ose f or each degree n pro jectiv e A -mo dules ′ G n and 133 ′′′ G n endo w ed with surjectiv e A - mo dule maps ′ G n − → ′ M n and ′′′ G n − → ′′′ M n . The latter map can b e lifted to an A - mo dule map ′′′ G n − → ′′ M n , leading to a surjectiv e map from the exact triple of A -mo dules ′ G n → ′ G n ⊕ ′′′ G n → ′′′ G n to the exact triple of C -como dules ′ M n → ′′ M n → ′′′ M n . Applying the construction of Lemma 1.1.3, one can obtain a surjectiv e map from a n exact triple of A -ﬂat C -como dules ′ P n → ′′ P n → ′′′ P n to t he exact t riple of C -como dules ′ M n → ′′ M n → ′′′ M n . Now consider three complexes o f C -como dules ′ L • , ′′ L • , a nd ′′′ L • whose terms ar e ( i ) L n = ( i ) P n − 1 ⊕ ( i ) P n and the diﬀeren tial d n ( i ) L : ( i ) L n − → ( i ) L n +1 maps ( i ) P n in to itself by the identit y map and v anishes in the restriction t o ( i ) P n − 1 and in the pro jection to ( i ) P n +1 . There are natural surjectiv e morphisms o f complexes ( s ) L • − → ( s ) M • constructed as in the pro of of Theorem 5.4. T a k en together, they form a surjectiv e map from the exact triple of complexes ′ L • → ′′ L • → ′′′ L • on to the exact triple of complexes ′ M • → ′′ M • → ′′′ M • . Let ′ K • → ′′ K • → ′′′ K • b e the k ernel of this map of exact t r iples of complexes; then the complexes ( s ) L • are 0-ﬂat , while the complexes ( s ) K • are ( m − 1)- ﬂat. Therefore, the total complex L • of the exact triple ′ L • → ′′ L • → ′′′ L • is 0-coacyclic, while the total complex K • of the exact triple ′ K • → ′′ K • → ′′′ K • is ( m − 1)-coacyclic. There is a surjectiv e morphism of complexes L • − → M • with the k ernel K • . No w let ′ K • → ′ L • → ′ M • and ′′ K • → ′′ L • → ′′ M • b e exact triples o f complexes of C -como dules where the complexes ′ K • , ′ L • , ′′ K • , a nd ′′ L • are ( m − 1)-coacyclic, and supp ose that there is a morphism of complexes ′ M • − → ′′ M • . Let us construct f or the complex M • = cone( ′ M • → ′′ M • ) an exact triple o f complexes K • → L • → M • with ( m − 1) - coacyclic complexes K • and L • . D enote by ′′′ L • the complex ′ L • ⊕ ′′ L • ; there is the em b edding of a direct summand ′ L • − → ′′′ L • and the surjectiv e morphism of complexes ′′′ L • − → ′′ M • whose comp onents are the comp osition ′ L • − → ′ M • − → ′′ M • and the surjectiv e morphism ′′ L • − → ′′ M • . These tw o morphisms form a comm utative square with the morphisms ′ L • − → ′ M • and ′ M • − → ′′ M • . The ke rnel ′′′ K • of the morphism ′′′ L • − → ′′ M • is the middle term of a n exact triple of complexes ′′ K • − → ′′′ K • − → ′ L • . Since the complexes ′′ K • and ′ L • are ( m − 1)-coa cyclic, the complex ′′′ K • is also ( m − 1)-coacyclic. Set L • = cone( ′ L • − → ′′′ L • ) a nd K • = cone( ′ K • − → ′′′ K • ); then there is a n exact tr iple of complexes K • → L • → M • with the desired prop erties. Ob viously , if certain complexes of C -como dules M • α can b e presen ted as quotien t complexes of ( m − 1)- coacyclic complexes by ( m − 1)- coacyclic sub complexes, then their direct sum L M • α can b e also presen ted in this w a y . Finally , let M • − → ′ M • b e a homotop y equiv alence of m -ﬂat complexes of C -como dules, and supp o se that t here is an exact triple ′ K • → ′ L • → ′ M • with ( m − 1)-coa cyclic complexes ′ K • and ′ L • . Let us construct an exact triple of com- plexes K • → L • → M • with ( m − 1) - coacyclic complexes K • and L • . Consider the cone of the morphism M • − → ′ M • ; it is con tractible, and therefore isomorphic to the 134 cone of the iden tit y endomorphism of a complex of C -como dules N • with zero diﬀer- en tial. The complex N • is m - ﬂat, so it can b e presen ted as the quotien t complex of a complex of A -ﬂat C -como dules P • b y its ( m − 1)- ﬂa t sub complex Q • . Hence the com- plex cone( M • → ′ M • ) is isomorphic to the quotien t complex of a 0-ﬂat con tractible complex cone(id P • ) by an ( m − 1)-ﬂat contractible sub complex cone(id Q • ). As we ha v e pro v en, for the co cone ′′ M • of the mo r phism ′ M • − → cone( M • → ′ M • ) there exists an exact triple ′′ K • → ′′ L • → ′′ M • with ( m − 1)-coacyclic complexes ′′ K • and ′′ L • . The complex ′′ M • is isomorphic t o the direct sum of the complex M • and the co cone of the iden tity endomorphism of the complex ′ M • . (Indeed, there is a term- wise split exact triple of complexes cone(id ′ M • )[ − 1] − → ′′ M • − → M • and the complex cone(id ′ M • )[ − 1] is contractible .) The latter co cone can b e presen ted as the quotien t complex of an ( m − 1)-ﬂat contractible complex ′ P • b y an ( m − 1 )-ﬂat con tractible sub complex ′ Q • , e. g., b y taking ′ P • = cone(id ′ L • )[ − 1] and ′ Q • = cone(id ′ K • )[ − 1]. No w supp ose that there are exact triples ′′ K • → ′′ L • → ′′ M • and ′ Q • → ′ P • → ′ N • with ( m − 1 )-coacyclic complexes ′′ K • , ′′ L • , ′ Q • , and ′ P • for certain complexes ′′ M • = M • ⊕ ′ N • and ′ N • . Let us construct an exact triple K • → L • → M • with ( m − 1)-coacyclic complexes K • and L • (in fa ct, w e will hav e K • = ′′ K • and our construction with obvious mo diﬁcations will work for the kernel M • of a surjectiv e morphism of complexes ′′ M • − → ′ N • ). Set ′′′ M • = M • ⊕ ′ P • ; then there is a surjectiv e morphism of complexes ′′′ M • − → ′′ M • with the k ernel ′ Q • . Let ′′′ L • b e the ﬁb ered pro duct of the complexes ′′′ M • and ′′ L • o v er ′′ M • ; then there are exact triples of complexes ′′ K • − → ′′′ L • − → ′′′ M • and ′ Q • − → ′′′ L • − → ′′ L • . It follows from the latter exact triple that the complex ′′′ L • is ( m − 1 )-coacyclic. F urthermore, there is an injectiv e morphism of complexes M • − → ′′′ M • with the cok ernel ′ P • . Let L • b e the ﬁbered pro duct o f t he complexes M • and ′′′ L • o v er ′′′ M • ; then there are exact triples of complexes ′′ K • − → L • − → M • and L • − → ′′′ L • − → ′ P • . It follows from the latter exact triple that the complex L • is ( m − 1)-coacyclic. P art (a) is prov en; the pro ofs of par ts (b) and (c) are completely analogous.  Remark. It follows from part (a) of Theorem that (in the same assumptions) any co- acyclic complex of coﬂa t C -como dules is coacyclic with resp ect t o the exact catego r y of coﬂat C - como dules. Indeed, for an y complex of C -como dules M • coacyclic with resp ect to the exact category of A -ﬂat C -como dules the complex R 2 ( M • ) is coacyclic with resp ect to the exact category of coﬂat C - como dules, and f o r any complex o f coﬂat C -como dules M • the cone of the morphism M • − → R 2 ( M • ) is coacyclic with resp ect to the exact category of coﬂat C -como dules (b y Lemma 1.2.2). Analog ously , if C is a ﬂat righ t A -mo dule then an y coacyclic complex o f C / A -coﬂat left C -como dules is coacyclic with resp ect to the exact category of C / A -coﬂat left C -como dules. F or copro- jectiv e C -como dules, coinjectiv e C -contramo dules, (quite) C / A -injectiv e C - como dules, and (quite) C / A -pro jectiv e C -contramo dules even stronger results are provided b y Re- mark 5.5, Theorem 5.4, and Theorem 5.5. 135 7.3. Derived functors of pull-bac k and push-forw ard. Let C − → D b e a map of corings compatible with a k -algebra map A − → B . Assume that C is a ﬂat right A -mo dule and D is a ﬂat righ t B -mo dule. Then the functor mapping the quotien t category o f the homotop y category of complexes of D /B -coﬂat left D -como dules b y its in tersection with the thick subcategory of coacyclic complexes to the co deriv ed category of left D -como dules is an equiv alence of tria ng ulated categories by Lemma 2.6. Indeed, for any complex of left D -como dules N • there is a morphism from N • in to a complex of D / B -coﬂat D -como dules R 2 ( N • ) with a coacyclic cone, whic h w as constructed in 2 .5. Comp ose the functor N • 7− → C N • acting f r o m the homotopy catego ry of left D -como dules to the homotopy category of left C -como dules with the lo calization f unctor Hot ( C – como d ) − → D co ( C – como d ) and restrict it to the full sub category of complexes of D /B -coﬂat D -como dules. By Theorem 7.2.1(a), this r estriction factorizes throug h the co deriv ed category of left D -como dules. Let us denote the righ t deriv ed functor so obtained by N • 7− → R C N • : D co ( D – como d ) − − → D co ( C – como d ) . According to Lemma 6.5.2, this deﬁnition of a righ t deriv ed functor do es not dep end on the c hoice of a subcatego r y of a djusted complexes. Assume that C is a ﬂat left and right A -mo dule, A has a ﬁnite w eak homolog ical dimension, and D is a ﬂat righ t B - mo dule. Then the functor mapping the quo- tien t category of the homotop y category of complexes of A -ﬂat C -como dules b y its in tersection with the thic k sub category of coacyclic complexes to the co deriv ed cat- egory of C -como dules is an equiv alence of triangulated cat ego ries b y Lemma 2.6. Indeed, for an y complex of C -como dules M • there is a morphism into M • from a complex of A -ﬂat C -como dules L 1 ( M • ) with a coacyclic cone, which was constructed in 2.5. Comp o se the functor M • 7− → B M • acting fr om the homotop y category o f left C -como dules to the ho mo t o p y category of left D -como dules with the lo calization functor Hot ( D – como d ) − → D co ( D – como d ) and restrict it to the full sub category o f complexes of A -ﬂat C -como dules. It follows fro m Theorem 7 .2.2(a) that this restric- tion factorizes thro ugh the co deriv ed categor y of left C - como dules. Let us denote the left derive d functor so obtained by M • 7− → L B M • : D co ( C – como d ) − − → D co ( D – como d ) . According to Lemma 6.5.2, this deﬁnition of a left deriv ed functor do es not dep end on the c hoice of a subcatego r y of a djusted complexes. Analogously , assume that C is a pro jectiv e left A -mo dule and D is a pro jectiv e left B -mo dule. Then the left deriv ed functor Q • 7− → C L Q • : D ctr ( D – contra ) − − → D ctr ( C – contra ) is deﬁned by restricting the functor Q • 7− → C Q • to the full sub category of complexes of D /B -coinjectiv e left D - contramo dules. 136 Assume that C is a pro jectiv e left and a ﬂat right A -mo dule, A has a ﬁnite left homological dimension, and D is a pro jectiv e left B - mo dule. Then the righ t deriv ed functor P • 7− → B R P • : D ctr ( C – contra ) − − → D ctr ( D – contra ) is deﬁned b y restricting t he functor P • 7− → B P • to the f ull sub category of complexes of A -injectiv e left C - contramo dules. Prop erties of the ab ov e-deﬁned deriv ed functors will b e studied (in the greater generalit y of semimo dules and semicon tramo dules) in Section 8. In particular, the functor N • 7− → R C N • is righ t adjoint to the functor M • 7− → L B M • when the latter is deﬁned; the functor Q • 7− → C L Q • is left adjoint to the functor P • 7− → B R P • when the latter is deﬁned; the equiv alences of categories D co ( C – como d ) ≃ D ctr ( C – contra ) and D co ( D – como d ) ≃ D ctr ( D – contra ), when they are deﬁned, thansform the functor N • 7− → R C N • in to the functor Q • 7− → C L Q • ; and there are formulas connecting our deriv ed functors with the deriv ed functors Ctrtor, Coto r and Co ext. 7.4. F aithfully ﬂat/pro jectiv e base ring c hange. 7.4.1. The main ideas of the following are due to Ko n tsevic h and Rosenberg [33]. Let C be a coring o v er a k -algebra A and A − → B b e a k -a lgebra morphism. The coring B C B o v er the k -algebra B is constructed in the follow ing w ay . As a B - B -bimo dule, B C B is equal to B ⊗ A C ⊗ A B . The com ultiplication in B C B is deﬁned as the comp osition B ⊗ A C ⊗ A B − → B ⊗ A C ⊗ A C ⊗ A B − → B ⊗ A C ⊗ A B ⊗ A C ⊗ A B = ( B ⊗ A C ⊗ A B ) ⊗ B ( B ⊗ A C ⊗ A B ) of the map induced by the comultiplication in C and the map induced by the map A − → B . The counit in B C B is deﬁned as the comp osition B ⊗ A C ⊗ A B − → B ⊗ A B − → B of the map induced b y the counit in C and t he map induced b y t he m ultiplication in B . The coring B C B is a univers al initia l ob ject in t he category o f corings D o v er B endo w ed with a map C − → D compat ible with the map A − → B . As alw ays , B is called a faithfully ﬂat righ t A -mo dule if it is a ﬂat righ t A -mo dule a nd for any nonzero left A -mo dule M the tensor pro duct B ⊗ A M is nonzero. Assuming the former condition, the latter one holds if and only if the map M = A ⊗ A M − → B ⊗ A M is inj ective for any left A -mo dule M . Therefore, B is a faithfully ﬂa t right A -mo dule if and only if t he map A − → B is injectiv e and its cok ernel A/B is a ﬂat righ t A -mo dule. Analogously , the ring B is called a faith- fully pro jectiv e left A -mo dule if it is a pro jectiv e generator of the category of left A -mo dules, i. e., it is a pro jec tiv e left A -mo dule and for an y nonzero left A -mo dule P the mo dule Hom A ( B , P ) is nonzero. Assuming the f ormer condition, the latter one holds if and only if the map Hom A ( B , P ) − → Hom A ( A, P ) = P is surjectiv e fo r any left A -mo dule P . Therefore, B is a faithfully pro jectiv e left A - mo dule if and o nly if the map A − → B is injectiv e and its cok ernel A/B is a pro jectiv e left A -mo dule. 137 If the coring C is a ﬂat righ t A - mo dule and the ring B is a faithfully ﬂat righ t A -mo dule, then the functors M 7− → B M and N 7− → C N a r e m utually in v erse equiv- alences b etw een the ab elian catego r ies of left C -como dules and left B C B -como dules. Analogously , if C is a pro jective left A -mo dule and B is a faithfully pro jectiv e left A -mo dule, then the functors P 7− → B P and Q 7− → C Q are m utually inv erse equiv- alences b et w een the ab elian categories of left C -contramo dules and left B C B -con- tramo dules. Both asse rtions follow from the next general Theorem, whic h is t he particular case of Barr – Bec k Theorem [36] for ab elian catego ries and exact functors. Theorem. If ∆ : B − → A is an exact functor b etwe en ab elian c ate gories mapping nonzer o ob j e cts to nonzer o obje cts and Γ : A − → B is a functor left (r esp., right) adjoint to ∆ , then the natur a l functor fr om the c ate gory B to the c ate gory of mo d ules over the monad ∆Γ (r esp ., c omo dules over the c omona d ∆Γ ) over the c ate gory A is an e quivalenc e of ab eli a n c ate gories.  T o pro v e the ﬁrst assertion, it suﬃces to apply Theorem to the functor ∆ : C – como d − → B – mo d mapping a C -como dule M to the B -mo dule B ⊗ A M and the functor Γ : B – mo d − → C – como d righ t adjoin t to ∆ mapping a B -mo dule U to the C -como dule C ⊗ A U . T o pro v e the second assertion, apply Theorem to the functor ∆ : C – contra − → B – mo d mapping a C -contr amo dule P to t he B -mo dule Hom A ( B , P ) and the functor Γ : B – mo d − → C – contra left adjoint to ∆ mapping a B -mo dule V to the C -contramo dule Hom A ( C , V ). 7.4.2. Let C b e a coring ov er a k -algebra A and A − → B b e a k -algebra morphism. Assume that C is a ﬂat left and righ t A -mo dule and B is a fa ithfully ﬂat left and right A -mo dule. Then it follows from Prop osition 7.1.3(a) that for an y righ t C -como dule N and an y left C -como dule M there is a natural map N  C M − → N B  B C B B M , whic h is an isomorphism, a t least, when one of the A -mo dules N and M is ﬂat or one of the B C B -como dules N B and B M is quasicoﬂat. Analogously , assume that C is a pro jectiv e left and a ﬂat rig ht A -mo dule and B is a fa it hf ully pro jectiv e left a nd a fa it hf ully ﬂat right A -mo dule. The n it follows from Prop o sition 7.1 .3(b-c) that for an y left C -como dule M and an y left C -contramo dule P there is a natural map Cohom B C B ( B M , B P ) − → Cohom C ( M , P ), whic h is an isomorphism, at least, when the A -mo dule M is pro jectiv e, the A -mo dule P is injectiv e, the B C B -como dule B M is quasicopro jectiv e, or the B C B -contramo dule B P is quasicoinjectiv e. Remark. In general the map N  C M − → N B  B C B B M is not an isomorphism, ev en under the stro ngest of o ur assumptions on A , B , and C . F o r example, let C = A and B C B = B ⊗ A B ; t hen N  C M = N ⊗ A M , while N B  B C B B M is t he k ernel o f the pair of maps N ⊗ A B ⊗ A M ⇒ N ⊗ A B ⊗ A B ⊗ A M induced by the map A − → B . The sequence 0 − → N ⊗ A M − → N ⊗ A B ⊗ A M − → N ⊗ A B ⊗ A B ⊗ A M is exact if one of tw o A -mo dules M and N is ﬂat or admits a B -mo dule structure, but in 138 general the map N ⊗ A M − → N ⊗ A B ⊗ A M is not injectiv e. Indeed, let k be a ﬁeld, A = k [ x ] b e the algebra of p olynomials in one v ariable, and B = k [ x, ∂ x ] b e the algebra of diﬀeren tial op erators in the aﬃne line. Let M = k = N b e one-dimensional A -mo dules with the trivial action of x . Then the map N ⊗ A M − → N ⊗ A B ⊗ A M is zero, since m ⊗ 1 ⊗ n = m ⊗ ( ∂ x x − x∂ x ) ⊗ n = 0 in N ⊗ A B ⊗ A M . Assume t ha t C is a pro jectiv e left a nd a ﬂat righ t A -mo dule a nd B is a faithfully pro jectiv e left and a faithfully ﬂa t rig h t A -mo dule. The n the equiv alences b et w een the categories C – como d and B C B – como d and b et w een the categories C – contra and B C B – contra tr ansform t he functors Ψ C and Φ C in to the functors Ψ B C B and Φ B C B . Indeed, one ha s Hom B C B ( B C B , B M ) = Hom C ( C B , M ) = Hom A ( B , Hom C ( C , M )) and B C B ⊙ B C B B P = B C ⊙ C P = B ⊗ A ( C ⊙ C P ). Alternativ ely , the same isomorphisms can b e constructed a s in 7 .1.4 using Prop ositions 5.2.1(e) and 5.2.2(e). 7.4.3. Let C b e a coring ov er a k -algebra A and A − → B b e a k -algebra morphism. Ob viously , if C is a ﬂat righ t A -mo dule and B is a faithfully ﬂat right A -mo dule, then a complex of left C -como dules M • is coacyclic if an y o nly if the complex of left B C B -como dules B M • is coacyclic. So the functor M • 7− → B M • induces an equiv alence of the co derive d categories o f left C -como dules and left B C B -como dules. If C is a pro- jectiv e left A -mo dule and B is a faithfully pro jectiv e left A -mo dule, then a complex of left C - contramo dules P • is contraacyclic if a nd only if the complex of B C B -contra- mo dules B P • is con traacyclic. So the functor P • 7− → B P • induces an equiv alence of the con traderiv ed categor ies of left C -contramo dules and left B C B -contramo dules. If C is a ﬂat left a nd right A -mo dule, B is a faithf ully ﬂat left and righ t A -mo dule, and A and B hav e ﬁnite w eak homological dimensions, then the equiv alences of cat- egories D co ( como d – C ) ≃ D co ( como d – B C B ) and D co ( C – como d ) ≃ D co ( B C B – como d ) transform the deriv ed functor Cotor C in to the deriv ed functor Cotor B C B . If C is a pro jec tiv e left and a ﬂat righ t A -mo dule, B is a faithfully pro jectiv e left and a fa ithfully ﬂat right A -mo dule, and A and B ha v e ﬁnite left homolog ical dimen- sions, then the equiv alences o f categories D co ( C – como d ) ≃ D co ( B C B – como d ) and D ctr ( C – contra ) ≃ D ctr ( B C B – contra ) transform the deriv ed functor Co ext C in to the deriv ed functor Co ext B C B . In the same assumptions, the same equiv alences of cate- gories t r a nsform t he m utually inv erse functors R Ψ C and L Φ C in to the m utually inv erse functors R Ψ B C B and L Φ B C B . If C is a ﬂat righ t A -mo dule, B is a f aithfully ﬂa t rig h t A -mo dule, and A and B hav e ﬁnite left homological dimens ions, then the ab ov e equiv alence of categories transfor ms the functor Ext C in to the functor Ext B C B . If C is a pro jectiv e left A -mo dule, B is a faithfully pro jectiv e left A -mo dule, and A and B hav e ﬁnite left homological dimensions, then the ab o v e equiv alences of categories transform the functors Ext C and Ctrtor C in to the functors Ext B C B and Ctrtor B C B . These isomorphisms of f unctors can b e deduced from t he uniqueness/univ ersality assertions o f Lemmas 2.7 and 6.5.2 or deriv ed from the preserv ation/reﬂection results 139 of the next Remark. Besides, t hey ar e particular cases o f the muc h more general isomorphisms constructed in Section 8. Remark. In the strongest of the ab ov e ﬂa tness/pro jectivit y and homological dimen- sion assumptions, almost all the pro p erties of como dules and contramodules o v er corings considered in this b o ok a re preserv ed b y the pa ssages from a coring C to the coring B C B and bac k. This applies to the prop erties o f coﬂatness, copro jectivit y , coinjectivit y , relativ e coﬂatness, relativ e copro jectivity , relativ e coinjectivit y , injectiv- it y , pro jectivit y , contraﬂatness, relative injectivit y , relative pro jectivity , relative con- traﬂatness. All of this follow s from the facts that an A -mo dule M is ﬂat if and only if the B -module B ⊗ A M is ﬂat , a n A - mo dule M is pro j ective if and only if the B - mo dule B ⊗ A M is pro jectiv e, and an A -mo dule P is injectiv e if and only if the B -mo dule Hom A ( B , P ) is injectiv e. Indeed, supp ose t ha t the left B - mo dule B ⊗ A M is ﬂat. Any ﬂat left B -mo dule is a ﬂat left A - mo dule, since the ring B is a ﬂa t left A -mo dule. Consider the tensor pro duct of complex es ( A → B ) ⊗ A · · · ⊗ A ( A → B ) ⊗ A M , where the n um b er of factors A − → B is at least equal to the weak homological di- mension of A . This complex is exact eve rywhere except its righ tmost term, since the map A − → B is injectiv e and B / A is a ﬂat right A - mo dule. Since all terms of this complex, except p ossibly the leftmost one, a r e ﬂat left A -mo dules, the leftmost term A is also a ﬂat left A - mo dule. Alternativ ely , one can consider the complex M − → B ⊗ A M − → B ⊗ A B ⊗ A M − → · · · with the a lternating sums of the maps induced b y the map A − → B as the diﬀerentials; this complex o f left A -mo dules is acyclic, since the induced complex of left B -mo dules is con tractible. Notice that the assumption of ﬁnite weak homological dimension of the ring A is necess ary for this argumen t, since otherwise the ring B can b e a bsolutely ﬂat while the ring A is not (see R emark 8.4.3). Assuming only that C is a ﬂat right A -mo dule and B is a faith- fully ﬂat righ t A -mo dule, the righ t B C B -como dule N B is coﬂat if a right C -como dule N is coﬂat, etc. On the other hand, eve n under the strongest of the ab o v e assump- tions there are more quite C / A -injectiv e C -comodules than quite B C B /B -injectiv e B C B -como dules and there ar e more quite C / A -projectiv e C -contr amo dules than quite B C B /B -projectiv e B C B -contramo dules; i. e., quite relativ e injectivit y and quite rel- ativ e pro jectivity is not preserv ed b y the equiv alences of categories M 7− → B M and P 7− → B P in general. Analogously , t here are more quasicoﬂat C -como dules than qua- sicoﬂat B C B -como dules. Indeed, consider the case when C = A and B C B = B ⊗ A B . Then all C -como dules are coinduced and all C - contramo dules are induced, while a B C B -como dule is quite B C B /B -injectiv e, or a B C B -contramo dule is quite B C B /B -pro- jectiv e, if and only if the corresp onding A -mo dule is a direct summand of an A - mo dule admitting a B -mo dule structure. F or example, if A = k [ x ] and B = k [ x, ∂ x ] as in Remark 7.4.2, then the one-dimensional A -mo dule M with the trivial action of x is not the direct summand of a ny A - mo dule admitting a B -mo dule structure, since the equation xm = 0 w ould imply m = − x∂ x m . A t the same time, an y pro jectiv e left 140 A -mo dule is a direct summand of a pro jectiv e left B -module a nd any injectiv e left A -mo dule is a direct summand of an injectiv e left B -mo dule. It follows, in partic- ular, that the cokerne l o f a n injectiv e morphism of quite C / A -injectiv e C -como dules is not alw a ys quite C / A -injectiv e and the k ernel of a surjectiv e morphism of quite C / A -proj ective C -contramo dules is not alw ay s quite C / A -projectiv e. 7.5. R emarks on Morita morphisms. 7.5.1. A Morita morphism from a k -alg ebra A to a k - algebra B is an A - B -bimo dule E suc h t ha t E is a ﬁnitely generated pro jectiv e right B -mo dule. F or an y Morita morphism E from A to B , set E ∨ = Ho m B op ( E , B ); then E ∨ is a B - A -bimo dule and a ﬁnitely generated pro jectiv e left B -mo dule. T o an y k -algebra morphism A − → B , one can assign a Morita morphism E = B = E ∨ from A to B . Equiv alen tly , a Morita morphism fro m A to B can b e deﬁned as a pair consisting of an A - B -bimo dule E and a B - A - bimo dule E ∨ endo w ed with an A - A -bimo dule morphism A − → E ⊗ B E ∨ and a B - B -bimo dule morphism E ∨ ⊗ A E − → B suc h that the t w o comp ositions E − → E ⊗ B E ∨ ⊗ A E − → E and E ∨ − → E ∨ ⊗ A E ⊗ B E ∨ − → E ∨ are equal to the iden tit y endomorphisms of E and E ∨ . F or an y Morita morphism E fro m A to B the f unctor N 7− → A N = E ⊗ B N = Hom B ( E ∨ , N ) from t he category of left B -mo dules to the category of left A - mo dules has a left adjoin t functor M 7− → B M = E ∨ ⊗ A M and a righ t adjoint functor P 7− → B P = Hom A ( E , P ). Ana lo gously , the functor N 7− → N A = N ⊗ B E ∨ = Ho m B op ( E , N ) from the cat ego ry of right B -mo dules to the category o f right A -mo dules has a left adjoin t functor M 7− → M B = M ⊗ A E and a rig ht adjo in t f unctor P 7− → P B = Hom B op ( E ∨ , P ). Let C b e a coring o v er a k -algebra A and E b e a Morit a morphism from A to B . Then there is a coring structure o n the B - B - bimo dule B C B = E ∨ ⊗ A C ⊗ A E deﬁned in the following wa y [17]. The comultiplication in B C B is the comp osition E ∨ ⊗ A C ⊗ A E − → E ∨ ⊗ A C ⊗ A C ⊗ A E − → E ∨ ⊗ A C ⊗ A E ⊗ B E ∨ ⊗ A C ⊗ A E of the map induced by the comu ltiplication in C and the map induced by the map A − → E ⊗ B E ∨ . The counit in B C B is t he comp osition E ∨ ⊗ A C ⊗ A E − → E ∨ ⊗ A E − → B , where the ﬁrst map is induced by the counit in C . All the results of 7.1 – 7.3 can b e generalized to t he situation of a Morita morphism E from a k - algebra A to a k -algebra B a nd a morphism B C B − → D of coring s o v er B . In particular, for an y left C -como dule M t here is a natural D -como dule structure on the B -mo dule B M = E ∨ ⊗ A M , and analogously for rig h t como dules and left con tramo dules. F or an y righ t C -como dule M ′ and a n y left C -como dule M ′′ there is a natural map M ′  C M ′′ − → M ′ B  D B M ′′ compatible with the map M ′ ⊗ A M ′′ − → M ′ B ⊗ B B M ′′ , etc. All the results o f 7.4 can b e generalized to the case of a Morita morphism E from a k -a lgebra A to a k - algebra B . In particular, E ∨ is a (faithfully) ﬂat righ t A -mo dule if and only if E ⊗ B E ∨ is a (faithfully) ﬂat rig h t A -mo dule, etc. 141 7.5.2. One w ould lik e to deﬁne a Morita morphism from a coring C t o a coring D as a pair consisting of a C - D -bicomo dule E and a D - C -bicomo dule E ∨ endo w ed with maps C − → E  D E ∨ and E ∨  C E − → D satisfying appropria te conditions. This w orks ﬁne fo r coalgebras o v er ﬁelds, but in the coring situation it is not clear how to deal with the problems of nonasso ciativit y of the cotensor pro duct. That is why w e restrict ourselv es to the sp ecial case of coﬂat/copro jectiv e Mor it a morphisms. Notice that, assuming D to b e a ﬂat right B -mo dule, a k -linear functor Λ : C – como d − → D – como d is isomorphic to a f unctor of the form M 7− → K  C M for a certain D - C -bicomo dule K if and only if it pr eserv es cok ernels of the morphisms coinduced fro m morphisms of A -mo dules, k ernels of A - split morphisms, a nd inﬁnite direct sums. Analogously , assuming D to b e a pro j ectiv e left B -mo dule, a k -linear functor Λ : C – contra − → D – contra is isomorphic to a functor of the form P 7− → Cohom C ( K , P ) for a certain C - D -bicomo dule K if a nd only if it preserv es k ernels of the morphisms induced from morphisms of A -mo dules, cok ernels of A -split morphisms, and inﬁnite direct pro ducts. Indeed, let us comp ose our functor Λ with the induction functor A – mo d − → C – contra and with the for g etful functor D – contra − → B – mo d ; then the functor A – mo d − → B – mo d so obta ined has the form U 7− → Hom A ( K , U ) for an A - B - bimo dule K . This follows fro m a theorem of W atts ab out represen ta bilit y of left exact pro duct-preserving co v aria nt functors on the cat ego ry of mo dules ov er a ring, whic h is a part icular case of the abstract adjoint functor existence theorem [36]. The morphism o f functors Hom A ( C , Hom A ( C , U )) − → Hom A ( C , U ) induces a left C -coaction in K , while the functorial D -contramo dule structures on the B -mo dules Hom A ( K , U ) induce a right D - coa ction in K . Since the f unctor Λ sends the exact sequence s Hom A ( C , Hom A ( C , P )) − → Hom A ( C , P ) − → P − → 0 to exact sequences, it is isomorphic to the functor P 7− → Cohom C ( K , P ). Let C b e a coring ov er a k - algebra A and D b e a cor ing o v er a k -a lg ebra B . Assum e that C is a ﬂat right A -mo dule and D is a ﬂat righ t B -module. A right c oﬂat Morita morphism from C to D is a pair consisting of a D - coﬂat C - D -bicomo dule E and a C -coﬂat D - C -bicomo dule E ∨ endo w ed with a C - C -bicomo dule morphism C − → E  D E ∨ and a D - D - bicomo dule morphism E ∨  C E − → D suc h t ha t the tw o comp o sitions E − → E  D E ∨  C E − → E a nd E ∨ − → E ∨  C E  D E ∨ are equal to the identit y endomorphisms of E and E ∨ . A right coﬂa t Morita morphism ( E , E ∨ ) f rom C to D induces an exact functor M 7− → D M = E ∨  C M from the category of left C -como dules to the category of left D - como dules and an exact functor N 7− → C N = E  D N from the category of left D -como dules to the category of left C - como dules; t he former functor is left adjoin t to the latter one. Conv ersely , any pair of adjoint exact k -linear functors preserving inﬁnite direct sums b et w een the cat ego ries of left C -como dules and left D -como dules is induced by a righ t coﬂat Morita morphism. Analogously , assume that C is a pro j ective left A - mo dule a nd D is a pro jectiv e left B -mo dule. A left c opr oje ctive Morita mo rp hism from C to D is deﬁned as a pair 142 consisting of a C -coprojectiv e C - D -bicomo dule E and a D - copro jectiv e D - C -bicomo d- ule E ∨ endo w ed with a C - C -bicomo dule morphism C − → E  D E ∨ and a D - D -bi- como dule morphism E ∨  C E − → D satisfying the same conditions as ab o v e. A left copro jectiv e Morita morphism ( E , E ∨ ) from C to D induces an exact functor P 7− → D P = Cohom C ( E , P ) fro m the category of left C -contramo dules to the category of left D - contramo dules and an exact functor Q 7− → C Q = Cohom D ( E ∨ , Q ) from the category of left D - contramo dules to the category of left C - contramo dules; the former functor is righ t adjoint to the latter o ne. Con v ersely , any pair of adjoint exact k -linear functors preserving inﬁnite pro ducts b et w een the categories o f left C -contramo dules and left D -contramo dules is induced b y a left copro jectiv e Morita morphism. All the results of 7.1 – 7 .3 can b e extended to the situation of a left copro jectiv e and right coﬂat Morita mo r phism from a coring C to a coring D . In pa r ticular, f or an y righ t C - como dule M and an y left D - contramo dule Q the comp ositions ( M  C E ) ⊙ D Q − → ( M  C E ) ⊙ D Cohom C ( E , Cohom D ( E ∨ , Q )) − → M ⊙ C Cohom D ( E ∨ , Q ) and M ⊙ C Cohom D ( E ∨ , Q ) − → ( M  C E  D E ∨ ) ⊙ C Cohom D ( E ∨ , Q ) − → ( M  C E ) ⊙ D Q o f the maps induced by the morphisms E ∨  C E − → D and C − → E  D E ∨ and the natural “ev aluation” maps are m utually inv erse isomorphisms b etw een the k -mo dules M D ⊙ D Q and M ⊙ C C Q . F or an y left D -contr a mo dule Q there are natural isomorphisms of left C -como dules Φ C ( C Q ) = C ⊙ C C Q ≃ C D ⊙ D Q ≃ E ⊙ D Q ≃ E  D ( D ⊙ D Q ) = C (Φ D Q ) b y Prop osition 5.2.1(e), etc. Ho w ever, one sometimes has t o imp ose the homological dimension conditions on A and B where t hey were not previously needed and strengthen the quasicoﬂatness (quasicopro jectivit y , quasicoinjectivit y) conditions to coﬂatness (copro jectivit y , coinjectivit y) conditions. 7.5.3. A right c oﬂat Morita e quivalenc e b etw een corings C and D is a right coﬂat Morita morphism ( E , E ∨ ) f rom C to D suc h that the bicomo dule morphisms C − → E  D E ∨ and E ∨  C E − → D are isomorphisms; it can b e also considered as a right coﬂat Morita morphism ( E ∨ , E ) from D to C . L e f t c oﬂat Morita e quivalenc e s and left c opr o- je ctive Morita e quivalenc es are deﬁned in the analo gous wa y . A right coﬂat Morita equiv alence b et w een corings C and D induces a n equiv alence of the categories of left C -como dules and left D -como dules, and, assuming that C is a ﬂat right A - mo dule and D is a ﬂat righ t B -mo dule, an y equiv alence b etw een these tw o k -linear categories comes from a right coﬂat Morit a equiv alence. Analogously , a left copro j ective Morita equiv alence b et w een corings C and D induces a n equiv alence of the categories of left C -contramo dules and left D -contramo dules, a nd, assuming that C is a pro jectiv e left A -mo dule and D is a pro jectiv e left B -mo dule, an y equiv alence b etw een these t w o k -linear categories comes fro m a left copro jective Morita equiv alence. Let C b e a coring o v er a k -alg ebra A and ( E , E ∨ ) b e a Morita mor phism from A to B . If C is a ﬂat rig h t A -mo dule and E ∨ is a fa it hfully ﬂat rig h t A -mo dule, then the pair of bicomo dules E = C B = C ⊗ A E and E ∨ = B C = E ∨ ⊗ A C is a right coﬂat 143 Morita equiv alence b et w een the corings C a nd B C B . Analogously , if C is a pro jec tiv e left A -mo dule and E is a fa ithfully pro jectiv e left A - mo dule, then the same pair of bicomo dules E = C B and E ∨ = B C is a left copro jective Morita equiv alence b et w een the corings C a nd B C B . This is a reform ulation of the results of 7.4.1 in the case of a Morita morphism of k -algebras. All the results of 7.4.3 can b e generalized to the situatio n of a Morita equiv alence, satisfying appropriate coﬂatness/copro jectivit y conditions, b et w een corings C and D . The same applies to the results o f 7.4.2, with ho mological dimension conditions added when necessary and the quasicoﬂatness (quasicopro jectivit y , quasicoinjectivit y) con- ditions strengthened to coﬂat ness (copro jectivit y , coinjectivit y) conditions. Remark. When the rings A and B are semisimple, one can consider Morita mor - phisms from t he coring C to the coring D without any coﬂatness/copro jectivit y con- ditions imp o sed. Moreov er, for an y Morita morphism ( E , E ∨ ) fr o m C to D the left C -como dule E is copro jectiv e and the righ t C -como dule E ∨ is copro j ective . In par- ticular, a n y Morita equiv alence b et w een C a nd D is left and right copro jectiv e. On the other hand, without suc h conditions on the r ing s A a nd B not ev ery righ t coﬂat Morita equiv alence b et w een C and D is a left coﬂat Morita equiv alence. F o r example, when C is a ﬁnite- dimensional coalgebra ov er a ﬁeld k , B is the algebra ov er k dual to C , a nd D = B , t he righ t coﬂat Mor it a equiv alence b et w een C and D inducing the equiv alence of categories C – como d ≃ B – mo d is not left coﬂat, since this equiv alence of categories do es not preserv e coﬂatness of como dules. 144 8. Functoriality in the Semialgebra 8.1. Compatible morphisms. Let C − → D b e a map of corings compatible with a k -algebra map A − → B . Let S b e a semialgebra o v er the coring C and T b e a semialgebra o v er the coring D . 8.1.1. A map S − → T is called compatible with the maps A − → B and C − → D if the biaction maps A ⊗ k S ⊗ k A − → S a nd B ⊗ k T ⊗ k B − → T form a comm uta tiv e diagram with the maps S − → T and A ⊗ k S ⊗ k A − → B ⊗ k T ⊗ k B (that is the map S − → T is an A - A - bimo dule mo r phism), the bicoaction maps S − → C ⊗ A S ⊗ A C and T − → D ⊗ B T ⊗ B D form a comm utative diagram with the maps S − → T and C ⊗ A S ⊗ A C − → D ⊗ B T ⊗ B D (that it the induced map B ⊗ A S ⊗ A B − → T is a D - D -bicomo dule morphism), and furthermore, the semim ultiplication maps S  C S − → S a nd T  D T − → T and the semiunit maps C − → S and D − → T form comm utativ e diagra ms with the maps C − → D , S − → T , and S  C S − → T  D T . Let S − → T b e a map of semialgebras compatible with a map of corings C − → D and a k -algebra map A − → B . Let M b e a left S -semimo dule and N b e a left T -semimo dule. A map M − → N is called compatible with the maps A − → B , C − → D , and S − → T if it is compatib e with the maps A − → B and C − → D a s a map fro m a C -como dule to a D - como dule and the semiaction maps S  C M − → M and T  D N − → N form a comm utative diagram with the maps M − → N and S  C M − → T  D N . Analogo usly , let P b e a left S -semicontramo dule a nd Q b e a left T -semicontramo dule. A map Q − → P is called compatible with the maps A − → B , C − → D , and S − → T if it is compatib e with the maps A − → B and C − → D as a map fro m a D -contramo dule to a C -contramo dule and the semicon traaction maps P − → Cohom C ( S , P ) and Q − → Cohom D ( T , Q ) form a comm utativ e diagram with the maps Q − → P and Cohom D ( T , Q ) − → Cohom C ( S , P ). Let M ′ − → N ′ b e a map from a right S -semimo dule to a righ t T - semimo dule compatible with the maps A − → B , C − → D , and S − → T , and let M ′′ − → N ′′ b e a map from a left S -semimodule to a left T -semim o dule compatible with the maps A − → B , C − → D , and S − → T . Assume that the triple cotensor pro ducts M ′  C S  C M ′′ and N ′  D T  D N ′′ are asso ciativ e. Then there is a natural map of k -mo dules M ′ ♦ S M ′′ − → N ′ ♦ S N ′′ . Analogously , let M − → N b e a map fr o m a left S -semimo dule t o a left T - semimo dule compatible with the maps A − → B , C − → D , and S − → T , and let Q − → P b e a map from a left T -semic ontramo d- ule to a left S -semicontramo dule compatible with the maps A − → B , C − → D , and S − → T . Assume that the t r iple cohomomorphisms Cohom C ( S  C M , P ) and Cohom D ( T  D N , Q ) are asso ciative. Then there is a natural map of k - mo dules SemiHom T ( N , Q ) − → SemiHom S ( M , P ). 145 8.1.2. Let S − → T b e a map o f semialgebras compatible with a map of cor ing s C − → D and a k - algebra map A − → B . Assume t ha t C is a ﬂat righ t A - mo dule and either S is a coﬂat righ t C - como dule, or S is a ﬂat righ t A -mo dule and a C / A - coﬂat left C -como dule and A has a ﬁnite w eak homological dimension, or A is absolutely ﬂat . Then for any left T - semimo d- ule N there is a natural S -semimo dule structure on the left C -como dule C N . It is constructed as follo ws: the comp osition S  C C N − → T  D N − → N of the map induced b y the maps S − → T and C N − → N with the T -semiaction in N is a map f r o m a C -como dule to a D -como dule compatible with the maps A − → B and C − → D , hence there is a C -como dule map S  C C N − → C N . Analogously , assume that C is a pr o jectiv e left A - mo dule and either S is a copro jectiv e left C -como dule, or S is a pro jectiv e left A -mo dule and a C / A -coﬂat righ t C -como dule a nd A has a ﬁnite left ho mo lo gical dimension, o r A is semisimple. Then for an y left T - semicontramo d- ule Q there is a natural S -semicontramo dule structure on the left C -contramo dule C Q . Indee d, the comp osition Q − → Cohom D ( T , Q ) − → Cohom C ( S , C Q ) is a map from a D -contramo dule to a C -contramo dule compatible with the maps A − → B and C − → D , hence a C -contramo dule map C Q − → Cohom C ( S , C Q ). Assuming that D is a ﬂat righ t B -mo dule, C is a ﬂat right A -mo dule, and S is a coﬂat righ t C -como dule, for a ny D -coﬂat right T - semimo dule N there is a natural S - semimo dule structure on t he coﬂat right C - como dule N C and f or an y D -coinjectiv e left T -semi- contramo dule Q there is a natural S -semicontramo dule structure on the coinjectiv e left C - contramo dule C Q pro vided that B is a ﬂat righ t A -mo dule. Assume that C is a ﬂat right A -mo dule, S is a coﬂat righ t C - como dule, D is a ﬂat righ t B - mo dule, and T is a coﬂat right D -como dule. Then the functor N 7− → C N from the category of left T -semimo dules to the category of left S -semimo dules has a left adjoint functor M 7− → T M , whic h is constructed as f o llo ws. F or induced left S -semimodules, o ne has T ( S  C L ) = T  D B L ; to compute the T -semimodule T M for an arbitra r y left S - semimo dule M , o ne can represen t M as the cokernel of a morphism of induced S -semimo dules. Both k -mo dules Hom S ( M , C N ) and Hom T ( T M , N ) are isomorphic to the k -mo dule of all maps of semimodules M − → N compatible with the maps A − → B , C − → D , and S − → T . There are also a few situatio ns when the functor M 7− → T M is deﬁned on the full sub category of induced S -semimo dules. Under a nalogous assumptions, t he functor M 7− → M T left adjoin t to the f unctor N 7− → N C acts from the category of right S - semimo dules to the category of right T -semimo dules. No w a ssume that C is a ﬂat left and right A -mo dule, S is a ﬂat left A - mo dule and a coﬂat righ t C -como dule, A has a ﬁnite w eak homolog ical dimension, D is a ﬂat righ t B - mo dule, and T is a coﬂat right D -como dule. Then the functor N 7− → C N can be constructed in a diﬀerent w ay: when M is a ﬂat left A -mo dule, one has 146 T M = T C ♦ S M , where T C = T  D B C is a T - S -bisemimo dule with the right S -semi- mo dule structure pro vided by the ab o v e construction. T o compute the T -semimo dule T M for a n arbitrary left S -semimo dule M , o ne can r epresen t M as the cokerne l of a morphism of A -ﬂat S - semimo dules. Assuming only t hat C is a ﬂat right A -mo dule, S is a coﬂat right C -como dule, D is a ﬂat righ t B -mo dule, and T is a coﬂat right D -como dule, the functor M 7− → T M can b e deﬁned b y the for m ula T M = T C ♦ S M for an y M whenev er B is a ﬂat right A -mo dule. If C is a ﬂat left and right A - mo dule, S is a coﬂat left and right C -como dule, D is a ﬂat righ t B -mo dule, and T is a coﬂat righ t D - como dule, the f unctor M 7− → T M is given by the formula T M = T C ♦ S M on the full sub category of C -coﬂat S -semimo dules M . F urthermore, assume that C is a pr o jectiv e left A -mo dule, S is a copro jective left C -como dule, D is a pro jec tiv e left B -mo dule, and T is a copro jectiv e left D -como dule. Then the functor Q 7− → C Q from the category of left T -semicontramodules to the category of left S -semicontramo dules has a righ t adjo in t functor P 7− → T P , whic h is constructed as follo ws. F or coinduced left S -semicontramo dules, one has T Cohom C ( S , R ) = Cohom D ( T , B R ); to compute the T -semicontramo dule T P for an a rbitrary left S -semicontramo dule P , one can represen t P as the k ernel of a morphism o f coinduced S -semicontramo dules. Both k -mo dules Hom S ( C Q , P ) a nd Hom T ( Q , T P ) are isomorphic to t he k -mo dule of all maps o f semicon tramo dules Q − → P compatible with the maps A − → B , C − → D , a nd S − → T . There are also a few situations when the functor P 7− → T P is deﬁned on the full sub category of coinduced S -semicontramo dules. No w assume that C is a pro jectiv e left and a ﬂat righ t A -mo dule, S is a copro jectiv e left C -como dule and a ﬂat righ t A - mo dule, A has a ﬁnite left homolo g ical dimension, D is a pro jec tiv e left B -mo dule, and T is a copro j ective left D -como dule. Then the functor P 7− → T P can b e constructed in a diﬀeren t wa y: w hen P is an injectiv e left A -mo dule, T P = SemiHom S ( C T , P ); to compute the T -sem icontramo dule T P for an arbitrary left S - semicontramo dule P , one can represen t P as the ke rnel of a morphism of A -injectiv e S -semicontramo dules. Assuming only that C is a pro jectiv e left A -mo dule, S is a copro jectiv e left C -como dule, D is a pro jec tiv e left B - mo dule, and T is a copro jective left D -como dule, the functor P 7− → T P can b e deﬁned by the form ula T P = SemiHom S ( C T , P ) for any P whenev er B is a pro jectiv e left A -mo dule. If C is a pro j ective left and a ﬂat right A -mo dule, S is a copro jectiv e left and a coﬂat righ t C - como dule, D is a pro jective left B -mo dule, and T is a copro jectiv e left D -como dule, the functor P 7− → T P is giv en by the fo r m ula T P = SemiHom S ( C T , P ) on the full sub category of C -coinjectiv e S -semicontramo dules P . Assume that C is a pro jectiv e left A -mo dule, S is a copro jec tiv e left C -como dule, D is a pro jectiv e left B -mo dule, and T is a copro jectiv e left D - como dule. Then for an y right S -semimo dule M and any left T - semicontramo dule Q there is a natural isomorphism M T ⊚ T Q ≃ M ⊚ S C Q . Moreov er, b oth k -mo dules a re isomorphic to 147 the cok ernel of the pair o f maps ( M  C S ) B ⊙ D Q ⇒ M B ⊙ D Q one of which is induced b y t he S - semiaction in M and the other is deﬁned in terms of the morphism ( M  C S ) B − → M B  D T , the T -sem icon traaction in Q , and the natural “ev aluation” map ( M B  D T ) ⊙ D Cohom D ( T , Q ) − → M B ⊙ D Q . This is clear for M ⊚ S C Q , and to construct this isomorphism for M T ⊚ T Q it suﬃces to represen t M as the cokerne l of the pair of morphisms o f induced S -semimo dules M  C S  C S ⇒ M  C S . In the ab ov e situations when M T = M ♦ S C T , this isomorphism can b e also constructed by represen ting M T as the cok ernel o f the pair of T -semimodule morphisms M  C S  C C T ⇒ M  C C T and using the isomorphisms M  C C T ≃ M B  D T . 8.1.3. Let S − → T b e a map o f semialgebras compatible with a map of cor ing s C − → D and a k - algebra map A − → B . Prop osition. (a) L et M b e a left S -s e mimo dule and N b e a right T -sem i m o dule. Then the semitensor pr o duct T M = T C ♦ S M c an b e endowe d with a left T -semi- mo dule s tructur e via the c onstruction of 1.4.4 and the map of semitensor pr o d ucts N C ♦ S M − → N ♦ T T M induc e d by the ma ps of se m imo dules N C − → N and M − → T M is an isomorphism, at l e ast, in the fol lowing c ases: • D is a ﬂat right B -mo dule, T is a c o ﬂat right D -c omo dule, N is a c oﬂat right D -c omo dule, C is a ﬂat left A -mo dule, S is a ﬂat left A -m o dule and a C / A -c oﬂat right C -c o m o dule, the ring A has a ﬁnite we ak homolo gic al dimen - sion, and M is a ﬂat left A -mo dule, or • D is a ﬂat righ t B - m o dule, T is a c oﬂat right D -c omo dule, N is a c oﬂat right D -c omo dule, C is a ﬂat le ft A -mo dule, S is a c oﬂat left C -c omo dule, and M is a c oﬂat left C -c omo dule, or • D is a ﬂat righ t B - m o dule, T is a c oﬂat right D -c omo dule, N is a c oﬂat right D -c omo dule, C is a ﬂat right A -mo dule, S is a c oﬂat right C -c omo dule, and B is a ﬂat right A -mo dule, or • D is a ﬂat left B -mo dule, T is a ﬂat left B - mo dule a nd a D /B -c oﬂat rig h t D -c omo dule, the rin g B has a ﬁnite we ak homolo g i c al dimension, C is a ﬂat left A -mo d ule, S is a c oﬂat left C -c omo dule, an d M is a semiﬂat lef t S -semi- mo dule, or • D is a ﬂat lef t B -mo dule, T is a c oﬂat le ft D -c om o dule, B C is a c oﬂat le ft D -c omo dule, C is a ﬂat le ft A -mo dule, S is a c oﬂat left C -c omo dule, and M is a s e miﬂat left S -semim o dule. When the ring A ( r esp., B ) is absolutely ﬂat, the C / A -c oﬂatness (r esp . , D /B -c oﬂat- ness) assumption c an b e dr o pp e d. (b) L et P b e a left S -se m ic ontr amo dule and N b e a left T -s e mimo dule. Then the mo dule of semihomo m orphisms T P = SemiHom S ( C T , P ) c an b e end o we d with a left T -semic ontr amo dule structur e via the c onstruction of 3.4.4 and the map of the semihomomorp hism mo dules SemiHom T ( N , T P ) − → SemiHom S ( C N , P ) induc e d by 148 the maps o f sem imo dules and semic ontr amo dules C N − → N and T P − → P is an isomorphism, at le ast, in the fol lowing c ases: • D is a pr oje ctive left B -mo dule, T is a c opr oje ctive left D -c omo dule, N is a c opr oje ctive lef t D -c om o dule, C is a ﬂat right A -mo dule, S is a ﬂat right A -mo dule and a C / A -c op r oje ctive left C -c omo dule, the ring A has a ﬁnite left homolo g ic al d imension, and P is an inje ctive left A -mo dule, or • D is a pr oje ctive left B -mo dule, T is a c opr oje ctive left D - c omo d ule, N is a c opr oje ctive left D -c omo dule, C is a ﬂat right A -mo dule, S is a c o ﬂat right C -c omo dule, a n d P is a c o i n je ctive left C -c omo dule, or • D is a pr oje ctive left B - mo dule, T i s a c op r oje ctive left D -c omo dule, N is a c o- pr oje ctive le f t D -c o m o dule, C is a pr oje ctive left A -mo d ule, S is a c opr oje ctive left C -c o mo dule, and B is a pr oje ctive left A -mo dule, or • D is a ﬂat right B -mo dule, T is a ﬂat right B -mo dule and a D /B -c op r oje ctive left D -c omo dule, the ring B has a ﬁn ite left homolo gic al dimension, C is a ﬂat right A -mo dule, S is a c oﬂat right C -c omo dule, and P is a semiinj e ctive left S -semic ontr amo d ule, or • D is a ﬂat right B -mo dule, T is a c oﬂat right D -c omo dule, C B is a c oﬂat right D -c omo dule, C is a ﬂat right A -mo dule, S is a c oﬂat right C -c omo dule, and P is a semiinje ctive left S -sem ic ontr a mo dule. When the ring A (r esp., B ) is semisimp l e , the C / A -c opr oje ctivity (r esp . , D /B -c o pr o- je ctivity) assumption c an b e d r opp e d. (c) L et M b e a left S -semi m o dule and Q b e a left T -se mic ontr amo dule. Then the map of semiho m omorphism mo dules SemiHom T ( T M , Q ) − → SemiHom S ( M , C Q ) induc e d by the m ap of s emimo d ules M − → T M and the ma p of semic ontr amo dules Q − → C Q is an is o morphism, at le ast, in the fol lowing c ases: • D is a ﬂat right B -mo dule, T is a c oﬂat right D -c omo dule, Q is a c oinje c- tive left D -c ontr amo d ule, C is a pr oje ctive left A -mo dule, S is a pr oje ctive left A -mo dule and a C / A -c oﬂat right C -c omo d ule, the rin g A has a ﬁ n ite left homolo g ic al d imension, and M is a pr oje ctive left A -mo dule, or • D is a ﬂat right B -mo dule, T is a c oﬂat right D -c omo dule, Q is a c oinje ctive left D -c ontr amo d ule, C is a pr oj e ctive left A -mo dule, S is a c opr o j e ctive left C -c omo dule, a n d M is a c opr oje ctive left C -c omo dule, or • D is a ﬂat right B -mo dule, T is a c oﬂat right D -c omo dule, Q is a c oinje ctive left D -c ontr a mo dule, C is a ﬂat righ t A -m o dule, S is a c oﬂat right C -c om o dule, and B is a ﬂat right A -mo dule, o r • D is a pr oje ctive left B - m o dule, T is a pr oje ctive left B -mo dule and a D /B -c oﬂat right D -c omo dule, the ring B has a ﬁnite le ft homo lo gic al d imen- sion, C is a pr o j e ctive left A -mo dule, S is a c opr oje ctive lef t C -c omo dule, and M is a semipr oje ctive left S -semimo dule, or 149 • D i s a pr oje ctive left B -mo dule, T is a c opr oje ctive left D -c omo dule, B C is a c opr oje ctive left D -c omo dule, C is a pr oje c tive left A -mo dule, S is a c opr oje c- tive left C -c o m o dule, and M is a semipr oje ctive left S -semim o dule. When the ring A (r esp., B ) i s sem isimple, the C / A -c oﬂatness (r esp., D /B -c oﬂatness) assumption c an b e dr opp e d . Pr o of . P art (a) : under our assumptions, there is a natural isomorphism of rig h t S -semimodules N C ≃ N ♦ T T C . F or an y left S -semimo dule M and righ t T -sem imo d- ule N for whic h the iterated semitensor pro ducts ( N ♦ T T C ) ♦ S M and N ♦ T ( T C ♦ S M ) are deﬁned and t he triple cotensor pro duct N  D T C  C M is asso ciative , the map ( N ♦ T T C ) ♦ S M − → N ♦ T ( T C ♦ S M ) induced by the bisemimo dule maps S − → T C − → T compatible with the maps A − → B , C − → D , and S − → T f orms a comm utativ e diagram with the maps N  D T C  C M − → ( N ♦ T T C ) ♦ S M and N  D T C  C M − → N ♦ T ( T C ♦ S M ). Indeed, the map N  D T C  C M − → N  D ( T C ♦ S M ) is equal t o the comp osition of the map N  D T C  C M − → N  D T  D ( T C ♦ S M ) induced b y the maps T C − → T and M − → T C ♦ S M and the map N  D T  D ( T C ♦ S M ) − → N  D ( T C ♦ S M ) induced by the left T -semiaction in T C ♦ S M . T o c hec k this, one can notice that the diagram in question is obtained by ta king the cotensor pro duct with N of the diagram of maps T C  C M − → T  D ( T C ♦ S M ) − → T C ♦ S M and comp ose the lat ter diagram with the surjectiv e map T C  C S  C M − → T C  C M induced by the left S -semiaction in M . On the o ther hand, the comp osition of maps N  D T C  C M − → ( N ♦ T T C ) ♦ S M − → N  D ( T C ♦ S M ) is equal to the comp osition o f the same map N  D T C  C M − → N  D T  D ( T C ♦ S M ) and the ma p N  D T  D ( T C ♦ S M ) − → N  D ( T C ♦ S M ) induced b y t he r ig h t T -semiaction in N , since b oth comp ositions ar e equal t o the comp osition of the map N  D T C  C M − → N  C M induced b y the comp osition N  D T C − → N ♦ T T C − → N with the map N  C M − → N  D ( T C ♦ S M ) induced by the map M − → T C ♦ S M . It remains to apply Prop osition 1.4 .4. The pro ofs of parts (b) a nd (c) a re completely analogo us.  8.1.4. Let S − → T b e a map o f semialgebras compatible with a map of cor ing s C − → D and a k - algebra map A − → B . Assume that C is a pro jectiv e left a nd a ﬂat right A -mo dule, S is a copro jectiv e left and a coﬂat righ t C - como dule, D is a pro jec tiv e left and a ﬂat rig ht B -module, and T is a copro j ective left and a coﬂat rig ht C -como dule. Then fo r a n y left T - semi- contramo dule Q the na tural map of C - como dules Φ C ( C Q ) − → C (Φ D Q ) is an S -semi- mo dule morphism Φ S ( C Q ) − → C (Φ T Q ). Indeed, Φ S ( C Q ) = S ⊚ S C Q ≃ S T ⊚ T Q ≃ C T ⊚ T Q as a left S -semimo dule a nd C (Φ T Q ) = C ( T ⊚ T Q ), so there is an S -semi- mo dule morphism Φ S ( C Q ) − → C (Φ T Q ); it coincides with the C -como dule morphism Φ C ( C Q ) − → C (Φ D Q ) deﬁned in 7.1.4. Analogously , for an y left T -semimodule N the natural map of C -contra mo dules C (Ψ D N ) − → Ψ C ( C N ) is an S -semicontramo dule 150 morphism C (Ψ T N ) − → Ψ S ( C N ). Indeed, Ψ S ( C N ) = Hom S ( S , C N ) ≃ Hom T ( S T , N ) ≃ Hom T ( T C , N ) as a left S - semicontramo dule and C (Ψ T N ) = C Hom T ( T , N ). Assume that C is a pro jectiv e left A -mo dule, S is a copro jec tiv e left C -como dule, D is a pro jectiv e left B -mo dule, T is a copro jectiv e left D -como dule, and B is a pro jectiv e left A - mo dule. Then t he equiv alence of categories of C - coprojectiv e left S -semimodules and C -projectiv e left S -semicontramo dules and the equiv alence of cat- egories of D -coprojectiv e left T -semimo dules and D -projectiv e left S -semicontramo d- ules transform the functor N 7− → C N in to the functor Q 7− → C Q . Indeed, the a b ov e argumen t sho ws that for any D -pro jectiv e left T -semicontramo dule Q the isomor- phism Φ C ( C Q ) ≃ C (Φ D Q ) preserv es the S - semimo dule structures. Assume that C is a ﬂat righ t A -mo dule, S is a coﬂat righ t C -como dule, D is a ﬂat righ t B - mo dule, T is a coﬂat right D - como dule, and B is a ﬂat righ t A - mo dule. Then the equiv alence of categories of C -injectiv e left S -semimo dules a nd C -coinjec- tiv e left S -semicontramo dules and the equiv alence of categor ies of D - inj ective left T -semimo dules a nd D -coinjectiv e left S -semicontramo dules transform the functor N 7− → C N in to the f unctor Q 7− → C Q . Indeed, the ab ov e ar gumen t sho ws that for an y D - injectiv e left T -sem imo dule N the isomorphism C (Ψ D N ) ≃ Ψ C ( C N ) preserv es the S -semicontramo dule structures. Assume that C is a pro jectiv e left a nd a ﬂat right A -mo dule, S is a copro jectiv e left C -como dule and a ﬂat righ t A -mo dule, D is a pro jectiv e left and a ﬂat right B -mo dule, T is a copro jectiv e left D -como dule and a ﬂat righ t B -module, and the rings A and B hav e ﬁnite left homo lo gical dimensions. Then the equiv alence of categories of C / A - injectiv e left S -semimo dules and C / A -projective left S -semicontra- mo dules and the equiv alence of categories of D /B -injectiv e left T -semimodules and D /B -proj ective left T - semicontramo dules transform the functor N 7− → C N in to the functor Q 7− → C Q . Indeed, the ab ov e argumen t sho ws that f or any D /B -projec- tiv e left T -semicontramo dule Q the isomorphism Φ C ( C Q ) ≃ C (Φ D Q ) preserv es the S -semimodule structures. The analogous result holds when S is a pro jectiv e left A -mo dule and a coﬂat rig h t C -como dule and T is a pro jectiv e left B -mo dule and a coﬂat right D -como dule; it can b e prov en b y applying the a b ov e ar g umen t to the isomorphism C (Ψ D N ) ≃ Ψ C ( C N ) for a D /B - injectiv e left T -semimodule N . Finally , assume t ha t the rings A and B are semisimple. Then the equiv alence o f categories of C - injectiv e left S -semimo dules a nd C -pro jectiv e left S -semicontramo d- ules and the equiv alence of categories o f D -injective left T -semimo dules a nd D -pro- jectiv e left T -semicontramo dules transform the functor N 7− → C N in to the functor Q 7− → C Q . One can sho w this using the semialgebra analogues of the assertions of 7.1.2 related to quasicoﬂat como dules and quasicoinjectiv e con tramo dules. 151 8.2. Complexes, adjusted to pull-bac ks and push-forw ards. Let S − → T b e a map of semialgebras compatible with a map of corings C − → D and a k -alg ebra map A − → B . The fo llo wing result generalizes Theorem 6 .3. Theorem 1. (a) Assume that D is a ﬂat right B -m o dule, T is a c oﬂat right D -c omo dule and a D /B -c oﬂat ( D /B -c opr oje ctive) left D -c omo dule, and the ring B has a ﬁnite we ak (left) hom olo gic al dimension. Then the functor m apping the q uo- tient c ate gory of the homotopy c ate gory of c omplexes of D /B -c oﬂat ( D /B -c opr oje c- tive) left T -semimo dules by its interse ction w ith the thick sub c a te gory of D -c o acyclic c omplex e s in to the semiderive d c ate gory o f left T -semimo dules is an e quivalen c e of triangulate d c ate g o ri e s. (b) Assume that D is a pr oje ctive left B -mo dule, T is a c opr oje ctive left and a D /B -c oﬂat righ t D -c om o dule, and the ring B has a ﬁnite left homolo gic al dimen - sion. T h en the functor mapping the quotient c ate gory of the hom o topy c ate gory of c omplex e s o f D /B -c oinje ctive left T -semi c ontr amo dules by its interse ction with the thick sub c ate gory of D -c ontr aacyclic c omplexes into the semiderive d c ate gory of left T -semi c ontr amo dules is an e quivale nc e of triangulate d c ate gories. Pr o of . T o prov e part (a) f o r D /B -coﬂat T -semimodules, use Lemma 1.3.3, the con- struction of the morphism of complex es L • − → R 2 ( L • ) from the pro of of Theo- rem 2.6, and Lemma 2.6. T o pr ov e pa r t (a) f o r D /B -coprojectiv e T -semimodules, use L emma 3.3.3 ( b). T o prov e part (b), use Lemma 3.3.3(a) and the construction of the morphism of complexes L 2 ( R • ) − → R • from the pro of of Theorem 4.6.  A complex of S - semimo dules is called quite S / C / A -semiﬂat ( quite S / C / A -semi- pr oje ctive ) if it b elongs to t he minimal tria ngulated sub catego r y of the homotopy category of complexes of S -semimo dules con taining the complexes induced fro m com- plexes of A -ﬂat ( A -projective ) C -como dules and closed under inﬁnite direct sums. Analogously , a complex of S -semicontramo dules is called quite S / C / A -sem iinje ctive if it b elongs to the minimal triangulat ed subcategory of the homotop y category o f complexes of S -semicontramo dules con taining the complexes coinduced from com- plexes of A -injectiv e C -contramo dules and closed under inﬁnite pro ducts. Under appropriate a ssumptions on S , C , a nd A , any quite S / C / A -semiﬂat complex of A -ﬂat S -semimodules is S / C / A - semiﬂat in the sense of 2.8, and ana logously fo r birelativ e semipro jectivit y and semiinjectivit y in the sense of 4.8. An y quite S / C / A -semiﬂat complex of right S -semimo dules is S / C / A -contraﬂat, any quite S / C / A - semiprojectiv e complex of left S -semimo dules is S / C / A -projectiv e, and any quite S / C / A -semiinjec- tiv e complex of left S - semicontramo dules is S / C / A -injectiv e in the sense of 6.4. Theorem 2. (a) Assume that C is a ﬂat ( pr oje ctive) le f t and a ﬂat right A -mo dule, C is a ﬂat (pr oje ctive) left A -mo dule and a c oﬂat right C -c omo dule, and the ring A has a ﬁnite we a k (left) homo lo gic al dimensi o n. Then the functor mapp ing the quotient c ate- gory o f the homo topy c ate gory of quite S / C / A -semiﬂat (quite S / C / A -sem ipr oje ctive) 152 c omplex e s of left S -semimo dules by its minimal triangulate d sub c ate gory c ontaining c omplex e s induc e d fr om c o acyclic c omplexes of A -ﬂat ( A -p r oje ctive) S -semimo dules and close d under inﬁnite dir e ct sums into the semide riv e d c ate gory of left S -semi- mo dules is an e quivalenc e of triangulate d c ate gorie s . (b) Assume that C is a pr oje ctive left a nd a ﬂat rig h t A -mo d ule, C is a c opr oje ctive left C -c om o dule an d a ﬂat right A -mo dule, an d the ring A has a ﬁnite left homolo gic al dimension. Th e n the functor mapping the quotient c ate gory of the hom o topy c ate- gory o f quite S / C / A -semii n je ctive c omplexes o f l e ft S -se m ic ontr amo dules by its min- imal triangulate d sub c ate gory c ontaining c omplexes of c oinduc e d fr o m c ontr a a cyclic c omplex e s o f A -inje ctive C -c ontr amo dules and close d under inﬁnite pr o ducts into the semiderive d c ate gory of left S -semic on tr amo d ules is an e quivalenc e of triangulate d c ate go ries. Pr o of . Pro of of part (a): for a n y complex of S -semimo dules K • there is a natural morphism in to K • from a quite S / C / A -semiﬂat complex of S -semimo dules L 3 L 1 ( K • ) with a C -coacyclic cone. Hence it fo llo ws from Lemma 2.6 that the semideriv ed cat- egory of S - semimo dules is equiv alen t to the quotien t categor y of the homotopy cate- gory of quite S / C / A -semiﬂat complexes of S -semimo dules b y its inters ection with the thic k sub category of C -coa cyclic complexes. It remains to show that an y C -coacyclic quite S / C / A -semiﬂat complex of S -semimo dules b elongs to the minimal triangulated sub category con t aining the complexes induced from coacyclic complexes of A -ﬂat S -semimodules a nd closed under inﬁnite direct sums. Indeed, if a complex of A - ﬂat left S -semimo dules M • is C -coacyclic, then the total complex L 3 ( M • ) of the bar bi- complex · · · − → S  C S  C M • − → S  C M • up to the homoto p y equiv alence can b e ob- tained fr o m complexes of S -semimo dules induced from coacyclic complexes of A -ﬂa t C -como dules using the op erations of cone and inﬁnite direct sum. So the same applies to a C -coacyclic complex of S -semimodules M • homotop y equiv alen t to a complex of A -ﬂat S -semimo dules. On the other hand, if a complex of S -semimo dules M • is induced from a complex of C -como dules, t hen the cone of the morphism of complexes L 3 ( M • ) − → M • is a con tractible complex of S - semimo dules, since it is isomorphic to the cotensor pro duct ov er C of the bar complex · · · − → S  C S  C S − → S  C S − → S , whic h is contractible as a complex of left S -semimo dules with right C -como dule struc- tures, and a certain complex of left C -como dules. So the same applies to an y complex of S -semimo dules M • that up to the homotop y equiv alence can b e obtained from complexes of S -semimo dules induced fro m complexes of C -como dules using the op er- ations of cone and inﬁnite direct sum. P art (a ) f o r quite S / C / A -semiﬂat complexes is pro v en; the pro ofs o f part (a) for quite S / C / A -semiprojectiv e complexes and part (b) are completely analogous.  Theorem 3. ( a ) Assume that C is a ﬂat right A -mo dule, S is a c oﬂat right C -c omo dule, D is a ﬂat rig ht B -mo dule, and T is a c oﬂat right D -c omo dule. Then 153 the functor M • 7− → T M • maps quite S / C / A -semiﬂat (quite S / C / A -semi p r oje ctive) c omplex e s o f left S -semimo dules to quite T / D /B -se miﬂat (quite T / D /B -semipr oje c- tive) c omplexes of left T -semimo dules . Assume additional ly that C and S ar e ﬂat left A -mo dules and the ring A has a ﬁnite we ak homolo gic al dimension. Then the same functor map s C - c o acyclic quite S / C / A -s emiﬂat c omplexes of left S -semimo dules to D -c o acyclic c omplexes of left T -semim o dules. (b) Assume that C is a pr oje ctive left A -mo dule, S is a c opr oje ctive left C -c o m o dule, D is a pr oje ctive left B -mo dule, and T is a c op r oje ctive left D -c omo dule. Then the func tor P • 7− → T P • maps quite S / C / A -sem i inje ctive c omplexes of left S -semi- c ontr a m o dules to quite T / D /B -semiinje ctive c omplexes o f left T -se m ic ontr amo dules. Assume a d ditional ly that C and S ar e ﬂat right A -mo dules and the ring A has a ﬁnite left hom olo gic al dimension. Then the same functor maps C -c ontr aacyclic quite S / C / A -semiinje ctive c ompl e x es of left S -semic ontr amo dules to D -c ontr aacyclic c om - plexes of lef t T -semic ontr amo dules. (c) Assume that C is a pr oje ctive left and a ﬂat right A -mo dule, S is a c opr oje ctive left and a c oﬂat right C -c omo dule, A has a ﬁnite left homol o gic al dimens i o n, D is a pr oje ctive left and a ﬂat right B -mo d ule, T is a c opr oje ctive left and a c o ﬂat right D -c omo dule, and B has a ﬁnite left homolo gi c al dimension . Th e n the functor M • 7− → T M • maps S / C / A -pr oje ctive c omplexes of left S -sem imo dules to T / D /B -pr o j e ctive c omplex e s left T -semimo dules and the functor P • 7− → T P • maps S / C / A -inje ctive c omplex e s of left S -semic ontr amo dules to T / D /B -inje c tive c o mplexes of left T -semi- c ontr a m o dules. The same functors m ap C -c o acyclic S / C / A -pr oje ctive c omplexes of left S -semimo dules to D -c o acyclic c omplexes of left T -semimo dules and C -c ontr a- acyclic S / C / A -inje ctive c omple xes of left S -semi c ontr amo dules to D -c ontr aacyclic c omplex e s of left T -semic ontr a m o dules. Pr o of . P art (a): the functor M 7− → T M maps the S -semimo dule induced from a C -como dule L to the T -semimodule induced from the D -como dule B L . The ﬁrst assertion f ollo ws immediately; to prov e the second one, use Theorem 7.2.2(a) a nd Theorem 2(a ). The pro of of pa r t (b) is completely analogous. P art (c): t he ﬁrst assertion follows from the adjoin tness of functors M • 7− → T M • and N • 7− → C N • , the adjointnes s of functors P • 7− → T P • and Q • 7− → C Q • , a nd the second a ssertions of Theorem 7.2.1(a ) and (b). The second assertion follows from the ﬁrst assertions of Theorem 7 .2.1(a-b), b ecause a complex of left S -semimo dules M • is S / C / A -pro- jectiv e and C -coacyclic if and only if the complex Hom S ( M • , L • ) is acyclic for all complexes of C / A -injectiv e left S -semimo dules L • , and a complex of left S -semicon- tramo dules P • is S / C / A -injectiv e and C -contraacyclic if and only if the complex Hom S ( R • , P • ) is acyclic for all complexes of C / A -pro j ectiv e left S -semicontramo d- ules R • (and analogously for complexes of T - semimo dules and T -semicontramo dules). 154 This follo ws from Theorem 6.3 and the results of 6.5, since a complex of S -semimo d- ules is C -coacyclic iﬀ it represen ts a zero ob ject of the semideriv ed category of S - semi- mo dules, a nd a complex of S -semicontramo dules is C -contra a cyclic iﬀ it represen ts a zero ob ject of the semideriv ed category of S -semicontramo dules.  8.3. Derived functors of pull-bac k and push-forw ard. Let S − → T b e a map of semialgebras compatible with a map of corings C − → D and a k -algebra map A − → B . Assume that C is a ﬂat right A -mo dule, S is a coﬂat righ t C - como dule, D is a ﬂat righ t B -mo dule, T is a coﬂat right D -como dule and a D / B -coﬂat left D -como dule, and B has a ﬁnite w eak homological dimension. The right deriv ed functor N • 7− → R C N • : D si ( T – simo d ) − − → D si ( S – simo d ) is deﬁned by comp osing the functor N • 7− → C N • acting from the homotop y category of left T -semimo dules to the homotopy category of left S -semimo dules with the lo cal- ization functor Hot ( S – simo d ) − → D si ( S – simo d ) and r estricting it to the full sub cate- gory of complexes of D /B -coﬂat T -sem imo dules. By Theorems 8.2 .1(a) and 7.2.1(a ) , this restriction factor izes throug h the semideriv ed category o f left T -semimo dules. Assume that C is a ﬂat left and r ig h t A - mo dule, S is a ﬂat left A - mo dule and a coﬂat right C -como dule, A has a ﬁnite weak homological dimension, D is a ﬂa t right B -mo dule, a nd T is a coﬂat right D - como dule. The left deriv ed functor M • 7− → L T M • : D si ( S – simo d ) − − → D si ( T – simo d ) is deﬁned by comp osing the f unctor M • 7− → T M • acting from the homotopy cat- egory of left S -semimo dules to the homotop y category of left T -semimodules with the lo calization functor Hot ( T – simo d ) − → D si ( T – simo d ) and restricting it to the full subcategory of quite S / C / A -semiﬂat complexes of S - semimodules. By Theo- rems 8.2.2(a) and 8.2 .3(a), this restriction f a ctorizes through the semideriv ed catego r y of left S -semimo dules. Analogously , assume that C is a pro jective left A -mo dule, S is a copro jectiv e left C -como dule, D is a pro jectiv e left B -mo dule, T is a copro jec tiv e left D -como dule and a D /B -coﬂat righ t D -como dule, and B has a ﬁnite left homological dimension. The left deriv ed functor Q • 7− → C L Q • : D si ( T – sicntr ) − − → D si ( S – sicntr ) is deﬁned b y comp osing the functor Q • 7− → C Q • with the lo calization functor Hot ( S – sicntr ) − → D si ( S – sicntr ) and restricting it to the f ull sub category o f complexes of D /B -coinjectiv e T -semicontramo dules. By Theorems 8.2.1(b) and 7.2 .1(b), this restriction factorizes through the semideriv ed category of left T -semicontramo dules. According to Lemma 6.5.2, this deﬁnition of a left deriv ed functor do es not dep end on the c hoice of a subcatego r y of a djusted complexes. 155 Assume that C is a pro jectiv e left a nd a ﬂat right A -mo dule, S is a copro jectiv e left C -como dule and a ﬂat righ t A - mo dule, A has a ﬁnite left homolo g ical dimension, D is a pro jectiv e left B -mo dule, and T is a copro jective left D -como dule. P • 7− → R T P • : D si ( S – sicntr ) − − → D si ( T – sicntr ) is deﬁned b y comp osing the functor P • 7− → T P • with the lo calization func- tor Hot ( T – simo d ) − → D si ( T – simo d ) and r estricting it to the full sub category of quite S / C / A -semiﬂatcomplexe s of S -semicontramo dules. By Theorems 8.2 .2(b) and 8.2.3(b), this restriction factorizes through the semideriv ed category of left S -semicontramo dules. Acc ording t o Lemma 6.5.2, this deﬁnition of a righ t derive d functor do es not dep end on the c hoice of a sub category of adjusted complexes. Notice that in the assumptions of Theorem 8.2.3(c) ab o v e and Coro lla ry 1(c) b elow one can also deﬁne the left deriv ed functor M • 7− → L T M • in terms o f S / C / A -projectiv e complexes of left S - semimo dules and t he righ t derive d functor P • 7− → R T P • in terms of S / C / A -injectiv e complexes of left S -semicontramo dules. The deriv ed functors N • 7− → R C N • and Q • 7− → C L Q • in the categories of semi- mo dules and semicon tramo dules agree with the deriv ed functors N • 7− → R C N • and Q • 7− → C L Q • in the catego r ies of como dules and contramodules, so our notation is not am biguous. Remark 1. Under the assumptions that C is a ﬂat righ t A -mo dule, S is a coﬂat righ t C -como dule, D is a ﬂat right B -mo dule, T is a coﬂat righ t D -como dule, and B has a ﬁnite left homolog ical dimension, one can deﬁne the deriv ed functor N • 7− → R C N • in terms of injectiv e complexes of left T -semim o dules (see Remark 6.5). Corollary 1. (a) The d erive d functor M • 7− → L T M • is left adjoint to the deriv e d functor N • 7− → R C N • whenever b oth functors ar e deﬁne d by the ab ove c onstruction. (b) T he derive d functor P • 7− → T R P • is ri g ht adj o i nt to the derive d functor Q • 7− → C L Q • whenever b oth functors ar e deﬁne d by the a b ove c onstruction. (c) Assume that C is a pr oje ctive left and a ﬂat right A -mo dule, S i s a c opr oj e c- tive left and a c oﬂat rig h t C -c om o dule, A ha s a ﬁnite le f t homo lo gic al dimension, D is a pr o je ctive left a nd a ﬂat right B -m o dule, T is a c opr oje ctive left and a c o ﬂat right D -c omo dule, and B ha s a ﬁnite left homol o gic al dim ension. Th en for any ob- je cts M • in D si ( simo d – S ) and Q • in D si ( T – sicntr ) ther e is a na tur al i s omorphism CtrT or T ( M • L T , Q • ) ≃ CtrT or S ( M • , C L Q • ) in the d e ri v e d c ate g ory of k -mo dules. Pr o of . In the assum ptions of part (c), one can prov e somewhat stronger ver- sions of the assertions (a) and (b): for an y M • in D si ( S – simo d ) and N • in D si ( T – simo d ), there is a na tural isomorphism Ext T ( L T M , N ) ≃ Ext S ( M , R C N ) and for any P • in D si ( S – sicntr ) and Q • in D si ( T – sicntr ) there is a natural isomorphism Ext T ( Q • , T R P • ) ≃ Ext S ( C L Q • , P • ) in the deriv ed category o f k - mo dules. T o obtain the ﬁrst isomorphism, it suﬃces to represen t the ob ject M • b y an S / C / A -projectiv e 156 complex of left S -semimo dules a nd the ob ject N • b y a complex of D /B -injectiv e left T -semimo dules, and use Lemma 5.3.2(a ), Theorem 7.2.1( a ), and Theorem 8.2.3( c). In the second case, one can represen t the ob ject P • b y an S / C / A -injectiv e complex of left S -semicontramo dules and the ob ject Q • b y a complex o f D /B - projectiv e left T -semi- contramo dules, and use Lemma 5.3 .2 (b), Theorem 7.2.1 (b), and Theorem 8.2.3(c). T o verify part (c), it suﬃces to represen t the ob ject M • b y a quite S / C / A -semiﬂat complex of right S -semimo dules and the o b ject Q • b y a complex of D /B -projectiv e left S -semicontramo dules, and use Lemma 5.3.2 (b), Theorem 7.2.1(b), and Theo- rem 8.2.3(a). Finally , parts (a) and (b) in their w eak er assumptions follo w from the next Lemma.  Lemma. L et H 1 and H 2 b e c ate go rie s, S 1 and S 2 b e lo c a lizing classes of morphisms in H 1 and H 2 , and F 1 and F 2 b e ful l sub c ate g o ries in H 1 and H 2 . Assume that for any obje ct X ∈ H 1 ther e ex ists an obje ct U ∈ F 1 to gether w i th a m o rphism U − → X fr om S 1 and for any obje ct Y ∈ H 2 ther e ex i s ts an ob je ct V ∈ F 2 to gether with a morphism Y − → V fr om S 2 . L et Σ : H 1 − → H 2 b e a functor an d Π : H 2 − → H 1 b e a functor ri g h t adjoint to Σ . Assume that the mo rp h ism Σ( t ) b e l o ngs to S 2 for any morphism t ∈ F 1 ∩ S 1 and the morphism Π( s ) b elongs to S 1 for any morphism s ∈ F 2 ∩ S 2 . Then the rig h t derive d functor R Π : H 2 [ S − 1 2 ] − → H 1 [ S − 1 1 ] deﬁne d b y r estricting Π to F 2 is right adjoint to the left derive d functor L Σ : H 1 [ S − 1 1 ] − → H 2 [ S − 1 2 ] deﬁne d by r estricting Σ to F 1 . Pr o of . The functors F i [( F i ∩ S i ) − 1 ] − → H i [ S − 1 i ] are equiv alences o f categories b y Lemma 2.6, so the deriv ed functors L Σ and R Π can b e deﬁned. F or an y o b- jects U ∈ F 1 and V ∈ F 2 w e hav e to construct a bijection b et w een the sets Hom H 1 [ S − 1 1 ] ( U, Π V ) and Hom H 2 [ S − 1 2 ] (Σ U, V ), functorial in U and V . Any elemen t of the ﬁrst set can b e repres en ted by a fraction U ← U ′ → Π V in H 1 with the morphism U ′ − → U b elonging to S 1 . By assumption, one can choose U ′ to b e a n ob ject o f F 1 . Assign to this fraction the elemen t of the second set represen ted b y the fra ction Σ U ← Σ U ′ → V . By assumption, the morphism Σ U ′ − → Σ U b elongs to S 2 . Analogously , an y elemen t of the second set can b e represen ted by a fraction Σ U → V ′ ← V in H 2 with the morphism V − → V ′ b elongning to S 2 , and one can c ho ose V ′ to b e an ob ject of F 2 . Assign to this fraction the elemen t of the ﬁrst set represen ted by the f r a ction U → Π V ′ ← Π V . The comp ositions of these t w o maps b et w een sets o f morphisms are identities, since the square f ormed b y the morphisms U ′ − → U , U − → Π V ′ , U ′ − → Π V , and Π V − → Π V ′ and the square formed b y the morphisms Σ U ′ − → Σ U , Σ U − → V ′ , Σ U ′ − → V , and V − → V ′ are comm utativ e sim ultaneously .  Let R b e a semialgebra ov er a coring E o v er a k -algebra F , a nd T − → R b e a map of semialgebras compatible with a map of cor ing s D − → E and a k -algebra ma p 157 B − → F . Then t he comp osition provides a map of semialgebras S − → R compatible with a map of corings C − → E and a k -algebra map A − → B . Corollary 2. (a) Ther e is a natur al isomorphism R C ( R D L • ) ≃ R C L • for any obje ct L • in D si ( R – simo d ) w henever b oth functors L • 7− → R D L • and N • 7− → R C N • ar e deﬁne d by the ab ove c onstruction. (b) Ther e is a natur al isomorphism L R ( L T M • ) ≃ L R M • for any o bje ct M • in D si ( S – simo d ) w h enever b oth functors M • 7− → L T M • and N • 7− → L R N • ar e deﬁne d by the ab ove c onstruction. (c) T h er e is a natur al isomorph ism C L ( D L K • ) ≃ C L K • for any obje ct K • in D si ( R – sicntr ) w henever b oth functors K • 7− → D L K • and Q • 7− → C L Q • ar e deﬁne d by the ab ove c onstruction. (d) Ther e i s a natur al is o morphism R R ( T R P • ) ≃ R R P • for any obje c t P • in D si ( S – sicntr ) whenever b oth functors P • 7− → T R P • and Q • 7− → R R Q • ar e deﬁn e d by the ab ove c onstruction. Pr o of . P art ( a ) follows from the ﬁrst assertion of Theorem 7.2.1(a ) , part (b) follows from the ﬁrst assertion of Theorem 8.2.3(a), part (c) follo ws from the ﬁrst assertion of Theorem 7.2.1(b), part (d) follo ws from the ﬁrst assertion o f Theorem 8.2.3(b).  Recall that a complex of C -coﬂat right S -semimo dules is called quite semiﬂat if it b elongs to the minimal triangulated sub category o f the homotopy categor y of right S -semimodules con taining the complexes of S -semimo dules induced from complexes of coﬂat right C -como dules a nd closed under inﬁnite direc t sums (see 2.9). This deﬁnition presumes that C is a ﬂat righ t A -mo dule and S is a coﬂa t righ t C -como dule. Corollary 3. (a) Assume that C is a ﬂat left and right A -mo dule, S is a c oﬂat left and right C -c omo dule, A has a ﬁnite we ak homolo gic al dim ension, D is a ﬂat left and right B -m o dule, T is a c oﬂat left and right D -c omo dule, and B ha s a ﬁnite we ak homolo gic al dime nsion. Then for any obje cts M • in D si ( S – simo d ) and N • in D si ( simo d – T ) ther e is a natur al isomorphism SemiT or T ( N • , L T M • ) ≃ SemiT or S ( N • R C , M • ) in D ( k – mo d ) . (b) Under the assumptions of C or ol la ry 1(c), for any obje cts P • in D si ( S – sicntr ) and N • in D si ( T – simo d ) ther e is a natur al isomo rp hism SemiE xt T ( N • , T R P • ) ≃ SemiExt S ( R C N • , P • ) in D ( k – mo d ) . (c) Unde r the assumptions of Cor ol l a ry 1(c), for any obje cts M • in D si ( S – simo d ) and Q • in D si ( T – sicntr ) ther e is a natur al isomorphism SemiE xt T ( L T M • , Q • ) ≃ SemiExt S ( M • , C L Q • ) in D ( k – mo d ) . Pr o of . P art (a): represen t the ob ject M • b y a quite semiﬂat complex of S -semi- mo dules and the ob ject N • b y a semiﬂat complex of D -coﬂat T -semimo dules, and use the second case of Prop osition 8.1.3(a). Alternatively , represen t M • b y a quite S / C / A -semiﬂat complex o f A -ﬂat S -semimodules and N • b y a complex of D -coﬂat 158 T -semimo dules, and use Theorem 7.2 .1(a), Theorem 8.2.3(a), the result of 2.8, and the ﬁrst case of Prop osition 8 .1.3(a); or represen t M • b y a quite semiﬂat complex of semiﬂat S -semimo dules a nd N • b y a complex of D /B -coﬂat T -semimodules, and use the same Theorems, the result of 2 .8 , a nd the fourth case o f Prop osition 8.1.3(a). P art (b): represen t the ob j ect P • b y a semiinjectiv e complex of C -coinjectiv e S -semicontramo dules (having in mind Lemma 6 .4(c) or R emark 6.4) a nd the ob ject N • b y a semipro jectiv e complex of D -coprojectiv e T - semimo dules, a nd use the second case o f Prop osition 8.1 .3(b). Alternativ ely , represen t P • b y a quite S / C / A - semiin- jectiv e complex of A -injectiv e S - semicontramo dules a nd N • b y a complex of D -co- projectiv e T -semimo dules, and use Theorem 7.2.1(a), Theorem 8.2.3(b), the result of 4.8, and the ﬁrst case of Prop osition 8.1.3 ( b); o r represen t P • b y a semiinjectiv e complex of semiinjectiv e S -semicontramo dules and N • b y a complex of D /B -copro- jectiv e T -semim o dules, and use the same Theorems, the result of 4 .8, and the fourth case o f Prop osition 8.1.3 (b). P art (c): represe nt the ob ject M • b y a semipro jectiv e complex of C -coprojec- tiv e S -semicontramodules (having in mind Lemma 6.4(b) or Remark 6.4) and the ob ject Q • b y a semiin jectiv e complex o f D -coinjective T - semicontramo dules, and use the second case of Prop osition 8.1.3(c). Alternativ ely , represen t M • b y a quite S / C / A -semiprojectiv e complex o f A -projectiv e S -semimo dules and Q • b y a complex of D -coinjectiv e T -semicontramodules, and use Theorem 7.2 .1 (b), Theorem 8.2.3( a ), the result of 4.8, and the ﬁrst case of Prop osition 8.1.3(c); or represen t M • b y a semipro jectiv e complex of semipro jectiv e S -semimo dules and Q • b y a complex of D /B -coinjectiv e T -semicontramo dules, and use the same Theorems, the result of 4.8, and t he fourth case of Prop osition 8.1.3(c).  Remark 2. Supp ose that tw o ob jects ′ M • in D si ( simo d – S ) and ′ N • in D si ( simo d – T ) are endow ed with a morphism ′ M • L T − → ′ N • , or , which is the same, a morphism ′ M • − → ′ N • R C , and tw o ob jects ′′ M • in D si ( S – simo d ) and ′′ N • in D si ( T – simo d ) are en- do w ed with a mor phism L ′′ T M • − → ′′ N • , or, whic h is the same, a morphism ′′ M • − → R ′′ C N • . Then the t wo morphisms SemiT or S ( ′ M • , ′′ M • ) − → SemiT or T ( ′ N • , ′′ N • ) in D (k– mo d ) pro vided b y the comp ositions SemiT or S ( ′ M • , ′′ M • ) − → SemiT or S ( ′ N • R C , ′′ M • ) ≃ SemiT or T ( ′ N • , L ′′ T M • ) − → SemiT or T ( ′ N • , ′′ N • ) and SemiT or S ( ′ M • , ′′ M • ) − → SemiT or S ( ′ M • , R ′′ C N • ) ≃ SemiT or T ( ′ M • L T , ′′ N • ) − → SemiT or T ( ′ N • , ′′ N • ) coincide with each other. Indeed , let us represe n t the o b jects ′ M • and ′ N • b y complexes of right S -semimo dules and T -semim o dules in suc h a w ay that the adjoint mor- phisms ′ M • L T − → ′ N • and ′ M • − → ′ N • R C could b e r epresen ted by a map of complexes of semimo dule s ′ M • − → ′ N • compatible with the maps A − → B , C − → D , and S − → T . Applying to the complexes of ′ M • and ′ N • sim ultaneously the con- structions from the pro of of Theorem 2.6, one can construct a map of quite semi- ﬂat complexes of right semimodules L 3 R 2 L 1 ( ′ M • ) − → L 3 R 2 L 1 ( ′ N • ) represen ting 159 the same adjoint morphisms in the semideriv ed categories o f left semimo dules. So one can assume ′ M • a nd ′ N • to b e quite semiﬂat complexes . Analogously , repre- sen t the morphisms L ′′ T M • − → ′′ N • and ′′ M • − → R ′′ C N • in the semideriv ed cate- gories of left semimo dules by a map of quite semiﬂat complexes of left semimo dules ′′ M • − → ′′ N • compatible with the maps A − → B , C − → D , and S − → T . Then b oth comp ositions in question ar e represen ted b y the same map of complexes of k - mo dules ′ M • ♦ S ′′ M • − → ′ N • ♦ T ′′ N • . F urthermore, supp o se that tw o ob- jects M • in D si ( S – simo d ) a nd N • in D si ( T – simo d ) are endo w ed with a morphism L T M • − → N • , or, whic h is the same, a morphism M • − → R C N • , and t w o ob- jects P • in D si ( S – sicntr ) a nd Q • in D si ( T – sicntr ) are endo w ed with a morphism Q • − → T R P • , o r, which is the same, a morphism C L Q • − → P • . Then the tw o mor- phisms SemiExt T ( N • , Q • ) − → SemiExt S ( M • , P • ) in D (k– mo d ) provided b y the com- p ositions SemiExt T ( N • , Q • ) − → SemiExt T ( N • , R T P • ) ≃ SemiExt S ( R C N • , P • ) − → SemiExt S ( M • , P • ) and SemiExt T ( N • , Q • ) − → SemiExt T ( L T M • , Q • ) ≃ SemiExt S ( M • , C L Q • ) − → SemiExt S ( M • , P • ) coincide with eac h other. Corollary 4. Under the assumptions of Cor o l lary 1 (c), the mutual ly inverse e quiva- lenc es of c a te gories R Ψ S : D si ( S – simo d ) − → D si ( S – sicntr ) and L Φ S : D si ( S – sicntr ) − → D si ( S – simo d ) and the mutual ly inv e rs e e quiva lenc es of c ate gories R Ψ T : D si ( T – simo d ) − → D si ( T – sicntr ) and L Φ T : D si ( T – sicntr ) − → D si ( T – simo d ) tr ansform the derive d functor N • 7− → R C N • into the derive d functor Q • 7− → C L Q • . Pr o of . T o construct t he isomorphism L Φ S ( C L Q • ) ≃ R C ( L Φ T Q • ), represen t the ob- ject Q • b y a complex of D / B -projectiv e C -contra mo dules, and use Lemma 5.3.2, Theorem 7 .2 .1(b), and t he results of 7.1.4 and 8.1.4 . T o construct the isomor- phism C L ( R Ψ T N • ) ≃ R Ψ S ( R C N • ), r epresen t the ob ject N • b y a complex of D /B -injec- tiv e C -como dules, and use Lemma 5.3.2, Theorem 7.2.1(a), and the results of 7.1 .4 and 8.1.4. T o show that these isomorphisms agree, it suﬃces to chec k tha t for any adjoin t morphisms L Φ T Q • − → N • and Q • − → R Ψ T N • in the semideriv ed cate- gories of T -semimodules and T -semicontramodules the comp ositions L Φ S ( C L Q • ) − → R C ( L Φ T Q • ) − → R C N • and C L Q • − → C L ( R Ψ T N • ) − → R Ψ S ( R C N • ) are adjoint morphisms in the semideriv ed categories of S -semimo dules a nd S -semicontramo dules. Here o ne can represen t N • b y a semipro jectiv e complex o f D -coproj ective left T -semimo dules and Q • b y a semiinjectiv e complex o f D -coinjectiv e left T -semicontramo dules (ha ving in mind Lemmas 5.2 and 6.4) , and use a result o f 7.1.4.  Th us w e ha v e constructed three functors b etw een the semideriv ed catego ries D si ( S – simo d ) ≃ D si ( S – sicntr ) and D si ( T – simo d ) ≃ D si ( T – sicntr ): the functor de- scrib ed in Corollary 4, and tw o functors adjo int to it f rom the left and from the righ t, describ ed in Corollary 1. 160 Remark 3. One can show that the isomorphisms of deriv ed f unctors from Corol- lary 6 .6 are compatible with the c hange-of-semialgebra isomorphisms f rom Corol- laries 1, 3 , a nd 4 in the follow ing w ay . T o c hec k that the comp ositions of isomorphisms SemiExt T ( L T M • , R Ψ T ( N • )) − → Ext T ( L T M • , N • ) − → Ext S ( M • , R C N • ) and SemiExt T ( L T M • , R Ψ T ( N • )) − → SemiExt S ( M • , C L ( R Ψ T N • )) − → SemiExt S ( M • , R Ψ S ( R C N • )) − → Ext S ( M • , R C N • ) coincide, represen t the ob ject M • b y a semipro- jectiv e complex of semipro jectiv e left S -semimo dules and the ob ject N • b y a com- plex of D /B -injectiv e left T -semimodules, and use the result of 4.8. T o c heck that the compositions of isomorphisms CtrT or S ( M • , C L Q • ) − → CtrT or T ( M • L T , Q • ) − → SemiT or T ( M • L T , L Φ T ( Q • )) a nd CtrT or S ( M • , C L Q • ) − → SemiT or S ( M • , L Φ S ( C L Q • )) − → SemiT or S ( M • , R C ( L Φ T Q • )) − → SemiT or T ( M • L T , L Φ T ( Q • )) coincide, represen t the ob ject M • b y a quite semiﬂat complex of semiﬂat right S -semimo dules and the o b- ject Q • b y a complex of D /B -projectiv e left T -semicontramodules, and use the r esult of 2.8. Commutativit y of the resp ectiv e diagrams on the lev el of ab elian categories is straigh tforw ard to v erify under our assumptions on the terms of the complexes represen ting the ob jects M • . 8.4. R emarks on Morita morphisms. 8.4.1. Let C b e a coring ov er a k -algebra A and D b e a coring ov er a k -alg ebra B suc h that C is a ﬂat rig h t A - mo dule and D is a ﬂat righ t B -mo dule. Let ( E , E ∨ ) b e a right coﬂat Morita morphism from C to D and T be a semialgebra o v er the coring D suc h that T is a coﬂat righ t D -como dule. In this case, the semialgebra C T C o v er t he coring C is constructed in the follo wing w a y . As a C - C - bicomo dule, C T C is equal to E  D T  D E ∨ . The semim ultiplication in C T C is deﬁned a s the comp osition E  D T  D E ∨  C E  D T  D E ∨ − → E  D T  D T  D E ∨ − → E  D T  D E ∨ of the morphism induced b y the morphism E ∨  C E − → D and the mo r phism induced b y the semim ultiplication in T . The semiunit in C T C is deﬁned as the comp osition C − → E  D E ∨ − → E  D T  D E ∨ of the morphism induced b y the morphism C − → E  D E ∨ and t he morphism induced by the semiunit in T . F or example, if C − → D is a map of corings compatible with a k -alg ebra map A − → B suc h that B is a ﬂat right A -mo dule and C B is a coﬂat right D -como dule, one can ta k e E = C B and E ∨ = B C . Then the algebra C T C is a univers al ﬁnal ob ject in the category of semialgebras S ov er C endo w ed with a map S − → T compatible with the maps A − → B and C − → D . The semialgebra C T C = C B  D T  D B C can b e also deﬁned, e. g., when ( E , E ∨ ) is a Morita morphism from a k -a lg ebra A to a k -alg ebra B and B C B = E ∨ ⊗ A C ⊗ A E − → D is a morphism of coring s o v er B suc h that E ∨ is a ﬂat right A -mo dule, B C = E ∨ ⊗ A C is a D /B -coﬂat left D -como dule, T is a ﬂat right B -mo dule and a D /B -coﬂat left D - como dule, and the rings A and B ha v e ﬁnite we ak homological dimensions. 161 All the results of 8.1 – 8.3 can b e extended to the situation of a left copro j ectiv e and righ t coﬂat Morita morphism ( E , E ∨ ) from a coring C to a coring D and a morphism S − → C T C of semialgebras ov er C . In particular, when C is a ﬂat rig h t A -mo dule, D is a ﬂat right B -mo dule, S is a coﬂat righ t C -como dule, T is a coﬂat righ t D -como dule, and ( E , E ∨ ) is a righ t coﬂat Morita morphism, the functor N 7− → C N = E  D N from the category of left T - semimo dules to the category of left S -semimo dules has a left adjoin t functor M 7− → T M = T C ♦ S M . Analogously , when C is a pro jectiv e left A -mo dule, D is a pro jectiv e left B -mo dule, S is a copro jectiv e left C -como dule, T is a copro jectiv e left D -como dule, and ( E , E ∨ ) is a left copro jectiv e Morita morphism, the functor Q 7− → C Q = Cohom D ( E ∨ , Q ) fr o m the category of left T -semicon- tramo dules to the category of left S -semicontramo dules has a right adjoint functor P 7− → T P = SemiHom S ( C T , P ), etc. How ev er, one sometimes has to imp o se the homological dimension conditions on A and B where they were not previously needed. 8.4.2. Assume that C is a ﬂat righ t A -mo dule and D is a ﬂat righ t B - mo dule. A righ t D -como dule K is called faithful ly c oﬂat if it is a coﬂat D -como dule and for an y nonzero left D -como dule M the cotensor pro duct K  D M is nonzero. A righ t coﬂat Morita morphism ( E , E ∨ ) from C to D is called right fa i thful ly c oﬂat if t he righ t D -como dule E is faithfully coﬂat . A righ t coﬂat Morita morphism ( E , E ∨ ) is righ t faithfully coﬂat if and only if the r igh t D -como dule E ∨  C E is faithfully coﬂat and if and only if t he morphism E ∨  C E − → D is surjectiv e and its k ernel is a coﬂat rig ht D -como dule. Indeed, the cotensor pro duct E  D M is nonzero if and only if the mo r phism E ∨  C E  D M − → M is nonzero; this holds for an y nonzero left D -como dule M if and only if the morphism E ∨  C E  D M − → M is surjectiv e for a n y left D -como dule M , and it remains to use the results of (the pro o f of ) Lemma 1.2 .2. Let ( E , E ∨ ) b e a righ t faithfully coﬂat Morita morphism f r om C to D and T b e a semialgebra o v er the coring D suc h tha t T is a coﬂat righ t D - como dule. Then the functor N 7− → C N is an equiv alence of the ab elian categor ies of left T -semimo dules and left C T C -semimo dules. This follow s f r o m Theorem 7.4.1 applied to the functor ∆ : T – simo d − → C – como d mapping a T -sem imo dule N to the C -como dule C N and the functor Γ : C – como d − → T – simo d left adjoint to ∆ mapping a C - como dule M to the T - semimo dule T  D D M . No w assume that C is a pro jectiv e left A -mo dule and D is a pro jectiv e left B -mo dule. A left D -como dule K is called faithful ly c opr oje ctive if it is a copro jec- tiv e D -como dule a nd for an y nonzero left D -contramo dule P the cohomomorphism mo dule Cohom D ( K , P ) is nonzero. A faithfully copro jectiv e D -como dule is fa it h- fully coﬂat. A left copro jective Morita morphism ( E , E ∨ ) f rom C to D is called left faithful ly c opr oje ctive if the left D - como dule E ∨ is faithfully copro jectiv e. A left co- pro jectiv e Morita morphism ( E , E ∨ ) is left faithfully copro jectiv e if and o nly if the left 162 D -como dule E ∨  C E is fait hfully coﬂat and if and only if the mo r phism E ∨  C E − → D is surjectiv e and its ke rnel is a copro jectiv e left D -como dule. Let ( E , E ∨ ) b e a left faithfully copro jective Morita morphism from C to D and T b e a semialgebra o v er the coring D suc h that T is a copro jectiv e left D -como dule. Then the functor Q 7− → C Q is an equiv alence of the abelian categories o f left T -sem i- contramo dules and left C T C -semicontramo dules. This follows from Theorem 7.4 .1 applied to the f unctor ∆ : T – sicntr − → C – contra mapping a T -semicontramo dule Q to t he C -contramo dule C Q and t he functor Γ : C – contra − → T – sicntr right adjo in t to ∆ mapping a C -contramo dule P to the T -semicontramo dule Cohom D ( T , D P ). 8.4.3. Assume that C is a ﬂat righ t A -mo dule and D is a ﬂat righ t B -module. Let ( E , E ∨ ) b e a rig h t coﬂat Morita morphism f rom C to D and T b e a semialgebra ov er the coring D suc h that T is a coﬂa t righ t D -como dule. Then the functor N • 7− → C N • maps D - coa cyclic complexes of T -semimodules to C -coacyclic complexes of C T C -semi- mo dules and the semideriv ed catego ry of left C T C -semimo dules is a lo calization of the semideriv ed category of left T -semimodules b y the k ernel of the functor induced by N • 7− → C N • (as one can che c k by computing the f unctor M • 7− → C ( L T M • ) on the semi- deriv ed category of left C T C -semimo dules). The triangulated categories D si ( T – simo d ) and D si ( C T C – simo d ) are equiv alent when ( E , E ∨ ) is a right coﬂat Morita equiv alence, or more generally when the morphism E ∨  C E − → D is an isomorphism. Analogously , a ssume that C is a ﬂa t righ t A -mo dule and D is a pro j ective left B -mo dule. Let ( E , E ∨ ) b e a left copro jectiv e Mor it a morphism from C to D and T b e a semialgebra ov er the coring D suc h that T is a copro jectiv e left D -como dule. Then the functor Q • 7− → C Q • maps D - contr aacyclic complexes o f T -semicontra- mo dules to C -contraacyclic complexes of C T C -semicontramo dules and the semideriv ed category of left C T C -semicontramo dules is a lo calization of the semideriv ed category of left T - semicontramo dules b y the k ernel of the functor induced b y Q • 7− → C Q • . The triangulated categories D si ( T – sicntr ) and D si ( C T C – sicntr ) are equiv alen t when ( E , E ∨ ) is a left copro jective Morita equiv alence, o r more g enerally when the morphism E ∨  C E − → D is an isomorphism. Remark. The semideriv ed catego ries of left T -semimodules and left C T C -semimo d- ules can b e diﬀeren t ev en when ( E , E ∨ ) is a right fait hfully coﬂat Morita morphism and the ab elian categories of left T - semimo dules and left C T C -semimo dules are equiv alen t. Indeed, let A = B = k b e a ﬁeld and F b e a ﬁnite-dimensional algebra ov er k . Let D = F ∗ and C = End( F ) ∗ b e the coalgebras ov er k dual to the ﬁnite-dimensional k -alg ebras F and End( F ). Then there is a coalgebra morphism C − → D dual to the algebra em b edding F − → End( F ) related to the a ction of F in itself by left m ultiplications. Since End( F ) is a free left F -mo dule, C is a cofree right D -como dule. Set E = C = E ∨ ; this is a righ t faithfully copro jectiv e Morita morphism from C t o D . No w put T = D ; then the semideriv ed category o f left T -semim o dules coincides 163 with the co deriv ed category of left D -como dules. A t the same time, the coalgebra C is semisimple a nd a complex of C -como dules is coacyclic if and only if it is a cyclic, so the semideriv ed catego r y of left C T C -como dules is equiv alen t to the con v en tional deriv ed categor y of left D -como dules. When F is a F rob enius algebra, End ( F ) is a free left and righ t F -mo dule, so ( E , E ∨ ) is a left and r igh t faithfully copro jective Morita morphism, but the categories D si ( T – simo d ) and D si ( C T C – simo d ) are still not equiv alen t when the homological dimension of F is inﬁnite. Alternatively , one can consider the right copro jective Morita morphism from the coalgebra C = k to the coalgebra D = F ∗ with E = F ∗ and E ∨ = F and the same semialgebra T = D o v er D ; then the semialgebra C T C o v er C is isomorphic to the alg ebra F ov er k ; the category D si ( T – simo d ) is the co deriv ed category of F ∗ -como dules a nd the catego r y D si ( C T C – simo d ) is the derived category of F - mo dules. Assume that C is a ﬂat left and right A -mo dule, D is a ﬂat left a nd rig h t B -mo dule, the rings A and B ha v e ﬁnite we ak homolog ical dimensions, T is a coﬂat left and righ t D - como dule, and ( E , E ∨ ) is a left and righ t coﬂat Morit a morphism from C to D . Then whenev er the f unctor N • 7− → C N • induces an equiv alence of the semi- deriv ed categories of left T - semimo dules and left C T C -semimo dules and the functor N • 7− → N • C induces an equiv alence of the semideriv ed categories of rig h t T -semi- mo dules and right C T C -semimo dules, t hese equiv alences of categories transform the functor SemiT or T in to the functor SemiT or C T C . Assume that C is a pro jectiv e left a nd a ﬂa t right A - mo dule, D is a pro jectiv e left and a ﬂat right B - mo dule, the r ing s A and B hav e ﬁnite left homological dimensions, T is a copro jectiv e left and a coﬂat right D - como dule, and ( E , E ∨ ) is a left copro jectiv e and rig ht coﬂa t Morita morphism from C to D . Then whenev er the functor N • 7− → C N • induces an equiv alence of the semideriv ed categories of left T -semimo dules and left C T C -semimo dules and the functor Q • 7− → C Q • induces an equiv alence of the semideriv ed categories o f left T -semic ontramo dules and left C T C -semicontramo dules, these equiv alences of categories tra nsform the f unctor SemiExt T in to the functor SemiExt C T C and the equiv alences o f categories R Ψ T and L Φ T in to the equiv alences of categories R Ψ C T C and L Φ C T C . The same applies to the functors Ext T , Ext T , and CtrT or T , under the relev an t assumptions. 8.4.4. Here are some further partial results ab out equiv alence of the semideriv ed categories related to T and C T C . The problem is, essen tially , to ﬁnd conditions un- der whic h a complex of left D - como dules N • is coacyclic whenev er the complex of C -como dules C N • is coacyclic, or a complex of left D -contramo dules Q • is con tra- acyclic whenev er the complex of C - contramo dules C Q • is contraacyc lic. Consider the f ollo wing g eneral setting. Let A and B b e exact categories with exact functors of inﬁnite direct sum, ∆ : B − → A b e an exact functor preserving inﬁnite direct sums and suc h that a complex C • o v er B is a cyclic if the complex ∆( C • ) ov er A 164 is con tractible, and Γ : A − → B b e an exact functor left adjoin t to ∆. Clearly , if a complex C • is coacyclic then the complex ∆( C • ) is coacyclic; we w ould lik e to kno w when t he con v erse holds. First, if a complex C • is coacyclic whenev er the complex ∆( C • ) is contractible, then a complex C • is coacyclic if and only if the complex ∆( C • ) is coacyclic. In- deed, consider the bar bicomplex · · · − → Γ∆Γ∆( C • ) − → Γ∆( C • ) − → C • whose diﬀeren tials are the alternating sums of morphisms induced by the adjunction mor- phism Γ∆ − → Id. The t o tal complex of this bicomplex contructed b y taking inﬁnite direct sums along the diag onals b ecomes con tractible af t er applying the functor ∆; the con t r acting homotop y is induced by the adjunction morphism Id − → ∆Γ. By assumption, it f ollo ws that the total complex itself is coacyclic o v er B . On the other hand, if the complex ∆( C • ) is coacyclic o v er A , then ev ery complex (Γ∆) n ( C • ) is coacyclic o v er B , since the functors ∆ and Γ are exact and preserv e inﬁnite direct sums. The tot al complex of the bicomplex · · · − → Γ∆Γ∆( C • ) − → Γ∆( C • ) is ho- motop y equiv alen t to a complex o btained fr om the complexes (Γ∆) n ( C • ) using the op erations of shift, cone, and inﬁnite direct sum; hence the complex C • is coacyclic. By the same a r gumen t, a complex C • is acyclic if a nd only if the complex ∆( C • ) is acyclic, so if the exact category B has a ﬁnite homological dimension, then a complex C • is coacyclic if and only if the complex ∆( C • ) is coacyclic. This is the trivial case. Finally , let us sa y that an exact functor ∆ : B − → A has a ﬁnite relative homolog ical dimension if the category B with the exact catego ry structure formed b y the exact triples in B that split after applying ∆ ha s a ﬁnite homological dimension. W e claim that when the functor ∆ has a ﬁnite relative homological dimension, a complex C • o v er B is coacyclic if and only if the complex ∆( C • ) is coacyclic, in our assumptions. Indeed, consider again the bar bicomplex · · · − → Γ∆Γ∆( C • ) − → Γ∆( C • ) − → C • . One can assume that the category B contains images of idemp oten t endomorphisms, as passing to the Karoubian closure do esn’t c hange coacyclicit y . One can also assume that the complex C • is b ounded from ab ov e, as an y acyclic complex b ounded f r om b elo w is coacyclic. The complex · · · − → Γ∆Γ∆( X ) − → Γ∆( X ) is split exact in high enough (negativ e) degrees for an y ob ject X ∈ B , since it is exact and the complex of homomorphisms from it to an ob j ect Y ∈ B computes Ext( X , Y ) in the relativ e exact category . Let d b e an in teger not smaller than the relativ e homological dimension; denote by Z ( X ) the image of the morphism (Γ∆) d +1 ( X ) − → (Γ∆) d ( X ). Then the total complex of t he bicomplex · · · − → (Γ∆) d +2 ( C • ) − → (Γ∆) d +1 ( C • ) − → Z ( C • ) is contractible, while the total complex of the bicomplex (Γ∆) d ( C • ) / Z ( C • ) − → (Γ∆) d − 1 ( C • ) − → · · · − → Γ∆( C • ) − → C • is coacyclic. If the complex ∆( C • ) is coacyclic, the total complex of the bicomplex · · · − → Γ∆Γ∆( C • ) − → Γ∆( C • ) is a lso coacyclic; th us the complex C • is coacyclic. 165 8.4.5. Let S b e a semialgebra ov er a coring C and T b e a semialgebra ov er a cor ing D . Assume that C is a ﬂat right A - mo dule, D is a ﬂat r ig h t B -mo dule, S is a coﬂat right C -como dule, and T is a coﬂat righ t D - como dule. A right semiﬂat Morita m o rphism from S to T is a pair consisting o f a T -semiﬂat S - T -bisemimo dule E and an S -semiﬂat T - S -bisemimo dule E ∨ endo w ed with an S - S -bisemimodule mo r phism S − → E ♦ T E ∨ and a T - T - bisemimo dule morphism E ∨ ♦ S E − → T suc h that the tw o comp ositions E − → E ♦ T E ∨ ♦ S E − → E and E ∨ − → E ∨ ♦ S E ♦ T E ∨ − → E ∨ are equal to t he iden tity endomorphisms of E a nd E ∨ . A right semiﬂat Morita morphism ( E , E ∨ ) from S to T induces an exact functor M 7− → T M = E ∨ ♦ S M from the cat ego ry of left S -semimo d- ules to the cat ego ry of left T - semimo dules and an exact functor N 7− → S N = E ♦ T N from the category of left T - semimo dules to the category o f left S -semimo dules; the former functor is left adjoin t to the lat t er one. Con v ersely , any pa ir of adjoin t exact k -linear functors preserving inﬁnite direct sums b et w een the category of left S -semi- mo dules and left T -semimodules is induced by a right semiﬂat Morita morphism. Indeed, an y exact k -linear functor S – simo d − → T – simo d preserving inﬁnite direct sums is the functor of semitensor pr o duct with an S - semiﬂat T - S -bisemimodule; this can b e pro v en a s in 7.5.2. Analogously , assume that C is a pro jectiv e left A -mo dule, D is a pro jective left B -mo dule, S is a copro jec tiv e left C -como dule, and T is a copro jectiv e left D -como dule. A left semipr oje c tive Morita morphism f rom S to T is deﬁned as a pair consisting of an S - semiprojectiv e S - T -bisemim o dule E and a T -semiprojectiv e T - S -bisemimo dule E ∨ endo w ed with an S - S -bisemimodule mo r phism S − → E ♦ T E ∨ and a T - T -bisemimo dule morphism E ∨ ♦ S E − → T satisfying the same conditions as ab o v e. A left semipro jectiv e Morit a morphism ( E , E ∨ ) from S to T induces an exact functor P 7− → T P = SemiHom S ( E , P ) from the category of left S -semi- contramo dules to the category of left T - semicontramo dules and an exact functor Q 7− → S Q = SemiHom T ( E ∨ , Q ) from the category of left T -semic ontramo dules to the category of left S -semicontramo dules; the former functor is right adjoin t to the latter one. Con ve rsely , a ny pair of adjoint exact k -linear functors preserving inﬁnite pro ducts b etw een the catego r y of left S -semicontramo dules and left T -semicontra- mo dules is induced by a left semipro jectiv e Morita morphism. Inde ed, an y exact k -linear functor S – sicntr − → T – sicntr preservin g inﬁnite pro ducts is the functor of semihomomorphisms from a n S - semiprojectiv e S - T -bisemimodule. A righ t semiﬂat Morita morphism ( E , E ∨ ) fr o m S to T is called a right semiﬂat Morita e quivalenc e if the bisemimo dule morphisms S − → E ♦ T E ∨ and E ∨ ♦ S E − → T are isomorphisms; then it can b e also considered as a righ t semiﬂat Morit a morphism ( E ∨ , E ) fro m T to S . L eft semipr oje ctive Morita e q uivalenc es are deﬁned in the analogous w a y . A righ t semiﬂat Morita equiv alence b etw een semialgebras S and T induces an equiv alence of the ab elian categories of left S -semimo dules and left T -semi- mo dules, a nd in the relev an t ab o v e assumptions any equiv alence b et w een t hese t w o 166 k -linear categories comes f r om a righ t semiﬂat Morita equiv alence. Analogo usly , a left semipro jectiv e Morita equiv alence b et w een S a nd T induces an equiv alence of the ab elian categories of left S -semicontramo dules and left T -semicontramo dules, and in the relev an t ab ov e assumptions a n y equiv alence b etw een these tw o k -linear catego r ies comes from a left semipro jectiv e Morita equiv alence. Assume that the coring C is a ﬂat right A -mo dule a nd the coring D is a ﬂat righ t B -mo dule. Let T b e a semialgebra o v er D such that T is a coﬂat righ t D - como dule and ( E , E ∨ ) b e a right faithfully coﬂat Morita morphism f rom C to D . Then the pair of bisemimo dules E = C T and E ∨ = T C is a r igh t semiﬂat Morita equiv alence b et w een the semialgebras T a nd C T C . Analogously , assume that C is a pro jectiv e left A -mo dule and D is a pr o jectiv e left B -mo dule. Let T b e a semialgebra ov er D suc h that T is a copro jectiv e left D -como dule and ( E , E ∨ ) b e a left faithfully copro jectiv e Morita morphism from C to D . Then the same pair of bisemimo dules E and E ∨ is a left semipro jective Morita equiv alence b et w een T and C T C . All the results of 8.1 can b e exte nded to the case o f a left semipro jectiv e and righ t semiﬂat Morita morphism ( E , E ∨ ) from a semialgebra S to a semialgebra T . In particular, for any left T -semimodule N there are natural isomorphisms of left S -semicontramo dules Ψ S ( S N ) = Hom S ( S , S N ) ≃ Hom T ( T S , N ) ≃ Hom T ( E ∨ , N ) ≃ SemiHom T ( E ∨ , Hom T ( T , N )) = S (Ψ T N ) b y Prop osition 6.2.2(d), etc. Ho w ev er, one sometimes has to strengthen the coﬂatness (copro j ectivit y , coinjectivit y) conditio ns to the semiﬂatness (semipro jectivit y , semiinjectivit y) conditions. The ﬁrst assertions of Theorem 8.2.3(a), (b) and (c) do not hold for Morita mor- phisms of semialgebras, though. The derive d functors M • 7− → L T M • and P • 7− → T R P • still can b e deﬁned in terms of S / C - pro jectiv e ( = quite S / C -semiﬂat) complexes of S -semimodules and S / C -injectiv e ( = quite S / C -semiinjectiv e) complexes of S -semi- contramo dules. The right deriv ed functor N • 7− → R S N • can b e deﬁned in terms of injectiv e complexes of T -semimo dules and the left deriv ed functor Q • 7− → S L Q • can b e deﬁned in terms of pro jectiv e complexes of T -semicontramo dules (see Remark 6.5). The results of Corollaries 8.3.2–8.3.4 do not hold for Mor it a morphisms o f semialge- bras, as one can see in the example of the Morita equiv alence related to a F ro b enius algebra from Remark 8.4.3 considered as a Morita morphism in the inv erse direc- tion. The men tioned results remain v alid f or left semipro jectiv e and righ t semiﬂat Morita morphisms from S to T when the categories of C - como dules and C -contra- mo dules ha v e ﬁnite homological dimensions, or the Morita morphism of semialgebras is induce d b y a left copro jectiv e and r ig h t coﬂat Morita morphism of corings, or more generally when the functors N • 7− → S N • , N • 7− → N • S , and Q • 7− → S Q • map D -coacyclic and D -contraacyclic complexes to C - coacyclic and C -contraacyclic complexes. Morita equiv alences of semialgebras do not induce equiv alences of the semideriv ed categories of semimodules and semicon tramo dules, except in r a ther sp ecial cases. A 167 righ t semiﬂat Morita equiv alence b etw een S and T do es induce an equiv alence of t he semideriv ed categories of left S -semimo dules and left T -semimodules when the cate- gories of left C -como dules and left D -como dules ha v e ﬁnite homological dimensions, or when the Morita equiv alence comes f rom a righ t fait hf ully coﬂa t Morita morphism of corings and one of the conditions o f 8.4.3 – 8.4.4 is satisﬁed. 8.4.6. A short summary: one encoun ters no problems generalizing the results of 7.1 – 7.4 and 8.1 – 8.3 to the case of a Morita morphism of k - algebras and related maps of corings and semialgebras. The problems are manageable when one con- siders Morita morphisms of corings. And there are sev ere problems with Morit a morphisms/equiv a lences of semialgebras, whic h do not alw a ys respect the esse n tial structure of “an ob ject split in t w o ha lv es” (see In tro duction). 168 9. Closed Model Ca tegor y Str uctures By a close d m o del c ate gory w e mean a mo del categor y in the sense of Hov ey [26 ]. 9.1. Complexes of como dules and contramodules. Let C b e a coring o v er a k -alg ebra A . Assume that C is a pro jectiv e left and a ﬂat right A -mo dule a nd the ring A has a ﬁnite left homolo g ical dimension. Theorem. (a) The c ate gory of c omplexe s o f left C -c omo dules has a clos e d mo del c ate go ry structur e wi th the fol low ing pr op erties. A morp h ism is a we ak e q uivalenc e if an d only if its c on e is c o acyclic. A morphism is a c oﬁbr ation if and only if it is inje ctive and its c okernel is a c omplex of A -pr oje ctive C -c omo dules. A morphism is a ﬁbr ation if and only i f it is surje ctive and its kernel is a c om plex of C / A -inje ctive C -c omo dules. An ob je ct is simultane ously ﬁbr ant a nd c oﬁbr ant if and only if it is a c omplex of c opr oje ctive left C -c omo dules. (b) Th e c ate gory of c o m plexes of left C -c ontr am o dules h a s a close d mo del c ate gory structur e wi th the fol lowi n g pr op erties. A m o rphism is a we ak e quivalenc e if and only if its c on e is c ontr aacyclic. A morphi s m is a c o ﬁbr ation if and only if it is in je ctive and its c okernel is a c omplex of C / A -pr oje ctive C -c ontr a mo dules. A morphism is a ﬁbr ation if and on ly if it is surje ctive and its kernel is a c omplex of A -inje c tive C -c ontr a mo dules. An obje ct is simultane ously ﬁbr ant and c oﬁbr ant i f an d only if it is a c omplex of c oinj e ctive left C -c ontr amo dules. Pr o of . P art (a): the category of complexes o f left C - como dules has a rbitrary limits and colimits, since it is an a b elian category with inﬁnite direct sums and pro d- ucts. The tw o-out- o f-three pro p ert y of w eak equiv alences follo ws from the o ctahe- dron axiom, since coacyclic complexes form a tr ia ngulated sub catego ry of the ho- motop y category of left C -como dules. The retraction prop erties are clear, since the classes of pro j ective A -mo dules, C / A -injectiv e C - como dules, a nd coacyclic complexes of C -como dules ar e closed under direct summands. It is also clear that a morphism is a trivial coﬁbration if and only if it is injectiv e and its cokerne l is a coacyclic complex of A -pro j ectiv e C -como dules, and a morphism is a trivial ﬁbration if and only if it is surjectiv e and its k ernel is a coacyclic complex o f C / A -injectiv e C -como dules. No w let us v erify the lift ing pr o p erties. Lemma 1. L et U , V , X , a n d Y b e four o bje cts of an ab elian c ate gory A , U − → V b e an inje ctive morph i s m with the c okernel E , and X − → Y b e a surje c tive morphism with the kernel K . Supp ose that Ext 1 A ( E , K ) = 0 . T h en for any two morphisms U − → X and V − → Y forming a c om m utative squar e with the ab ove two morphisms ther e exists a morphism V − → X forming two c ommutative triangles with the gi v en four morphisms. 169 Pr o of . Let us ﬁrst ﬁnd a morphism V − → X making a comm utativ e triangle with the morphisms U − → X and U − → V . The obstruction to extending the morphism U − → X f r o m U to V lies in the group Ext 1 A ( E , X ). Since the comp o sition U − → X − → Y admits an extension to V , our elemen t of Ext 1 A ( E , X ) b ecomes zero in Ext 1 A ( E , Y ) and therefore comes from the group Ext 1 A ( E , K ). Now let us mo dify the obtained mo r phism so that the new morphism V − → X forms also a comm utativ e triangle with the morphisms V − → Y and X − → Y . The diﬀerence b et w een the giv en morphism V − → Y and the comp osition V − → X − → Y is a morphism V − → Y a nnihilating U , so it comes from a morphism E − → Y . W e need to lift the latter to a morphism E − → X . The obstruction to this lies in Ext 1 A ( E , K ).  T o v erify the condition of Lemma 1, consider a n extension E • − → M • − → K • of a complex of A - projectiv e left C -como dules K • b y a complex of C / A -inj ective left C -como dules E • . By Lemma 5.3.1(a), this extension is term-wise split, so it comes from a morphism of complexes of C -como dules K • − → E • [1]. Now supp ose that one of the complexes K • and E • is coacyclic. Then any mor phism K • − → E • [1] is homotopic to zero b y a result of 5.5, hence the extension of complexe s E • − → M • − → K • is split. The liﬁng prop erties are prov en. It remains to construct the functor ia l fa cto r izations. These constructions use tw o building blo ck s: one is Lemma 3 .1 .3(a), the other one is the fo llowing L emma 2. Lemma 2. (a) Ther e exists a (not always additive) functor a s signing to any left C -c omo dule an inje ctive morphism fr om it in to a C / A -in je ctive left C -c omo dule with an A -p r oje ctive c okernel. (b) Ther e exists a (not always additive) functor assigning to any left C -c ontr amo d- ule a surje ctive morp h ism onto i t fr om a C / A -pr oj e ctive left C -c ontr amo d ule w i th an A -inje ctive kernel. Pr o of . P art (a): let M b e a left C -como dule and P ( M ) − → M b e the surjectiv e mor- phism on to it fro m an A -projectiv e C -como dule P ( M ) constructed in Lemma 3.1.3(a). Let K b e k ernel of the map P ( M ) − → M and let P ( M ) − → C ⊗ A P ( M ) b e the C -coaction map. Set J ( M ) to b e the cok ernel of the comp osition K − → P ( M ) − → C ⊗ A P ( M ). Then the comp osition of maps P ( M ) − → C ⊗ A P ( M ) − → J ( M ) factor- izes thro ugh t he surjection P ( M ) − → M , so t here is a natural injectiv e mor phism o f C -como dules M − → J ( M ). The C -como dule J ( M ) is C / A -injectiv e as the coke rnel of an injectiv e map of C / A -injective C -como dules K − → C ⊗ A P ( M ). The cok ernel of the map M − → J ( M ) is isomorphic to the cok ernel of the map P ( M ) − → C ⊗ A P ( M ) and hence A -projectiv e. P ar t (a) is prov en; the construction of the surjectiv e morphism of C -contramo dules F ( P ) − → P in part (b) is completely analogous.  Let us ﬁrst decomp ose functorially an arbitrary morphism of complexes of left C -como dules L • − → M • in to a coﬁbration follow ed b y a ﬁbration. This can b e done 170 in either of tw o dual w a ys. Let us start with a surjectiv e morphism P + ( M • ) − → M • on to the complex M • from a complex of A -projectiv e left C -como dules P + ( M • ) constructed a s in the pro of of Theorem 2.5. Let K • b e the ke rnel of the morphism L • ⊕ P + ( M • ) − → M • and let K • − → J + ( K • ) b e an inj ective morphism from the complex K • in to a complex of C / A - injectiv e left C - como dules J + ( K • ) constructed in the analog ous wa y using Lemma 2. The cok ernel of this morphism is a complex of A -projectiv e C - como dules. Let E • denote the ﬁb ered copro duct o f L • ⊕ P + ( M • ) and J + ( K • ) ov er K • . There is a natur a l morphism of complexes E • − → M • whose comp osition with the morphism J + ( K • ) − → E • is zero and comp osition with the morphism L • ⊕ P + ( M • ) − → E • is equal to o ur morphism L • ⊕ P + ( M • ) − → M • . The morphism L • − → M • is equal to the comp osition L • − → E • − → M • . The cokerne l of the morphism L • − → E • is an extension of t he coke rnel of the morphism K • − → J + ( K • ) and the complex P + ( M • ), hence a complex o f A -projectiv e C -como dules. The kernel o f the morphism E • − → M • is isomorphic to J + ( K • ), which is a complex of C / A -injectiv e C -como dules. Another wa y is to start with an injectiv e morphism L • − → J + ( L • ) a nd consider the coke rnel of t he morphism L • − → M • ⊕ J + ( L • ). No w let us construct a factorization of the morphism L • − → M • in to a coﬁbration follo w ed b y a trivial ﬁbration. Represen t the kerne l of the morphism E • − → M • as the quotien t complex of a complex of A -pro jectiv e left C - como dules E • 1 b y a complex of C / A -injectiv e C -como dules; represen t the latter complex as the quotien t complex of a complex E • 2 with the analogous prop erties, etc. The complexes E • i are also complexes of C / A -injectiv e C -como dules a s extensions o f complexes of C / A -injec- tiv e C -como dules. F or d large enough, the k ernel Z • of the morphism E • d − → E • d − 1 will b e a complex of A - pro jectiv e C -como dules. Actually , E • i and Z • are complexes of copro j ectiv e C -como dules, as a C / A -injectiv e A -projectiv e left C -como dule Q is copro jectiv e (since the injection of C - como dules Q − → C ⊗ A Q splits, Q − → C ⊗ A Q − → C ⊗ A Q / Q b eing an exact triple of A -projectiv e C -como dules). Let K • b e the total complex of the bicomplex Z • − → E • d − → · · · − → E • 1 − → E • . Then the morphism L • − → M • factorizes through K • in a natural w ay , the k ernel of the mo r phism K • − → M • is a coacyclic complex of C / A -injectiv e C -como dules, and t he cok ernel of the morphism L • − → K • is a complex of A - pro jectiv e C -como dules. Notice that the complex K • is the cone of the natural morphism L 1 (k er( E • → M • )) − → E • , where L 1 denotes the functor f r o m the pro of of Theorem 4.5. Finally , let us construct a factorization of the morphism L • − → M • in to a trivial coﬁbration follow ed by a ﬁbra t io n. Represen t the cok ernel of the morphism L • − → E • as a sub complex of a complex of C / A -injectiv e left C -como dules 1 E • suc h that the quo- tien t complex is a complex of A -projectiv e C -como dules; represen t this quotien t com- plex as a sub complex of a complex 2 E • with the analog ous prop erties, etc. The com- plexes i E • are also complexes of A -projectiv e C -como dules as extensions of complexes of A -pro jectiv e C -como dules (so they ar e complexes of copro jectiv e C - como dules). Let 171 K • b e the to tal complex of the bicomplex E • − → 1 E • − → 2 E • − → · · · , constructed b y taking inﬁnite direct sums along the diagonals. The n the morphism L • − → M • factorizes through K • in a na tural wa y , the cok ernel of the morphism L • − → K • is a coacyclic complex of A -projectiv e C - como dules, and the k ernel of the morphism K • − → M • is a complex of C / A -injectiv e C -como dules. The class of C / A -injectiv e C -como dules is closed under inﬁnite direct sums b y Lemma 5.3.2(a). P art (a) is prov en; the pro of of part (b) is completely analogous.  Remark. It fo llows from the pro of of L emma 2 that a n y C / A -injectiv e left C -como dule can b e obta ined from coinduced C -como dules b y taking extensions, cok- ernels o f injectiv e morphisms, a nd direct summands. Analogously , any C / A -proj ec- tiv e left C -contramo dule can b e obt a ined from induced C - contramo dules by taking extensions, k ernels of surjectiv e morphisms, and direct summands. Let C and D b e t w o corings satisfying t he ab o v e assumptions and C − → D b e a map of corings compatible with a k -alg ebra map A − → B . Then the pair of adjoin t functors M • 7− → B M • and N • 7− → C N • is a Quillen adjunction [26] from the catego ry of complexes of left C -como dules to the category of complexes of left D -como dules; the pair of adjoin t functors Q • 7− → C Q • and P • 7− → B P • is a Q uillen adjunction from the category of complexes of left D -contramo dules t o the category o f complexes of left C -contramo dules. The same applies t o the case of a Morita morphism ( E , E ∨ ) from A to B and a morphism B C B − → D of corings ov er B . The pair of adjoint functors Φ C and Ψ C applied to complexe s term-wise is not a Quillen equiv alence, a nd not ev en a Quillen adjunction, b et w een t he mo del category of complexes o f left C -contramo dules and the mo del category of complexes of left C -como dules. This pair o f f unctor s is a Quillen equiv alence, ho w ev er, when C is a coring ov er a semisimple ring A . In general, the mo del categor ies of complexes of left C -como dules and left C -contramo dules can b e connected by a chain of three Quillen equiv alences (see Remark 9 .2.2). 9.2. Complexes of semimodules and semicon tramo dules. Let S b e a semial- gebra ov er a coring C o v er a k -algebra A . Assume that C is a pro jectiv e left and a ﬂat right A -mo dule, S is a copro jectiv e left and a coﬂat righ t C -como dule, and the ring A has a ﬁnite left homolo g ical dimension. A left S -semimo dule L is called S / C / A -pr oje ctive if the functor of S -semimo dule homomorphisms from L maps exact tr iples of C / A -injectiv e left S -semimo dules to exact triples. An A -proj ective left S -semimo dule L is called S / C / A -sem i p r oje ctive if the functor of semihomomorphisms from L ov er S maps exact triples of C / A -coinjec- tiv e left S -semicontramo dules to exact triples. Analogously , a left S -semicontramo d- ule Q is called S / C / A -inj e ctive if the functor o f S -semicontramo dule homomorphisms in to Q maps exact tr iples of C / A -projectiv e left S -semicontramo dules to exact triples. An A -injectiv e left S - semicontramo dule Q is called S / C / A -sem iinje ctive if the functor 172 of semihomomorphisms into Q o v er S maps exact triples of C / A -coprojectiv e left S -semimodules to exact triples. As in Lemma 6.4, it follows from Prop osition 6.2.2(c) that an A -pr o jectiv e left S -semimodule is S / C / A -pro j ectiv e if and only if it is S / C / A - semiprojectiv e. Analo- gously , it follows fro m Prop osition 6.2 .3(c) that an A -injectiv e left S -semicontramo d- ule is S / C / A -injectiv e if and only if it is S / C / A - semiinjectiv e. It will b e shown b elow that an y S / C / A - projectiv e left S - semimo dule is A -projectiv e and any S / C / A - injectiv e left S -semicontramo dule is A -injectiv e. Theorem. (a) The c ate gory of c omplexes of left S -semi m o dules has a close d mo del c ate go ry structur e wi th the fol low ing pr op erties. A morp h ism is a we ak e q uivalenc e if a nd only if its c one is C -c o ac ycli c . A morphism is a c oﬁbr ation if and only i f it is inje ctive and its c okernel is an S / C / A -pr oje ctive c omplex of S / C / A -pr oje ctive S -semimo dules. A morph ism is a ﬁb r ation if and on l y if it is surje ctive and its kernel is a c omplex of C / A -inje ctive S -semimo dules. An obje ct is si m ultane ously ﬁbr ant and c oﬁbr ant if and only if it is a se m ipr oje ctive c om plex of semipr oje ctive left S -semi- mo dules. (b) Th e c ate gory of c omplexes of le f t S -s e mic ontr amo dules has a c lose d mo del c at- e gory structur e with the fol lowing pr op erties. A morphism is a we ak e quivalenc e if and only if its c one is C -c ontr aacyclic. A morphism is a c oﬁ br ation if and only if it is inje ctive and its c okernel is a c om plex of C / A -pr oje ctive S -semic ontr amo dules. A morphism is a ﬁbr ation if and only if it is inje ctive and its c okernel is an S / C / A -in- je ctive c om plex of S / C / A -inje ctive S -semic on tr amo d ules. An obje c t is simultane ously ﬁbr ant and c oﬁbr ant if and only if it is a sem iinje ctive c omp lex of semiinje ctive left S -semic ontr amo dules. Pr o of . P art (a): existence of limits and colimits, the t w o-out-of- three prop erty of w eak equiv alences, and the retraction prop erties are v eriﬁed as in the pro o f o f The- orem 9.1. It is clear that a morphism is a trivial coﬁbra t io n if and only if it is injectiv e and its cok ernel is a C -coacyclic S / C / A -projectiv e complex of S / C / A - projec- tiv e S -semimo dules, and a mo r phism is a trivial ﬁbratio n if and only if it is surjectiv e and its k ernel is a C - coacyclic complex of C / A -injectiv e S -semimo dules. T o c hec k the lifting prop erties, use Lemma 9.1.1. Consider an extension E • − → M • − → K • of an S / C / A -projectiv e complex o f S / C / A -pro jectiv e left S -semimo dules K • b y a complex of C / A -injective left S - semimo dules E • . By the next Lemma 1, this extension is term- wise split, so it comes fro m a morphism of complexes of S -semimo dules K • − → E • [1]. No w supp ose that one of the complexes K • and E • is C -coacyclic. Then an y mor - phism K • − → E • [1] is homotopic to zero b y a result of 6.5 , hence the extension o f complexes E • − → M • − → K • is split. So after Lemma 1 is pro v en it will remain to construct the functorial factor izations. 173 Lemma 1. (a) A left S -semimo dule L is S / C / A -p r oje ctive if a nd o nly if for any C / A -inje ctive left S -semim o dule M the k -mo d ules Ext i S ( L , M ) of Y one da e x tensions in the ab el i a n c ate gory of left S -semimo dules va n ish for al l i > 0 . The functor of S -semimo dule homomorphisms into a C / A -inje ctive S -semim o dule maps exact triples of S / C / A -pr oje ctive left S -semimo dules to exact triples. The classes of S / C / A -pr o- je ctive left S -semimo dules a nd S / C / A -pr oje ctive c omplexes of S / C / A -pr oje c tive l e ft S -semimo dules ar e close d under ex tensions and k e rn els o f surje ctive morphisms. (b) A left S -semic on tr amo dule Q is S / C / A -inje ctive if and only if fo r an y C / A -pr o- je ctive left S -semic o n tr amo d ule P the k -m o dules Ext S ,i ( P , Q ) of Y one da extensi o ns in the ab e lian c ate gory o f lef t S -s e mic ontr amo dules vanish for a l l i > 0 . The functor of S -semic ontr amo dule homomo rp hisms fr om a C / A -pr oje ctive S -semic ontr amo dule maps ex a c t triples of S / C / A -inje ctive left S -sem ic ontr amo dules to exact triples . The classes of S / C / A -inje ctive left S -semi c ontr amo dules and S / C / A -i n je ctive c omplexes of S / C / A -inje ctive left S -semic ontr amo dules ar e close d under extension s an d c oker- nels of inje ctive morphisms. Pr o of . P art (a): the forgetful functor S – simo d − → C – como d preserv es injectiv e ob- jects, since it is righ t adjoint t o the exact functor of induction. Let us sho w that an y C / A - injectiv e left S -semimo dule M is a subsemimodule of an injectiv e S -semi- mo dule (it will f ollo w that the category of left S -semimo dules has enoug h injectiv es). The construction of Lemma 3.3.2 ( b) assigns to a C / A -coinjectiv e left S - semicontra- mo dule P an injectiv e map from it in to a semiinjectiv e S -semicontramo dule I ( P ) with a C / A -coinjectiv e cok ernel. Indeed, the cok ernel of the map P − → I ( P ) is a C / A -coinjectiv e C -contramo dule b y Lemma 3.1.3(b), so I ( P ) is a C / A -coinjectiv e C -contramo dule as an extension of t w o C / A -coinjectiv e C -contramo dules and a coin- jectiv e C -contramo dule as an A -injectiv e and C / A - coinjectiv e C -contramo dule. Hence I ( P ) = Cohom C ( S , I ( P )) is a semiinjectiv e S -semicontramo dule. The cok ernel of the comp o sition of injectiv e morphisms P − → Cohom C ( S , P ) − → Cohom C ( S , I ( P )) is an extension o f the cokerne l of the morphism Cohom C ( S , P ) − → Cohom C ( S , I ( P )) and the cokernel of the mor phism P − → Cohom C ( S , P ); the former is C / A -coinjec- tiv e since the coke rnel of the morphism P − → I ( P ) is, and the latter is C / A -coin- jectiv e as a C -contramo dule direct summand of Cohom C ( S , P ). Hence the cok ernel of the morphism P − → I ( P ) is C / A -coinjectiv e. Applying these observ ations to the S - semicontramo dule P = Φ S ( M ) and using Lemmas 5 .3.2(b) and 5.3.1(c), w e conclude that M − → Φ S I (Ψ S M ) is an injectiv e morphism of S -semimo dules when- ev er M is a C / A - injectiv e left S -semimo dule. Now the f unctor Φ S maps semiinjectiv e S -semicontramo dules to injective S -semimo dules b y Prop osition 6.2.2(a). So an y C / A -injectiv e left S -semimo dule M has an injectiv e right resolution; b y the construction or by Lemma 5.3.1(a), this resolution is exact with resp ect to the exact category of C / A -injectiv e S -semimo dules. Applying to this resolution the functor of S -semimo dule homomorphisms from an S / C / A -pro j ectiv e left S - semimo dule L , 174 w e obta in the desired v anishing Ext i S ( L , M ) = 0 for all i > 0. The remaining assertions fo llow (to v erify the assertions related to complexes , notice that the class of acyclic complexes o f k -mo dules is closed under extensions and coke rnels o f injectiv e morphisms). P art (a) is prov en; the pr o of of pa r t (b) is completely analogous and based on the construction o f a semicon tra mo dule pro jectiv e resolution. Alternativ ely , one can argue as in the pro of o f Lemma 5.3.1(a-b).  The analogous results for S / C / A - semiprojectiv e ( complexes of ) left S -semimo dules and S / C / A -semiinjectiv e (complexes of ) left S -semicontramo dules can b e obtained b y considering the deriv ed functor SemiExt ∗ S ( M , P ), deﬁned as the cohomology of the ob ject SemiExt S ( M , P ) of D ( k – mo d ). F o r an A -projectiv e S -semimo dule M and a C / A -coinjectiv e S -semicontramo dule P or a C / A -coprojectiv e S -semimo dule M and an A -injective S -semicontramo dule P it is computed b y the cobar-complex Cohom C ( M , P ) − → Coho m C ( S  C M , P ) − → · · · , hence SemiExt i S ( M , P ) = 0 f o r i > 0 and SemiExt 0 S ( M , P ) = SemiHom S ( M , P ). Lemma 2. (a) Ther e exists a (not always additive) functor a s signing to any left S -semimo dule an inje ctive mo rp hism fr om it into a C / A -inje ctive S -semimo dule with an S / C / A -pr oje ctive c okerne l . F urthermor e, ther e exists a functor assigning to any c omple x of lef t S -semim o dules an inje ctive morphis m f r om it into a c omplex of C / A -inje ctive S -s e mimo dules such that the c okernel is a C -c o acyclic S / C / A -pr o- je ctive c ompl e x of S / C / A -pr oje ctive S -semim o dules. (b) Th er e exists a (not always additive) func tor assigning to any left S -semic on- tr amo d ule a s urje ctive morphi s m o nto it fr om a C / A -pr oje ctive S -sem ic ontr amo d- ule with an S / C / A -inje ctive kernel. F urthermor e, ther e exists a functor assigning to any c om plex of left S -semic ontr amo dules a surje ctive morphism onto it fr om a c omplex of C / A -pr oje ctive S -semic on tr amo d ules s uch that the kernel is a C -c ontr a- acyclic S / C / A -inje ctive c omplex of S / C / A -inje ctive S -semi c ontr amo dules. Pr o of . P art (a): mo dify the construction of the sec ond assertion of Lemma 1.3.3, replacing the injectiv e morphism of C -como dules M − → G ( M ) = C ⊗ A M with the injectiv e morphism of C - como dules M − → J ( M ) constructed in Lemma 9.1.2(a). In other w ords, for any left S -semimo dule M denote by K ( M ) t he k ernel of the mor- phism S  C M − → M and b y Q ( M ) the cok ernel of the comp osition K ( M ) − → S  C M − → S  C J ( M ). The comp osition of maps S  C M − → S  C J ( M ) − → Q ( M ) factorizes through the surjection S  C M − → M , so t here is a natural injectiv e morphism o f S -semimo dules M − → Q ( M ). The cok ernel of this morphism is isomorphic to S  C ( J ( M ) / M ), whic h is an S / C / A -projectiv e S -semimo dule b e- cause J ( M ) / M is an A - projectiv e C -como dule. As in the pro of of Lemma 1.3.3, the S -semimo dule morphism M − → Q ( M ) can b e lif t ed to a C - como dule mor- phism M − → S  C J ( M ). Let J ( M ) denote the inductiv e limit of the se quence 175 M − → S  C J ( M ) − → Q ( M ) − → S  C J ( Q ( M )) − → Q ( Q ( M )) − → · · · ; it is the de- sired C / A -injectiv e S -semimodule into whic h M maps injectiv ely with an S / C / A -pro- jectiv e cok ernel. Indeed, J ( M ) is C / A - injectiv e b y Sublemma 3.3.3 .B a nd t he coke rnel of the morphism M − → J ( M ) is S / C / A -proj ectiv e b y the next Sublemma. Sublemma . (a ) L et 0 = U • 0 − → U • 1 − → U • 2 − → · · · b e an inductive system of c omplex e s o f left S - s e mimo dules such that the suc c essive c okernels coke r( U • i − 1 → U • i ) a r e S / C / A -pr oje ctive c o m plexes o f S / C / A -pr oje ctive S -semimo dules. Th en the inductive limit lim − → U • i is an S / C / A -pr oje ctive c omplex of S / C / A -pr oje ctive S -semi- mo dules. (b) L et 0 = U • 0 ← − U • 1 ← − U • 2 ← − · · · b e a pr oje ctive system of c omplexes of left S -sem ic ontr a mo dules such that the suc c essive kernels ke r ( U • i → U • i − 1 ) ar e S / C / A -inje ctive c omplexes of S / C / A -inj e ctive S -semim o dules. The n the pr oje ctive limit lim ← − U • i is an S / C / A -inje ctive c omplex of S / C / A -inje ctive S -semimo dules. Pr o of . The forgetful functor S – simo d − → A – mo d preserv es inductiv e limits, sinc e it preserv es cok ernels and inﬁnite direct sums, so one has Hom S (lim − → U • i , M • ) = lim ← − Hom S ( U • i , M • ) for any complex of left S -semimo dules M • . Analog ously , the forgetful functor S – sicntr − → A – mo d preserv es pro jectiv e limits, since it preserv es k ernels and inﬁnite pro ducts, so one has Hom S ( P • , lim ← − U • i ) = lim ← − Hom S ( P • , U • i ) for an y complex of left S -semicontramo dules P • . As the pro jectiv e limits of sequences of surjectiv e maps preserv e exact triples and acyclic complexes, the assertions of Sublemma follow from Lemma 1.  The ﬁrst statemen t of Lemma 2(a) is pro v en. T o prov e the second one, con- sider the f unctor a ssigning to a complex of left S -semimo dules M • the injective map fro m it in to the complex J + ( M • ), whic h is constructed in terms of the functor M 7− → J ( M ) as in the pro of of Theorem 2 .5. By Sublemma, the cok ernel of the mor- phism M • − → J + ( M • ) is an S / C / A -projectiv e complex of S / C / A - projectiv e S -semi- mo dules, since a complex of S -semimo dules induced from a complex of A -projectiv e C -como dules b elongs to this class. Set 0 J • = J + ( M • ), 1 J • = J + (cok er( M • → 0 J • )), etc. The complexes i J • are complexes of C / A - injectiv e S -semimo dules and the com- plexes coke r( M • → 0 J • ), cok er( i − 1 J • → i J • ) are S / C / A -projectiv e complexes of S / C / A -projectiv e S -semimo dules. The complexes i J • for i > 0 a re also S / C / A -pro - jectiv e complexes of S / C / A -projectiv e S -semimo dules as extensions of complexes with these prop erties. Let K • b e the to tal complex of the bicomplex 0 J • − → 1 J • − → · · · , constructed b y taking inﬁnite direct sums along the diagonals. Then K • is a com- plex of C / A -injectiv e S -semimo dules and the cokerne l of the injectiv e morphism M • − → K • is a C -coacyclic (and ev en S - coacyclic) S / C / A -projectiv e complex of S / C / A -projectiv e S -semimo dules. T o chec k the latter prop erties, one can apply Sub- lemma to the canonical ﬁltration of the complex 0 J • / M • − → 1 J • − → 2 J • − → · · · 176 The pro of of Lemma 2(b) is completely analogous and based on the mo diﬁcation of the construction of the second assertion of Lemma 3.3.3(a) using the surjectiv e morphism of C -contramo dules F ( P ) − → P from Lemma 9.1.2(b) in place of the morphism G ( P ) = Hom A ( C , P ) − → P .  In the sequel w e will denote by M 7− → J ( M ) the functor constructed in Lemma 2 rather than its more simplistic v ersion from Lemmas 1 .3.3 and 3.3.3. Lemma 3. (a) Ther e exists a (not always additive) functor a s signing to any left S -semimo dule a surje ctive ma p on to it fr o m an S / C / A -pr oje ctive S -semimo dule w ith a C / A -inje ctive kernel. F urthermor e, ther e exists a functor assi g ning to any c omplex of left S -sem i mo dules a surje ctive map onto it fr om an S / C / A -pr oje c tive c omplex of S / C / A -pr oje ctive S -semimo dules such that the kernel is a C -c o acyclic c o m plex of C / A -inje ctive S -semimo dules. (b) Ther e exists a (not always additive) functor as s igning to any left S -semic ontr a- mo dule an inje ctive ma p fr om it into an S / C / A -inje ctive S -semic ontr amo dule with a C / A -pr oje ctive c okernel. F urthermor e, ther e exists a functor as s i g ning to any c omplex of left S -semic ontr amo dules an inje ctive map fr om it in to an S / C / A -inje ctive c omp le x of S / C / A -inje ctive S -semic ontr amo dules such that the c oke rnel is a C -c ontr aacyclic c omplex of C / A -pr oje ctive S -sem i c ontr amo dules. Pr o of . P art (a): fo r any left S -semimo dule L , consider the inj ective morphism L − → J ( L ) from Lemma 2 and denote by K ( L ) its cok ernel. The functor M 7− → P ( M ) of Lemmas 1.3.2 and 3.3.2 assigns to a C / A -injectiv e left S - semimo dule M a surjectiv e mo r phism o n to it fro m the S -semimo dule P ( M ) induced from a copro- jectiv e C - como dule P ( M ) suc h that the k ernel is a C / A -injectiv e S -semimo dule (see the pro of of Lemma 1). Denote by F ( L ) the k ernel of the comp osition P ( J ( L )) − → J ( L ) − → K ( L ). The comp osition of maps F ( L ) − → P ( J ( L )) − → J ( L ) factor- izes through the injection L − → J ( L ), so there is a natural surjectiv e morphism of S -semimodules F ( L ) − → L . The S -semimo dules P ( J ( L )) and K ( L ) are S / C / A -pro- jectiv e, hence the S -semimo dule F ( L ) is also S / C / A -projectiv e. The k ernel of the morphism F ( L ) − → L is C / A -injectiv e, since it is isomorphic to the k ernel of the morphism P ( J ( L )) − → J ( L ). This pro v es the ﬁrst statemen t of part (a). No w consider the functor assigning to any complex of left S -semimo dules L • the surjectiv e map onto it from the complex F + ( L • ). The complex F + ( L • ) is S / C / A -pro- jectiv e as the k ernel of a surjectiv e morphism of S / C / A -projectiv e complexes; it is a lso a complex of S / C / A - pr o jectiv e a nd A - pro jectiv e S -semimo dules. Set F • 0 = F + ( L • ), F • 1 = F (ker( F • 0 → L • )), etc. The complexes k er( F • 0 → L • ), k er( F • i → F • i − 1 ) are complexes of C / A -injectiv e S -semimo dules, hence the complexes F • i for i > 0 are also complexes of C / A -injectiv e S - semimo dules as extensions of complexes o f C / A -injectiv e S -semimodules. F or d large enoug h, the k ernel Z • of the morphism F • d − 1 − → F • d − 2 will be a complex of A - projectiv e S -semimodules, and consequen tly a complex of 177 C -coprojectiv e S -semimo dules. Let E • b e the tota l complex of the bicomplex · · · − − → S  C S  C Z • − − → S  C Z • − − → F • d − 1 − − → F • d − 2 − − → · · · − − → F • 0 , constructed b y ta king inﬁnite direct sums a lo ng the diagonals. Then the complex E • is a complex of S / C / A -projectiv e S -semimo dules, and it is an S / C / A - projectiv e complex since it is homoto py equiv alen t to a complex obtained from suc h complexes using the op erations of cone and inﬁnite direct sum. The k ernel of the morphism E • − → L • is a complex of C / A -injectiv e S -semimodules, and it is C -coacyclic since it con t a ins a C -contractible subcomplex o f S -semimo dules suc h that the quotien t complex is the total complex of a ﬁnite exact complex of complexes of S -semimo d- ules. P art (a) is prov en; the pro o f of part (b) is completely analogous.  Let us show t ha t any S / C / A -projectiv e left S - semimo dule L is A - projectiv e. Con- sider the surjectiv e mor phism F ( L ) − → L fro m Lemma 3. Since its ke rnel is C / A -injectiv e, we hav e an extension of an S / C / A -projective left S -semimo dule by a C / A -injectiv e left S -semimo dule, whic h is alw a ys trivial b y Lemma 1. Therefore, L is a direct summand o f F ( L ), while F ( L ) is A -projectiv e by the construction. Analogously , any S / C / A -injectiv e left S -semicontramo dule is A -injectiv e. Let us sho w that any S / C / A -projectiv e C / A - injectiv e left S -semimo dule M is a di- rect summand of the S -semimo dule induced fro m the C -como dule coinduced from a pro jectiv e A - mo dule; in particular, a left S -semimo dule is sim ultaneously S / C / A - pr o - jectiv e and C / A -injectiv e if and only if it is semipro jectiv e. Consider the exact triple K − → S  C M − → M , where K = k er( S  C M → M ). If a n S -semimo dule M is C / A -injectiv e, then so is the S -semimo dule S  C M , since C / A -injectivit y is equiv alen t to C / A -coprojectivity ; then the S -semimo dule K is C / A -injectiv e a s a C -como dule direct summand of S  C M . If the S -semimo dule M is also S / C / A -projectiv e, then our exact triple splits o v er S and M is a direct summand o f the induced S -semi- mo dule S  C M . Since the C -como dule M is A -projectiv e and C / A -injectiv e, it is a direct summand of the C -como dule coinduced from a pro jectiv e A -mo dule. Anal- ogously , an y S / C / A -injectiv e C / A -projectiv e left S -semicontramo dule P is a direct summand of the S - semicontramo dule coinduced from the C -contr amo dule induced from a n injectiv e A - mo dule; in particular, a left S -semicontramo dule is sim ultane- ously S / C / A -injectiv e and C / A -projective if and only if it is semiinjectiv e. In other w ords, M is a direct summand of a direct sum of copies of t he S -semimodule S and P is a direct summand of a pro duct of copies of the S -semicontramo dule Hom k ( S , k ∨ ). An S / C / A -projectiv e complex of C -coprojectiv e left S -semimo dules M • is homo- top y equiv alen t to a complex obtained from complexes of S -semimo dules induced from complexes of C -coprojectiv e C - como dules using the op erations of cone and inﬁnite di- rect sum. In particular, the complex M • it is a semipro jectiv e. Indeed, the total com- plex of the bicomplex · · · − → S  C S  C M − → S  C M − → M is contractible, b eing 178 a C -coacyclic S / C / A - projectiv e complex of C / A -injectiv e left S -semimo dules. Analo- gously , an S / C / A -injectiv e complex of C -coinjectiv e left S -semicontramo dules P • is homotop y equiv alen t t o a complex obt a ined from complexes of S -semicontramo dules coinduced from complexes o f C -coinjectiv e C -contramo dules using the o p erations of cone and inﬁnite pro duct. In particular, the complex P • is semiinjectiv e. Finally w e turn to the construction of functorial factorizations. As in the pro of of Theorem 9.1, let us ﬁrst decompose an arbitrary morphism of complexes of left S -semimodules L • − → M • in to a coﬁbration follow ed b y a ﬁbratio n. This can b e done in either of tw o dual w a ys. Let us start with an injectiv e morphism from the complex L • in to the complex J + ( L • ) constructed in Lemma 2. Let K • b e the cok ernel o f the morphism L • − → M • ⊕ J + ( L • ) a nd let F + ( K • ) − → K • b e the surjectiv e morphism onto the complex K • from the complex F + ( K • ) constructed in Lemma 3. Let L • − → E • − → F + ( K • ) b e the pull-bac k of the exact triple L • − → M • ⊕ J + ( L • ) − → K • with respect to the morphism F + ( K • ) − → K • . Then the morphism L • − → M • is equal to the comp osition L • − → E • − → M • . The cok ernel F + ( K • ) of the morphism L • − → E • is an S / C / A -projectiv e complex o f S / C / A - pro- jectiv e S -semimo dules. The k ernel of the morphism E • − → M • is an extension of the complex J + ( L • ) and the k ernel o f the morphism F + ( K • ) − → K • , hence a complex of C / A -injectiv e S -semimodules. Another w ay is to start with the surjectiv e morphism F + ( M • ) − → M • and consider the k ernel of the morphism L • ⊕ F + ( M • ) − → M • . No w let us construct a factorization of the morphism L • − → M • in to a coﬁbration follo w ed b y a trivial ﬁbration. Represen t the kerne l of the morphism E • − → M • as the quotien t complex of an S / C / A -projectiv e complex of S / C / A -proj ective left S -semimodules P • b y a C -coacyclic complex of C / A -injectiv e S - semimo dules. Then the complex P • is also a complex of C / A -injectiv e S -semimo dules (so it is ev en a semipro jectiv e complex of semipro jectiv e S -semimodules). Let K • b e the cone of the morphism P • − → E • . Then the morphism L • − → M • factorizes through K • in a natural w a y , t he k ernel of t he morphism K • − → M • is a C -coacyclic complex of C / A -injectiv e S -semimo dules, and the cokerne l of the morphism L • − → K • is an S / C / A -projectiv e complex o f S / C / A -pro jectiv e S -semimo dules. It remains to construct a factorization o f the morphism L • − → M • in to a triv- ial coﬁbration follow ed b y a ﬁbra t io n. Represen t the cok ernel o f the morphism L • − → E • as a sub complex of a complex of C / A -inj ective S -semimo dules Q • suc h that the quotien t complex is a C -coa cyclic S / C / A -pro j ectiv e complex of S / C / A -pro- jectiv e S -semimo dules. Then the complex Q • is a lso an S / C / A -projectiv e complex of S / C / A -projectiv e S -semimo dules (hence a semipro jectiv e complex of semipro jectiv e S -semimodules). Let K • b e t he co cone of the morphism E • − → Q • . Then the mor- phism L • − → M • factorizes through K • in a natural w ay , the k ernel of the morphism K • − → M • is a complex of C / A -injectiv e S -semimo dules, and the coke rnel of the 179 morphism L • − → K • is a C -coacyclic S / C / A -projectiv e complex of S / C / A -projectiv e S -semimodules. P art (a) of Theorem is pro v en; t he pro of o f part (b) is completely analogous.  Remark 1. One can obtain descriptions of S / C / A -pro j ectiv e complexes of S / C / A -projectiv e S -semimo dules, C -coacyclic S / C / A -projectiv e complexes of S / C / A -projectiv e S -semimo dules, etc., from the pro of of the ab ov e Theorem. Namely , let M • b e an S / C / A -projectiv e complex of S / C / A -projectiv e left S -semi- mo dules; decomp ose the morphism 0 − → M • in to a coﬁbration 0 − → K • follo w ed b y a trivial ﬁbration K • − → M • b y the ab ov e construction (this can b e a lso obtained directly from Lemma 3). Then the complex M • is a direct summand of K • and therefore can b e obta ined from complexes o f S -semimo dules induced fr o m complexes of A -projective C -como dules using the o p erations of cone, inﬁnitely iterated exten- sion in the sense of inductiv e limit, and k ernel of surjectiv e morphism. Let M • b e a C -coacyclic S / C / A - pro jectiv e complex of S / C / A -projectiv e left S -semimo dules; decomp ose the morphism 0 − → M • in to a trivial coﬁbration 0 − → K • follo w ed b y a ﬁbration K • − → M • b y the ab ov e construction. Then the complex M • is a direct summand of K • and therefore up to the homotop y equiv alence can b e obtained from the total complexes of exact triples of S / C / A -projective complexes of S / C / A - projectiv e S - semimo dules using the op erations of cone and inﬁnite direct sum. The analogo us results hold for complexes of left S -semicontramo dules. Let S and T b e t w o semialgebras satisfying the ab o v e assumptions and S − → T b e a map of semialgebras compatible with a map of corings C − → D and a k - algebra map A − → B . Then the pair of a dj o in t functors M • 7− → T M • and N • 7− → C N • is a Quillen adjunction from the category of complexes of left S -semimo dules to the categor y of complexes of left T - semimo dules; the pair of adjoint functors Q • 7− → C Q • and P • 7− → T P • is a Quillen a djunction from the category of complexes of left T - semi- contramo dules to the category of complexes of left S -semicontramo dules. These results follow from Theorems 7.2.1 and 8.2.3(c). They also hold in the case o f a left copro jectiv e and righ t coﬂat Morit a morphism ( E , E ∨ ) f rom C to D and a morphism S − → C T C of semialgebras ov er C . The pair of adjoint functors Φ S and Ψ S applied to complexes t erm- wise is not a Quillen equiv alence, a nd not ev en a Quillen adjunction, b et w een t he mo del category of complexes of left S - semicontramo dules and t he mo del category of complexes of left S -semimodules. Instead, this pair of adjoint functors b et w een closed mo del categories has the following prop erties. The functor Φ S maps t r ivial coﬁbrat io ns o f complexes of left S -semicontramo dules to we ak equiv alences of complexes o f left S -semimo dules. The functor Ψ S maps trivial ﬁbrations of complexes of left S -semimo dules to w eak equiv alences of complexes o f left S -semicontramo dules. F or a coﬁbrant complex of left S -semicontramo dules P • and 180 a ﬁbran t complex of left S -semimo dules M • , a morphism Φ S ( P • ) − → M • is a w eak equiv alence if and only if the corresp onding morphism P • − → Ψ S ( M • ) is a we ak equiv alence. F urthermore, the functor Φ S maps coﬁbrant complexes to ﬁbran t ones, while the functor Ψ S maps ﬁbrant complexes to coﬁbrant ones. The restrictions o f the functors Φ S and Ψ S are m utually inv erse equiv alences b et w een the full sub categories formed b y coﬁbran t complexes of left S -semicontramo dules and ﬁbrant complexes of righ t S - semimo dules. These restrictions of functors also send we ak equiv a lences to w eak equiv alences. Remark 2. One can connect the a b o v e mo del categories of complexes of left S - semi- mo dules and left S -semicontramo dules by a c hain of three Quillen adjunctions by considering other mo del category structures on these tw o catego r ies. The ab o v e mo del catego ry structures on the category of complexes o f left S - semimo dules can b e called the semipro jectiv e mo del structure, and the mo del category structure on the category of complexes o f left S -semicontramo dules can b e called the semiinjectiv e mo del structure. In addition to these, there is also the injectiv e mo del structure on the categor y of complexes of left S -semimodules, and the pro j ective mo del structure on t he category of complexes of left S -semicontramo dules. In these alternative mo del structures, w eak equiv alences ar e still morphisms with C -coacyclic or C -contra acyclic cones, resp ectiv ely . A morphism of complexes of semimo dules is a coﬁbration if a nd only if it is injectiv e, and a morphism of complexes of semicon tramo dules is a ﬁbration if and only if it is surjectiv e. A morphism of complexes o f semimo dules is a ﬁbration if a nd only if it is surjectiv e and its ke rnel is an injectiv e complex o f injectiv e semi- mo dules in the sense of Remark 6 .5 a nd the pro of of Lemma 1 ab ov e; a mor phism of complexes of semicon tramo dules is a coﬁbration if and only if it is injectiv e and its cok ernel is an pro jectiv e complex of pro jectiv e semicon tramo dules. One c heck s that these a mo del catego ry structures in the w ay analogous to (and m uc h simpler than) t he pro of of Theorem ab ov e. The functors Φ S and Ψ S are a Quillen equiv a- lence b et w een the inj ective and the pro jectiv e mo del category structures; the iden tity functors are Quillen equiv alences b et w een the semipro jectiv e a nd the injectiv e mo del structures, and b etw een the semiinjectiv e and the pro jectiv e mo del structures. 181 10. A Constru ction of Semialgebras 10.1. C onstr uction of como dules and contramodules. 10.1.1. Let A and B b e asso ciativ e k -algebras. F or an y pro jectiv e ﬁnitely-generated left A -mo dule U and any left A -mo dule V there is a natural isomorphism Hom A ( U, A ) ⊗ A V ≃ Hom A ( U, V ) given by the for- m ula u ∗ ⊗ v 7− → ( u 7→ h u, u ∗ i v ) . In particular, for a n y A - B -bimo dule C a nd an y pro jectiv e ﬁnitely-generated left B - mo dule D there are natural isomorphism s Hom A ( C ⊗ B D , A ) ≃ Hom B ( D , Hom A ( C , A )) ≃ Hom B ( D , B ) ⊗ B Hom A ( C , A ). It follows that there is a tensor an ti-equiv alence b et w een the tensor category of A - A -bimo dules that are pro jective and ﬁnitely-generated as left A - mo dules and the tensor category of A - A -bimo dules that are pro jectiv e and ﬁnitely-generated as righ t A -mo dules, giv en by the m utually-in v erse functors C 7− → Hom A ( C , A ) and K 7− → Ho m A op ( K , A ). Therefore, noncomm utativ e ring structures on a rig h t- pro jectiv e and ﬁnitely A - A -bimo dule K cor r esp o nd bijectiv ely to coring structures on the left- pro jectiv e a nd ﬁnitely-generated A - A -bimo dule Hom A op ( K , A ). On the other hand, for an y coring C o v er A there is a natural structure of a k -algebra on Hom A ( C , A ) together with a morphism of k -algebras A − → Hom A ( C , A ). F urthermore, let K b e a k -a lgebra endow ed with a k -algebra map A − → K suc h that K is a ﬁnitely-generated pro jective righ t A -mo dule, and let C = Hom A op ( K , A ) b e the corresp onding coring ov er A . Then the nat ural isomorphism N ⊗ A C = Hom A op ( K , N ) fo r a right A -mo dule N pro vides a bijectiv e corresp ondence b etw een the structures of r igh t K -mo dule and righ t C -como dule on N . Analogously , the natu- ral isomorphism Hom A ( C , P ) = K ⊗ A P for a left A -mo dule P pro vides a bijectiv e cor- resp ondence b etw een the structures of left K -module and left C -contramo dule on P . In other words, there are isomor phisms of ab elian categories como d – C ≃ mo d – K and C – contra ≃ K – mo d . 10.1.2. Here is a generalization of the situation w e j ust describ ed. Let C b e a coring o v er a k -algebra A and K b e a k -alg ebra endo w ed with a k -algebra map A − → K . Supp ose that w e are give n a pairing φ : C ⊗ A K − → A whic h is an A - A - bimo dule map satisfying the follo wing conditions of compatibility with the com ultiplication in C and the m ultiplication in K and with the counit o f C and the unit of K . First, the comp osition C ⊗ A K ⊗ A K − → C ⊗ A C ⊗ A K ⊗ A K − → C ⊗ A K − → A of the map induced b y the com ultiplication in C , t he ma p induced b y the pairing φ , and the pairing φ itself should b e equal to the comp osition C ⊗ A K ⊗ A K − → C ⊗ A K − → A of the ma p induced b y the mu ltiplication in A and the pairing φ . Second, the comp osition C = C ⊗ A A − → C ⊗ A K − → A of the map coming from the unit of K with the pairing φ should b e equal t o t he counit of C . Equiv alently , the map K − → Hom A ( C , A ) induced by φ should b e a morphism of k -algebras. 182 Then for an y rig ht C - como dule N the comp osition N ⊗ A K − → N ⊗ A C ⊗ A K − → N of the map induced by t he C -coaction in N and the map induced b y t he pairing φ deﬁnes a structure of right K -mo dule on N . Analo g ously , for an y left C -contramo dule P the comp osition K ⊗ A P − → Hom A ( C , P ) − → P of the map giv en b y the form ula k ′ ⊗ p 7− → ( c 7→ φ ( c, k ′ ) p ) a nd the C - con traaction map deﬁnes a structure of left K -mo dule on P . So the pairing φ induces faithful f unctors ∆ φ : como d – C − → mo d – K and ∆ φ : C – contra − → K – mo d . In particular, a pairing φ prov ides the coring C with a structure of left C -como dule endo w ed with a r ig h t action of the k -a lgebra K b y C -como dule endomorphisms. Moreo v er, the data of a right a ction of K by endomorphisms of the left C -como dule C a greeing with the right action of A in C is equiv alen t to the da ta of a pairing φ . 10.1.3. When C is a pro jectiv e left A - mo dule, the functor ∆ φ has a left adj o in t functor Γ φ : K – mo d − → C – contra . This functor sends the induced left K -mo dule K ⊗ A V to the induced left C -contra mo dule Hom A ( C , V ) f o r an y left A -mo dule V ; to compute Γ φ ( M ) for an a r bit r a ry left K -mo dule M , one can respresen t M as the cok ernel of a morphism of K -mo dules induced from A -mo dules. Analogously , when C is a ﬂat r ig h t A -mo dule, the functor ∆ φ has a righ t adj o in t functor Γ φ : mo d – K − → como d – C . This functor sends the coinduced righ t K -mo dule Hom A op ( K , U ) to the coinduced righ t C -como dule U ⊗ A C for an y r ig h t A -mo dule U ; t o compute Γ φ ( N ) for an arbitrar y right K -module N , one can represen t N as the k ernel of a morphism of K -mo dules coinduced from A -mo dules. Without any conditions on the coring C , the comp osition o f functors Ψ C : C – como d − → C – contra and ∆ φ : C – contra − → K – mo d has a left a dj o in t functor K – mo d − → C – como d mapping a left K -mo dule M to the left C -como dule C ⊗ A M . Ana lo gously , the comp osition of the functors Φ C op : contra – C − → como d – C and ∆ φ : como d – C − → mo d – K has a right adjoin t functor mo d – K − → contra – C mapping a right K -mo dule N to the righ t C - contr amo dule Hom K op ( C , N ). So one can compute the comp ositions of functors Φ C Γ φ and Ψ C op Γ φ in this w a y . 10.1.4. It is easy to see that the functor ∆ φ is fully faithful whenev er for any righ t A -mo dule N the map N ⊗ A C − → Hom A op ( K , N ) given b y the formula n ⊗ c 7− → ( k ′ 7→ nφ ( c, k ′ )) is injectiv e. In particular, when A is a semisimple ring, the functor ∆ φ is fully faithful if the map C − → Hom A op ( K , A ) induce d b y the pairing φ is injectiv e, i. e., the pair ing φ is nondegenerate in C . 10.2. C onstr uction of semialgebras. 10.2.1. Assume that a coring C ov er a k -algebra A is a ﬂat left A -mo dule. Let K b e a k -a lg ebra endo w ed with a k -algebra map A − → K and a pairing φ : C ⊗ A K − → A satisfying the conditions of 10.1 .2, and let R b e a k - algebra endo w ed with a k -algebra map f : K − → R suc h that R is a ﬂat left K -mo dule. Then the tensor pro duct C ⊗ K R 183 is a coﬂat left C - como dule endo w ed with a right a ction of the k - algebra K (and ev en of the k -a lgebra R ) b y left C -como dule endomorphisms. Supp ose that t here exists a structure o f righ t C -como dule on C ⊗ K R inducing the existing structure of right K -mo dule and such that the follow ing three maps are righ t C -como dule morphisms: (i) the left C -coaction map C ⊗ K R − → C ⊗ A ( C ⊗ K R ), (ii) the se miunit map C = C ⊗ K K − → C ⊗ K R , and (iii) the semim ultiplication map ( C ⊗ K R )  C ( C ⊗ K R ) ≃ C ⊗ K R ⊗ K R − → C ⊗ K R , where the isomorphism in (iii) is the in v erse of the natura l isomorphism of Prop osition 1.2.3 ( a ) and the map b eing comp osed is induced b y the m ultiplication in R . Then the semiunit a nd semim ultiplication maps (ii) and (iii) deﬁne a semialgebra structure o n the C - C - bico- mo dule S = C ⊗ K R . Notice that the maps (i- iii) alw ays pres erv e the righ t K -mo dule structures. If the functor ∆ φ is fully faithful, then a rig h t C -como dule structure inducing a giv en righ t K -mo dule structure on C ⊗ K R is unique prov ided that it exists, and the maps (i-iii) preserv e this structure automatically . If the functor ∆ φ is an equiv alence of categories, then a unique righ t C -como dule structure with the desired prop erties exists on C ⊗ K R . The asso ciativit y of semim ultiplication in S follows from the asso ciativity of m ul- tiplication in R a nd the commu tativity of diagrams of asso ciativit y isomorphisms of cotensor pro ducts. 10.2.2. By Pro p osition 1.2.3(a), for an y righ t C -como dule N there is a natural iso- morphism N  C S ≃ N ⊗ K R , hence ev ery right S -semimo dule has a natural structure of righ t R -mo dule. So there is a faithful exact functor ∆ φ,f : simo d – S − → mo d – R whic h agrees with t he functor ∆ φ : como d – C − → mo d – K . Moreov er, t he category of righ t S -semimo dules is isomorphic to the category of k -mo dules N endo w ed with a rig h t C -como dule and righ t R -mo dule structures satisfying the following compati- bilit y conditions: ﬁrst, the induced righ t K -mo dule structures should coincide, and second, the a ctio n map N ⊗ K R − → N should b e a morphism of rig h t C -como dules, where the right C - como dule structure o n N ⊗ K R is provided b y the isomorphism N ⊗ K R = N  C S . When the functor ∆ φ is fully faithful, the category simo d – S is simply describ ed as t he full subcatego ry of the category of right R -mo dules consist- ing o f t hose mo dules whose rig h t K -mo dule structure comes from a right C -como dule structure. Analogously , if C is a pro jec tiv e left A -mo dule and R is a pro jectiv e left K -mo dule, then S is a copro jectiv e left C -como dule and by Propo sition 3.2.3.2(a) fo r an y left C -contramo dule P there is a natural isomorphism Cohom C ( S , P ) ≃ Ho m K ( R, P ), hence an y left S -semicontramo dule has a natural structure of left R -mo dule. So there is a faithful exact functor ∆ φ,f : S – sicntr − → R – mo d whic h agrees with the functor ∆ φ : C – como d − → K – mo d . Moreo v er, the category of left S -semicontramo dules is 184 isomorphic to the category of k -mo dules P endow ed with a left C -contramo dule and a left R -mo dule struc tures satisfying the follo wing compatibilit y conditions: ﬁrst, the induced left K - mo dule structures should coincide, and second, the action map P − → Hom K ( R, P ) should b e a morphism of C -contramo dules, where the left C - con- tramo dule structure o n Hom K ( R, P ) is pro vided b y the isomorphism Hom K ( R, P ) = Cohom C ( S , P ). 10.2.3. When K is a pro jectiv e ﬁnitely-generated right A -mo dule and the pairing φ corresp onds to an isomorphism C ≃ Hom A op ( K , A ), the isomorphisms of categories ∆ φ : como d – C ≃ mo d – K and ∆ φ : C – contra ≃ K – mo d transform the functor of con- tratensor pro duct ov er C in to the functor of tensor pro duct o v er K : N ⊙ C P ≃ N ⊗ K P . Indeed, Hom A ( C , P ) = K ⊗ A P . When in addition R is a pro jectiv e left K -mo dule, the isomorphisms of categories ∆ φ,f : simo d – S ≃ mo d – R and ∆ φ,f : S – sicntr ≃ R – mo d transform the functor of contratensor pro duct ov er S in to the functor of tensor pro d- uct o v er R : N ⊚ S P ≃ N ⊗ R P . Indeed, N  C S = N ⊗ K R . 10.2.4. The functor ∆ φ,f has a r ig h t adjoin t functor Γ φ,f : mo d – R − → simo d – S , whic h agr ees with the f unctor Γ φ : mo d – K − → como d – C . The functor Γ φ,f is con- structed a s follow s. Let N b e a right R -mo dule; it has an induced righ t K -mo dule structure. Consider the comp osition ∆ φ (Γ φ ( N )  C S ) = ∆ φ Γ φ ( N ) ⊗ K R − → N ⊗ K R − → N of the isomorphism of Proposition 1.2.3(a), the map induced b y the adjunction map ∆ φ Γ φ ( N ) − → N , a nd the righ t action map. By adjunction, this comp osition corresp onds to a right C -como dule morphism Γ φ ( N )  C S − → Γ φ ( N ), whic h prov ides a right S -semimo dule structure on Γ φ ( N ). Analogously , if C is a pro jec tiv e left A -mo dule and R is a pro jectiv e left K -mo dule, then the functor ∆ φ,f has a left adjoint functor Γ φ,f : R – mo d − → S – sicntr , whic h agrees with the functor Γ φ : K – mo d − → C – contra . The functor Γ φ,f is constructed as fo llows. Let P b e a left R -mo dule; it has an induced left K -mo dule struc- ture. Consider the comp osition P − → Hom K ( R, P ) − → Hom K ( R, ∆ φ Γ φ ( P )) = ∆ φ (Cohom C ( S , Γ φ ( P ))) of the action ma p, the map induced b y the adjunction map P − → ∆ φ Γ φ ( P ), and the isomorphism of Prop osition 3.2.3.2( a ). By adjunc- tion, this comp osition corresp onds to a left C -contra mo dule morphism Γ φ ( P ) − → Cohom C ( S , Γ φ ( P )), whic h provides a left S -semicontramo dule structure on Γ φ ( P ). Notice that the semialgebra S has a structure of left S -semimo dule endo w ed with a righ t a ction of the k -algebra R b y left S -semimo dule endomorphisms. So when C is a ﬂat right A - mo dule and S turns out to b e a coﬂat r igh t C -como dule, there is a functor S – simo d − → R – mo d mapping a left S -semimo dule M to the left R -mo dule Hom S ( S , M ). This functor has a left adjoin t functor mapping a left R -mo dule M to the left S -semimo dule S ⊗ R M = C ⊗ K M . In t he case when C is a pro jectiv e left A -mo dule and R is a pro jectiv e left K -mo dule, the former functor is isomorphic to ∆ φ,f Ψ S , and consequen tly the latter functor is isomorphic to Φ S Γ φ,f . Analogously , 185 when C is a pro jectiv e righ t A -mo dule and S turns out to b e a copro jectiv e righ t C -como dule, the functor Ψ S op Γ φ,f maps a r igh t R - mo dule N to the righ t S -semicon- tramo dule Hom R op ( S , N ) = Hom K op ( C , N ). Let us p oint out that no explicit description of the c ate gory of left S -s e m imo dules is in gener al available . W e only describ ed the categories of right S -semimo dules and left S - semicontramo dules, and constructed certain functors acting to a nd fro m the category of left S -semimo dules. 10.2.5. The f o llo wing observ ations w ere inspired by [2, section 5]. Supp ose that there is a comm utative diagra m of k -alg ebra maps A − → K , K − → R , A ′ − → K ′ , K ′ − → R ′ , A − → A ′ , K − → K ′ , R − → R ′ . L et C b e a coring o v er A and C ′ b e a coring o v er A ′ endo w ed with a map of corings C − → C ′ compatible with the k -a lgebra map A − → A ′ . Assume that C is a ﬂat left A -mo dule, C ′ is a ﬂat left A ′ -mo dule, R is a ﬂat left K - mo dule, and R ′ is a ﬂat left K ′ -mo dule. Let φ : C ⊗ A K − → A and φ ′ : C ′ ⊗ A ′ K ′ − → A ′ b e t w o pa ir ings satisfying the conditions of 10.1.2 a nd forming a comm utativ e diagram with the maps C ⊗ A K − → C ′ ⊗ A ′ K ′ and A − → A ′ . F urthermore, supp ose tha t the natural map K ⊗ A A ′ − → K ′ is an isomorphism. Assume tha t t here is a structure of right C -como dule on C ⊗ K R and a structure of right C ′ -como dule on C ′ ⊗ K ′ R ′ satisfying the conditions of 10.2.1, so tha t C ⊗ K R is a semialgebra ov er C and C ′ ⊗ K ′ R ′ is a semialgebra o v er C ′ . Then the natural map f r o m the right C ′ -como dule C ⊗ K R ⊗ A A ′ to the rig ht C ′ -como dule C ′ ⊗ K ′ R ′ is a morphism of righ t K ′ -mo dules. If it is also a morphism of righ t C ′ -como dules, then the map C ⊗ K R − → C ′ ⊗ K ′ R ′ is a map of semialgebras compatible with the map of corings C − → C ′ and t he k -algebra map A − → A ′ . Supp ose that there is a comm utative diagra m of k -alg ebra maps A − → K , K − → R , A − → K ′ , K ′ − → R ′ , K − → K ′ , R − → R ′ . Let C and C ′ b e tw o corings o v er A and C ′ − → C b e a morphism of corings o v er A . As sume that C and C ′ are ﬂat left A - mo dules, R is a ﬂat left K -mo dule, and R ′ is a ﬂat left K ′ -mo dule. Let φ : C ⊗ A K − → A and φ ′ : C ′ ⊗ A K ′ − → A b e t w o pairings satisfying the conditions of 10.1.2 and forming a comm utative diag ram with the maps C ′ ⊗ A K − → C ⊗ A K and C ′ ⊗ A K − → C ′ ⊗ A K ′ . F urthermore, supp ose that the natural map K ′ ⊗ K R − → R ′ is an isomorphism. Ass ume tha t there is a structure of righ t C -como dule on C ⊗ K R and a structure o f right C ′ -como dule on C ′ ⊗ K ′ R ′ satisfying the conditions of 10.2 .1, so that C ⊗ K R is a semialgebra o v er C a nd C ′ ⊗ K ′ R ′ is a semialgebra ov er C ′ . In this case, if the righ t K -mo dule map C ′ ⊗ K ′ R ′ = C ′ ⊗ K ′ K ′ ⊗ K R ≃ C ′ ⊗ K R − → C ⊗ K R is a mor phism of righ t C -como dules, then it is a map of semialgebras compatible with the morphism C ′ − → C o f corings o v er A . 10.3. E n t wining structures. An imp ortant particular case of the ab o v e construc- tion of semialgebras w as considered in [15]. Namely , it was noticed that there is a set of data from whic h one can construct b oth a coring and a semialgebra. 186 10.3.1. Let C b e a coring ov er a k -a lg ebra A and A − → B b e a morphism of k -alg ebras. A right entwining structu r e for C and B ov er A is an A - A -bimo dule map ψ : C ⊗ A B − → B ⊗ A C satisfying the following equations: (i) the comp osition C ⊗ A B ⊗ A B − → B ⊗ A C ⊗ A B − → B ⊗ A B ⊗ A C − → B ⊗ A C of tw o maps induced b y the map ψ and the map induced by the multiplic ation in B is equal to the comp osition C ⊗ A B ⊗ A B − → C ⊗ A B − → B ⊗ A C of the map induced b y the m ultiplication in B and the map ψ ; (ii) the map ψ forms a comm utativ e triangle with the maps C − → C ⊗ A B and C − → B ⊗ A C coming fro m the unit of B ; (iii) t he comp osition C ⊗ A B − → C ⊗ A C ⊗ A B − → C ⊗ A B ⊗ A C − → B ⊗ A B ⊗ A C of the ma p induced b y the comultiplic ation in C and t w o maps induced b y the map ψ is equal to the comp osition C ⊗ A B − → B ⊗ A C − → B ⊗ A C ⊗ A C of the map ψ and the map induced b y the comultiplication in C ; a nd (iv) the map ψ forms a comm utativ e tria ng le with the maps C ⊗ A B − → B and B ⊗ A C − → B coming fro m the counit of C . A lef t entwining structur e for C and B ov er A is deﬁned as an A - A -bimo dule map ψ # : B ⊗ A C − → C ⊗ A B satisfying the opp osite equations. Notice that whenev er a map ψ : C ⊗ A B − → B ⊗ A C is in v ertible the map ψ is a rig ht ent wining structure if and o nly if the map ψ # = ψ − 1 is a left en t wining structure. 10.3.2. A (right) entwine d mo dule o v er a r igh t en t wining structure ψ : C ⊗ A B − → B ⊗ A C is a k -mo dule N endow ed with a right C -como dule and a righ t B - mo dule structures suc h that the corresp onding right A -mo dule structures coincide and the follo wing equation holds: the comp osition N ⊗ A B − → N ⊗ A C ⊗ A B − → N ⊗ A B ⊗ A C − → N ⊗ A C of the ma p induced by the C -coaction in N , the map induced b y the map ψ , and the map induced b y the B -action in N is equal to the comp osition N ⊗ A B − → N − → N ⊗ A C o f the B -action map and the C -coaction map. A (left) entwine d c ontr amo dule ov er a right en tw ining structure ψ is a k - mo dule P endo w ed with a left C -contr amo dule and a left B -mo dule structures suc h that the corresp onding left A -mo dule structures coincide and the follow ing equation holds: the comp osition Hom A ( C , P ) − → Hom A ( C , Hom A ( B , P )) = Hom A ( B ⊗ A C , P ) − → Hom A ( C ⊗ A B , P ) = Hom A ( B , Hom A ( C , P )) − → Hom A ( B , P ) of the map induced b y the B -action in P , the map induced b y t he map ψ , and the map induced b y the C -con traaction in P is equal to the comp osition Hom A ( C , P ) − → P − → Hom A ( B , P ) of the C -con traaction map and the B -action map. (Left) entwine d mo dules and ( r igh t) entwine d c ontr amo dules ov er a left en twining structure are deﬁned in the a nalogous w a y . 10.3.3. Let ψ : C ⊗ A B − → B ⊗ A C b e a righ t ent wining structure. Deﬁne a coring D ov er B as the left B -mo dule B ⊗ A C endo w ed with the follow ing right action of B , com ultiplication, and counit. The rig h t B - action is t he comp osition ( B ⊗ A C ) ⊗ A B − → B ⊗ A B ⊗ A C − → B ⊗ A C of the map induced b y the map ψ and the m ultiplication in B . The comultiplication is the map B ⊗ A C − → B ⊗ A C ⊗ A C = ( B ⊗ A C ) ⊗ B ( B ⊗ A C ) 187 induced by t he comultiplic ation in C . The counit is the map B ⊗ A C − → B ⊗ A A = B coming fr o m the counit of C . One has to use the equation (i) on the en t wining map ψ to c hec k that the righ t action of B is asso ciat ive, the equation (ii) to c hec k that the righ t action of B agrees with the existing righ t action of A , and the equations (iii) and ( iv) to c hec k that the com ultiplication and counit are righ t B - mo dule maps. Analogously , for a left ent wining structure ψ # : B ⊗ A C − → C ⊗ A B one deﬁnes a coring D # = C ⊗ A B o v er B . When ψ # = ψ − 1 are t w o inv erse maps sat ysfying the en t wining structure equations, the maps ψ a nd ψ # themselv es a re m utually inv erse isomorphisms D # ≃ D b et w een the corresp onding corings ov er B . 10.3.4. Let ψ : C ⊗ A B − → B ⊗ A C b e a right en twining structure. Deﬁne a semi- algebra S o v er C as the left C - como dule C ⊗ A B endo wed with the follo wing right coaction of C , semim ultiplication, and semiunit. The right C -coaction is the com- p osition C ⊗ A B − → C ⊗ A C ⊗ A B − → ( C ⊗ A B ) ⊗ A C of the map induced b y the com ultiplication in C and the map induced by the map ψ . The semim ultiplication is the map ( C ⊗ A B )  C ( C ⊗ A B ) = C ⊗ A B ⊗ A B − → C ⊗ A B induced b y the multi- plication in B . The semiunit is the map C = C ⊗ A A − → C ⊗ A B coming from the unit of B . The multiple cot ensor pro ducts N  C S  C S  C · · ·  C S and the m ultiple cohomomorphisms Cohom C ( S  C · · ·  C S , P ) are asso ciativ e for any r ig h t C -como dule N and any left C - contr a mo dule P by Prop o sitions 1.2.5(e) and 3.2.5(h). Analogously , for a left ent wining structure ψ # : B ⊗ A C − → C ⊗ A B one deﬁnes a semialgebra S # = B ⊗ A C o v er C . When ψ # = ψ − 1 are tw o in v erse maps sat ysfying the en t wining structure equations, the maps ψ a nd ψ # themselv es a re m utually inv erse isomorphisms S ≃ S # b et w een the corresp onding semialgebras ov er C . 10.3.5. An en twined mo dule o v er a righ t ent wining structure ψ is the same that a righ t D - como dule and the same that a righ t S -semimo dule; in other w ords, the cor- resp onding categories are isomorphic. Analogously , a n ent wined mo dule ov er a left en t wining structure ψ # is the same that a left D # -como dule and the same that a left S # -semimo dule. Similar assertions apply to contramo dules: an en tw ined con tramo d- ule ov er a righ t en twining structure ψ is the same that a left D - contr a mo dule and the same that a left S -semicontramo dule; analogo usly f o r a left en t wining structure. F or an y en t wined mo dule N o v er a righ t ent wining structure ψ there is a nat- ural injective morphism N − → N ⊗ B D ≃ N ⊗ A C fro m N in to a n en twined mo dule whic h as a C -como dule is coinduced f rom an A -mo dule. Analogously , for an y left en t wined contramodule P o v er ψ there is a natural surjectiv e morphism Hom A ( C , P ) ≃ Hom B ( D , P ) − → P onto P f rom an en t wined contramo dule whic h as a C -contra mo dule is induced fro m an A -mo dule. So w e obtain, in the entwining struc- tur e c ase , a f unctor ia l injection fro m an arbitrary S -semimo dule in to a C / A -injectiv e S -semimodule and a f unctor ia l surjection on to an arbitr a ry S -semicontramo dule from 188 a C / A -projectiv e S - semicontramo dule constructed in a w a y m uc h simpler than that of Lemmas 1.3.3 and 3.3.3 (cf. [1, 2]) . When the ring A is semisimple, there is also a functorial surjection on to an ar bit r a ry D -como dule N from a B - pro jectiv e D -como dule N  C S ≃ N ⊗ A B and a functorial injection from an arbitrary D -contramo dule P in to a B -injectiv e D -contramo dule Cohom C ( S , P ) ≃ Hom A ( B , P ); these a r e m uch simpler constructions than those of Lemmas 1.1.3 and 3.1.3. When B is a ﬂat righ t A -mo dule, t he construction of the semialgebra S = C ⊗ A B corresp onding to an en t wining structure ψ b ecomes a particular case of the construc- tion of the semialgebra S = C ⊗ K R corresp onding to a pairing φ (take K = A , R = B , a nd the o nly p ossible φ ). 10.3.6. When ψ # = ψ − 1 are tw o in v erse en twining structures, there is an explicit description of b o th the categor ies o f left and righ t como dules o v er D # ≃ D and b oth the categories of left and r igh t semimo dules ov er S ≃ S # . When ψ is in v ertible, the multiple cotensor products N  C S  C · · ·  C S  C M and the multiple cohomomorphisms Cohom C ( S  C · · ·  C S  C M , P ) are asso cia- tiv e for any righ t C -como dule N , left C - como dule M , and left C - contramo dule P by Prop ositions 1.2.5 (f ) and 3.2.5(j), so the functors of semitensor pro duct and semiho- momorphism ov er S are every where deﬁned. 10.4. Semipro duct and semimorphis ms. Let ψ : C ⊗ A B − → B ⊗ A C b e a righ t en t wining structure; supp o se t hat ψ is an in v ertible map. Let S = C ⊗ A B and D = B ⊗ A C b e the corresp onding semialgebra o v er C and coring o v er B . One deﬁnes [43] the semipr o duct N ⊗ C B M of a right en t wined mo dule N o v er ψ and a left ent wined mo dule M o v er ψ − 1 as the image of the comp osition of maps N  C M − → N ⊗ A M − → N ⊗ B M . Analogously , one deﬁnes the k -mo dule of semim o rp hisms Ho m C B ( M , P ) from a left en twined mo dule M o v er ψ − 1 to a left ent wined contramo dule P ov er ψ as the image of the comp osition of maps Hom B ( M , P ) − → Hom A ( M , P ) − → Cohom C ( M , P ). There is a na tural map of semialgebras S − → B compatible with the map C − → A of corings o v er A . Hence f o r any ent wined mo dules N o v er ψ and M ov er ψ − 1 there is a natural injectiv e ma p from the pair of morphisms N  C S  C M ⇒ N  C M to the pair of morphisms N ⊗ A B ⊗ A M ⇒ N ⊗ A M . Therefore, we ha v e a natural surjectiv e map N ♦ S M − → N ⊗ C B M , which is an isomorphism if and only if the map N ♦ S M − → N ⊗ B M is injective . Analogously , for any en t wined mo dule M o v er ψ − 1 and ent wined con tramo dule P o v er ψ there is a natural surjectiv e map from the pair of mor phisms Hom A ( M , P ) ⇒ Hom A ( B ⊗ A M , P ) to the pair of morphisms Cohom C ( M , P ) ⇒ Cohom C ( S  C M , P ). So w e get a natural injectiv e map Hom C B ( M , P ) − → SemiHom S ( M , P ), whic h is an isomorphism if a nd only if the map Hom B ( M , P ) − → SemiHom S ( M , P ) is surjectiv e. 189 Consider the na t ural injectiv e morphism of ent wined mo dules N − → N ⊗ B D = N ⊗ A C . T aking the semitensor pro duct of this mor phism with M ov er S , we obtain the map N ♦ S M − → ( N ⊗ A C ) ♦ S M ≃ N ⊗ B M that w e are interes ted in. Hence the natura l map N ♦ S M − → N ⊗ C B M is an isomor phism whenev er the semitensor pro duct with M maps A -split injections of right S -semimo dules to injections or N has suc h prop erty with resp ect to left S - semimo dules. This includes the cases when N or M is an S -semimo dule induced f rom a C -como dule. Analogously , consider the nat ur a l surjectiv e morphism of en t wined contramodules Hom A ( C , P ) = Hom B ( D , P ) − → P . The map Hom B ( M , P ) − → SemiHom S ( M , P ) can b e o btained by taking the semihomomorphisms ov er S from M to the morphism Hom A ( C , P ) − → P or b y taking the semihomomorphisms ov er S from the morphism M − → C ⊗ A M to P . Th us the natural map Hom C B ( M , P ) − → SemiHom S ( M , P ) is an isomorphism whenev er the functor of semihomomorphisms f r o m M maps A -split surjections of left S -semicontramo dules to surjections or the functor of semihomo- morphisms in to P maps A -split inj ections o f left S -semimo dules to surjections. This includes the cases when M is an S - semimo dule induced from a C -como dule or P is an S -semicontramo dule coinduced from a C -contramo dule. In the same w a y one constructs a natural injective map N ⊗ C B M − → N  D M and sho ws that it is an isomorphism whenev er the cotensor pro duct with N or M ov er D maps surjections of D - como dules to surjections, in particular, whe n o ne of the D -como dules M or N is quasicoﬂat. Analogously , there is a natural surjectiv e map Cohom D ( M , P ) − → Hom C B ( M , P ), which is an isomorphism whenev er the functor of cohomomorphisms fro m M o v er D maps injections of left D - contramo dules to injections or the functor of cohomomorphisms in to P o v er D maps surjections of left D -como dules to injections, in pa rticular, when M is a quasicopro jectiv e D -como dule or P is a quasicoinjectiv e D -contramo dule. 190 11. Re la t ive Nonhomogeneous Koszul Duality 11.1. Gr aded semialgebras. 11.1.1. All the constructions of Sections 1–10 can b e carried out with the category of k -mo dules replaced b y the category o f graded k - mo dules. So one w ould consider a graded k -a lg ebra A , a coring ob jec t C in t he tensor category of graded A - A -bimo dules, a ring ob ject S in a tensor category of graded C - C -bico- mo dules, assume A to hav e a ﬁnite graded homological dimension, consider graded S -semimodules and graded S -semicontramo dules. All of our deﬁnitions and results can b e tra nsfered to the gra ded situation without an y diﬃculties. All the functors so obtained comm ute with t he shift of grading in mo dules. F urthermore, there a re two f o rgetful functors Σ and Π from the category o f graded k -mo dules k – mo d gr to the cat ego ry k – mo d , the functor Σ sending a graded k - mo dule to the inﬁnite direct sum of its comp onents and the functor Π sending it to their inﬁnite pro duct. F or any gra ded semialgebra S ov er a graded coring C ov er a gra ded k -alg ebra A , there are natural structures of a k -algebra on Σ A , of a coring o v er Σ A on Σ C , and of a semialgebra ov er Σ C on Σ S . F o r an y g raded S -semimo dule M there is a natura l structure of a Σ S -semimo dule o n Σ M and for a n y gr a ded S -semicontra- mo dule P there is a natural structure of a Σ S -semicontramo dule on Π P . The functors of semitensor pro duct and semihomomorphism deﬁned in the graded setting are related to their ungraded analog ues b y the form ulas Σ( N ♦ gr S M ) ≃ Σ N ♦ Σ S Σ M and Π SemiHom gr S ( M , P ) ≃ SemiHom Σ S (Σ M , Π P ). The functors N 7− → C N and M 7− → T M comm ute with the forgetful f unctors Σ and the functors Q 7− → C Q and P 7− → T P comm ute with the forgetful functors Π. The corresp onding deriv ed functors SemiT or , SemiExt, etc., hav e the analogous prop erties. How ev er, the functors Hom S , Hom S , CtrT or S , Ψ S , a nd Φ S and their derive d functors ha v e no prop erties of compatibilit y with the functors of fo rgetting the grading. Th us one has to b e aw are of the distinction b et w een Hom S and Hom gr S , Φ S and Φ gr S , etc. 11.1.2. Assume that A is a nonnegativ ely gra ded k -algebra, C is a nonnegativ ely graded coring ov er A , and S is a nonnegativ ely graded semialgebra ov er C . L et S – simo d ↑ and simo d ↑ – S denote the categories of nonnegativ ely graded S -semimo dules, and S – contra ↓ denote the category of nonp o sitiv ely gra ded S -semicontramodules. All the constructions of Sections 1 –4 in their graded v ersions preserv e the cat ego ries of como dules a nd semimo dules graded b y nonnegativ e in tegers and the categories of con tramo dules and semicon tra mo dules graded b y nonp ositive in tegers. All the def- initions and results of these sections can b e t r a nsfered to the describ ed situatio n of b ounded grading a nd no problems o ccur. In particular, one can apply Lemma 2.7 to deﬁne the functors SemiT or and SemiExt in the b ounded grading case. Moreo v er, the functors so obtained ag ree with the functors SemiT or S gr and SemiExt gr S deﬁned 191 in terms of complexes with unbounded gra ding . This is so b ecause t he construc- tions of resolutions agree. F or the same reasons, in the assumptions of 6 .3 the func- tors D si ( S – simo d ↑ ) − → D si ( S – simo d gr ) and D si ( S – sicntr ↓ ) − → D si ( S – sicntr gr ) a re fully faithful, and the functor CtrT or deﬁned b y applying Lemma 6 .5.2 in the b ounded grading case agr ees with the functor CtrT or S gr . But the functors Ψ gr S and Φ gr S do no t preserv e the b ounded grading. 11.2. D iﬀerential semialgebras. 11.2.1. Let B b e a graded k -algebra endo w ed with an o dd deriv atio n d B of degree 1 and D b e a graded coring o v er B . A homogeneous map d D : D − → D of degree 1 is called a c o derivation o f D with r esp e ct to d B if the biaction map B ⊗ k D ⊗ k B − → D and t he comultiplic ation map D − → D ⊗ B D are morphisms in the category of graded k -mo dules endo w ed with endomorphisms of degree 1, where the endomorphisms of the tensor pro ducts are deﬁne d b y the usual sup er-Leibniz rule d ( xy ) = d ( x ) y + ( − 1) | x | xd ( y ) (the degree of a homogeneous elemen t x b eing denoted by | x | ). In this case, it follow s that the counit map D − → B satisﬁes the same condition. In the particular case when B is concen tr a ted in the degree 0 and d B = 0 , t he condition on the biaction map simply means that d D is a B - B -bimo dule morphism. No w assume that B is a DG- a lgebra o v er k , i. e., d 2 B = 0. A DG-c oring o v er B is a gr a ded coring D ov er the graded ring B endo w ed with a co deriv ation d D : D − → D with resp ect to d B suc h that d 2 D = 0 . Let D b e a DG- coring o v er a DG- algebra B . Then the cohomology H ( D ) is endo w ed with a natura l structure of a graded coring o v er the graded a lgebra H ( B ) pro vided t ha t the natural maps H ( D ) ⊗ H ( B ) H ( D ) − → H ( D ⊗ B D ) a nd H ( D ) ⊗ H ( B ) H ( D ) ⊗ H ( B ) H ( D ) − → H ( D ⊗ B D ⊗ B D ) are isomorphisms. A map of D G-corings C − → D compatible with a morphism of D G-algebras A − → B induces a map of graded corings H ( C ) − → H ( D ) compatible with the morphism of graded algebras H ( A ) − → H ( B ) whenev er b oth DG- cor ing s C and D satisfy the a b ov e tw o conditions. Here a map C − → D from a DG-coring C o v er a DG-algebra A t o a D G-coring D o v er a DG- algebra B is called compatible with a morphism o f DG -algebras A − → B o v er k if the map of graded corings C − → D is compatible with the morphism of graded algebras A − → B and t he map C − → D is a morphism of complexes. 11.2.2. Co deriv ations of a g raded coring D of degree − 1 with resp ect to co deriv a- tions of a graded k -algebra B of degree − 1 a r e deﬁned in the same w ay as ab o v e. No w let A b e an ungraded k -algebra. A q uasi - diﬀer ential c oring D ∼ o v er A is a graded coring ov er A endow ed with a co deriv ation ∂ o f degree 1 (with resp ect to the zero deriv ation of the k - algebra A , whic h is considered as a graded k - algebra concen t r a ted in degree 0) such that ∂ 2 = 0 a nd the cohomolog y of ∂ v anish. If D ∼ is a quasi-diﬀeren tial coring o v er a k - algebra A , then the cok ernel D ∼ / im ∂ o f the 192 deriv ation ∂ has a natura l structure of graded coring ov er A . A quasi-diﬀ er ential structur e on a graded coring D is the data of a quasi-diﬀerential coring D ∼ together with an isomorphism of graded corings D ∼ / im ∂ ≃ D . W e will denote the g r ading of a quasi-diﬀeren tial coring D ∼ b y low er indices, ev en though the diﬀeren t ia l ra ises the degree. This terminolog y and notation is explained by the following construction (cf. 0 .4.4). W e will use Swee dler’s notation [45 ] p 7− → p (1) ⊗ p (2) for the com ultiplication map of a coring D o v er A ; here p ∈ D and p (1) ⊗ p (2) ∈ D ⊗ A D . A CDG-c oring D ov er a k -algebra A is a gra ded coring ov er A endo w ed with a co deriv a tion d of degree − 1 (with resp ect to the zero deriv ation of A ) and an A - A -bimo dule map h : D 2 − → A satisfying the equations d 2 ( p ) = h ( p (1) ) p (2) − p (1) h ( p (2) ) a nd h ( d ( p )) = 0 for all p ∈ D , where the map h is considered to b e extende d b y zero to the comp onen ts D i with i 6 = 2. Giv en a CDG -coring D ov er a k -algebra A a nd a CDG- cor ing E o v er a k - algebra B , a morphism of CDG -corings D − → E compatible with a morphism of k -algebras A − → B is a pair ( g , a ), where g : D − → E is a map of g raded corings compatible with a morphism of k -alg ebras A − → B and a : D 1 − → B is an A - A - bimo dule ma p satisfying the equations d ( g ( p )) = g ( d ( p )) + a ( p (1) ) g ( p (2) ) + ( − 1) | p | g ( p (1) ) a ( p (2) ) and h ( g ( q )) = h ( q ) + a ( d ( q )) + a ( q (1) ) a ( q (2) ) hold for all p ∈ D and q ∈ D 2 (where the map a is extended by zero to t he comp onen ts D i with i 6 = 1). Comp osition of morphisms of CDG-coring s is deﬁned b y the rule ( g ′ , a ′ )( g ′′ , a ′′ ) = ( g ′ g ′′ , a ′ g ′′ + a ′′ ); identit y morphisms are the morphisms (id , 0) . So the category of CDG-corings is deﬁned. Notice that tw o CDG - corings of the fo rm ( D , d ′ , h ′ ) and ( D , d ′′ , h ′′ ) ov er a k -alg ebra A with d ′′ ( p ) = d ′ ( p ) + a ( p (1) ) p (2) + ( − 1) | p | p (1) a ( p (2) ) and h ′′ ( q ) = h ′ ( q ) + a ( d ′ ( q )) + a ( q (1) ) a ( q (2) ), where a : D 1 − → A is an A - A -bimo dule map, are alw ay s naturally isomorphic to eac h other, the isomorphism b eing give n by (id , a ) : ( D , d ′ , h ′ ) − → ( D , d ′′ , h ′′ ). The category of DG- cor ing s (ov er ungraded k -algebras considered a s D G -algebras concen t r a ted in degree zero) has D G-corings D o v er k -algebras A a s ob jects and maps of DG-coring s D − → E compatible with morphisms o f k -algebras A − → B as morphisms. The category o f quasi-diﬀeren t ia l corings can b e deﬁned as the full sub category of the cat ego ry of DG-cor ing s whose ob jects are the D G-corings with acyclic diﬀerentials. One can also consider t he category of DG- corings ( ov er ungraded k -alg ebras) with co deriv ations of degree − 1. There is an o b vious faithful, but no t fully faithful functor from the latter category to the category of CDG-corings, assigning the CDG- coring ( D , d, h ) with h = 0 to a DG -coring ( D , d ) and the morphism of CDG-corings ( g , 0) to a map of DG-coring s g : D − → E compatible with a morphism of k -alg ebras A − → B . There is a natural fully fait hf ul functor fro m the category of CDG -corings to the category of quasi-diﬀerential corings, whose image consists o f the quasi-diﬀeren tial 193 corings D ∼ o v er A for whic h the counit ma p D ∼ 0 − → A can b e presen ted as the com- p osition of the co deriv ation comp onent ∂ 0 : D ∼ 0 − → D ∼ 1 and some A - A - bimo dule map δ : D ∼ 1 − → A . In other w or ds, a quasi-diﬀeren tia l coring comes from a CDG-coring if and only if the counit map D ∼ 0 /∂ − 1 D ∼ − 1 − → A can b e extended to an A - A -bimo dule map D ∼ 1 − → A , where the A - A -bimo dule D ∼ 0 /∂ − 1 D ∼ − 1 is em b edded in to the A - A - bimo dule D ∼ 1 b y the map ∂ 0 . In particular, the categories of quasi- diﬀeren tial corings and CDG -corings o v er a ﬁeld A = k ( quasi-diﬀer ential c o algeb r as and C DG-c o algebr as o v er k ) are naturally equiv a len t. Let us ﬁrst construct the inv erse functor. Giv en a quasi-diﬀeren tial coring ( D ∼ , ∂ ) and a map δ : D ∼ 1 − → A as ab ov e, set D = D ∼ / im ∂ and deﬁne d and h b y the form ulas d ( p ) = δ ( p (1) ) p (2) + ( − 1) | p | p (1) δ ( p (2) ) and h ( q ) = δ ( q (1) ) δ ( q (2) ) for p ∈ D ∼ and q ∈ D ∼ 2 , where the map δ is extended by zero to the comp onents D ∼ i with i 6 = 1 and r ∈ D denotes the image of an elemen t r ∈ D ∼ . T o a map of quasi-diﬀeren tial corings g : D ∼ − → E ∼ endo w ed with maps δ D : D ∼ 1 − → A and δ E : E ∼ 1 − → B with the ab ov e prop erty , compatible with a morphism of k -a lgebras f : A − → B , one assigns the morphism of CDG-corings ( g , δ E g − f δ D ), where g : D − → E denotes the induced morphism on the cok ernels of the co deriv ations ∂ . Con v ersely , to a CDG-cor ing ( D , d , h ) o v er a k -alg ebra A one assigns the quasi- diﬀeren tial coring ( D ∼ , ∂ ) ov er A whose graded comp onen ts are the A - A - bimo dules D ∼ i = D i ⊕ D i − 1 , the co deriv ation ∂ is giv en b y the form ula ∂ ( τ p + ∂ q ) = ∂ p , and the com ultiplication is giv en b y the form ula τ p + ∂ q 7− → τ p (1) ⊗ τ p (2) + ( − 1) | p (1) | τ d ( p (1) ) ⊗ ∂ p (2) + ( − 1) | p (2) | h ( p (1) ) ∂ p (2) ⊗ ∂ p (3) + ∂ q (1) ⊗ τ q (2) + ( − 1) | q (1) | τ q (1) ⊗ ∂ q (2) + ( − 1) | q (1) | ∂ d ( q (1) ) ⊗ ∂ q (2) , where τ p + ∂ q = ( p, q ) is a formal notatio n for an elemen t of L i ( D i ⊕ D i − 1 ). T o a morphism o f CDG -corings ( g , a ) : D − → E , the morphism of quasi-diﬀerential corings L i ( D i ⊕ D i − 1 ) − → L i ( E i ⊕ E i − 1 ) giv en b y the formula τ p + ∂ q 7− → τ g ( p ) + a ( p (1) ) ∂ g ( p (2) ) + ∂ g ( q ) is assigned. F or a quasi- diﬀeren tial coring ( D ∼ , ∂ ) o v er a k -alg ebra A endow ed with a map δ : D ∼ 1 − → A with the ab o v e prop erty and the correspo nding CDG-coring ( D , d, h ), the nat ur a l morphism of quasi-diﬀerential corings D ∼ − → L i ( D i ⊕ D i − 1 ) o v er A is giv en by the form ula p 7− → τ p + δ ( p (1) ) ∂ p (2) for p ∈ D ∼ . This morphism is an isomorphism, since the induced morphism o f the cok ernels of the co deriv ations ∂ is an isomorphism. 11.2.3. Let B b e a gra ded k -algebra endo w ed with a deriv ation d B of degree 1 and D b e a graded coring ov er B endo w ed with a co deriv ation ∂ D with resp ect t o d B . Let T b e a graded semialgebra o v er D . A homogeneous map d T : T − → T of degree 1 is called a semiderivation of T with r esp e ct to d D and d B if the biaction map B ⊗ k T ⊗ k B − → T , the bicoaction map T − → D ⊗ B T ⊗ B D , and the semim ultiplication map T  D T − → T a r e morphisms in the category of graded k -mo dules endo w ed with endomorphisms of degree 1. In this case, it follows that the semiunit map D − → T satisﬁes the same condition. In the particular case when B and D a re concen tra ted 194 in degree 0 and d B = 0 = d D , the conditions on the biaction a nd bicoaction map simply mean that d T is a D - D -bicomo dule morphism. Let B b e a D G -algebra ov er k and D b e a DG- coring ov er B . A DG-se mialgebr a o v er D is a graded semialgebra ov er the graded coring D endo w ed with a semideriv a- tion d T with resp ect to d D and d B suc h that d 2 T = 0 . Let T b e a DG-semialgebra ov er a D G-coring D . Then the cohomology H ( T ) is endo w ed with a natural structure of g raded semialgebra ov er the graded coring H ( D ) pro vided that (i) the natural maps from the tensor pro ducts o f cohomology to the cohomology of the tensor pro ducts are isomorphisms for the tensor pro ducts D ⊗ B D , D ⊗ B D ⊗ B D , D ⊗ B T , T ⊗ B D , D ⊗ B D ⊗ B T , T ⊗ B D ⊗ B D , D ⊗ B T ⊗ B D , T ⊗ B T , D ⊗ B T ⊗ B T , T ⊗ B T ⊗ B D , T ⊗ B D ⊗ B T , D ⊗ B T ⊗ B D ⊗ B T , T ⊗ B D ⊗ B T ⊗ B D , T ⊗ B T ⊗ B T , T ⊗ B D ⊗ B T ⊗ B T , T ⊗ B T ⊗ B D ⊗ B T ; (ii) the m ultiple cotensor pro ducts H ( T )  H ( D ) · · ·  H ( D ) H ( T ) are asso ciativ e, where the gra ded H ( D )- H ( D )- bicomo dule structure on H ( T ) is w ell-deﬁned in view of (i); and (iii) the nat ural maps H ( T  D T ) − → H ( T )  H ( D ) H ( T ), H ( D ⊗ B T  D T ) − → H ( D ) ⊗ H ( B ) H ( T )  H ( D ) H ( T ), H ( T  D T ⊗ B D ) − → H ( T )  H ( D ) H ( T ) ⊗ H ( B ) H ( D ), and H ( T  D T  D T ) − → H ( T )  H ( D ) H ( T )  H ( D ) H ( T ), whic h are w ell-deﬁned in view of (i) and (ii), a re isomorphisms. A map of D G-semialgebras S − → T compatible with a map of DG -corings C − → D and a mor phism o f DG- algebras A − → B induces a map of graded semialgebras H ( S ) − → H ( T ) compatible with the map o f gra ded corings H ( C ) − → H ( D ) and the morphism of graded k -algebras H ( A ) − → H ( B ) whenev er b oth DG -semialgebras S and T satisfy the ab o v e three conditions. Here a map S − → T from a DG- semialgebra S o v er a DG-coring C to a D G-semialgebra T o v er a DG- coring D is called compatible with a map of DG-cor ing s C − → D and a morphism o f DG- algebras A − → B if the map of graded semialgebras S − → T is compatible with the map of gra ded corings C − → D and the morphism of g r a ded k -alg ebras A − → B , and t he maps S − → T and C − → D are morphisms of complexes. 11.3. One-sided SemiT or. L et S b e a semialgebra o v er a coring C ov er a k -alg ebra A . W e will consider t w o situations separately . 11.3.1. Assume that C is a ﬂat right A - mo dule and S is a coﬂat rig ht C - como dule. Consider the functor of semitensor pro duct ov er S o n the Carthesian pro duct of the homotop y cat ego ry of complexes of C -coﬂat right S -semimo dules and the homotopy category of complexes of left S -semimo dules. The semideriv ed category of C -coﬂat righ t S -semimo dules is deﬁned as the quotien t category of t he homotopy catego r y of C -coﬂat righ t S -semimo dules b y the thic k sub category of complexes of right S -semi- mo dules that as complexes of C -como dules are coacyclic with resp ect to the exact category of coﬂat r ig h t C - como dules. A complex of left S -semimo dules M • is called 195 semiﬂat r elative to C if the complex N • ♦ S M • is acyclic for any C -contractible complex of C -coﬂat righ t S -semimo dules N • (cf. 2.8). The left deriv ed functor SemiT or S on the Carthesian pro duct of the semideriv ed category of C -coﬂat right S -semimo dules and the semideriv ed category of left S -semi- mo dules is deﬁned b y restricting the functor of semitensor pro duct to the Carthesian pro duct of the homo t o p y category of C -coﬂat right S -semimodules and the ho mo - top y categor y of complexes of left S -semimo dules semiﬂat relativ e to C , or to the Carthesian pro duct of the homo t o p y category of semiﬂat complexes of righ t S -semi- mo dules a nd the ho mot op y category of left S -semimo dules. This deﬁnition of a deriv ed functor is a particular case of b oth Lemmas 2.7 and 6.5.2. If N • is a com- plex of C -coﬂat rig h t S -semimo dules and M • is a complex of left S -se mimo dules, then the tota l complex of the bar bicomplex · · · − → N •  C S  C S  C M • − → N •  C S  C M • − → N •  C M • , constructed b y taking inﬁnite direct sums along the diagonals, represen ts the ob ject SemiT or S ( M • , N • ) in D ( k – mo d ). When the semi- unit map C − → S is injectiv e and its coke rnel is a ﬂat rig h t A - mo dule ( and hence a coﬂat right C -como dule b y Lemma 1.2.2 ), o ne can also use the reduced bar bicomplex · · · − → N •  C S / C  C S / C  C M • − → N •  C S / C  C M • − → N •  C M • . In the case when S is a graded semialgebra one analogously deﬁnes the deriv ed functor SemiT or S gr acting from the Carthesian pro duct of the semideriv ed category of C -coﬂa t graded righ t S -semimo dules and the semideriv ed category of gr aded left S -semimodules to the deriv ed cat ego ry of graded k -mo dules. 11.3.2. Assume that C is a ﬂat righ t A -mo dule, S is a ﬂa t righ t A -mo dule and a C / A -coﬂat left C -como dule, and the ring A has a ﬁnite w eak homological dimension. Consider the functor of semitensor pro duct ov er S on the Carthesian pro duct of the homotopy category of complexes of A -ﬂa t righ t S -semimo dules and the homotop y category of complexes of C / A -coﬂat left S -semimo dules. The semide riv ed category of A -ﬂat righ t S -semimo dules ( C / A - coﬂat left S -semimo dules) is deﬁned a s the quo- tien t category of the homot o p y category of A -ﬂat right S -semimo dules ( C / A -coﬂat left S -semimo dules) by the thick sub category of complexes of S -semimo dules that as complexes of C -como dules a re coacyclic with resp ect to the exact category of A -ﬂat right C -como dules ( C / A -coﬂat left C -como dules). A complex o f C / A -coﬂat left S -semimo dules M • is called semiﬂat r elative to A if the complex of k -mo dules N • ♦ S M • is acyclic f o r an y complex of right S -semimo dules N • that as a complex of right C -como dules is coacyclic with resp ect to the exact category of A -ﬂat righ t C -como dules. A complex of A -ﬂat right S -semimodules N • is called S / C / A -semiﬂat if the complex of k -mo dules N • ♦ S M • is acyclic for any C -contractible complex of C / A -coﬂat left S -semimo dules M • (cf. 2.8). The left deriv ed functor SemiT or S on the Carthesian pro duct of the semideriv ed category of A -ﬂat righ t S -semimo dules and the semideriv ed category of C / A -coﬂat 196 left S -semimo dules is deﬁned by restricting the functor of semitensor pro duct to the Carthesian pro duct of the homotopy cat ego ry of A -ﬂat right S -semimo dules and the homotop y category of complexes of C / A -coﬂat left S -semimo dules semiﬂat relative to A , or to the Carthesian pro duct o f the homotopy catego ry of S / C / A - semiﬂat com- plexes o f A - ﬂat r ig h t S -semimodules and the homotopy category of C / A -coﬂat left S -semimodules. This deﬁnition of a deriv ed f unctor is a particular case of b o th Lem- mas 2.7 and 6.5.2. If N • is a complex of A - ﬂa t rig h t S - semimo dules and M • is a com- plex of C / A -coﬂat left S - semimo dules, then the t o tal complex of the ba r bicomplex · · · − → N •  C S  C S  C M • − → N •  C S  C M • − → N •  C M • , constructed by tak- ing inﬁnite dir ect sums along the diagonals, represen ts t he ob jec t SemiT o r S ( M • , N • ) in D ( k – mo d ). When the se miunit map C − → S is injectiv e and its cok ernel is a ﬂat righ t A - mo dule (the cok ernel is a C / A -coﬂat left C -como dule b y Lemma 1.2 .2), one can also use the reduced bar bicomplex · · · − → N •  C S / C  C S / C  C M • − → N •  C S / C  C M • − → N •  C M • . In the case when S is a graded semialgebra one analogously deﬁnes the deriv ed functor SemiT or S gr acting fro m the Carthesian pro duct of the semideriv ed category of A -ﬂat graded right S -semimo dules and the semideriv ed category of C / A -coﬂat graded left S -semimo dules to the deriv ed category of graded k -mo dules. 11.4. K oszul semialgebras and corings. 11.4.1. Let S b e a semialgebra ov er a coring C o v er a k -algebra A . Supp o se t ha t S is endo w ed with an a ug men tation, i. e., a mo r phism S − → C of semialgebras o v er C ; let S + b e the k ernel o f this map. W e will denote by Bar • ( S , C ) the reduced bar complex · · · − → S +  C S +  C S + − → S +  C S + − → S + − → C . It can b e also deﬁned as the coring L ∞ n =0 S  C n + o v er t he k -algebra A (the “cotensor coring ” of the C - C -bicomo d- ule S ) endo w ed with the unique g rading suc h that the comp onent S + is situated in degree − 1 and the unique co deriv ation (with resp ect to the zero deriv at io n of A ) of degree 1 whose comp onent mapping S +  C S + to S + is equal to the semim ultiplication map S +  C S + − → S + . So Bar • ( S , C ) is a D G -coring ov er t he k -algebra A considered as a DG-alg ebra concen trated in degree 0 . No w let S b e a graded semialgebra ov er a coring C o v er a k - a lgebra A , where A and C are considered as a graded k -algebra and a gra ded coring concen tra t ed in degree 0; assume a dditio nally tha t S is concen trated in nonnegative degrees, C is the comp onent of degree 0 in S , and the augmen tation map S − → C is simply the pro jection of S to its comp o nen t of degree 0. In this case there is a graded v ersion Bar • gr ( S , C ) of the ab ov e ba r complex, whic h is a bigraded ob ject with the grading denoted b y upp er indices coming from the cotensor p o w ers of S + and the grading denoted b y lo w er indices coming from the grading of S + itself. Notice that the comp onen t Bar i n ( S , C ) can b e only nonzero when 0 6 − i 6 n . 197 Let C and D b e corings ov er a k -alg ebra A . Supp ose tha t we are giv en t w o maps C − → D and D − → C that are morphisms of cor ing s o v er A suc h that the comp o sition C − → D − → C is the iden tity; let D + b e t he cok ernel of the map C − → D . Assume that the m ultiple cotensor pro ducts D  C · · ·  C D , where D is endo w ed with a C - C - bi- como dule structure via the morphism D − → C , a re associative. W e will denote b y Cob • ( D , C ) the reduced cobar complex C − → D + − → D +  C D + − → D +  C D +  C D + − → · · · It can b e also deﬁned as the semialgebra L ∞ n =0 D  C n + o v er the coring C (the “ cot ensor semialgebra” of the C - C -bicomo dule D ) endo w ed with the unique grading suc h that the comp onent D + is situated in degree 1 and t he unique semideriv ation (with respect to d C = 0 and d A = 0 ) of degree 1 whose component mapping D + to D +  C D + is equal to the comultiplication map D + − → D +  C D + . So Cob • ( D , C ) is a DG-semialgebra ov er the coring C ov er t he k -algebra A , where A and C are considered as a DG -algebra and a DG-cor ing concen trated in degree 0. No w let D b e a graded coring ov er a k -algebra A considered as a graded k - algebra concen t r a ted in degree 0 and C b e a coring o v er A ; assume additionally that D is concen t r a ted in nonnegative degrees, C is the comp onen t of degree 0 in D , and the maps C − → D and D − → C are simply the embedding of and the pro jection to the comp onen t of degree 0. In this case there is a graded v ersion Cob • gr ( D , C ) of the ab ov e cobar complex, whic h is a bigraded ob ject with the grading denoted by upp er indices coming from the cotensor p ow ers of D + and the grading denoted by low er indices coming from the grading of D + itself. Notice that the comp o nent Cob i n ( D , C ) can b e only nonzero when 0 6 i 6 n . 11.4.2. Let C b e a coring ov er a k -algebra A . Assume that C is a ﬂat righ t A -mo dule. A gra ded semialgebra S ov er C is called right c oﬂat Ko s zul if (i) S is nonnega- tiv ely graded and the semiunit homomorphism is an isomorphism C ≃ S 0 ; (ii) the comp onen ts S i are ﬂat rig h t A -mo dules; (iii) the cohomology H i n Bar • gr ( S , C ) are only nonzero on the diagonal − i = n ; and (iv) whenev er the comp onent Bar • n ( S , C ) is a complex of A -ﬂat righ t C -como dules, so the diagonal cohomology H − n n Bar • gr ( S , C ) can b e endo w ed with a right C -como dule structure as the k ernel of a morphism in the category of right C -como dules, it is a coﬂat righ t C -como dule. When the ring A has a ﬁnite w eak homological dimension, there is an analogous deﬁnition of a rig ht ﬂat and left r elatively c oﬂat Koszul semialgebra S ov er C . One imp oses the same conditions (i-iii) a nd replaces ( iv) with the condition (iv ′ ) the diagonal cohomology H − n n Bar • gr ( S , C ) is a C / A -coﬂat left C -como dule f or all n . A graded coring D ov er the k -algebra A endo wed with a morphism D − → C of corings ov er A is called a right c oﬂat Koszul c o ring over C if (i) D is nonnegative ly graded and the morphism D − → C v a nishes on the comp onents of p ositive degree in D and induces a n isomorphism D 0 ≃ C ; (ii) whenev er a comp onen t D n is a ﬂat righ t A -mo dule, it is a coﬂat right C -como dule; (iii) whenev er all the mu ltiple cotensor 198 pro ducts en tering into the construction of the comp o nen t Cob • n ( D , C ) are asso ciativ e, so this comp onent is w ell-deﬁned, the cohomology H i Cob • n ( D , C ) is only nonzero on the diagona l i = n ; and (iv) in the assumptions of (iii), the diagonal cohomology H n Cob • n ( D , C ) is a ﬂat r ig h t A -mo dule. When the ring A has a ﬁnite w eak homological dimension, there is an analogous deﬁnition of a right ﬂat and left r elativel y c oﬂat Koszul c oring D o v er C . One im- p oses the same conditions (i-ii), (iv), and replaces (iii) with the condition (iii ′ ) the comp onen t D n is a C / A -coﬂat left C -como dule for all n . 11.4.3. The ob jects of the category of right coﬂat Koszul semialgebras are righ t coﬂat Koszul semialgebras S o v er corings C o v er k -algebras A such t ha t C is a ﬂat righ t A -mo dule. Morphisms are maps of graded semialgebras S − → S ′ compatible with maps of corings C − → C ′ and mo r phisms of k -algebras A − → A ′ . Imp osing the additional assumption that A has a ﬁnite w eak homological dimension, one analo- gously deﬁnes t he category of right ﬂat and left r elat ively coﬂat K oszul semialgebras. The ob jects of the catego ry of righ t coﬂat Ko szul cor ing s are righ t coﬂat K oszul corings D ov er corings C ov er k - algebras A suc h that C is a ﬂat righ t A -mo dule. Morphisms are maps of graded corings D − → D ′ compatible with morphisms of k -alg ebras A − → A ′ . Imp osing the additional assumption that A has a ﬁnite w eak homological dimension, one analo g ously deﬁnes the category o f righ t ﬂat and left relativ ely Koszul corings. Theorem. T he c ate gory of right c oﬂat Koszul semialg ebr as is e quivalent to the c at- e gory of right c oﬂat Ko szul c orings. Analo gously, the c ate gory of right ﬂat and left r elatively c oﬂat Koszul semia l g e br as is e quivalent to the c a te gory of right ﬂat and left r elatively c oﬂat Koszul c orings. In b oth c ases, the mutual ly inverse e quivalenc es ar e pr ovide d by the functor assigni n g to a Kos zul se m ialgebr a S the c orin g of c oh o molo gy of the gr ade d DG-c o ring Bar • gr ( S , C ) and the functor assigni n g to a Koszul c oring D the semialgebr a of c ohomolo gy of the gr ade d DG-semialg e b r a Cob • gr ( D , C ) . Pr o of . The assertions of Theorem follo w from Prop ositions 1 and 2 b elo w. T o c hec k the conditions of 11.2 needed for the coring of cohomology and the semialgebra of cohomology to b e deﬁned, use Lemma 1.2.2 and Prop osition 1.2.5.  Let C b e a cor ing ov er a k -a lg ebra A . Prop osition 1. (a) Assume that C is a ﬂat right A -mo dule. Then a gr ade d se m i- algebr a S over C is right c oﬂat Ko s zul if and only if (i) S is nonne gatively gr ade d and the semiunit map is an isomorphism C ≃ S 0 ; (ii) for any n > 1 the natur al map fr om the quotient k -mo dule of the c otensor p ower S  C n 1 by the s um of the ker- nels of its maps to c otensor pr o ducts S  C i − 1 1  C S 2  C S  C n − i − 1 1 , i = 1 , . . . , n − 1 to the c omp onent S n is an isomorphism ; (iii) the lattic e of submo dules of the k -mo dule S  C n i gener ate d by these n − 1 kernels is distributive; (iv) al l the quotient mo dules 199 of e m b e dde d submo dules b elongin g to the mentione d lattic e ar e ﬂat right A -mo dules in their natur al right A -mo dule structur es; and (v) al l the q uotient mo dules of em- b e dde d submo d ules b elonging to this lattic e ar e c oﬂat right C -c omo dules in their right C -c omo dule structur es that ar e wel l-deﬁn e d in view of (iv). (b) Assume that C is a ﬂat rig h t A -mo dule and A ha s a ﬁnite we ak hom olo gic al dimension. Then a gr ade d semialgebr a S over C is right ﬂat and le f t r elatively c oﬂat Koszul if a n d on ly if it satisﬁes the c onditions (i-iv) of (a) and the c o n dition (v ′ ) al l the quotient mo dules of emb e dde d submo dules b elonging to the la ttic e under c onsider- ation a r e C / A -c oﬂat left C -c omo d ules in their natur al left C -c omo d ule structur es. Prop osition 2. (a) Assume that C is a ﬂat right A -mo dule. Then a gr ad e d c o ring D endowe d with a morph i s m D − → C o f c orings ov e r A is a right c o ﬂat Koszul c orin g over C if and only if (i) D is nonp os itively gr ade d and the morphism D − → C vanishes on the c om p onents of p ositive de gr e es in D and induc es an isomorphism D 0 ≃ C ; (ii) for any n > 1 the natur al map fr om the c omp onent D n to the interse ction of images of the maps fr om c otensor pr o ducts D  C i − 1 1  C D 2  C D  C n − i − 1 1 , i = 1 , . . . , n − 1 to the c otensor p ower D  C n 1 is an isomo rp h ism; (iii) the lattic e of submo dules of the k -mo dule D  C n 1 gener ate d by these n − 1 image s is distributive; (iv) al l the quotient mo dules of the emb e dde d submo dules b elonging to the mentione d lattic e ar e ﬂat right A -mo dules in their natur al right A -m o dule structur es; and (v) al l the quotient mo dules of emb e dde d submo dules b elonging to this lattic e ar e c oﬂat right C - c omo d ules in their right C -c omo dule structur es that ar e wel l-deﬁn e d in view of (iv). (b) Assume that C is a ﬂat rig h t A -mo dule and A ha s a ﬁnite we ak hom olo gic al dimension. Then a gr ade d c oring D endowe d with a morphism D − → C of c orings over A is a righ t ﬂat an d le f t r elatively c oﬂat Koszul c ori n g o ver C if a n d only if it satisﬁes the c onditions (i-iv ) of (a) and the c ondition (v ′ ) a l l the quotient mo dules of emb e dde d submo dules b e longing to the lattic e under c onsider ation ar e C / A -c oﬂat left C -c omo dules in their natur al le ft C -c om o dule structur es. Pr o of of Pr op ositions 1 and 2. Both Prop ositions follo w b y induction in the in ter- nal degree n from Lemma 1.2.2, Prop osition 1.2.5, and the next Lemma 1 (parts (a) ⇐ ⇒ (c), (a) ⇐ ⇒ (c*)), and the ﬁnal a ssertion) and Lemma 2.  Lemma 1. L et W b e a k -mo dule an d X 1 , . . . , X n − 1 ⊂ W b e a c ol le ction o f submo d ules such that any pr op er subset X 1 , . . . , b X k , . . . , X n − 1 gener ates a d i stributive lattic e of submo dules in W . Then the fol low ing c onditions ar e e quivale n t: (a) the c ol le ction of submo dules X 1 , . . . , X n − 1 gener ates a di s tributive lattic e of submo dules in W ; 200 (b) the fol lowing c omple x of k -mo dules K • ( W ; X 1 , . . . , X n − 1 ) is e x act 0 − → X 1 ∩ · · · ∩ X n − 1 − → X 2 ∩ · · · ∩ X n − 1 − → X 3 ∩ · · · ∩ X n − 1 /X 1 − → · · · − → n − 1 \ s = i +1 X s  X i − 1 t =1 X t − → · · · − → X n − 1 / ( X 1 + · · · + X n − 3 ) − → W / ( X 1 + · · · + X n − 2 ) − → W / ( X 1 + · · · + X n − 1 ) − → 0 , wher e we denote Y / Z = Y / Y ∩ Z ; (c) the fol lowing c omplex of k -mo dules B • ( W ; X 1 , . . . , X n − 1 ) W − → M t W /X t − → · · · − → M t 1 < ··· 2 S n ; the comp onen ts of t he diﬀeren tial in this complex are the zero map T 1 − → S 0 and the pro jection T 1 − → S 1 . There is a natural morphism of complexes of g raded right S -semimo dules X • − → Y • whose comp o nen ts a re the pro jections T +  C S − → T 1  C S 0 ≃ T 1 and S − → S 0 ⊕ S 1 . All the three complexe s X • , Y • , and k er( X • → Y • ) a re complexes of C -coﬂat righ t S -semimo dules ( A -ﬂat right S -semimo dules). Let us sho w that the complex k er( X • → Y • ) is is coacyclic with resp ect to the exact category o f coﬂat graded righ t C -como dules ( A -ﬂat righ t C -como dules). Indeed, denote by Z • the k ernel of the map from the reduced bar resolution of the right T -semimo dule C (written down ab o v e) to C itself. The complex of graded T -semimo dules Z • has a nat ural endomorphism z of in ternal degree 1 a nd cohomological degree 0 induced by the endomorphism of the reduced bar resolution acting by the iden tity on the cotensor factors T + and b y the na t ur a l inj ections T n − 1 → T n on the cotensor fa ctors T . Since Z • is a con- tractible complex of coﬂat graded right C -como dules ( A -ﬂat rig ht C -como dules), the endomorphism z is injectiv e, and its cok ernel is a complex o f coﬂat right C -como dules ( A -ﬂat right C -como dules), t his cok ernel is coacyclic with resp ect to the exact cate- gory o f coﬂat g raded righ t C -como dules ( A -ﬂat righ t C -como dules). No w the k ernel k er( X • → Y • ) is isomorphic a s a complex of right C -como dules to the k ernel of a sur- jectiv e morphism from cok er( z ) to the con tractible t w o-term complex of coﬂat righ t C -como dules ( A -ﬂat right C - como dules) T 1 − → T 1 . Since the semitensor pro duct X • ♦ S C is isomorphic to Ba r • gr ( T , C ), it represen ts the ob ject SemiT or S gr ( C , C ) in the deriv ed category of graded k -mo dules (see 11.3). On t he other hand, since X • is a b ounded from ab o v e complex whose terms considered as one- term complexes are semiﬂat complexes of graded righ t S - semimo dules ( S / C / A - semi- ﬂat complexes of gr a ded right S -semimo dules), X • is a semiﬂat complex of graded righ t S - semimo dules ( S / C / A -semiﬂat complex of graded righ t S -semimo dules). The cone of the mor phism X • − → Y • is coacyclic with resp ect to the exact category o f 203 coﬂat graded right C - como dules ( A -ﬂat graded rig h t C -como dules), so the semitensor pro duct X • ♦ S C represen ts also the ob jec t SemiT or T gr ( Y • , C ) in the derive d categor y of graded k - mo dules. In the semideriv ed category o f graded C -coﬂat ( A -ﬂa t) rig h t S -semimo dules there is a distinguished triangle C ( − 1)[1] − → Y • − → C − → C ( − 1)[2] (where the n um b er in round brac k ets denotes the shift of in ternal grading M (1) n = M n +1 ). It f o llo ws from the induced long exact sequence of cohomolog y of the o b jects SemiT or S gr ( − , C ) b y induction in the in ternal degree that Bar • gr ( S , C ) ha s nonzero coho mo lo gy on the diagonal − i = n only if a nd only if Bar • gr ( T , C ) has nonzero cohomolog y on the di- agonal − i = n only . Assume that this is so; then there are short exact seque nces 0 − → H − n +1 n − 1 Bar • gr ( S , C ) − → H − n n Bar • gr ( T , C ) − → H − n n Bar • gr ( S , C ) − → 0. F urther- more, the diagonal cohomology H − n n Bar • gr ( T , C ) and H − n n Bar • gr ( S , C ) a re ﬂat right A -mo dules b y Lemma 11.4 .3.2(a), and so are endo w ed with C - C -bicomo dule struc- tures. The ma ps H − n n Bar • gr ( T , C ) − → H − n n Bar • gr ( S , C ) in the short exact sequences ab ov e are induced b y the mor phism of semialgebras T − → S , hence they ar e mor- phisms of C - C -bicomo dules. Let us describ e the comp ositions H − n n Bar • gr ( T , C ) − → H − n n Bar • gr ( S , C ) − → H − n − 1 n +1 Bar • gr ( T , C ), whic h will b e denoted by ∂ n . Let t : C − → T 1 b e the natural injection. Conside r the endomorphism ∂ X of in ternal degree 1 and cohomological degree − 1 of the complex of g r a ded right S -semimo dules X • that is deﬁned b y t he follo wing for mulas: the comp onent S maps to T +  C S b y t  id, the comp onen t T +  C S maps to T +  C T +  C S by t  id  id − id  t  id, etc. Consider also the endomorphism ∂ Y of in ternal degree 1 and coho mo lo gical degree − 1 of the complex of graded right S - semimo dules Y mapping S 0 ⊕ S 1 to T 1 b y the comp osition of the pro jection S 0 ⊕ S 1 → C and the embedding t . Then the endomorphisms ∂ X and ∂ Y form a comm utativ e diagram with the morphism X • − → Y • . Since the endomorphism ∂ Y represen ts in the semideriv ed category of C -coﬂat ( A -ﬂat) graded S -semimo dules the comp osition of morphisms Y • − → C − → Y • (1)[ − 1] f rom the distinguished triangle a b ov e, the desired maps ∂ n are induced b y the endomorphism ∂ Bar of the bar complex Bar • gr ( T , C ) = X • ♦ S C that is induced b y the endomorphism ∂ X of the complex X • . The endomorphism ∂ Bar maps the com- p onen t C t o T + b y t , the comp onen t T + to T +  C T + b y t  C id − id  C t , etc. Since ∂ Bar is an endomorphism o f complexes of C - C -bicomo dules, ∂ n are also endomor- phisms of C - C -bicomo dules. Hence the maps H − n +1 n − 1 Bar • gr ( S , C ) − → H − n n Bar • gr ( T , C ) in the short exact seque nces ab ov e are mo r phisms of C - C -bicomo dules. Now it fol- lo ws easily by induction using Lemma 1.2.2 that all H − n n Bar • gr ( T , C ) are coﬂat r ig h t C -como dules ( C / A -coﬂat left C -como dules) if a nd o nly if all H − n n Bar • gr ( S , C ) are coﬂat righ t C -como dules ( C / A -coﬂat left C -como dules).  204 A semialgebra S ∼ o v er a coring C endo w ed with a righ t coﬂat (r igh t ﬂat and left relatively coﬂat) increasing ﬁltratio n F is called a right c oﬂat ( right ﬂat and left r elatively c oﬂat ) nonhomo gene ous Koszul semialgeb r a ov er C if the equiv alen t conditions of Theorem are satisﬁed for it, i. e., the graded semialgebras L n F n S ∼ and L n F n S ∼ /F n − 1 S ∼ are right coﬂat (right ﬂat a nd left relativ ely coﬂat) Koszul semialgebras ov er C . 11.6. Poincare–Birkhoﬀ–Witt theorem. Let C b e a coring ov er a k -algebra A ; assume that C is a ﬂat r ig h t A -mo dule. A quasi-diﬀeren tia l coring D ∼ o v er A con- cen trated in the nonnegativ e degrees and endo w ed with an isomorphism C ≃ D ∼ 0 is called right c oﬂat ( right ﬂat and le ft r elatively c oﬂat ) Kosz ul ov er C if the graded coring D ∼ / im ∂ is rig h t coﬂat (right ﬂat and left relativ ely coﬂat) K o szul o v er C . Lemma. L et T b e a right c oﬂat (right ﬂat and left r elatively c oﬂat) Koszul se m i- algebr a over C and E b e the quadr atic dual right c oﬂat (right ﬂat and le f t r elatively c oﬂat) Ko szul c oring ove r C . Then a C - C -bic omo dule morphism C − → T 1 ≃ E 1 c an b e extende d to a gr ad e d T - T -bisemimo dule morphism T − → T of de gr e e 1 (i. e., r ep- r esents a “c entr al element” of T ) i f and only if it c an b e extende d to a c o derivation E − → E of de gr e e 1 of the c oring E (with r esp e ct to the zer o c o derivation of A ). Both the T - T -bisemim o dule mo rp h ism and the c o derivation of E with the given c omp onen t C − → T 1 ≃ E 1 ar e unique if they exist; the c o derivation always has a zer o squar e. Pr o of . Both conditions hold if and o nly if t he diﬀerence o f the tw o maps T 1 ≃ C  C T 1 − → T 1  C T 1 and T 1 ≃ T 1  C C − → T 1  C T 1 induced b y o ur map C → T 1 factorizes through the injection E 2 − → E 1  C E 1 ≃ T 1  C T 1 .  The ob jects of the category of righ t coﬂat no nho mo g eneous Ko szul semialgebras are rig h t coﬂat nonhomogeneous K o szul semialgebras ( S ∼ , F ) o ve r corings C o v er k -alg ebras A suc h that C is a ﬂat righ t A - mo dule. Morphisms are maps of semi- algebras S ∼ − → S ∼ ′ compatible with maps of corings C − → C ′ and morphisms of k -alg ebras A − → A ′ whic h map the ﬁltrat ion comp onen ts F n S ∼ in to the ﬁltratio n comp onen ts F ′ n S ∼ ′ . Imp osing the additional assumption that A has a ﬁnite w eak homological dimension, one analo g ously deﬁnes the category o f righ t ﬂat and left relativ ely coﬂat nonhomo g eneous Koszul semialgebras. The ob jects of the category of righ t coﬂat Koszul quasi-diﬀerential corings are righ t coﬂat Koszul quasi-diﬀeren tia l corings D ∼ o v er corings C o v er k -algebras A suc h that C is a ﬂat righ t A - mo dule. Morphisms are maps o f graded corings D ∼ − → D ∼ ′ compatible with morphisms of k - algebras A − → A ′ and making a commu tative diagram with the co deriv ations ∂ and ∂ ′ . Imp o sing the additional assumption that A has ﬁnite w eak homological dimension, o ne analog ously deﬁnes the category of rig h t ﬂat and left relative ly coﬂat Koszul quasi-diﬀeren tial cor ing s. 205 Theorem. The c ate gory of right c oﬂat (right ﬂat a nd left r elatively c oﬂat) nonho- mo gene ous Koszul semialgeb r as is e quivalent to the c ate gory of right c oﬂat (right ﬂat and left r elatively c o ﬂat) Koszul quasi-diﬀer en tial c o ri n gs. If a ﬁ l ter e d semia l g e - br a S ∼ over a c oring C and a quasi-diﬀ er ential c oring D ∼ c orr e sp ond to e ach other under this duality, then the gr ade d sem ialgebr a T = L n F n S ∼ and the gr ade d c or- ing D ∼ ar e quadr atic dual right c oﬂat (right ﬂat and left r elatively c oﬂat) Koszul semialgebr a and c oring over C ; the gr ade d semialgebr a S = L n F n S ∼ /F n − 1 S ∼ and the gr ade d c oring D = D ∼ / im ∂ ar e quadr atic dual ri g ht c oﬂat (right ﬂat and left r elatively c oﬂat) K o szul semialgebr a and c o ring ove r C ; the r elate d isomo rphisms F 1 S ∼ ≃ D ∼ 1 and F 1 S ∼ /F 0 S ∼ ≃ D ∼ 1 /∂ 0 D ∼ 0 ar e c omp atible with e ach other; and the inje ction F 0 S ∼ − → F 1 S ∼ c orr e sp onds to the c o derivation c omp onent ∂ 0 : D ∼ 0 − → D ∼ 1 under the iso m orphisms F 0 S ∼ ≃ C ≃ D ∼ 0 and F 1 S ∼ ≃ D ∼ 1 . Pr o of . It follows from Lemma that the category of right coﬂat (righ t ﬂat and left relativ ely coﬂat) Koszul semialgebras T endo w ed with a T - T -bisemimo dule morphism T − → T of degree 1 is equiv alent to the category o f right coﬂat (r ig h t ﬂat and left relativ ely coﬂat) Koszul corings E endo w ed with a co deriv ation o f degree 1. It remains to pro v e t ha t semialgebras T with maps T − → T of degree 1 coming from righ t coﬂat (r ig h t ﬂat and left relativ ely coﬂat) nonhomogeneous Koszul semialgebras S ∼ corresp ond under this equiv alence to right coﬂat (r ig h t ﬂat and left relativ ely coﬂat) Koszul quasi-diﬀeren tial corings D ∼ = E and vice v ersa. Besides, we will ha v e to sho w that whenev er for a quasi-diﬀerential coring D ∼ the graded coring D ∼ / im ∂ is a righ t coﬂat ( left relativ ely coﬂat) Koszul coring ov er a coring C , the graded coring D ∼ is also a right coﬂat (left ﬂat and righ t relativ ely coﬂa t ) Koszul coring ov er C . According to the pro of of Theorem 11.5, fo r any righ t coﬂat (right ﬂat and left relativ ely coﬂat) nonhomogeneous Koszul semialgebra S ∼ there is a right coﬂat (right ﬂat and left relatively coﬂat) Koszul quasi-diﬀeren tial coring D ∼ . Indeed, set D ∼ = L n H − n n Bar • gr ( T , C ), where T = L n F n S ∼ ; then the endomorphism ∂ of the C - C - bi- como dule D ∼ induced b y the endomorphism ∂ Bar of t he reduced bar construction Bar • gr ( T , C ) is a co deriv ation of degree 1 (with resp ect to the zero co deriv a tion of A ) and its restriction to D ∼ 0 coincides with the injection D ∼ 0 ≃ T 0 − → T 1 ≃ D ∼ 1 . It also follo ws f rom this pro of that the r ig h t coﬂat (right ﬂa t and left relativ ely coﬂat) Koszul semialgebra S = L F n S ∼ /F n − 1 S ∼ is quadratic dual to the coring D = D ∼ / im ∂ , whic h is therefore right coﬂat (rig h t ﬂat and left relativ ely coﬂat) Koszul ov er C . Let us no w construct the nonhomogeneous Koszul semialgebra corresp onding to a righ t coﬂat (right coﬂat and left relativ ely coﬂat) Ko szul quasi-diﬀerential coring D ∼ o v er a coring C . Set D = D ∼ / im ∂ . Consider the big r aded coring K o v er the k - a lgebra A (whic h is considered as a bigraded k -alg ebra concen tra ted in the bidegree (0 , 0 )) with the comp onen ts K p,q = D ∼ q − p for p 6 0 , q 6 0 and K p,q = 0 otherwise. The coring K considered as a gr a ded coring in the total g rading p + q has a co deriv ation ∂ K (with respect to the zero co deriv ation of A ) mapping the comp onent 206 K p,q to K p,q +1 b y ∂ q − p ; one has ∂ 2 K = 0. There is a morphism of bigra ded cor ing s K − → D inducing an isomorphism of the corings of cohomology , where the coring D is placed in the bigrading D p, 0 = D − p and endo w ed with the zero diﬀeren tial. Denote by K + the cok ernel of the injection C ≃ K 0 , 0 − → K . Let R = L ∞ r =0 K  C r + b e the “cotensor semialgebra” of the bigra ded C - C -bicomo dule K + . By the deﬁnition, R is a trigraded semialgebra ov er the coring C (whic h is considered as a t r igraded coring concen trated in the tridegree (0 , 0 , 0)) with the gradings p and q inherited from t he bigrading of K + and the a dditional grading r by the n um b er of cotensor factors. W e will consider R as a g r aded semialgebra in t he total grading p + q + r . The semialgebra R is endow ed with three semideriv ations (with resp ect to the zero deriv ation of the coring C ) of total degree 1, whic h w e will no w introduce. Let ∂ R b e the only semideriv ation of R whic h preserv es K + ⊂ R (embedded as the part of degree r = 1) and whose restriction to K + is equal to − ∂ K . Let d R b e the only semideriv ation of R whic h maps K + to K +  C K + b y the comp osition of the com ultiplication map K + − → K +  C K + and t he sign automorphism of K +  C K + acting o n the comp onen t K p ′ ,q ′  C K p ′′ ,q ′′ as ( − 1) p ′ + q ′ . Fina lly , let δ R b e the only semideriv ation of R whose restriction to K + is the iden tity map of the compo nent K − 1 , − 1 ≃ C to the semiunit comp onen t R 0 , 0 , 0 = C a nd zero on all the remaining comp onen ts of K + . All the three diﬀeren tials are constructed so that they satisfy t he sup er-Leibniz rule in the parity p + q + r . The semideriv ations ∂ R , d R , a nd δ R ha v e tridegrees ( 0 , 1 , 0 ) , (0 , 0 , 1), and (1 , 1 , − 1), resp ectiv ely , in the trigrading ( p, q , r ). All the three semideriv ations ha v e zero squares, a nd they pa ir wise an ti-commu te. There is a rig h t coﬂa t (right ﬂat and left relatively coﬂat) increasing ﬁltrat io n F on the graded semialgebra R whose comp onen t F n R is the direct sum of all trigrading comp onen ts R p,q ,r with − p 6 n . This ﬁltration is compatible with the diﬀerentials ∂ R , d R , and δ R ; the semialgebra L n F n R /F n − 1 R with the diﬀeren tial induced b y ∂ R + d R + δ R is natura lly isomorphic to the semialgebra R with the diﬀerential ∂ R + d R . Consider the f o llo wing sign-mo diﬁed vers ion of cobar construction ′ Cob( D , C ). D e- ﬁne ′ Cob( D , C ) as the “tensor semialgebra” L r D  C r + of the C - C - bicomo dule D + and endo w it with the gra ding p coming from the grading D p = D − p of D + and the grading r b y the n um b er of cotensor factors. W e will consider ′ Cob( D , C ) as a graded semialgebra ov er C in the tota l grading p + r . Let d ′ Cob b e the only co deriv ation of ′ Cob( D , C ) whic h maps D + ⊂ ′ Cob( D , C ) to D +  C D + b y the comp o sition of the com ultiplication map D + − → D +  C D + and t he sign automorphism o f D +  C D + act- ing on the comp onen t D p ′  C D p ′′ as ( − 1) p ′ . Then one has d ′ 2 Cob = 0 . Notice that the diﬀeren tial d ′ Cob satisﬁes the sup er-Leibniz rule in the parit y p + r , while the diﬀeren- tial d Cob of the cobar construction Cob • gr ( D , C ) satisﬁes the sup er-Leibniz rule in the parit y r . The a ut o morphism of L r D  C r + acting on the comp onent D p 1  C · · ·  C D p r b y minus one to the p o w er P r s =1 p s ( p s + 1) / 2 + P 1 6 s 0 (where, as alw a ys, S = L n F n S ∼ /F n − 1 S ∼ ). Th us in b o t h cases the quotien t complex is coacyclic. F ourthly , it remains to ch ec k that for an y quasi-diﬀeren tial left D ∼ -como dule L the cone of the natural morphism of quasi-diﬀeren tial D ∼ -como dules L − → ΞΥ • ( L ) is coacyclic. First let us show that it suﬃces t o consider the case when L is a graded C -como dule endo w ed with a graded D ∼ -como dule structure via the em b ed- ding o f corings D ∼ 0 − → D ∼ . T o this end, consider the increasing ﬁltration of L b y quasi-diﬀeren tial D ∼ -subcomo dules G n ( L ) = ν − 1 L ( L i 6 n D ∼ i  C L ), where ν L : L − → D ∼  C L denotes the coaction map. The quotien t quasi-diﬀeren tial como dules G n ( L ) /G n − 1 ( L ) are are graded D ∼ -como dules originating from graded C -como dules; the ﬁltrat io n G induces a ﬁltration on the cone of the morphism L − → ΞΥ • ( L ) 215 whose comp onen ts are t he cones of the mor phisms G n ( L ) − → ΞΥ • ( G n ( L )); and the asso ciated quotient quasi-diﬀeren tial como dules o f the latter ﬁltration are the cones of the morphisms G n ( L ) /G n − 1 ( L ) − → ΞΥ • ( G n ( L ) /G n − 1 ( L )). No w assume that L is a graded C -como dule with the induced graded D ∼ -como dule structure, or ev en a complex of C - como dules with the induced quasi-diﬀerential D ∼ -como dule structure. In this case, the quasi-diﬀeren tial D ∼ -como dule ΞΥ • ( L ) = D  C S ∼  C L has an increasing ﬁltratio n by quasi-diﬀeren tial sub como dules giv en b y the fo r mula F n ΞΥ • ( L ) = P i + j 6 n D j  C F i S ∼  C L ⊂ D  C S ∼  C L . The cone of the morphism L − → ΞΥ • ( L ) has the induced ﬁltration F whose comp onents are the cones of the morphisms L − → F n ΞΥ • ( L ). The asso ciated quotien t como dules of the latter ﬁltration are coacyclic quasi-diﬀeren tia l D ∼ -como dules. Indeed, the comp o- nen t F 0 cone( L − → ΞΥ • ( L )) is isomorphic to the cone o f the iden tit y endomorphism of L , while the quotient quasi-diﬀeren tial como dules with n > 0 are isomorphic to cotensor pro ducts of p ositiv e-degree comp onen ts of the Koszul complex D  C S and the C -como dule L , endow ed with the quasi-diﬀeren tial D ∼ -como dule structures o rig- inating from their structures of complexes of C -como dules. Th us all these quotien t quasi-diﬀeren tial como dules are coacyclic. P art (b): w e will only construct a pair of adjoint functors b etw een the DG -category of complexes of righ t S ∼ -semimo dules a nd the DG-category of quasi-diﬀeren tial righ t D ∼ -como dules; t he rest of t he pro of is iden tical to that of part (a ). The functor Ξ assigns to a rig ht S -semimodule N the graded rig h t D - como dule N  C D endo w ed with a righ t D ∼ -como dule structure in terms of the follo wing maps N  C D n − → N  C D ∼ n . Consider the map N  C D n − → N  C D n equal to the diﬀerence o f the identit y map and the comp osition N  C D ∼ n − → N  C D ∼ 1  C D ∼ n − 1 − → N  C D ∼ n − 1 − → N  C D ∼ n of the map induced by the com ultiplication morphism, the map induced b y the semiaction morphism, and the map induced b y the morphism ∂ n − 1 . This diﬀerence factorizes through the surjection N  C D ∼ n − → N  C D n , hence the desired map. The functor Υ assigns to a quasi-diﬀerential rig ht D ∼ -como dule R the complex of righ t S ∼ -semi- mo dules Υ • ( R ) = R  C S ∼ with the terms Υ i ( R ) = R − i  C S ∼ and t he diﬀeren tials d i deﬁnes as ( − 1) i times the comp osition R − i  C S ∼ − → R − i − 1  C D ∼ 1  C S ∼ − → R − i − 1  C S ∼ of the map induced b y the coaction morphism and the map induced b y the semim ultiplicatio n morphism. P art (c): let us construct a pair of adjoin t functors betw een the DG -cate- gory of complexes of left S ∼ -semicontramo dules and the D G-category of quasi- diﬀeren tial left D ∼ -contramo dules. The f unctor Ξ assigns to a left S -semicontra- mo dule P the graded left D -contramo dule Cohom C ( D , P ) endo w ed with a left D ∼ -contramo dule structure in terms of t he following maps Cohom C ( D ∼ n , P ) − → Cohom C ( D n , P ). Consider the map Cohom C ( D ∼ n , P ) − → Cohom C ( D ∼ n , P ) equal to the diﬀerence o f the iden tit y map and the comp osition Cohom C ( D ∼ n , P ) − → Cohom C ( D ∼ n − 1 , P ) − → Cohom C ( D ∼ n − 1 , Cohom C ( D ∼ 1 , P )) − → Cohom C ( D ∼ n , P ) of 216 the map induced by the mor phism ∂ n − 1 , the map induced b y the semicon traac- tion morphism P − → Cohom C ( F 1 S ∼ , P ), and the map induced by the comu l- tiplication morphism D ∼ n − → D ∼ 1  C D ∼ n − 1 . This diﬀerence f actorizes through the injection Cohom C ( D n , P ) − → Cohom C ( D ∼ n , P ), hence the desired map. The functor Υ right adjoin t to Ξ assigns to a quasi-diﬀeren tial left D ∼ -contramo dule Q the complex of left S ∼ -semicontramo dules Υ • ( Q ) = Cohom C ( S ∼ , Q ) with t he terms Υ i ( Q ) = Cohom C ( S ∼ , Q − i ) and the diﬀerential deﬁned as the composition Cohom C ( S ∼ , Q ) − → Cohom C ( F 1 S ∼  C S ∼ , Q ) − → Cohom C ( S ∼ , Q ) of the map in- duced by the semim ultiplication morphism F 1 S ∼  C S ∼ − → S ∼ and the map induced b y the con traaction morphism Cohom C ( D ∼ 1 , Q ) − → Q . The r est of the pro of is a nalogous to that of part ( a ), with the exception of the argumen t related to the ﬁltration G (the ﬁrst step of the fourth pa rt of the pro of ). The problem here is that the decreasing ﬁltration G o f a graded D ∼ -con- tramo dule Q whose comp onen ts are the images G n Q of the contraaction maps Cohom C ( D ∼ / L i 6 n D ∼ i , Q ) − → Q is not in general se parated, i. e., the in tersec- tion of G n Q ma y b e nonzero ( see App endix A). What one should do is replace an arbitrary quasi-diﬀeren tial left D ∼ -contramo dule Q with the tota l quasi-diﬀeren tial con tramo dule R of its bar resolution · · · − → Cohom C ( D ∼  C D ∼  C D ∼ , Q ) − → Cohom C ( D ∼  C D ∼ , Q ) − → Cohom C ( D ∼ , Q ). Since the cone of the natural mor- phism of quasi-diﬀeren tial D ∼ -contramo dules R − → Q is con traacyclic, one can consider the quasi-diﬀeren tial D ∼ -contramo dule R instead o f Q . In addition to the ﬁlt r a tion G in tro duced a b ov e, consider also the decreasing ﬁltration ′ G o f a g r a ded D ∼ -contramo dule Q whose comp onen ts are the images ′ G n Q of the con traaction maps Cohom C ( D / L i 6 n D i , Q ) − → Q . It is clear that R ≃ lim ← − n R / ′ G n R . Next one can either sho w that ′ G is a ﬁltration by g raded D ∼ -sub- contramo dules and use the ﬁltration ′ G of R , or sho w that the ﬁltrations G and ′ G are commensurable, ′ G n R ⊂ G n R ⊂ ′ G n − 1 R , and use the ﬁltration G of R . (The quotient quasi-diﬀeren tial D ∼ -contramo dules G n R /G n +1 R originate from graded C -contra- mo dules, while the quotien t quasi-diﬀeren t ial D ∼ -contramo dules ′ G n R / ′ G n +1 R orig- inate f r o m complexes of C - contramo dules, whic h is also suﬃcien t.) Both assertions for an arbitrary graded D ∼ -contramo dule follow fro m the f act tha t the comp osition D ∼ i − → D ∼ 1  C D ∼ i − 1 − → D ∼ 1  C D i − 1 of the comultiplication map and the map in- duced b y the natural surjection D ∼ i − 1 − → D i − 1 is injectiv e and its cok ernel, b eing isomorphic to the cok ernel of the comultiplic ation map D i − → D 1  C D i − 1 , is a co- pro jectiv e left C - como dule. T o c hec k the latter, consider the comp osition of the map D ∼ i − → D ∼ 1  C D i − 1 in question with the map ∂ i − 1 . Alternativ ely , one can replace an arbitrary quasi-diﬀeren tial D ∼ -contramo dule Q with the cone of the morphism k er(Cohom C ( D ∼ , Q ) → Q ) − → Cohom C ( D ∼ , Q ) and use t he appropriate generalization of Lemma A.2 .3.  217 Remark. Not ice that no homological dimension condition on the k -alg ebra A is assumed in the ab o v e Theorem. In part icular, when C = A , so S ∼ is just a ﬁl- tered k -algebra, Theorem pro vides a description of certain semideriv ed categories of S ∼ -mo dules relativ e to F 0 S ∼ = A . A description of the conv en tional deriv ed cat- egory can also be obtained. Namely , in t he assumptions of pa rt ( a) of Theorem the con ven tional deriv ed category of left S ∼ -semimo dules is equiv alen t to the quo- tien t category of the co derive d category of quasi-diﬀeren tial left D ∼ -como dules b y its minimal triangulated sub catego ry con taining all the quasi-diﬀeren tial D ∼ -como dules originating from acyclic complexes of left C -como dules and closed under inﬁnite direct sums. This is so b ecause for an y acyclic complex of S ∼ -semimo dules M • the quasi- diﬀeren tial D ∼ -como dules F n Ξ( M • ) /F n − 1 Ξ( M • ) originate from acyclic complexes o f C -como dules, and con v ersely , for an y quasi-diﬀeren tial D ∼ -como dule L or ig inating from an acyclic complex of C -como dules the complex of S ∼ -semimo dules Υ • ( L ) is acyclic. The ana logous result holds for righ t S ∼ -semimo dules in the assumptions of part (b); and in the assumptions of par t (c) the conv en tional deriv ed catego ry of left S ∼ -semicontramo dules is equiv alen t to the quotien t category of the con traderiv ed category of quasi-diﬀeren tial left D ∼ -contramo dules b y its minimal triangulated sub- category con taining all the quasi-diﬀeren tial contramodules orig inating from acyclic complexes of left C -contramo dules and closed under inﬁnite pro ducts. 11.9. SemiT or and Cotor, SemiExt and Co ext . 11.9.1. Let ( D ∼ , ∂ ) b e a quasi-diﬀeren tial coring o v er a k -alg ebra A ; assume that D = D ∼ / im ∂ is a ﬂat left and righ t A - mo dule. Let N b e a quasi-diﬀeren tial right D ∼ -como dule and M b e a quasi-diﬀeren tial left D ∼ -como dule. Assume that one of the graded A -mo dules N and M is ﬂat. Then on the cotensor pro duct N  D M of the graded como dules N and M ov er the graded coring D there is a natural diﬀeren tia l with zero square, whic h is deﬁned as follows . Consider the ma p δ : N  D M − → N  D D ∼  D M giv en b y the for mula x  y 7− → − x (0)  x (1)  y + x  y ( − 1)  y (0) . This map factorizes through the injection N  D M − → N  D D ∼  D M giv en b y the for mula x  y 7− → ( − 1) | x (0) | x (0)  ∂ ( x (1) )  y = ( − 1) | x | x  ∂ ( y ( − 1) )  y (0) , hence t he desired map d : N  D M − → N  D M . Let us che c k that d 2 = 0, that is the image of d is con tained in N  D ∼ M . Set d ( x  y ) = x ′  y ′ . Consider the tw o elemen ts x ′  y ′ ( − 1)  y ′ (0) and x ′ (0)  x ′ (1)  y of t he cotensor pro duct N  D D ∼  D M ; w e ha v e t o c hec k t ha t these tw o elemen ts coincide. Consider the image of the former elemen t under the map N  D D ∼  D M − → N  D D ∼  D D ∼  D M giv en by the formula u  b  v 7− → ( − 1) | u (0) | u (0)  ∂ ( u (1) )  b  v and t he image of the latter elemen t under the map N  D D ∼  D M − → N  D D ∼  D D ∼  D M giv en b y the form ula u  b  v 7− → ( − 1) | u | + | b | u  b  ∂ ( v ( − 1) )  v (0) . The sum of these tw o elemen ts of N  D D ∼  D D ∼  D M is equal to t he image of t he elemen t δ ( x  y ) under the map N  D D ∼  D M − → N  D D ∼  D D ∼  D M induced b y t he 218 com ultiplication map D ∼ − → D ∼  D D ∼ . It remains to not ice that the Cart hesian square formed by the maps D ∼ − → D ∼ ⊕ D ∼ , D ∼ ⊕ D ∼ − → D ∼  D D ∼ , D ∼ − → D ∼ , and D ∼ − → D ∼  D D ∼ constructed in 11.7.1 remains Carthesian after taking the cotensor pro duct with N and M . W e will denote the complex w e hav e constructed b y N  • D M ; its terms are N  n D M = ( N  D M ) − n . No w assume that the ring A has a ﬁnite w eak homological dimension. In order to deﬁne the double-sided deriv ed functor of the functor  • D , we will show that the co deriv ed category of quasi-diﬀeren tial D ∼ -como dules is equiv alen t to the quotien t category o f the homo t o p y category of D -coﬂat quasi-diﬀeren tial D ∼ -como dules b y its in tersection with the thic k sub category of coacyclic quasi-diﬀeren tial D ∼ -como dules. The argumen t is analogo us to that of either Theorem 2.5 or Theorem 2.6. First let us construct fo r any quasi-diﬀeren tial left D ∼ -como dule K a morphism in to it from an A -ﬂat quasi-diﬀerential left D ∼ -como dule L 1 ( K ) with a coacyclic cone. Use the graded v ersion of Lemma 1.1 .3 to obrain a ﬁnite resolution 0 − → Z − → P d − 1 ( K ) − → · · · − → P 0 ( K ) − → K of a graded D ∼ -como dule K consisting of A -ﬂat graded D ∼ -como dules. The t o tal quasi-diﬀeren tial D ∼ -como dule of the com- plex o f quasi-diﬀeren tial D ∼ -como dules Z − → P d − 1 ( K ) − → · · · − → P 0 ( K ) is an A -ﬂat quasi-diﬀeren tia l D ∼ -como dule whose morphism in to K ha s a coacyclic cone. Indeed, t he total quasi-diﬀeren tia l D ∼ -como dule of an y acyclic complex of quasi- diﬀeren tial D ∼ -como dules b o unded from b elow is coacyclic, since it ha s an increas- ing ﬁltration by quasi-diﬀeren tial D ∼ -subcomo dules suc h that the asso ciated quotien t D ∼ -como dules are isomorphic to cones of iden tit y endomorphisms of certain quasi- diﬀeren tial D ∼ -como dules. No w let us construct for an y A -ﬂat quasi-diﬀeren tial left D ∼ -como dule L a mo r - phism fr o m it into a D -coﬂat quasi-diﬀeren tia l left D ∼ -como dule R 2 ( L ) with a coa- cyclic cone. Consider the cobar construction D ∼ ⊗ A L − → D ∼ ⊗ A D ∼ ⊗ A L − → · · · Notice that D ∼ is a coﬂat gra ded left D -como dule, since there is an exact triple o f left D -como dules D ( − 1) − → D ∼ − → D (where D ( − 1) i = D i − 1 ). Hence the tot al quasi- diﬀeren tial D ∼ -como dule of this cobar complex of quasi-diﬀeren tial D ∼ -como dules is a D -coﬂat quasi-diﬀeren t ial D ∼ -como dule suc h that the map into it from the quasi- diﬀeren tial D ∼ -como dule L has a coacyclic cone. It is easy to see that the cotensor pro duct of a quasi-diﬀeren tial r ig h t D ∼ -como dule and a quasi-diﬀeren tial left D ∼ -como dule is an acyclic complex whenev er o ne of the tw o quasi-diﬀeren tial D ∼ -como dules is coa cyclic and the other o ne is D -coﬂat. The derive d functor Cotor D ∼ q on the Cart hesian pro duct of co derived categories of righ t and left quasi-diﬀerential D ∼ -como dules is deﬁned by restricting the func- tor  • D to the Carthesian pro duct of the ho motop y category of quasi-diﬀeren tial righ t D ∼ -como dules and the homotopy category of D - coﬂat quasi-diﬀeren tial left D ∼ -como dules or to the Carthesian pro duct of the homotop y category of D -coﬂat quasi-diﬀeren tial right D ∼ -como dules and the homot o p y categor y of quasi-diﬀeren tial 219 left D ∼ -como dules, and comp osing it with the lo calization functor Hot ( k – mo d ) − → D ( k – mo d ). 11.9.2. Let ( D ∼ , ∂ ) b e a quasi-diﬀeren tial coring o v er a k -alg ebra A ; assume that D = D ∼ / im ∂ is a pro jectiv e left and a ﬂat righ t A -mo dule. Let M b e a quasi-diﬀeren tial left D ∼ -como dule and P b e a quasi-diﬀeren tial left D ∼ -contramo dule. Assume that either the graded A -mo dule M is pro jective , or the graded A -mo dule P is injectiv e. Then on the graded k - mo dule of cohomomorphisms Cohom D ( M , P ) fr o m the graded como dule M to the gr a ded con tramo dule P ov er the gra ded coring D there is a nat ur a l diﬀerential with zero square, whic h is deﬁned as follow s. Consider the map δ : Cohom D ( M , Cohom D ( D ∼ , P )) ≃ Cohom D ( D ∼  D M , P ) − → Cohom D ( M , P ) deﬁned b y the form ula f 7− → π P ◦ f − f ◦ ν M (where π P and ν M denote the con traaction a nd coaction morphisms). This map f actorizes through the surjection Cohom D ( D ∼  D M , P ) − → Cohom D ( M , P ) induced by the morphism ∂ : D − → D ∼ , hence the map d : Cohom D ( M , P ) − → Cohom D ( M , P ). W e will denote the complex w e hav e constructed b y Cohom • D ( M , P ); its terms are Cohom n D ( M , P ) = Cohom D ( M , P ) − n . Assume that the ring A has a ﬁnite left homological dimension. Then the co deriv ed category of quasi-diﬀeren tial left D ∼ -como dules is equiv alent to the quotien t category of the homotop y categor y of D -coprojectiv e quasi-diﬀeren tial left D ∼ -como dules b y it s in tersection with the thic k sub category of coacyclic quasi-diﬀeren tial D ∼ -como dules. Analogously , the contraderiv ed category of quasi-diﬀerential left D ∼ -contramo dules is equiv alen t to the quotient catego r y of the homotopy category of D -coinjectiv e quasi- diﬀeren tial left D ∼ -contramo dules by its in tersection with the thic k sub category of con traacyclic quasi-diﬀeren tial D ∼ -contramo dules. The double-sided deriv ed functor Co ext q D ∼ on the Carthesian pro duct of the co deriv ed category of quasi-diﬀerential left D ∼ -como dules and the contraderiv ed category of quasi-diﬀeren tial left D ∼ -con- tramo dules is deﬁned b y restricting the functor Cohom • D to the Carthesian pro duct of the homotopy category o f D -coprojectiv e quasi-diﬀeren tial left D ∼ -como dules and the homotop y category of quasi-diﬀeren tial left D ∼ -contramo dules or to the Carthesian pro duct of the homotopy category of quasi-diﬀerential left D ∼ -como dules and the homotop y category of D -coinjective quasi-diﬀeren tia l left D ∼ -contramo dules, and comp osing in with the lo calization functor Hot ( k – mo d ) − → D ( k – mo d ). 11.9.3. Let C b e a coring ov er a k -alg ebra A . Assume that C is a ﬂat left a nd rig ht A -mo dule and A has a ﬁnite weak homolog ical dimension. Let S ∼ b e a left and righ t coﬂat nonhomogeneous Koszul semialgebra o v er C , and D ∼ b e the left and rig ht coﬂat Koszul quasi-diﬀeren tial coring nonhomogeneous quadratic dual to S ∼ . 220 Corollary . (a) The e quivalenc es of c ate gories D si ( simo d – S ∼ ) ≃ D co ( qcmd – D ∼ ) and D si ( S ∼ – simo d ) ≃ D co ( D ∼ – qcmd ) tr ansform the derive d functor SemiT or S ∼ into the derive d functor Cotor D ∼ q . (b) Assume add itional ly that C is a pr oje ctive left A -mo dule, A has a ﬁnite left homolo g ic al dimens i o n, and S ∼ is a left c opr oje ctive noh nomo ge n e ous Koszul se m i- algebr a. Then the e quival e nc es of c ate gories D si ( S ∼ – simo d ) ≃ D co ( D ∼ – qcmd ) and D si ( S ∼ – sicntr ) ≃ D ctr ( D ∼ – qcntr ) tr ansform the derive d functor SemiExt S ∼ into the derive d functor Coext q D ∼ . Pr o of . P art (a): for an y complex of righ t S ∼ -semimo dules N • and an y quasi- diﬀeren tial left D ∼ -como dule L there is a natural isomorphism o f complexes of k -mo dules Ξ( N • )  • D L ≃ N • ♦ S ∼ Υ • ( L ). Indeed, b oth complexes a r e isomor- phic to the total complex of the bicomplex N i  C L j , one of whose diﬀeren tials is induced by the diﬀeren tial in N • and the o ther is equal to the comp osition N i  C L j − → N i  C D ∼ 1  C L j − 1 − → N i  C L j − 1 of the map induced by the D ∼ -coaction in L and the map induced by the S ∼ -semiaction in N i . No w let N • b e a semiﬂat com- plex of C -coﬂa t right S ∼ -semimo dules and M • b e a complex o f left S ∼ -semimo dules. Then there is an isomorphism Ξ( N • )  • D Ξ( M • ) ≃ N • ♦ S ∼ Υ • Ξ( M • ) and a quasi- isomorphism N • ♦ S ∼ Υ • Ξ( M • ) − → N • ♦ S ∼ M • . Analogously , for a complex of r ig h t S ∼ -semimo dules N • and a semiﬂat complex of C - coﬂat left S ∼ -semimo dules M • there is an isomorphism Ξ( N • )  • D Ξ( M • ) ≃ Υ • Ξ( N • ) ♦ S ∼ M • and a quasi-isomorphism Υ • Ξ( N • ) ♦ S ∼ M • − → N • ♦ S ∼ M • . It is easy to chec k that the square diagram formed b y these maps is comm utative . The pr o of of part ( b) is completely analogous.  Question. Can one construct a como dule-con tramo dule corresp ondence (equiv alence b et w een the co derive d and con traderiv ed categories) for quasi-diﬀeren tial como dules and con tramo dules? Also, is there a nat ur a l closed mo del categor y structure o n the category of quasi-diﬀeren tial como dules (contramodules)? 221 Appendix A. Contramodules over Coalgebras o ver Fie lds Let C b e a coasso ciativ e coalgebra with counit ov er a ﬁeld k . It is w ell-kno wn [45] that C is the union of its ﬁnite-dimensional sub coalg ebras and any C - como dule is a union of ﬁnite-dimensional como dules ov er ﬁnite-dimensional sub coalgebras of C . The dual assertion for C -contramo dules is not true: for the most common of non- semisimple inﬁnite- dimensional coalgebras C there exist C -contramo dules P such that the in tersection of t he images of Hom C ( C / U , P ) in P ov er all ﬁnite-dimensional sub coalgebras U ⊂ C is nonzero. A w eak er statemen t holds, how ev er: if the map Hom C ( C / U , P ) − → P is surjectiv e f o r a ny ﬁnite-dimensional sub coalgebra U of C , then P = 0. Besides, ev en tho ug h adic ﬁlt r a tions of contramo dules a r e not in general separated, they are a lw a ys c omplete . Using the relat ed tec hniques w e sho w that any con traﬂat C -contramo dule is pro jectiv e, generalizing the w ell-kno wn result that any ﬂat mo dule ov er a ﬁnite- dimensional algebra is pro jectiv e [5]. A.1. Coun terexamples. A.1.1. Let C b e the coalgebra for whic h the dual algebra C ∗ is isomorphic to the algebra of formal p ow er series k [[ x ]]. Then a C -contra mo dule P can b e equiv alen tly deﬁned as a k -ve ctor space endo w ed with the fo llowing op eratio n of summation of sequence s o f v ectors with formal coeﬃcien ts x n : for a ny elemen ts p 0 , p 1 , . . . in P , an elemen t of P denoted b y P ∞ n =0 x n p n is deﬁned. This op eration should satisfy the follo wing equations: P ∞ n =0 x n ( ap n + bq n ) = a P ∞ n =0 x n p n + b P ∞ n =0 x n q n for a , b ∈ k , p n , q n ∈ P (linearit y); P ∞ n =0 x n p n = p 0 when p 1 = p 2 = · · · = 0 (counit y); and P ∞ i =0 x i  P ∞ j =0 x j p ij  = P ∞ n =0 x n  P i + j = n p ij  for any p ij ∈ P , i, j = 0 , 1 , . . . (con- traasso ciativit y). Here the in terior summation sign in t he righ t hand side denotes the conv en tional ﬁnite sum of elemen ts of a vec tor space, while t he three other sum- mation signs refer to the contramo dule inﬁnite summation op eration. The fo llo wing examples o f C -contramo dules are reve aling. Let E denote t he free C -contramo dule generated b y the sequence of sym b ols e 0 , e 1 , . . . ; its elemen ts can b e represen ted as formal sums P ∞ i =0 a i ( x ) e i , where a i ( x ) are formal p o w er series in x suc h that the sequenc e of their orders of zero ord x a i ( x ) at x = 0 tends to inﬁnit y as i increases. Let F denote the free C -contramo dule generated by t he sequ ence of sym b ols f 1 , f 2 , . . . ; then C -contramo dule homomorphisms from F to E corresp o nd bijectiv ely to sequences of elemen ts of E that are images of the elemen ts f i . W e are in terested in the map g : F − → E sending f i to x i e i − e 0 ; in other w ords, an elemen t P ∞ i =1 b i ( x ) f i of F is mapp ed to the elemen t P ∞ i =1 x i b i ( x ) e i −  P ∞ i =1 b i ( x )  e 0 . It is clear from this form ula that the elemen t e 0 ∈ E do es not b elong to the image of g . Let P denote the cok ernel o f the morphism g and p i denote the images of the elemen ts e i in P . Then one has p 0 = x n p n in P ; in other w or ds, the elemen t p 0 b elongs to 222 the image of Hom C ( C / U , P ) under the con traaction map Hom C ( C , P ) − → P f or any ﬁnite-dimensional sub coalgebra U = ( k [[ x ]] /x n ) ∗ of C . No w let E ′ b e the free C - contramo dule generated b y the sym b ols e 1 , e 2 , . . . , P ′ denote the cok ernel of the map g ′ : F − → E ′ sending f i to x i e i , and p ′ i denote the im- ages of e ′ i in P ′ . Then the result of the contramo dule inﬁnite summation P ∞ n =1 x n p ′ n is nonzero in P ′ , ev en though ev ery elemen t x n p ′ n is equal to zero. Therefore, the c ontr a m o dule summ a tion op er ation c anno t b e understo o d as any kind of limit of ﬁnite p artial sums . Actually , t he C -contramo dule P ′ is just the direct sum of the con tra- mo dules k [[ x ]] /x n k [[ x ]] o v er n = 1, 2, . . . in the category of C - contramo dules. Notice that the elemen t P ∞ n =1 x n p ′ n also b elongs to the image of Hom C ( C / U , P ′ ) in P ′ for an y ﬁnite-dimensional sub coalg ebra U ⊂ C . Remark. In the ab ov e no t a tion, a C -contramo dule structure on a k -v ector space P is uniquely determined by the underlying structure of a mo dule o v er the algebra of p olynomials k [ x ]; the natural functor C – contra − → k [ x ]– mo d is fully f aithful. Indeed, for an y p 0 , p 1 , . . . in P the sequence q m = P ∞ n =0 x n p m + n ∈ P , m = 0, 1, . . . is the unique solution of the sy stem of equations q m = p m + xq m +1 . The image o f this functor is a full ab elian sub cat ego ry closed under k ernels, cok ernels, extensions, and inﬁnite pro ducts; it consists of all k [ x ]-mo dules P suc h that Ext i k [ x ] ( k [ x, x − 1 ] , P ) = 0 for i = 0, 1 . It f ollo ws that if D is a coalgebra for whic h the dual algebra D ∗ is isomorphic to a quotient algebra of the algebra of formal p ow er series k [[ x 1 , . . . , x m ]] in a ﬁnite num b er of (commuting) v a r iables by a closed ideal, then the natural f unctor D – contra − → k [ x 1 , . . . , x m ]– mo d is fully faithful. A.1.2. Now let us giv e an example of ﬁn i te-d i m ensional (namely , t w o-dimensional) con tramo dule P ov er a coalgebra C suc h tha t the inte rsection o f the images of Hom C ( C / U , P ) in P is nonzero. Not ice tha t for any coalgebra C there are natural left C ∗ -mo dule structures on any left C -como dule and an y left C - contramo dule; that is there are natural fa ithful functors C – como d − → C ∗ – mo d a nd C – contra − → C ∗ – mo d (where C ∗ is considered as an abstract algebra without any top ology). The functor C – como d − → C ∗ – mo d is fully faithful, while the functor C – contra − → C ∗ – mo d is fully faithful on ﬁnite-dimensional con tramo dules. Let V b e a v ector space and C b e the coalgebra suc h that the dual algebra C ∗ has the form k i 2 ⊕ i 2 V ∗ i 1 ⊕ k i 1 , where i 1 and i 2 are idemp oten t elemen ts with i 1 i 2 = i 2 i 1 = 0 and i 1 + i 2 = 1 . Then left C ∗ -mo dules are essen tially pairs of k -vec tor spaces M 1 , M 2 endo w ed with a map V ∗ ⊗ k M 1 − → M 2 , left C -como dules are pairs of ve ctor spaces M 1 , M 2 endo w ed with a map M 1 − → V ⊗ k M 2 , and left C -contramo dules are pair s of v ector spaces P 1 , P 2 endo w ed with a map Hom k ( V , P 1 ) − → P 2 . In particular, the functor C – contra − → C ∗ – mo d is not surjectiv e on morphisms of inﬁnite-dimensional ob jects, while the functor C – como d − → C ∗ – mo d is not surjectiv e on the isomorphism classes of ﬁnite-dimensional ob jects . (Neither is in general the functor C – contra − → C ∗ – mo d , 223 as one can see in the example o f a n analogous coalgebra with three idemp oten t linear functions instead of t w o a nd three ve ctor spaces instead of one; when k is a ﬁnite ﬁeld and C is the coun table direct sum of copies of the coalgebra k , there ev en exists a one-dimensional C ∗ -mo dule whic h comes fr om no C -como dule or C -contramo dule.) Let P b e the C - contramo dule with P 1 = k = P 2 corresp onding to a linear f unction V ∗ − → k coming fro m no elemen t o f V . Then the interse ction of the imag es o f Hom C ( C / U , P ) in P ov er all ﬁnite-dimensional subcoa lgebras U ⊂ C is equal to P 2 . More generally , for an y coalgebra C any ﬁnite- dimensional left C -como dule M has a na tural left C -contramo dule structure giv en b y the comp osition Hom k ( C , M ) ≃ C ∗ ⊗ k M − → C ∗ ⊗ k C ⊗ k M − → M of the map induced by the C -coaction in M and the map induced by the pairing C ∗ ⊗ k C − → k . The category of ﬁnite- dimensional left C -como dules is isomorphic to a full sub categor y of the category of ﬁnite-dimensional left C -contr a mo dules; a ﬁnite-dimensional C - contramo dule comes from a C -como dule if and only if it comes f rom a contramo dule ov er a ﬁnite-dimensional sub coalgebra of C . W e will see b elo w that ev ery irr e ducible C -contramo dule is a ﬁnite-dimensional con tramo dule o v er a ﬁnite-dimensional sub coalg ebra o f C ; it follows that the ab ov e functor prov ides a bijective corresp ondence b et w een irreducible left C -como dules and irreducible left C -contra mo dules. Compairing t he cobar complex for como dules with the bar complex for con tramo d- ules, one disco ve rs that for an y ﬁnite-dimensional left C -como dules L a nd M there is a natural isomorphism Ext C ,i ( L , M ) ≃ Ext i C ( L , M ) ∗∗ . In other w ords, t he Ext spaces b etw een ﬁnite-dimensional C - como dules in the category of a rbitrary C -contra- mo dules are the completions of t he Ext spaces in the category of ﬁnite-dimensional C -como dules with resp ect to t he proﬁnite-dimensional to p ology . A.2. Nak a y ama’s Lemma. A coalgebra is called c osimple if it has no nontrivial prop er sub coa lgebras. A coalgebra C is called c osem isimple if it is a union of ﬁnite- dimensional coa lg ebras dual t o semisimple k -alg ebras, or equiv alen tly , if ab elian the category of (left or right) C - como dules is semisim ple. An y cosemisimple coalgebra can b e decomp osed in to an (inﬁnite) direct sum of cosimple coalg ebras in a unique w a y . F or any coalgebra C , let C ss denote its maximal cosemisimple sub coalgebra; it con tains all other cosemisim ple sub coalgebras of C . Lemma 1. L et C b e a c o algebr a over a ﬁeld k and P b e a nonzer o left C -c ontr a- mo dule. The n the ima ge of the sp ac e Ho m k ( C / C ss , P ) under the c ontr aaction map Hom k ( C , P ) − → P is not e qual to P . Pr o of . Notice that the coalgebra without counit D = C / C ss is c o nilp otent , that is an y elemen t of D is annihilated b y t he iterated comultiplication map D − → D ⊗ n with a large enough n . W e will show that for an y con tramo dule P ov er a conilp oten t coal- gebra D surjectivit y of t he map Hom k ( D , P ) − → P implies v anishing of P . Indeed , assume that the contraaction map π P of a D -contramo dule P is surjectiv e. Let p 224 b e an elemen t of P ; it is equal to π P ( f 1 ) fo r a certain map f 1 : D − → P . Since the map π P is surjectiv e, the map f 1 can b e lifted to a certain map D − → Hom k ( D , P ), whic h supplies a map f 2 : D ⊗ k D − → P . So one constructs a sequenc e of maps f i : D ⊗ i − → P suc h that f i − 1 = π P , 1 ( f i ), where π P , 1 signiﬁes the application of π P at the ﬁrst tensor factor of D ⊗ i . Set g i = µ D , 2 ..i ( f i ) = f i ◦ µ D , 2 ..i , where µ D , 2 ..i denotes the tensor pro duct of the iden tity map D − → D with the iterated com ultiplication map D − → D ⊗ i − 1 . Then g i is a map D ⊗ D − → P deﬁned for each i > 2. W e ha v e π P , 1 ( g i ) = µ D , 1 ..i − 1 ( f i − 1 ) a nd µ D ( g i ) = µ D , 1 ..i ( f i ). Notice tha t by conilp ot ency of the coalgebra D the series P ∞ i =2 g i con v erges in the sense of p oint-wise limit of functions D ⊗ k D − → P , and ev en of functions D − → Hom k ( D , P ). (As alw a ys, w e presume the identiﬁcation Hom k ( U ⊗ k V , W ) = Hom k ( V , Hom k ( U, W )) when w e consider left con tramo dules.) Therefore, π P , 1  P ∞ i =2 g i  = P ∞ i =2 µ D , 1 ..i − 1 ( f i − 1 ) a nd µ D  P ∞ i =2 g i  = P ∞ i =2 µ D , 1 ..i ( f i ), hence π P , 1  P ∞ i =2 g i  − µ D  P ∞ i =2 g i  = f 1 . By the con traasso ciativit y equation, it follows that p = π P ( f 1 ) = 0.  Lemma 2. L et c o algebr a C b e the dir e c t sum of a family of c o algebr as C α . Then any left c ontr amo dule P over C is the pr o duct of a uniquely deﬁne d fami l y of left c ontr a m o dules P α over C α . Pr o of . Uniqueness and functoriality is clear, since the comp o nent P α can b e r ecov ered as the image of the pro jector corresp onding to the linear function on C that is equal to the counit on C α and v anishes on C β for all β 6 = α . Existence is obvious for a free C - contr a mo dule. No w supp o se that a C -contramo dule Q is the pro duct of C α -contramo dules Q α ; let us sho w that any sub contramodule R ⊂ Q is the pro duct of its images R α under the pro jections Q − → Q α . Let r α b e a family of elemen t s of R . Consider the linear map f : C − → R whose restriction to C α is equal to the comp osition C α − → k − → R of the counit of C α and the map sending 1 ∈ k to r α . Set r = π R ( f ). Then the image o f the elemen t r under the pro jection R − → R α is equal to the image of r α under this pro jection. Th us R is iden tiﬁed with the pro duct of R α . It remains to notice tha t any C -contramo dule is isomorphic to the quotien t con tramo dule of a free contramodule b y one of its sub contramo dules.  Corollary . F or any c o alg ebr a C and any nonzer o c ontr amo dule P over C ther e exists a ﬁnite-dim e nsional (and even c osimpl e ) sub c o algebr a U ⊂ C such that the image of Hom k ( C / U , P ) under the c ontr a action map Hom k ( C , P ) − → P is n o t e qual to P . Pr o of . By Lemma 1, the image of the map Ho m k ( C / C ss , P ) − → P is not equal to P . Denote this image by Q ; it is a sub contramo dule of P and the quotien t con tramo dule P / Q is a contramo dule ov er C ss . By Lemma 2, there exists a cosimple sub coalgebra C α of C ss suc h that P / Q has a nonzero quotien t whic h is a con tramo dule ov er C α .  Lemma 3. L et C 0 ⊂ C 1 ⊂ C 2 ⊂ · · · ⊂ C b e a c o algebr a w i th an incr e asing se quenc e of sub c o algebr a s . F or a C -c ontr amo dule P , denote by G i P the image of the c ontr aa c tion 225 map Hom k ( C / C i , P ) − → P . Then for any C -c o ntr amo dule P the n a tur al m a p P − → lim ← − i P /G i P is surje ctive. Pr o of . The assertion is obv ious for a free C -contra mo dule. Represen t an arbitrary C -contramo dule P as the quotien t con tramo dule Q / K of a free C -contramo dule Q . Since the maps G i Q − → G i P are surjectiv e, there ar e short exact sequences 0 − → K / K ∩ G i Q − → Q /G i Q − → P / G i P − → 0. Pas sing to t he pro jectiv e limits, we see that the map lim ← − i Q /G i Q − → lim ← − i P /G i P is surjectiv e.  When C = S i C i , it follows , in particular, that K ≃ lim ← − i K /G i K for a n y C -contra- mo dule K whic h is a sub contramo dule of a pr o jectiv e C -contramo dule. A.3. Con traﬂat con tramo dules. Lemma. L et C b e a c o algeb r a over a ﬁeld k . Then a left C - c ontr am o dule is c ontr aﬂat if an d only if i t is pr oje ctive. Pr o of . F or any C -contramo dule Q and an y subcoalgebra V ⊂ C denote by V Q = cok er(Hom C ( C / V , Q ) → Q ) ≃ Cohom C ( V , Q ) the maximal quotien t contramo dule of Q that is a con tramo dule o v er V . The k ey step is to construct fo r an y C ss -contra- mo dule R a pro jectiv e C -contramo dule Q suc h that C ss Q ≃ R . By Lemma A.2.2, R is a pro duct of con tramo dules ov er cosimple comp onents C α of C ss . An y contramo dule o v er C α is, in turn, a direct sum of copies o f the unique irreducible C α -contramo dule. Hence it suﬃces to consider the case of an irreducible C α -contramo dule R . Let e α b e an idemp oten t elemen t of the algebra C ∗ α suc h that R is isomorphic to C ∗ α e α . Consider the idempotent linear function e ss on C ss equal t o e α on C α and zero o n C β for all β 6 = α . It is w ell-kno wn that fo r any surjectiv e map of ring s A − → B whose k ernel is a nil ideal in A any idemp otent elemen t of B can b e lifted to an idemp oten t elemen t of A . Using this fact for ﬁnite-dimensional algebras and Zorn’s Lemma, o ne can sho w that an y idemp oten t linear f unction on C ss can b e extended to an idemp oten t linear function on C . Let e b e an idemp otent linear function on C extending e ss ; set Q = C ∗ e . Then one has C ss Q ≃ ( C ss C ∗ ) e ≃ C ss ∗ e ss ≃ R as desired. Now let P b e a con traﬂat left C -contramo dule. Consider a pr o jectiv e left C -contramo dule Q suc h that C ss Q ≃ C ss P . Since Q is pro jec tiv e, the map Q − → C ss P can b e lifted to a C -contramo dule morphism f : Q − → P . Since C ss (cok er f ) = coke r ( C ss f ) = 0, it follo ws fro m Lemma A.2.1 tha t the morphism f is surjectiv e; it remains to sho w that f is injective . F or any right como dule N o v er a sub coalgebra U ⊂ C there is a nat ura l isomorphism N ⊙ C P ≃ N ⊙ U U P , hence the U -contramo dule U P is con traﬂat. No w let U b e a ﬁnite-dimensional sub coalgebra; then U P is a ﬂat left U ∗ -mo dule. Denote b y K the ke rnel of the map U f : U Q − → U P . F or an y righ t U ∗ -mo dule N w e ha v e a short exact sequenc e 0 − → N ⊗ U ∗ K − → N ⊗ U ∗ U Q − → N ⊗ U ∗ U P − → 0. Since f or an y cosimple sub coalgebra U α ⊂ U the map U ∗ α ⊗ U ∗ U f = U α f is an isomorphism, w e 226 can conclude that the mo dule U ∗ α ⊗ U ∗ K = U ∗ α K is zero. It follow s that K = 0 and the map U f is an isomorphism. Finally , let K b e the k ernel of the map Q − → P . Since U f is an isomorphism, the sub con tramo dule K ⊂ Q is contained in the image of Hom k ( C / U , Q ) in Q for an y ﬁnite-dimensional sub coalgebra U ⊂ C . But the in tersection o f suc h images is zero, b ecause the Q is a pro jectiv e C -contramo dule.  Remark. Muc h more g enerally , one can deﬁne left contramodules ov er an arbitrary complete and separated top ological ring R where op en righ t ideals form a base of neigh b orho o ds of zero (cf. [6]). Namely , for any set X let R [ X ] denote the set of all formal linear comb inations of elemen ts of X with co eﬃcien ts co v erging to zero in R , i. e., the set of all formal sums P x ∈ X r x x , with r x ∈ R suc h that for any neigh b orho o d of zero U ⊂ R one has r x ∈ U for all but a ﬁnite n um b er of elemen ts x ∈ X . Then for any set X there is a natural map of “op ening the parenthe ses” R [ R [ X ]] − → R [ X ] assigning a formal linear combination to a formal linear combination of fo rmal linear com binations; it is well-deﬁn ed in view of our conditio n on R . There is also a map X − → R [ X ] deﬁned in terms of the unit elemen t of R ; tak en together, these tw o natural maps ma ke the functor X 7− → R [ X ] a monad o n the category of sets. Left con tramo dules o v er R a r e, b y the deﬁnition, mo dules o v er this monad. O ne can see that the category of left R -contr a mo dules is ab elian a nd there is a forgetful functor from it to the category of R - mo dules; this functor is exact and preserv es inﬁnite pro d- ucts. F o r example, when R = Z l is the t o p ological ring of l - a dic integers, the category of R -contramo dules is isomorphic to the category of w eakly l -complete ab elian groups in the sense of Jannsen [28], i. e., ab elian groups P suc h that Ext i Z ( Z [ l − 1 ] , P ) = 0 for i = 0, 1. When R is a to p ological algebra ov er a ﬁeld, the ab o v e deﬁnition of an R -contramo dule is equiv alen t to the deﬁnition giv en in D.5.2. Now if T is a top o- logical ring without unit satisfying the ab o v e condition, and T is pronilp oten t, that is for any neib orho o d of zero U ⊂ T there exists n suc h that T n ⊂ U , then an y left T -contramo dule P suc h that the contraaction map T [ P ] − → P is surjectiv e v anishes. Besides, any left con tramo dule o v er the top olog ical pro duct R of a family of rings (with units) R α satisfying the ab ov e condition is naturally the pro duct of a family of left R α -contramo dules. Fina lly , let R b e a top olo gical ring satisfying the ab ov e con- dition and endow ed with a decreasing ﬁltration R ⊃ G 1 R ⊃ G 2 R ⊃ · · · by closed left ideals suc h that a ny neighborho o d of zero in R con tains G i R fo r large i . F or an y left R -contramo dule P , denote b y G i P the image of t he con traaction ma p G i R [ P ] − → P ; then the natural map P − → lim ← − i P /G i P is surjectiv e. The pro ofs of these assertions are analogous to t ho se of Lemmas A.2.1– A.2.3. When op en tw o-sided ideals form a base of neighborho o ds of zero in R and the discrete quotien t rings of R ar e right Artinian, a left R - contramo dule P is pro jectiv e if and only if for a n y op en tw o-sided ideal J ⊂ R the cok ernel of the con traaction map J [ P ] − → P is a pro jectiv e left R/J - mo dule. The pro of of t his r esult is analogo us to that of Lemma A.3. It follows that the class of pro jective left R -contramo dules is closed under inﬁnite pro ducts 227 under these assumptions. F or a proﬁnite ring R (deﬁned equiv alen tly as a pro jectiv e limit of ﬁnite rings endo wed with the top ology of pro jec tiv e limit or as a proﬁnite ab elian gro up endo w ed with a con t in uous associative m ultiplication with unit) one can eve n obtain the como dule-con tramo dule corresp ondence. Namely , the co deriv ed category o f discrete left R -mo dules is equiv alen t to the con traderiv ed category of left R -contramo dules; this equiv alence is constructed in the w a y analogous t o 0 .2.6 with the role of a coalgebra C play ed b y the discrete R - R -bimo dule of con tin uous ab elian group homomorphisms R − → Q / Z . More generally , let R b e a top o logical ring where op en tw o-sided ideals f orm a base of neigh b orho ds of zero and the discrete quotien t rings are righ t Artinian. A pseudo-compact right R -mo dule is a top olog- ical righ t R -mo dule whose op en submo dules form a base of neigh b orho o ds of zero and discrete quotien t mo dules hav e ﬁnite length. The category of pseudo-compact righ t R -mo dules is a n a b elian category with exact functors of inﬁnite pro ducts and enough pro jectiv es; the pro jectiv e pseudo-compact right R -mo dules are the direct summands of inﬁnite pro ducts of copies of the pseudo-compact right R -mo dule R . There is a natura l an ti-equiv alence b et w een the con traderiv ed categories of pseudo- compact right R -mo dules a nd left R - contramo dules provided b y the deriv ed functors of pseudo-compact mo dule and con tramo dule homomo r phisms in to R . 228 Appendix B. Comp aris on with Arkhipov’s Ext ∞ / 2+ ∗ and Sevosty anov’s T or ∞ / 2+ ∗ Semi-inﬁnite cohomology of asso ciat ive a lgebras w as in tro duced b y S. Arkhip o v [1, 2]; later A. Sev ost y ano v studied it in [43]. The constructions of deriv ed functors SemiT or and SemiExt in this b o ok are based on three key ideas which w ere not kno wn in the ’9 0 s: namely , (i) the notion of a semialgebra and the functors of semitensor pro duct and semihomomorphisms; (ii) the constructions of adjusted ob j ects f r o m Lemmas 1.3.3 and 3 .3.3; and ( iii) the deﬁnitions of semideriv ed categories. W e ha v e discusse d alr eady Sevost y anov’s substitute f o r (i) in 10.4 and men tioned Arkhip ov’s substitute for (ii) in 10.3.5. Here w e consider Arkhip o v’s substitute for (i) and suggest an Arkhip ov and Sev o st y ano v-st yle subs titute for (iii). Combin ing these tog ether, w e o btain comparison results relating out SemiExt to Arkhip ov’s Ext ∞ / 2+ ∗ and our SemiT or to Sev ost y ano v’s T or ∞ / 2+ ∗ . Throughout this app endix w e will freely use the not a tion and remarks of 11 .1. B.1. Algebras R and R # . B.1.1. Let R b e a g raded asso ciativ e algebra o v er a ﬁeld k endow ed with a pair of subalgebras K and B ⊂ R . Assume that all the comp onents K i are ﬁnite dimensional, K i = 0 for i large negat ive, and B i = 0 for i large p ositive . Set C i = K ∗ − i and C = L i C i ; this is the coalgebra g r aded dual to the algebra K . The coalgebra structure on C exists due to the conditions imp osed on the grading of K . There is a natural pairing φ : C ⊗ k K − → k satisfying the conditions of 10.1.2. Notice that a structure o f g raded (left or rig h t) C -como dule on a graded k -v ector space M with M i = 0 f o r i ≫ 0 is the same that a structure of graded (left o r righ t) K -mo dule o n M . Analogously , a structure of graded (left or righ t) C -contramo dule on a graded k -vector space P with P i = 0 fo r i ≪ 0 is the same that a structure of graded (left or right) K -mo dule on P . Indeed, o ne has Hom gr k ( C , P ) ≃ K ⊗ k P . F urthermore, assume t ha t the m ultiplication map K ⊗ k B − → R is an isomorphism of graded v ector spaces. The algebra R is uniquely determined b y the algebras K and B and the “p ermutation” ma p B ⊗ k K − → K ⊗ k B obtained by restricting t he m ultiplication map R ⊗ k R − → R ≃ K ⊗ k B to the subspace B ⊗ k K ⊂ R ⊗ k R . T ransfering the tensor factors K to the o ther sides of this arrow , one o btains a map C ⊗ k B − → Hom gr k ( K , B ) giv en b y the f orm ula c ⊗ b 7− → ( k ′ 7→ ( φ ⊗ id B )( c ⊗ bk ′ )), where the graded Hom space in the righ t ha nd side is deﬁned, as alwa ys, as direct sum of the spaces of homogeneous maps of v arious degrees. By the conditions imp osed on the gradings of K and B , w e ha v e B ⊗ k C ≃ Hom gr k ( K , B ), so w e g et a ho mo g eneous map ψ : C ⊗ k B − → B ⊗ k C . One can c hec k that the map ψ is a rig ht en twining structure for the gra ded coalgebra C and the graded algebra B ov er k . 229 Con v ersely , if the map ψ corresp onding to a “p erm utation” map B ⊗ k K − → K ⊗ k B satisﬁes the en t wining structure equations, t hen the latter ma p can b e extended to an a sso ciativ e algebra structure o n R = K ⊗ k B with subalgebras K and B ⊂ R . Ho w ev er, not every homo g ene ous map C ⊗ k B − → B ⊗ k C c o mes f r om a homo gene ous map B ⊗ k K − → K ⊗ k B . In the describ ed situatio n the constructions of 10 .2 and 1 0.3 pro duce the same graded semialgebra C ⊗ K R = S ≃ C ⊗ k B . The pairing φ : C ⊗ k K − → k is non- degenerate in C , so the functor ∆ φ is fully faithful a nd in order t o sho w that the construction of 1 0.2 w orks one only has to ch ec k that t here exists a r ig h t C - como dule structure on C ⊗ K R inducing the giv en right K -mo dule structure. This is so b ecause S i = 0 for i ≫ 0 according to t he conditions imp osed on the g radings of K and B . B.1.2. Now supp ose t ha t we are giv en tw o g raded alg ebras R and R # with the same t w o g raded subalgebras K , B ⊂ R and K , B ⊂ R # suc h that the m ultiplication maps K ⊗ k B − → R and B ⊗ k K − → R # are isomorphisms of v ector spaces. Assume that dim k K i < ∞ for all i , K i = 0 fo r i ≪ 0, a nd B i = 0 fo r i ≫ 0. F urthermore, assume that the right en tw ining structure ψ : C ⊗ k B − → B ⊗ k C coming from the “p erm utation” map in R and the left ent wining structure ψ # : B ⊗ k C − → C ⊗ k B coming from the “p ermutation” map in R # are in v erse to each other. Then there ar e isomorphisms of g r a ded semialgebras S = C ⊗ K R ≃ C ⊗ k B ≃ B ⊗ k C ≃ R # ⊗ K C = S # , whic h allo w one to describ e left and r igh t S -semimodules and S -semicontramo dules in terms of left and r igh t R -mo dules and R # -mo dules. In particular, S has a natural structure of graded R # - R -bimo dule. By the g r a ded v ersion of the result o f 10.2.2, a structure of graded righ t S -semi- mo dule on a graded k - v ector space N with N i = 0 for i ≫ 0 is the same that a structure of graded right R -mo dule on N . A structure of graded left S -semimo dule on a g raded k - vector space M with M i = 0 f o r i ≫ 0 is the same that a structure o f graded left R # -mo dule on M . A structure of graded left S -semicontramo dule on a graded k -v ector space P with P i = 0 for i ≪ 0 is the same that a structure of graded left R -mo dule on P . In other words, there are isomorphisms of the corresponding categories of graded mo dules and ho mogeneous morphisms b et w een them. Besides, for any graded right R -mo dule N with N i = 0 for i ≫ 0 and an y graded left R -mo dule P with P i = 0 fo r i ≪ 0 t here is a na t ur a l isomorphism N ⊚ gr S P ≃ N ⊗ R P . Indeed, one has N ⊙ gr C P ≃ N ⊗ K P and ( N  gr C S ) ⊙ gr C P ≃ N ⊗ K R ⊗ K P . B.1.3. A gra ded K -mo dule M with M i = 0 for i ≫ 0 is injectiv e as a graded C -como dule if and only if it is injectiv e as a graded K -mo dule and if and only if it is injectiv e in the category of graded K -mo dules with the same restriction on t he grading. Analogously , a graded K -mo dule P with P i = 0 for i ≪ 0 is pro jectiv e as a graded C -contramo dule if a nd only if it is pro jectiv e as a graded K -module and if and 230 only if it is pro jectiv e in the category of graded K -mo dules with the same restriction on the grading. By the gra ded v ersion o f Prop osition 6.2.1 (a), for a ny graded right R - mo dule N with N i = 0 f or i ≫ 0 and an y K -injectiv e g raded left R # -mo dule M with M i = 0 for i ≫ 0 there are natural isomorphisms N ♦ gr S M ≃ N ⊚ gr S Ψ gr S ( M ) ≃ N ⊚ gr S Hom gr R # ( S , M ) . Analogously , for an y K -injectiv e graded r igh t R -mo dule N with N i = 0 for i ≫ 0 and an y graded left R # -mo dule M with M i = 0 for i ≫ 0 there is a natural isomorphism N ♦ gr S M ≃ M ⊚ gr S op Hom gr R op ( S , N ) The contratensor pro ducts in the right hand sides of these form ulas c annot b e in general replaced by the tensor pro duct ov er R and R # , as t he graded S -semicontra- mo dule Ψ gr S ( M ) do es no t hav e zero compo nen ts in large negativ e degrees. In this situation the contratens or pr o duct is a certain quotien t space of the tensor pro duct. By the g r aded v ersion of Pro p osition 6.2.3 (a), for a n y K -injectiv e gra ded left R # -mo dule M with M i = 0 for i ≫ 0 and any graded left R -mo dule P with P i = 0 for i ≪ 0 there are natural isomorphisms SemiHom gr S ( M , P ) ≃ Hom S gr (Ψ gr S ( M ) , P ) ≃ Hom S gr (Hom gr R # ( S , M ) , P ) . Here the homomorphisms o f graded S -semicontramo dules again c a n not b e replaced b y homomorphisms of graded left R -mo dules. The former homomorphism spaces a re certain subspaces of the latter ones. By the g r a ded ve rsion of Prop osition 6.2.2( a ), for any graded left R # -mo dule M with M i = 0 for i ≫ 0 and an y K -projectiv e gr a ded left R -mo dule P with P i = 0 for i ≪ 0 there are natural isomorphisms SemiHom gr S ( M , P ) ≃ Hom gr S ( M , Φ gr S ( P )) ≃ Hom gr R # ( M , S ⊗ gr R P ) . Here the homomor phisms of graded left S -semimo dules c an b e replaced b y the ho- momorphisms of graded left R # -mo dules, since the functor ∆ φ is fully faithful, and consequen tly so is the functor ∆ φ,f . All of these formulas except the last one ha v e ungra ded v ersions: N ♦ S M ≃ N ⊚ S Hom R # ( S , M ) , N ♦ S M ≃ M ⊚ S op Hom R op ( S , N ) , SemiHom S ( M , P ) ≃ Hom S (Hom R # ( S , M ) , P ) under the appropriate K -injectivit y conditions. B.2. Finite-dimensi onal c ase. When the subalgebra K ⊂ R is ﬁnite- dimensional, the algebra R # can b e constructed without any reference to the grading or t he com- plemen ta ry subalgebra B . 231 B.2.1. Let K b e a ﬁnite-dimensional k -algebra a nd C = K ∗ b e the coalgebra dual to K . Then the categories of left C - como dules and left C -contramo dules are isomorphic to the category of left K -mo dules and the category of righ t C -como dules is isomorphic to the category of right K -mo dules. The adjoint f unctors Φ C and Ψ C can b e therefore considered as adjoint endofunctors on the categor y of left K -mo dules deﬁned by the form ulas P 7− → C ⊗ K P and M 7− → K  C M ≃ Hom K ( C , M ). The restrictions o f these functors deﬁne an equiv a lence b et w een the categor ies of pro jectiv e and injectiv e left K -mo dules. By Prop o sition 1.2.3(a- b), the m utually inv erse equiv alences P 7− → C ⊗ K P a nd M 7− → K  C M b etw een the category of K - K -bicomo dules tha t are pro jectiv e a s left K -mo dules and the catego r y of K - K -bicomo dules t hat are injectiv e as left K -mo dules transforms the functor of tensor pro duct o v er K in the former category in to the functor of cotensor pro duct o v er C in the latter one. In o ther w ords, these tw o tensor categories ar e equiv alent, and therefore there is a corresp ondence b et w een ring o b jects in the former and the lat t er tensor category . B.2.2. Let K b e a ﬁnite-dimensional k -algebra and K − → R b e a morphism of k -alg ebras. By the ab o v e argumen t, if R is a pro jectiv e left K -mo dule, then the tensor pro duct S = C ⊗ K R has a natural structure of semialgebra ov er C . F urthermore, if S is an injectiv e right K -mo dule, then the cotensor pro duct R # = S  C K has a natural structure of k -alg ebra endo w ed with a k -algebra morphism K − → R # . In this case the semialgebra S can b e also obtained a s the tensor pro duct R # ⊗ K C . By the result of 10.2.2, a structure of right S -semimo dule on a k -v ector space N is the same that a structure of right R -mo dule on N . A structure of left S -semi- mo dule on a k -ve ctor space M is the same that a structure of left R # -mo dule on M . A structure of left S -semicontramo dule on a k -ve ctor space P is the same tha t a structure of left R -mo dule on P . In other w ords, the corresp onding categories are isomorphic. Besides, for an y right R - mo dule N and any left R -mo dule P there is a natural isomorphism N ⊚ S P ≃ N ⊗ R P (see 10.2.3 ). Remark. The case of a F rob enius algebra K is of sp ecial intere st. In this case the k -alg ebra R # is isomorphic t o the k - algebra R , but the k -a lgebra morphisms K − → R and K − → R # diﬀer b y the F rob enius automorphism of K . B.2.3. By Prop osition 6.2.1(a), for any right R -mo dule N and an y K -injectiv e left R -mo dule M there a r e natural isomorphisms N ♦ S M ≃ N ⊚ S Ψ S ( M ) ≃ N ⊗ R Hom R # ( S , M ) . Analogously , for any K -injectiv e r igh t R - mo dule N and any left R # -mo dule M there is a natural isomorphism N ♦ S M ≃ Hom R op ( S , N ) ⊗ R # M . 232 By Prop osition 6.2.3(a), for any K -injectiv e left R # -mo dule M and any left R -mo dule P there ar e natural isomorphisms SemiHom S ( M , P ) ≃ Hom S (Ψ S ( M ) , P ) ≃ Hom R (Hom R # ( S , M ) , P ) . By Prop osition 6.2.2(a), f or an y left R # -mo dule M and any K -proj ectiv e left R -mo dule P there ar e natural isomorphisms SemiHom S ( M , P ) ≃ Hom S ( M , Φ S ( P )) ≃ Hom R # ( M , S ⊗ R P ) . All of these formulas hav e ob vious graded v ersions. B.3. Semijectiv e complexes. Let S b e a graded semialgebra ov er a g raded coalge- bra C ov er a ﬁeld k . Supp ose that S i = 0 = C i for i > 0 and C 0 = k . Assume also that S is an injectiv e left and righ t graded C -como dule. Let C – como d ↓ and como d ↓ – C de- note the categories of C - como dules g raded b y nonp o sitiv e in tegers, C – contra ↑ denote the categor y of left C -como dules gr a ded by nonnegative in tegers, S – simo d ↓ , simo d ↓ – S , and S – sicntr ↑ denote the categories of graded S -semimo dules and S -semicontramo d- ules with analogously b ounded grading. An y a cyclic complex o v er C – como d ↓ is coacyclic with resp ect to C – como d ↓ . Analo- gously , a n y acyclic complex ov er C – contra ↑ is con traacyclic with resp ect to C – contra ↑ . Indeed, let K • b e an acyclic complex of no np ositiv ely g raded C -como dules. As b e- fore, w e denote b y upper indices the homological grading and b y low er indices the in ternal grading. In tro duce an increasing ﬁltrat io n on K • b y the complexes of graded sub como dules F n K j = L i > − n K j i . Then the acyclic complexes of tr ivial C -como dules F n K • /F n − 1 K • are clearly coacyclic. So w e hav e D si ( S – simo d ↓ ) = D ( S – simo d ↓ ) a nd D si ( S – sicntr ↑ ) = D ( S – sicntr ↑ ). A complex M • o v er C – como d ↓ is called inje ctive if the complex of homog eneous homomorphisms in to M • from an y acyclic complex o ver C – como d ↓ is acyclic. In this case the comple x of homogeneous homomo r phisms in to M • from an y acyclic complex ov er C – como d gr is also acyclic. Analogously , a complex P • o v er C – contra ↑ is called pr oje ctive if the complex of homogeneous homomorphisms from P • in to an y acyclic complex ov er C – contra ↑ is acyclic. In this case the complex of homogeneous homomorphisms from P • in to any a cyclic complex o v er C – contra gr is also acyclic. By Lemma 5.4, any complex of injectiv e ob jects in C – como d ↓ is injectiv e a nd any complex of pro jectiv e ob jects in C – contra ↑ is pro jectiv e. A complex M • o v er S – simo d ↓ is called quite S / C -pr oje ctive if the complex o f ho- mogeneous homomorphisms from M • in to any C - contr a ctible complex o v er S – simo d ↓ is acyclic. Equiv alen tly , M • should b elong to the minimal triangulat ed sub cate- gory of Hot ( S – simo d ↓ ) con taining the complexes of graded semimo dules induced from complexes ov er C – como d ↓ and closed under inﬁnite direct sums. Indeed, any quite S / C -projectiv e complex of graded S - semimo dules is homoto p y equiv alen t to the total complex of its bar resolution. 233 Analogously , a complex P • o v er S – sicntr ↑ is called quite S / C - i n je ctive if the com- plex of homogeneous homomorphisms into P • from an y C -contra ctible complex o v er S – sicntr ↑ is acyclic. Equiv alen t ly , P • should b elong to the minimal triangulated sub- category o f Hot ( S – sicntr ↑ ) containing the complexes of gra ded se micon tramo dules coinduced from complexes ov er C – contra ↑ and closed under inﬁnite pro ducts. A complex M • o v er S – simo d ↓ is called sem ije ctive if it is C - injectiv e a nd quite S / C -projectiv e. Analogously , a complex P • o v er S – sicntr ↑ is called semije ctive if it is C -proj ectiv e and quite S / C -injectiv e. Clearly , an y acyclic semijectiv e complex of semimo dules or semicon tramo dules is con tractible. By t he graded vers ion of Remark 6.4, an y semipro jectiv e complex of nonp ositiv ely graded C - injectiv e S -semimo dules is semijectiv e and an y se miinjectiv e complex of nonnegativ ely graded C -pr o jectiv e S -semicontramo dules is semijectiv e. Hence the homotop y category o f semijectiv e complexes ov er S – simo d ↓ or S – sicntr ↑ is equiv a- len t to the derive d category D ( S – simo d ↓ ) or D ( S – sicntr ↑ ), any semijectiv e complex is semipro jectiv e or semiinjectiv e, and one can use semijectiv e complexes to compute the deriv ed functors SemiT or S and SemiExt S . B.4. Explicit resolutions. Let us return to the situation of B.1.2, but mak e the stronger a ssumptions that dim k K i < ∞ for all i , K i = 0 for i < 0, K 0 = k , and B i = 0 for i > 0. Set C i = K ∗ − i and S = C ⊗ K R ≃ R # ⊗ K C = S # . B.4.1. F or any complex o f nonnegative ly graded left R -mo dules P • denote b y L 2 ( P • ) the total complex of the reduced relativ e bar complex · · · − − → R ⊗ B R/B ⊗ B R/B ⊗ B P • − − → R ⊗ B R/B ⊗ B P • − − → R ⊗ B P • . It do es not matter whether to construct this total complex by taking inﬁnite direct sums or inﬁnite pro ducts in the category of g raded R -mo dules, as the tw o total complexes coincide. The complex L 2 ( P • ) is a complex of K -projectiv e nonnegativ ely graded left R -mo dules quasi-isomorphic to the complex P • . F or an y complex of nonp ositiv ely graded left R # -mo dules M • denote by L 3 ( M • ) the total complex of the reduced relativ e bar complex · · · − − → R # ⊗ K R # /K ⊗ K R # /K ⊗ K M • − − → R # ⊗ K R # /K ⊗ K M • − − → R # ⊗ K M • , constructed b y taking inﬁnite direct sums along the dia gonals. The complex L 3 ( M • ) is a quite S / C -projectiv e complex of nonp o sitiv ely graded left S -semimo dules quasi- isomorphic to the complex M • . By 4.8 a nd B.1.3, the complex Hom gr R # ( L 3 ( M • ) , S ⊗ gr R L 2 ( P • )) represen ts the ob- ject SemiExt gr S ( M • , P • ) in D ( k – vect gr ). W e ha v e repro duced Arkhip o v’s explicit com- plex [1, 2] computing Ext ∞ / 2+ ∗ R ( M • , P • ). 234 B.4.2. F or any complex of nonp ositive ly graded left R # -mo dules M • denote by R 2 ( M • ) t he total complex of the reduced relativ e cobar complex Hom gr B ( R # , M • ) − − → Hom gr B ( R # /B ⊗ B R, M • ) − − → Hom gr B ( R # /B ⊗ B R # /B ⊗ B R, M • ) − − → · · · It do es not matter whether to construct this total complex by taking inﬁnite direct sums or inﬁnite pro ducts in the catego ry of gra ded R # -mo dules, as the t w o total complexes coincide. The complex R 2 ( M • ) is a complex o f K -injectiv e nonp ositiv ely graded left R # -mo dules quasi-isomorphic to the complex M • . F or an y complex of K -injectiv e nonp ositiv ely graded left R # -mo dules M • the com- plex L 3 ( M • ) deﬁned in B.4.1 is a semipro jectiv e complex of C -injective left S -semi- mo dules, since it is isomorphic to the total complex of the r educed bar complex · · · − − → S  C S / C  C S / C  C M • − − → S  C S / C  C M • − − → S  C M • and t he left C -como dule S / C is injectiv e in our assumptions. Let N • b e a complex of nonp o sitively graded righ t R -mo dules and M • b e a com- plex of nonp ositiv ely graded left R # -mo dules. By 10.4 , the complex N • ⊗ C B L 3 R 2 ( M • ) represen ts the ob ject SemiT or S gr ( N • , M • ) in D ( k – vect gr ). W e ha v e repro duced Sev o st y ano v’s explicit complex [43 ] computing T or R ∞ / 2+ ∗ ( N • , M • ). B.5. Explicit resolutions for a ﬁnite-dimensional subalgebra. Let us consider the situation of an asso ciativ e algebra R endo w ed with a pair of subalgebras K and B ⊂ R suc h that the multiplic ation map K ⊗ k B − → R is an isomorphism of v ector spaces and K is a ﬁnite-dimensional alg ebra. Let C = K ∗ b e the coalgebra dual to K . Then the construction of B.1.1 – B.1.2 is applicable, e. g., with R endo w ed by the trivial grading, a nd whenev er the en tw ining map ψ : B ⊗ k C − → C ⊗ k B turns out to b e inv ertible, this construction pro duces an alg ebra R # with subalgebras K and B and isomorphisms of semialgebras S = C ⊗ K R ≃ C ⊗ k B ≃ B ⊗ k C ≃ R # ⊗ K C = S # . B.5.1. F or an y complex of right R -mo dules N • denote by R 2 ( N • ) the total complex of the reduced relativ e cobar complex Hom B op ( R, N • ) − − → Hom B op ( R ⊗ B R/B , N • ) − − → Hom B op ( R ⊗ B R/B ⊗ B R/B , N • ) − − → · · · , constructed b y taking inﬁnite direct sums alo ng the diagonals. The complex R 2 ( N • ) is a complex of K -injectiv e righ t R -mo dules and the cone of the morphism N • − → R 2 ( N • ) is K -coacyclic (and eve n R -coacyclic). F o r any complex of left R # -mo dules M • the complex R 2 ( M • ) is constructed in the analogo us wa y . 235 F or an y complex of left R -mo dules P • denote by L 2 ( P • ) the total complex of the reduced relativ e bar complex · · · − − → R ⊗ B R/B ⊗ B R/B ⊗ B P • − − → R ⊗ B R/B ⊗ B P • − − → R ⊗ B P • , constructed by taking inﬁnite pro ducts along the diagonals. The complex L 2 ( P • ) is a complex of K -projectiv e left R - mo dules and the cone of t he morphism L 2 ( P • ) − → P • is K -contraacyclic (and ev en R -contraa cyclic). F or any complex of right R - mo dules N • denote b y L 3 ( N • ) the total complex of the reduced relativ e bar complex · · · − − → N • ⊗ K R/K ⊗ K R/K ⊗ K R − − → N • ⊗ K R/K ⊗ K R − − → N • ⊗ K R, constructed by taking inﬁnite direct sums along the diagonals. The complex L 3 ( N • ) is a quite S / C -pr o jectiv e complex of righ t S -semimodules and the cone o f the morphism L 3 ( N • ) − → N • is C -contractible. Whenev er N • is a complex of C -injectiv e rig h t S -semimodules, L 3 ( N • ) is a semipro jective complex of C -injectiv e righ t S -semimo d- ules, as it w as explained in B.4.2. F or a complex of left R # -mo dules M • the complex L 3 ( M • ) is constructed in t he analogous w a y . F or an y complex of left R -mo dules P • denote by R 3 ( P • ) the t otal complex of the reduced relativ e cobar complex Hom K ( R, P • ) − − → Hom K ( R/K ⊗ K R, P • ) − − → Hom K ( R/K ⊗ K R/K ⊗ K R, M • ) − − → · · · , constructed by taking inﬁnite pro ducts along t he diagonals. The complex R 3 ( P • ) is a quite S / C - inj ective complex of left S -semicontramo dules and the cone of the morphism P • − → R 3 ( P • ) is C - contractible. Whenev er P • is a complex o f C -pro- jectiv e left S -semicontramo dules, R 3 ( P • ) is a semiinjectiv e complex of C - projectiv e righ t S -semimodules, since it is isomorphic t o the tota l complex of the reduced cobar complex Cohom C ( S , P • ) − − → Cohom C ( S / C  C S , P • ) − − → Cohom C ( S / C  C S / C  C S , P • ) − − → · · · and t he righ t C -como dule S / C is injectiv e in our assumptions. B.5.2. One can use these resolutions in v arious w a ys to compute the derived functors SemiT or S , SemiExt S , Ψ S , Φ S , Ext S , Ext S , and CtrT or S . Sp eciﬁcally , for an y complex of right R - mo dules N • and an y complex of left R # -mo dules M • the ob ject SemiT or S ( N • , M • ) in D ( k – vect ) is represen ted by either of the four complexes N • ⊗ R Hom R # ( S , L 3 R 2 ( M • )) , L 3 ( N • ) ⊗ R Hom R # ( S , R 2 ( M • )) , Hom R op ( S , L 3 R 2 ( N • )) ⊗ R # M • , Hom R op ( S , R 2 ( N • )) ⊗ R # L 3 ( M • ) 236 according to the formulas of B.2.3 a nd the results of 2.8. F or an y complex of left R # -mo dules M • and a ny complex of left R -mo dules P • the ob jec t SemiExt S ( M • , P • ) in D ( k – vect ) is represen ted b y either of the four complexes Hom R (Hom R # ( S , L 3 R 2 ( M • )) , P • ) , Hom R (Hom R # ( S , R 2 ( M • )) , R 3 ( P • )) , Hom R # ( M • , S ⊗ R R 3 L 2 ( P • )) , Hom R # ( L 3 ( M • ) , S ⊗ R L 2 ( P • )) according to the formulas of B.2.3 and the results of 4.8. One can also use the constructions of 10.4 instead of the fo rm ulas B.2.3. F or an y complex of left R # -mo dules M • the ob ject Ψ S ( M • ) in D si ( S – sicntr ) is represen ted by the complex of left R - mo dules Hom R # ( S , R 2 ( M • )). F or any complex of left R -mo dules P • the ob ject Φ S ( P • ) in D si ( S – simo d ) is represen ted by the complex of left R # -mo dules S ⊗ R L 2 ( P • ). F or a ny complexes o f left R # -mo dules L • and M • the ob ject Ext S ( L • , M • ) in D ( k – vect ) is represen ted b y the complex Hom R # ( L 3 ( L • ) , R 2 ( M • )) . F or an y complexes of left R -mo dules P • and Q • the ob ject Ext S ( P • , Q • ) in D ( k – v ect ) is represen ted by the complex Hom R ( L 2 ( P • ) , R 3 ( Q • )) . F or an y complex of right R -mo dules N • and any complex of left R -mo dules P • the ob ject CtrT or S ( N • , P • ) in D ( k – v ect ) is represen ted b y the complex L 3 ( N • ) ⊗ R L 2 ( P • ) . These a ssertions follow from the results of 6.5. In the situation of B.2.2 (with no complemen tary subalgebra B ) one has to use the constructions of resolutions R 2 ( N ), R 2 ( M ), and L 2 ( P ) from the pro o fs of Theo- rems 2 .6 and 4.6 instead o f the constructions of B.5.1. 237 Appendix C. Semialgebras Associa ted to Harish-Chandra P airs b y Leonid Positse lski and Dmitriy Rumynin Recall that in 10.2 w e describ ed the cat ego ries of righ t semimo dules and left semi- con tramo dules o v er a semialgebra of t he form S = C ⊗ K R , but no satisfactory description of the category of left semimo dules ov er S was fo und. Here w e consider the situation when C and K are Hopf algebras ov er a ﬁeld k , and, under certain assumptions, construct a Morita equiv alence b etw een the semialgebras C ⊗ K R and R ⊗ K C . This includes the case of an algebraic Harish-Chandra pair ( g , H ) with a smo oth aﬃne alg ebraic gro up H . C.1. T wo semialgebras. C.1.1. Let K and C b e tw o Hopf algebras o v er a ﬁeld k with inv ertible antipo des s . Using Sw eedler’s notat ion [45], w e will denote the m ultiplications in K and C by x ⊗ y 7− → xy and the comultiplications by x 7− → x (1) ⊗ x (2) ; the units will b e denoted b y e and the counits b y ε , so that one has s ( x (1) ) x (2) = ε ( x ) e = x (1) s ( x (2) ) for x ∈ K or C . Let h , i : K ⊗ k C − → k b e a pairing b etw een K and C suc h that h xy , c i = h x, c (1) ih y , c (2) i and h x, c d i = h x (1) , c ih x (2) , d i for x , y ∈ K , c , d ∈ C ; b esides, o ne should ha v e h x, e i = ε ( x ) a nd h e, c i = ε ( c ). The pairing h , i will b e also presumed to b e compatible with the an tip o des, h s ( x ) , c i = h x, s ( c ) i . Finally , supp ose t ha t w e are g iv en an “adjo in t” righ t coa ctio n of C in K de- noted by x 7→ x [0] ⊗ x [1] , satisfying the equations x (1) y s ( x (2) ) = h x, y [1] i y [0] and ( xy ) [0] ⊗ ( xy ) [1] = x [0] y [0] ⊗ x [1] y [1] ; b esides, a ssume that e [0] ⊗ e [1] = e ⊗ e . This coaction should also satisfy the equations of compatibility with the squares o f an- tip o des ( s 2 x ) [0] ⊗ ( s 2 x ) [1] = s 2 ( x [0] ) ⊗ s − 2 ( x [1] ) and compatibilit y with t he pairing h s − 1 ( x [0] ) , c (2) i s ( c (1) ) c (3) x [1] = h s − 1 ( x ) , c i e . When the pairing h , i is nondegenerate in C , the latter f our equations fo llo w from the ﬁrst one and the previous assumptions. Indeed, one ha s h y , s 2 ( x ) [1] i ( s 2 x ) [0] = y (1) s 2 ( x ) s ( y (2) ) = s 2 (( s − 2 y ) (1) xs (( s − 2 y ) (2) )) = h s − 2 ( y ) , x [1] i s 2 ( x [0] ) = h y , s − 2 ( x [1] ) i s 2 ( x [0] ) and h s − 1 ( x [0] ) , c (2) ih y , s ( c (1) ) c (3) x [1] i = h s − 1 ( x [0] ) , c (2) ih y (3) , x [1] ih y (1) , s ( c (1) ) ih y (2) , c (3) i = h s − 1 ( y (3) xs ( y (4) )) , c (2) ih s ( y (1) ) , c (1) i h y (2) , c (3) i = h s ( y (1) ) y (4) s − 1 ( x ) s − 1 ( y (3) ) y (2) , c i = h s − 1 ( x ) , c i ε ( y ); analogo usly for the second and the third equations. C.1.2. Let R b e an a sso ciativ e a lg ebra ov er k endo w ed with a morphism of algebras f : K − → R and a r igh t coaction of the coalgebra C , whic h w e will denote by u 7− → u [0] ⊗ u [1] , u ∈ R . Assume tha t f is a morphism o f righ t C -como dules and the righ t coaction o f C in R satisﬁes the equations f ( x ) uf ( s ( x )) = h x, u [1] i u [0] and ( u v ) [0] ⊗ ( uv ) [1] = u [0] v [0] ⊗ u [1] v [1] for u , v ∈ R , x ∈ K . 238 Deﬁne a pairing φ r : C ⊗ k K − → k b y the fo rm ula φ r ( c, x ) = h s − 1 ( x ) , c i . The pairing φ r satisﬁes the conditions of 10.1.2; in particular, it induces a righ t action of K in C giv en b y the for mula c ← x = h s − 1 ( x ) , c (2) i c (1) . Assume that t he morphism of k -algebras f mak es R a ﬂat left K -mo dule. W e will now apply the construction of 10 .2 .1 in order to o bt a in a semialgebra structure on the tensor pro duct S r = C ⊗ K R . Deﬁne a rig h t C -como dule structure on C ⊗ K R by the form ula c ⊗ K u 7− → c (1) ⊗ K u [0] ⊗ c (2) u [1] . First let us c hec k that this coaction is w ell-deﬁned. W e hav e ( c ← x ) ⊗ u = h s − 1 x, c (2) i c (1) ⊗ u 7− → c (1) ⊗ K u [0] ⊗ h s ( − 1) x, c (3) i c (2) u [1] and c ⊗ f ( x ) u 7− → c (1) ⊗ K f ( x [0] ) u [0] ⊗ c (2) x [1] u [1] = ( c (1) ← x [0] ) ⊗ K u [0] ⊗ c (2) x [1] u [1] = c (1) ⊗ K u [0] ⊗ h c (2) , s − 1 ( x [0] ) i c (3) x [1] u [1] ; now h s − 1 ( x ) , d (2) i d (1) = h s − 1 ( x [0] ) , d (3) i d (1) s ( d (2) ) d (4) x [1] = h s − 1 ( x [0] , d (1) i d (2) x [1] for d ∈ C , x ∈ K . F urthermore, this right C -como dule structure agrees with the right K -mo dule structure on C ⊗ K R , since h s − 1 ( x ) , c (2) u [1] i c (1) ⊗ K u [0] = h s − 1 ( x (2) ) , c (2) i c (1) ⊗ K h s − 1 ( x (1) ) , u [1] i u [0] = ( c ← x (3) ) ⊗ K f ( s − 1 ( x (2) )) uf ( x (1) ) = c ⊗ K uf ( x ). It is easy to see that this rig h t coaction of C comm utes with the left coaction of C and that the semiunit map C − → C ⊗ K R is a morphism of righ t C -como dules. Finally , to show that the semim ultiplication map ( C ⊗ K R )  C ( C ⊗ K R ) − → C ⊗ K R is a morphism of righ t C -como dules, one deﬁnes a righ t C -como dule structure on C ⊗ K R ⊗ K R by the fo rm ula c ⊗ K u ⊗ K v 7− → c (1) ⊗ K u [0] ⊗ K v [0] ⊗ c (2) u [1] v [1] and c hec ks that b o t h the isomorphism C ⊗ K R ⊗ K R ≃ ( C ⊗ K R )  C ( C ⊗ K R ) and the map C ⊗ K R ⊗ K R − → C ⊗ K R are morphisms of rig ht C - como dules. C.1.3. Deﬁne a pairing φ l : K ⊗ k C − → k b y the form ula φ l ( x, c ) = h s ( x ) , c i . The pairing φ l induces a left action of K in C giv en by the formu la x → c = h s ( x ) , c (1) i c (2) . Assume that the morphism of k -a lgebras f mak es R a ﬂat righ t K -mo dule. W e will apply the opp o site v ersion of t he construction of 10.2.1 in order to obtain a semialgebra structure on S l = R ⊗ K C . Deﬁne a left C - como dule structure on R ⊗ K C b y the form ula u ⊗ K c 7− → c (1) s − 1 ( u [1] ) ⊗ u [0] ⊗ K c (2) . This coaction is w ell-deﬁned, since one has u ⊗ ( x → c ) = u ⊗ h s ( x ) , c (1) i c (2) 7− → h s ( x ) , c (1) i c (2) s − 1 ( u [1] ) ⊗ u [0] ⊗ K c (3) and uf ( x ) ⊗ c 7− → c (1) s − 1 ( u [1] x [1] ) ⊗ u [0] f ( x [0] ) ⊗ K c (2) = c (1) s − 1 ( x [1] ) s − 1 ( u [1] ) ⊗ u [0] ⊗ K ( x [0] → c (2) ) = h s ( x [0] ) , c (2) i c (1) s − 1 ( x [1] ) s − 1 ( u [1] ) ⊗ u [0] ⊗ K c (3) ; no w the identit y h s ( x ) , c (1) i c (2) = h s ( x [0] ) , c (2) i c (1) s − 1 ( x [1] ) follows from the iden tities h s − 1 ( x ) , d (2) i d (1) = h s − 1 ( x [0] ) , d (1) i d (2) x [1] and ( s 2 x ) [0] ⊗ ( s 2 x ) [1] = s 2 ( x [0] ) ⊗ s − 2 ( x [1] ). This left C -como dule structure agrees with the left K -mo dule structure on R ⊗ K C , since h s ( x ) , c (1) s − 1 ( u [1] )) i u [0] ⊗ K c (2) = h s ( x (1) ) , s − 1 ( u [1] ) i ⊗ K h s ( x (2) ) , c (1) i c (2) = h x (1) , u [1] i u [0] ⊗ K ( x (2) → c ) = x (1) us ( x (2) ) x (3) ⊗ K c = xu ⊗ K c . The rest is a nalogous to C.1.2; the left C -como dule structure on R ⊗ K R ⊗ K C is deﬁned b y the formula u ⊗ K v ⊗ K c 7− → c (1) s − 1 ( v [1] ) s − 1 ( u [1] ) ⊗ u [0] ⊗ K v [0] ⊗ K c (2) . C.1.4. According to 10.2.2, the category of r ig h t S r -semimo dules is isomorphic to the category of k -v ector spaces N endo w ed with righ t C -como dule and right R - mo dule 239 structures suc h that h s − 1 ( x ) , n (1) i n (0) = nf ( x ) and ( nr ) (0) ⊗ ( nr ) (1) = n (0) r [0] ⊗ n (1) r [1] for n ∈ N , x ∈ K , r ∈ R , where n 7− → n (0) ⊗ n (1) denotes the righ t C -coaction map and n ⊗ r 7− → nr denotes the righ t R -a ction ma p. Assuming that R is a pro jectiv e left K -mo dule, the category of left S r -semicontra- mo dules is isomorphic to the category of k -v ector spaces P endo w ed with left C -con- tramo dule and left R -mo dule structures such that π P ( c 7→ h s − 1 ( x ) , c i p ) = f ( x ) p and π P ( c 7→ r [0] g ( cr [1] )) = r π P ( g ) fo r p ∈ P , x ∈ K , c ∈ C , r ∈ R , g ∈ Hom k ( C , P ), where π P denotes the con traaction map and r ⊗ p − → r p denotes the left action map. The category o f left S l -semimo dules is isomorphic to the category o f k -v ector spaces M endo w ed with left C -como dule and left R -mo dule structures suc h that h s ( x ) , m ( − 1) i m (0) = f ( x ) m and ( r m ) ( − 1) ⊗ ( r m ) (0) = m ( − 1) s − 1 ( r [1] ) ⊗ r [0] m [0] for m ∈ M , x ∈ K , r ∈ R , where m 7− → m ( − 1) ⊗ m (0) denotes the left C - coaction map and r ⊗ m 7− → r m denotes the left R - action map. C.2. Morit a equiv alence. C.2.1. Let E b e a k -vec tor space endo w ed with a C - C -bicomo dule structure and a righ t C - mo dule structure satisfying the equation ( j c ) ( − 1) ⊗ ( j c ) (0) ⊗ ( j c ) (1) = j ( − 1) c (1) ⊗ j (0) c (2) ⊗ j (1) c (3) for j ∈ E , c ∈ C , where j 7− → j ( − 1) ⊗ j (0) ⊗ j (1) denotes the bicoaction map and j ⊗ c 7− → j c denotes the rig h t action map. In particular, E is a righ t Hopf mo dule [45 ] ov er C , hence E is isomorphic to the tensor pro duct E ⊗ k C as a righ t C -como dule and a righ t C -mo dule, where E is a k -ve ctor space whic h can b e deﬁned as the subspace in E cons isting of all i ∈ E suc h that i (0) ⊗ i (1) = i ⊗ e . One can see that E is a left C -subcomo dule in E , so E can b e iden tiﬁed with the tensor pro duct E ⊗ k C endo w ed with the bicoaction ( i ⊗ c ) ( − 1) ⊗ ( i ⊗ c ) (0) ⊗ ( i ⊗ c ) (1) = i ( − 1) c (1) ⊗ ( i (0) ⊗ c (2) ) ⊗ c (3) and the r ig h t action ( i ⊗ c ) d = i ⊗ cd . Besides , the isomorphism E ⊗ k C ≃ C ⊗ k E given by the formulas i ⊗ c 7− → i ( − 1) c ⊗ i (0) and c ⊗ i 7− → i (0) ⊗ s − 1 ( i ( − 1) ) c identiﬁes E with the tensor pro duct C ⊗ k E endo w ed with t he bicoaction ( c ⊗ i ) ( − 1) ⊗ ( c ⊗ i ) (0) ⊗ ( c ⊗ i ) (1) = c (1) ⊗ ( c (2) ⊗ i (0) ) ⊗ s − 1 ( i ( − 1) ) c (3) and t he righ t action ( c ⊗ i ) d = cd ⊗ i . C.2.2. The pairing s φ l and φ r induce left and right actions of K in E given by the form ulas x → j = h s ( x ) , j ( − 1) i j (0) and j ← x = h s − 1 ( x ) , j (1) i j (0) for j ∈ E , x ∈ K . Assume that these t w o actions satisfy the equation x [0] → ( j x [1] ) = j ← x, or equiv alently , x → j = ( j s − 1 ( x [1] )) ← x [0] . Let us construct an isomorphism E ⊗ K R ≃ R ⊗ K E . Set the map E ⊗ K R − → R ⊗ K E to b e giv en by the formula j ⊗ K u 7− → u [0] ⊗ K j u [1] and the map R ⊗ K E − → E ⊗ K R to b e giv en b y the formula u ⊗ K j 7− → j s − 1 ( u [1] ) ⊗ K u [0] for j ∈ E , u ∈ R . W e ha v e to c hec k that these maps a re w ell-deﬁned. One has x → ( j c ) = h s ( x ) , j ( − 1) c (1) i j (0) c (1) = h s ( x (2) ) , j ( − 1) ih s ( x (1) ) c (1) i j (0) c (1) = ( x (2) → j ) ( x (1) → c ). Therefore, h x (1) , d (1) i x (2) → ( j d (2) ) = ( x (3) → j ) ( x (2) → 240 ( s − 1 ( x (1) ) → d )) = ( x → j ) d . No w w e ha v e ( j ← x ) ⊗ u 7− → u [0] ⊗ K ( j ← x ) u [1] and j ⊗ f ( x ) u 7− → f ( x [0] ) u [0] ⊗ K j x [1] u [1] = f ( x [0](1) ) u [0] f ( s ( x [0](2) )) f ( x [0](3) ) ⊗ K j x [1] u [1] = h x [0](1) , u [1] i u [0] f ( x [0](2) ) ⊗ K j x [1] u [2] = u [0] ⊗ K h x [0](1) , u [1] i x [0](2) → ( j x [1] u (2) ) = u [0] ⊗ K ( x [0] → ( j x [1] )) u [1] . Analogously , one has ( j c ) ← x = h s − 1 ( x ) , j (1) c (2) i j (0) c (1) = h s − 1 ( x (2) ) , j (1) i h s − 1 ( x (1) ) , c (2) i j (0) c (1) = ( j ← x (2) )( c ← x (1) ). Therefore, h s − 1 ( x (1) ) , d (1) i ( j s − 1 ( d (2) )) ← x (2) = h x (1) , s − 1 ( d (1) ) i ( j s − 1 ( d (2) )) ← x (2) = ( j ← x (3) )(( s − 1 ( d ) ← s ( x (1) )) ← x (2) ) = ( j ← x ) s − 1 ( d ). Now we ha v e u ⊗ ( x → j ) 7− → ( x → j ) s − 1 ( u [1] ) ⊗ K u [0] and uf ( x ) ⊗ j 7− → j s − 1 ( x [1] ) s − 1 ( u [1] ) ⊗ K u [0] f ( x [0] ) = j s − 1 ( x [1] ) s − 1 ( u [1] ) ⊗ K f ( x [0](3) ) f ( s − 1 ( x [0](2) )) u [0] f ( x [0](1) ) = j s − 1 ( x [1] ) s − 1 ( u [2] ) ⊗ K h s − 1 ( x [0](1) ) , u [1] i f ( x [0](2) ) u [0] = h s − 1 ( x [0](1) ) , u [1] i ( j s − 1 ( x [1] ) s − 1 ( u [2] )) ← x [0](2) ⊗ K u [0] = ( j s − 1 ( x [1] )) ← x [0] ⊗ K u [0] . Chec king that these tw o maps are m utually in v erse is easy . C.2.3. Assume that R is a ﬂat left and righ t K -mo dule. T hen the vec tor space E ⊗ K R ≃ E  C ( C ⊗ K R ) is endo w ed with the structures of left C -como dule, righ t S r -semimo dule, and righ t R -mo dule. The v ector space R ⊗ K E ≃ ( R ⊗ K C )  C E is endo w ed with the structures of righ t C -como dule, left S l -semimo dule, and left R -mo dule. Let us che c k t ha t the isomorphism E ⊗ K R ≃ R ⊗ K E preserv es the C - C -bicomo dule structures. Indeed, one has ( j ⊗ K u ) ( − 1) ⊗ ( j ⊗ K u ) (0) ⊗ ( j ⊗ K u ) (1) = j ( − 1) ⊗ ( j (0) ⊗ K u [0] ) ⊗ j (1) u [1] and ( u ⊗ K j ) ( − 1) ⊗ ( u ⊗ K j ) (0) ⊗ ( u ⊗ K j ) (1) = j ( − 1) s − 1 ( u [1] ) ⊗ ( u [0] ⊗ K j (0) ) ⊗ j (1) , hence ( u [0] ⊗ K j u [1] ) ( − 1) ⊗ ( u [0] ⊗ K j u [1] ) ( − 0) ⊗ ( u [0] ⊗ K j u [1] ) (1) = j ( − 1) u [2] s − 1 ( u [1] ) ⊗ ( u [0] ⊗ K j (0) u [3] ) ⊗ j (1) u [4] = j ( − 1) ⊗ ( u [0] ⊗ K j (0) u [1] ) ⊗ j [1] u [2] . Set E ⊗ K R = E ≃ R ⊗ K E . The left and right actions of R in E commute; indeed, the left and the induced righ t actions o f R in R ⊗ K E are giv en b y the form ulas w ( u ⊗ K j ) = w u ⊗ K j and ( u ⊗ K j ) v = u v [0] ⊗ K j v [1] . It follows easily that E  C S r ≃ E ≃ S l  C E is an S l - S r -bisemimo dule. C.2.4. No w assume that the C - C - bicomo dule E can b e included into a Morita auto- equiv alence ( E , E ∨ ) of C . This means that a C - C -bicomo dule E ∨ is giv en together with isomorphisms of C - C -bicomo dules E  C E ∨ ≃ C ≃ E ∨  C E suc h that the t w o induced isomorphisms E  C E ∨  C E ⇒ E coincide and the tw o induced isomorphisms E ∨  C E  C E ∨ ⇒ E ∨ coincide (see 7.5). The Morita equiv alence ( E , E ∨ ) is unique if it exists, and it exists if and only if the left C -como dule E is one-dimensional. In the latter case, the bicomo dule E ∨ is constructed as follows. The left C -coaction in E has the f orm i ( − 1) ⊗ i (0) = c E ⊗ i for a certain elemen t c E ∈ C suc h that c E (1) ⊗ c E (2) = c E ⊗ c E and ε ( c E ) = 1. Set E ∨ = Hom k ( E , k ) and deﬁne a left coaction of C in E ∨ b y the formula ˇ ı ( − 1) ⊗ ˇ ı (0) = s ( c E ) ⊗ ˇ ı . T ak e E ∨ = E ∨ ⊗ k C and deﬁne the C - C -bicomo dule structure on E ∨ b y the formula ( ˇ ı ⊗ c ) ( − 1) ⊗ ( ˇ ı ⊗ c ) (0) ⊗ ( ˇ ı ⊗ c ) (1) = ˇ ı ( − 1) c (1) ⊗ ( ˇ ı (0) ⊗ c (2) ) ⊗ c (3) . Then one has E  C E ∨ ≃ E ⊗ k E ∨ ⊗ k C ≃ C and 241 E ∨  C E ≃ E ∨ ⊗ k E ⊗ k C ≃ C . There is also an isomorphism E ∨ ⊗ k C ≃ C ⊗ k E ∨ giv en b y the f o rm ulas analogous to C.2.1 . C.2.5. T aking the cotensor pro duct of the isomorphism E  C S r ≃ S l  C E with the C - C -bicomo dule E ∨ on the left, w e o btain an isomorphism S r ≃ E ∨  C S l  C E . D eﬁne a semialgebra structure on E ∨  C S l  C E in terms of the semialgebra structure on S l and the isomor phisms E  C E ∨ ≃ C ≃ E ∨  C E (see 8.4 .1). Using the facts that E  C S r ≃ S l  C E is an S l - S r -bisemimo dule and the isomorphism E  C S r ≃ S l  C E forms a commutativ e diag r a m with the maps E − → E  C S r and E − → S l  C E induced b y the semiunit morphisms of S r and S l , one can che c k that S r ≃ E ∨  C S l  C E is an isomorphism of semialgebras ov er C . It follow s that S l and S r are left and righ t coﬂat C -como dules. Set S r  C E ∨ = E ∨ ≃ E ∨  C S l ; then E ∨ is an S r - S l -bicomo dule and the pair ( E , E ∨ ) is a left and righ t coﬂat Morita equiv alence b et w een S l and S r (see 8.4.5). The category of left S r -semimo dules can b e no w describ ed. Namely , it is equiv alen t to the category of left S l -semimo dules; this equiv alence assigns to a left S r -semi- mo dule L the left S l -semimo dule E  C L a nd to a left S l -semimo dule M the left S r -semimo dule E ∨  C M . On the lev el of C -como dules, one has E  C L ≃ E ⊗ k L and E ∨  C M ≃ E ∨ ⊗ k M ; the left C - coaction in E ⊗ k L and E ∨ ⊗ k M is giv en by the form ulas ( i ⊗ l ) ( − 1) ⊗ ( i ⊗ l ) (0) = c E l ( − 1) ⊗ ( i ⊗ l (0) ) and ( ˇ ı ⊗ m ) ( − 1) ⊗ ( ˇ ı ⊗ m ) (0) = s ( c E ) m ( − 1) ⊗ ( ˇ ı ⊗ m (0) ). C.2.6. In particular, when the left and right actions o f K in C satisfy the equation x [0] → ( cx [1] ) = c ← x, or equiv alen tly , x → c = ( cs − 1 ( x [1] )) ← x [0] , one can tak e E = C = E ∨ . Th us the semialgebras S r and S l are isomorphic in this case, the isomorphism b eing giv en b y the form ulas c ⊗ K u 7− → u [0] ⊗ K cu [1] and u ⊗ K c 7− → cs − 1 ( u [1] ) ⊗ K u [0] for c ∈ C and u ∈ R . Remark. One can construct an isomorphism b et w een v ersions of the semialgebras S r and S l in sligh tly larger generality . Namely , let χ r , χ l : K − → k b e k -algebra ho- momorphisms satisfying the equations χ ( x [0] ) x [1] = χ ( x ) e and χ ( x [0](2) ) x [0](1) ⊗ x [1] = χ ( x (2) ) x (1)[0] ⊗ x (1)[1] . These equations hold automatically when the pairing h , i is non- degenerate in C (apply id ⊗ χ to the iden tit y h y , x [1] i x [0](1) ⊗ x [0](2) = ( h y , x [1] i x [0] ) (1) ⊗ h y , x [1] i x [0] ) (2) = ( y (1) xs ( y (2) )) (1) ⊗ ( y (1) xs ( y (2) )) (2) = y (1) x (1) s ( y (4) ) ⊗ y (2) x (2) s ( y (3) )). Deﬁne pairings φ r : C ⊗ k K − → k and φ l : K ⊗ k C − → k by the form ulas φ r ( c, x ) = χ r ( x (2) ) h s − 1 ( x (1) ) , c i and φ l ( x, c ) = χ l ( x (2) ) h s ( x (1) ) , c i , and mo dify the deﬁnitions of the rig h t and left actions of K in C accordingly , c ← x = φ r ( c (2) , x ) c (1) and x → c = φ l ( x, c (1) ) c (2) . Then the tensor pro ducts C ⊗ K R and R ⊗ K C are semi- algebras ov er C with the C - C -bicomo dule structures give n b y the same formulas a s in C.1.2 – C.1.3. Assuming that the mo diﬁed left and righ t actions of K in C satisfy 242 the ab o v e equation, the maps c ⊗ K u 7− → u [0] ⊗ K cu [1] and u ⊗ K c 7− → cs − 1 ( u [1] ) ⊗ K u [0] are mutually in v erse isomorphisms b etw een these tw o semialgebras. C.2.7. The opp osite c oring D op to a coring D o v er a k -a lg ebra B and the opp osite semialgebr a T op to a semialgebra T ov er D are deﬁned in the obvious w a y; D op is a coring o v er B op and T op is a semialgebra ov er D op . In the ab ov e assumptions, notice t he identit y s ( x → c ) = s ( c ) ← s ( x ) for x ∈ K , c ∈ C . Supp ose that the k -alg ebra R is endo w ed with an an ti-endomorphism s satisfying the equations f ( s ( x )) = s ( f ( x )) and c (1) ⊗ K ( s ( u )) [0] ⊗ c (2) ( s ( u )) [1] = c (1) ⊗ K s ( u [0] ) ⊗ u [1] c (2) ; the second equation follows f rom the ﬁrst one if the pairing h , i is nondegenerate in C . Then there is a map of semialgebras s : ( R ⊗ K C ) op − → C ⊗ K R compatible with the isomorphism of coalg ebras s : C op ≃ C ; it is deﬁned by the form ula s ( u ⊗ K c ) = s ( c ) ⊗ K s ( u ). Supp ose that w e are giv en a map s : E − → E satisfying the equation ( s ( j ) ) (0) ⊗ ( s ( j )) (1) = s ( j (0) ) ⊗ s ( j ( − 1) ). Then one has s ( x → j ) = s ( j ) ← s ( x ). The induced map s : R ⊗ K E − → E ⊗ K R given b y the form ula s ( u ⊗ K j ) = s ( j ) ⊗ K s ( u ) is a map of righ t semimo dules compatible with the isomorphism of coalgebras s : C op ≃ C and the map of semialgebras s : S l op − → S r , where the rig h t S l op -semimo dule structure on R ⊗ K E corresp onds to its left S l -semimo dule structure. No w assume that C is comm utativ e, K is co comm utativ e, a nd our data satisfy the equations s 2 ( u ) = u , s 2 ( j ) = j , s ( j c ) = s ( j ) s ( c ), and ( s ( u )) [0] ⊗ ( s ( u )) [1] = s ( u [0] ) ⊗ u [1] ; the latt er equation holds auto matically when the pairing h , i is nondegenerate in C . Then the comp osition of the isomorphism o f S l - S r -bisemimo dules E ⊗ K R ≃ R ⊗ K E and the map s : R ⊗ K E − → E ⊗ K R in an in v olution of the bisemimo dule E transforming its left S l -semimo dule and righ t S r -semimo dule structures in to eac h other in a w ay compatible with the isomorphism of coalgebras s : C op − → C and the isomorphism of semialgebras s : S l op ≃ S r . In particular, in the situation of C.2.6 the map s : R ⊗ K C − → C ⊗ K R b ecomes an inv olutiv e anti-automorphism of the semialgebra S l ≃ S r compatible with the a nti-automorphism s of the coalgebra C . C.3. Semitensor pro duct and semihomomorphisms , SemiT or and SemiExt. Let us return to the assumptions of C.1.1 – C.2.4. C.3.1. Let N b e a righ t C - como dule and M b e a left C -como dule. Then one can easily c hec k tha t the tw o injections N  C E ∨  C M ≃ N  C ( E ∨ ⊗ k C )  C M ≃ N  C ( E ∨ ⊗ k M ) − → N ⊗ k E ∨ ⊗ k M and N  C E ∨  C M ≃ N  C ( C ⊗ k E ∨ )  C M ≃ ( N ⊗ k E ∨ )  C M − → N ⊗ k E ∨ ⊗ k M coincide. Let N b e a rig ht S r -semimo dule and M b e a left S l -semimo dule (see C.1.4). Then the isomorphism ( N ⊗ K R )  C ( E ∨ ⊗ k M ) ≃ N  C ( C ⊗ K R )  C E ∨  C M ≃ N  C E ∨  C ( R ⊗ K C )  C M ≃ ( N ⊗ k E ∨ )  C ( R ⊗ K M ) induce d b y the isomorphism ( C ⊗ K R )  C E ∨ ≃ E ∨  C ( R ⊗ K C ) can b e computed as follows. 243 There is an isomorphism ( C ⊗ k R )  C E ∨ ≃ E ∨  C ( R ⊗ k C ) deﬁned by the same form ulas that the isomorphism S r  C E ∨ ≃ E ∨  C S l ( ⊗ K b eing replaced with ⊗ ). Hence the induced isomorphism ( N ⊗ k R )  C ( E ∨ ⊗ k M ) ≃ ( N ⊗ k E ∨ )  C ( R ⊗ k M ), whic h is give n by the simple f orm ula n ⊗ r ⊗ ˇ ı ⊗ m 7− → n ⊗ ˇ ı ⊗ r ⊗ m . The isomorphisms ( N ⊗ K R )  C ( E ∨ ⊗ k M ) ≃ ( N ⊗ k E ∨ )  C ( R ⊗ K M ) and ( N ⊗ k R )  C ( E ∨ ⊗ k M ) ≃ ( N ⊗ k E ∨ )  C ( R ⊗ k M ) form a comm utativ e diagram with the natural maps ( N ⊗ k R )  C ( E ∨ ⊗ k M ) − → ( N ⊗ K R )  C ( E ∨ ⊗ k M ) and ( N ⊗ k E ∨ )  C ( R ⊗ k M ) − → ( N ⊗ k E ∨ )  C ( R ⊗ K M ). This prov ides a description of the ﬁrst isomorphism whenev er the latter t w o maps ar e surjectiv e—in particular, when either N or M is a coﬂat C -como dule. T o compute the desired isomorphism in the general case, it suﬃces to represen t either N or M as t he k ernel of a morphism of C -coﬂat semimo dules (b oth sides of this isomorphism preserv e k ernels, since R is a ﬂat left a nd righ t K -mo dule). C.3.2. Let M b e a left C -como dule a nd P b e a left C -contramo dule. Then one can c hec k that the tw o surjections Hom k ( E ∨ ⊗ k M , P ) − → Cohom C ( E ∨ ⊗ k M , P ) ≃ Cohom C ( E ∨  C M , P ) a nd Hom k ( M , Hom k ( E ∨ , P )) − → Cohom C ( M , Hom k ( E ∨ , P )) ≃ Cohom C ( M , Cohom C ( E ∨ , P )) coincide. Let M b e a left S l -semimo dule and P b e a left S r -semicontramo dule. Assum- ing t ha t R is a pro jectiv e left K - mo dule, the isomorphism Cohom C ( E ∨ ⊗ k M , Hom K ( R, P )) ≃ Cohom C ( E ∨  C M , Cohom C ( C ⊗ K R, P )) ≃ Cohom C (( R ⊗ K C )  C M , Cohom C ( E ∨ P )) ≃ Cohom C ( R ⊗ K M , Hom k ( E ∨ , P )) induced b y t he isomorphism ( C ⊗ K R )  C E ∨ ≃ E ∨  C ( R ⊗ K C ) can b e computed as follows. The isomorphism Cohom C ( E ∨ ⊗ k M , Ho m k ( R, P )) ≃ Cohom C ( R ⊗ k M , Hom k ( E ∨ , P )) induced by the isomorphism ( C ⊗ k R )  C E ∨ ≃ E ∨  C ( R ⊗ k C ) is giv en b y the simple form ula g 7− → h , h ( r ⊗ m )( ˇ ı ) = g ( ˇ ı ⊗ m )( r ). The isomor- phisms Cohom C ( E ∨ ⊗ k M , Hom K ( R, P )) ≃ Cohom C ( R ⊗ K M , Hom k ( E ∨ , P )) and Cohom C ( E ∨ ⊗ k M , Hom k ( R, P )) ≃ Cohom C ( R ⊗ k M , Hom k ( E ∨ , P )) form a comm utativ e diagra m with the natural maps Cohom C ( E ∨ ⊗ k M , Hom K ( R, P )) − → Cohom C ( E ∨ ⊗ k M , Hom k ( R, P )) and Cohom C ( R ⊗ K M , Hom k ( E ∨ , P )) − → Cohom C ( R ⊗ k M , Hom k ( E ∨ , P )). This provide s a description o f the ﬁrst isomor- phism in the case when the latter tw o maps a r e injectiv e—in particular, when either M is a copro jective C -como dule, or P is a coinjectiv e C - contramo dule. T o compute the desired isomorphism in the general case, it suﬃces to either represen t M as the k ernel of a morphism of C -coproj ectiv e semimo dules, o r represen t P as the cok ernel of a morphism of C -coinjectiv e semicon tramo dules. C.3.3. Assume that the k -algebra R is endow ed with a Ho pf algebra structure u 7− → u (1) ⊗ u (2) , u 7− → ε ( u ) with in v ertible an tip o de s suc h tha t f : K − → R is a Hopf algebra morphism. Let N b e a right S r -semimo dule and M b e a left S l -semimo d- ule; assume that either N or M is a coﬂat C - como dule. Deﬁne righ t R -mo dule and righ t C - como dule structures on the tensor pro duct N ⊗ k E ∨ ⊗ k M by the formulas 244 ( n ⊗ ˇ ı ⊗ m ) r = nr (2) ⊗ ˇ ı ⊗ s − 1 ( r (1) ) m and ( n ⊗ ˇ ı ⊗ m ) (0) ⊗ ( n ⊗ ˇ ı ⊗ m ) (1) = ( n (0) ⊗ ˇ ı ⊗ m (0) ) ⊗ s − 1 ( m ( − 1) ) c E n (1) . Then the semitensor pro duct N ♦ S r E ∨ ♦ S l M (whic h is easily seen to b e asso ciative) is uniquely determined by these right R - mo dule and righ t C -como dule structures on N ⊗ k E ∨ ⊗ k M . Indeed, the subspace N  C ( E ∨ ⊗ k M ) ≃ N  C E ∨  C M ≃ ( M ⊗ k E ∨ )  C M of the space N ⊗ k E ∨ ⊗ k M can b e deﬁned by the equation n (0) ⊗ ˇ ı ⊗ m (0) ⊗ s − 1 ( m ( − 1) ) c E n (0) = n ⊗ ˇ ı ⊗ m ⊗ e . The isomorphism N ⊗ k R ⊗ k E ∨ ⊗ k M ≃ N ⊗ k E ∨ ⊗ k M ⊗ k R giv en by the form ulas n ⊗ r ⊗ ˇ ı ⊗ m 7− → n ⊗ ˇ ı ⊗ r (1) m ⊗ r (2) and n ⊗ ˇ ı ⊗ m ⊗ r 7− → n ⊗ r (2) ⊗ ˇ ı ⊗ s − 1 ( r (1) ) m transforms the pair of maps N ⊗ k R ⊗ k E ∨ ⊗ k M ⇒ N ⊗ k E ∨ ⊗ k M giv en b y the for mulas n ⊗ r ⊗ ˇ ı ⊗ m 7− → nr ⊗ ˇ ı ⊗ m , n ⊗ ˇ ı ⊗ r m into the pair of maps N ⊗ k E ∨ ⊗ k M ⊗ k R ⇒ N ⊗ k E ∨ ⊗ k M giv en b y the formulas n ⊗ ˇ ı ⊗ m ⊗ r 7− → nr (2) ⊗ ˇ ı ⊗ s − 1 ( r (1) ) m , ε ( r ) n ⊗ ˇ ı ⊗ m . This isomorphism also transforms the subspace ( N ⊗ k R )  C ( E ∨ ⊗ k M ) of N ⊗ k R ⊗ k E ∨ ⊗ k M , whic h can b e deﬁned b y the equation n (0) ⊗ r [0] ⊗ ˇ ı ⊗ m (0) ⊗ s − 1 ( m ( − 1) ) c E n (1) r [1] = n ⊗ r ⊗ ˇ ı ⊗ m ⊗ e , in to the subspace of N ⊗ k E ∨ ⊗ k M ⊗ k R deﬁned b y the equation n (0) ⊗ ˇ ı ⊗ r (2)[0](1) ( s − 1 ( r (1) )) [0] m (0) ⊗ r (2)[0](2) ⊗ s − 2 (( s − 1 ( r (1) )) [1] ) s − 1 ( m ( − 1) ) c E n (1) r (2)[1] = n ⊗ ˇ ı ⊗ m ⊗ r ⊗ e . Finally , the same isomorphism transforms the quotien t space N ⊗ K R ⊗ k E ∨ ⊗ k M of t he space N ⊗ k R ⊗ k E ∨ ⊗ k M into the quotien t space ( N ⊗ k E ∨ ⊗ k M ) ⊗ K R of the space N ⊗ k E ∨ ⊗ k M ⊗ k R , as one can c hec k using the isomorphism N ⊗ k K ⊗ k R ⊗ k E ∨ ⊗ k M ≃ N ⊗ k E ∨ ⊗ k M ⊗ k K ⊗ k R giv en b y the formulas n ⊗ x ⊗ r ⊗ ˇ ı ⊗ m 7− → n ⊗ ˇ ı ⊗ x (1) r (1) m ⊗ x (2) ⊗ r (2) and n ⊗ ˇ ı ⊗ m ⊗ x ⊗ r 7− → n ⊗ x (2) ⊗ r (2) ⊗ ˇ ı ⊗ s − 1 ( r (1) ) s − 1 ( x (1) ) m . C.3.4. Let M b e a left S l -semimo dule and P b e a left S r -semicontramo dule; assume that either M is a copro jectiv e C - como dule, or P is a coinjectiv e C -contramo dule. Deﬁne left R - mo dule and left C -contra mo dule structures on the space Hom k ( E ∨ ⊗ k M , P ) by the form ulas r g ( ˇ ı ⊗ m ) = r (2) g ( ˇ ı ⊗ s − 1 ( r (1) ) m ) and π ( h )( ˇ ı ⊗ m ) = π P ( c 7→ h ( s − 1 ( m ( − 1) ) c E c )( ˇ ı ⊗ m (0) )) for g ∈ Hom k ( E ∨ ⊗ k M , P ) a nd h ∈ Hom k ( C , Hom k ( E ∨ ⊗ k M , P )), where π P denotes the C -con traaction in P . Then the semihomomorphism space SemiHom S r ( E ♦ S l M , P ) is uniquely determined b y these R - mo dule and C -con- tramo dule structures o n Hom k ( E ∨ ⊗ k M , P ). This is established in the w a y analog o us to C.3.3 using the isomorphism Hom k ( R ⊗ k E ∨ ⊗ k M , P ) ≃ Hom k ( R, Hom k ( E ∨ ⊗ k M , P )) give n b y the form ulas g 7− → h , g ( r ⊗ ˇ ı ⊗ m ) = h ( r (2) )( ˇ ı ⊗ r (1) ), h ( r ) ( ˇ ı ⊗ m ) = g ( r (2) ⊗ ˇ ı ⊗ s − 1 ( r (1) ) m ). C.3.5. No w assume that C is commutativ e, K is co commu tative , and the equations ( s ( u )) [0] ⊗ ( s ( u )) [1] = s ( u [0] ) ⊗ u [1] , ε ( u [0] ) u [1] = ε ( u ) e , and u (1)[0] ⊗ u (2)[0] ⊗ u (1)[1] u (2)[1] = u [0](1) ⊗ u [0](2) ⊗ u [1] are satisﬁed fo r u ∈ R ; when the pairing h , i is nondegenerate in C , these equations hold automatically . Let N b e a righ t S r -semimo dule and M b e a left S l -semimo dule. Deﬁne righ t R -mo dule a nd righ t C -como dule structures on the tensor pro duct N ⊗ k M b y the form ulas ( n ⊗ m ) r = nr (2) ⊗ s − 1 ( r (1) ) m and ( n ⊗ m ) (0) ⊗ ( n ⊗ m ) (1) = ( n (0) ⊗ m (0) ) ⊗ 245 s − 1 ( m ( − 1) ) n (1) . These rig h t a ctio n and right coaction satisfy t he equations of C.1.4, so they deﬁne a rig ht S r -semimo dule structure on N ⊗ k M . The ground ﬁeld k , endo w ed with the trivial left R -mo dule and left C - como dule structures r a = ε ( r ) a and a ( − 1) ⊗ a (0) = e ⊗ a for a ∈ k , b ecomes a left S l -semimo dule. Then there is a natural isomorphism N ♦ S r E ∨ ♦ S l M ≃ ( N ⊗ k M ) ♦ S r E ∨ ♦ S l k . Indeed, let us ﬁrst assume that either N or M is a coﬂat C -como dule; notice that N ⊗ k M is then a coﬂat C -como dule, to o. The isomorphism N  C ( E ∨ ⊗ k M ) ≃ ( N ⊗ k M )  C E ∨ giv en by the form ula n ⊗ ˇ ı ⊗ m 7− → n ⊗ m ⊗ ˇ ı and the isomorphism ( N ⊗ K R )  C ( E ∨ ⊗ k M ) ≃ (( N ⊗ k M ) ⊗ K R )  C E ∨ constructed in C.3.3 transform the pair of maps whose cok ernel is N ♦ S r E ∨ ♦ S l M into the pair of maps whose cok ernel is ( N ⊗ k M ) ♦ S r E ∨ ♦ S l k . In the general case, represen t N or M as the k ernel of a morphism of C -coﬂat semimo dules; then the pair of ma ps whose cok ernel is N ♦ S r E ∨ ♦ S l M and the pair o f maps whose cok ernel is ( N ⊗ k M ) ♦ S r E ∨ ♦ S l k b ecome the k ernels o f isomorphic morphisms of pairs of maps. C.3.6. Let M b e a left S l -semimo dule and P b e a left S r -semicontramo dule. Deﬁne left R -mo dule a nd left C -contramo dule structures o n the space Hom k ( M , P ) by the form ulas r g ( m ) = r (2) g ( s − 1 ( r (1) ) m ) and π ( h )( m ) = π P ( c 7→ h ( s − 1 ( m ( − 1) ) c )( m (0) )) for g ∈ Hom k ( M , P ) and h ∈ Hom k ( C , Hom k ( M , P )). These left action and left con traaction satisfy the equations of C.1.4, so they deﬁne a left S r -semicontramo dule structure on Ho m k ( M , P ). Then there is a na t ur a l isomorphism SemiHom S r ( E ∨ ♦ S l M , P ) ≃ SemiHom S r ( E ∨ ♦ S l k , Hom k ( M , P )). C.3.7. Let N • b e a complex of right S r -semimo dules and M • b e a complex of left S l -semimo dules. Then there are na tural isomorphisms SemiT or S r ( N • , E ∨ ♦ S l M • ) ≃ SemiT or S l ( N • ♦ S r E ∨ , M • ) ≃ SemiT o r S r ( N • ⊗ k M • , E ∨ ♦ S l k ) in the deriv ed category of k - v ector spaces. The isomorphism b et w een the ﬁrst t w o ob jects is pro vided b y the results of 8.4.3, and the isomorphism b et w een either of the ﬁrst tw o ob jects and the third ob ject fo llows from C.3.5. Indeed, assume that the complex N • is semiﬂat. Then the complex N • ⊗ k M • is also semiﬂat, since ( N • ⊗ k M • ) ♦ S r L • ≃ ( N • ⊗ k M • ) ♦ S r E ∨ ♦ S l E ♦ S r L • ≃ ( N • ⊗ k M • ⊗ k ( E ♦ S r L • )) ♦ S r E ∨ ♦ S l k ≃ N • ♦ S r E ∨ ♦ S l ( M • ⊗ k ( E ♦ S r L • )) for any complex of left S r -semimo dules L • . C.3.8. Let M • b e a complex of left S l -semimo dules and P • b e a complex of left S r -semicontramo dules. Then there ar e natural isomorphisms SemiExt S r ( E ∨ ♦ S l M • , P • ) ≃ SemiExt S l ( M • , SemiHom S r ( E ∨ , P • )) ≃ SemiExt S r ( E ∨ ♦ S l k , Ho m k ( M • , P • )). C.4. H arish-Chandra pairs. 246 C.4.1. Let ( g , H ) b e an algebraic Harish-Chandra pair ov er a ﬁeld k , t ha t is H is an algebraic group, whic h we will assume to b e aﬃne, g is a Lie a lgebra into whic h the Lie a lgebra h o f the algebraic group H is embedded, a nd an action of H by Lie algebra automorphisms of g is giv en. Tw o conditions should b e satisﬁed: h is an H -submo dule in g where H acts by the adjo int action of H in h , a nd the action of h in g obtained b y diﬀeren tiating the action of H in g coincides with the a djoin t action of h in g . Notice that the dimension of h is presumed to b e ﬁnite, though the dimension of g may b e inﬁnite. Set K = U ( h ) and R = U ( g ) to b e the univ ersal en v eloping algebras of h and g , and f : K − → R to b e the morphism induced by the em b edding h − → g . Let C = C ( H ) b e the coa lg ebra of functions on H . Then C , K , a nd R a re Hopf algebras; the adjoin t action of H in h and the giv en action of H in g pro vide us with righ t coactions x 7− → x [0] ⊗ x [1] and u 7− → u [0] ⊗ u [1] of C in K and R ; a nd there is a natural pairing h , i : K ⊗ k C − → k suc h that the equations of C.1.1 – C.1.2 and C.3.5 are satisﬁed. C.4.2. So w e obtain t w o opp osite semialgebras S l = S l ( g , H ) and S r = S r ( g , H ) suc h that the categories of left S l -semimo dules and righ t S r -semimo dules are isomorphic to the category of Harish-Chandra mo dules ov er ( g , H ). Recall that a Harish-Chandra mo dule N ov er ( g , H ) is a k -v ector space endo wed with g -mo dule and H -module structures suc h that the t w o induced h -mo dule structures coincide a nd the action map g ⊗ k N − → N is a morphism of H -mo dules. The a ssertion f o llo ws from C.1.4; indeed, it suﬃces to notice that the equations of C.1.4 hold whenev er t hey hold for x and r b elonging to some sets of generators of the alg ebras K and R . Analogously , the category of left S r ( g , H )-semicontramo dules is isomorphic to the category o f k -v ector spaces P endo w ed with g -mo dule and C ( H )- contramo dule struc- tures suc h that the tw o induced h -mo dule structures coincide and the action map P − → Hom k ( g , P ) is a morphism of C ( H )-contr a mo dules. Here a left C -contramo d- ule structure induces an h -mo dule structure by the fo rm ula xp = − π P ( c 7→ h x, c i p ) for p ∈ P , x ∈ h , c ∈ C ; for a left C - como dule M and a left C -contramo d- ule P , the left C - contramo dule structure on Hom k ( M , P ) is deﬁned b y the fo rm ula π ( g )( m ) = π P ( c 7→ g ( s − 1 ( m ( − 1) ) c )( m (0) )) for m ∈ M , g ∈ Hom k ( C , Hom k ( M , P )). C.4.3. No w assume that the algebraic group H is smo oth (i. e., reduced). Let E b e the C - C -bicomo dule and right C - mo dule of diﬀerential top forms on H , with the bicomo dule structure coming from the action of H on itself by left and righ t shifts and t he mo dule structure giv en b y the multiplication o f top forms with functions. Let E ∨ b e the C - C -bicomo dule of to p p olyv ector ﬁelds on H . Then the equation of C.2.1 is clearly satisﬁed, and one can see that ( E , E ∨ ) is a Morita a uto equiv alence of C . The left C -como dules E and E ∨ can b e iden tiﬁed with the top exterior p ow ers of the v ector spaces Hom k ( h , k ) and h , resp ective ly; c E is the mo dular character of H . The 247 in v erse elemen t anti-automorphism of H induces a map s : E − → E satisfying the equations of C.2.7. Let us show that the equation of C.2.2 holds for E . First let us c hec k that it suﬃces to pro v e the desired equation for all x b elonging to a set of g enerato r s of the algebra K . Indeed, one ha s ( xy ) [0] → ( j ( xy ) [1] ) = x [0] y [0] → ( j x [1] y [1] ) = x [0](1) y [0] s ( x [0](2) ) x [0](3) → ( j x [1] y [1] ) = h x [0](1) , y [1] i y [0] x [0](2) → ( j x [1] y [2] ) = y [0] → ( x [0](2) → ( h x [0](1) , y [1] i j x [1] y [2] )) = y [0] → (( x [0] → ( j x [1] )) y [1] ) f o r j ∈ E , x , y ∈ K , since x (2) → ( h x (1) , c (1) i j c (2) ) = x (2) → ( j ( s − 1 ( x (1) ) → c )) = ( x (3) → j )( x (2) → ( s − 1 ( x (1) ) → c )) = ( x → j ) c for j ∈ E , x ∈ K , c ∈ C . So it remains to ch ec k that the equation ho lds f or x ∈ h ⊂ K . F or h ∈ h , let r h and l h denote the left- and r ig h t-in v aria nt v ector ﬁelds on H cor- resp onding to h . Then one has r h = h [1] l h [0] , hence ω ← h = Lie r h ω = Lie l h [0] ( ω h [1] ) = h [0] → ( ω h [1] ) for ω ∈ E , where Lie v ω denotes the Lie deriv ativ e of a to p form ω along a v ector ﬁeld v . C.4.4. Th us there is a left and right coﬂat Morita equiv alence ( E ( g , H ) , E ∨ ( g , H )) b et w een the semialgebras S r ( g , H ) and S l ( g , H ). The f unctors o f semitensor pro duct with E ( g , H ) and E ∨ ( g , H ) pro vide m utually in v erse equiv alences b et w een the cate- gory of left S r -semimo dules and the category of left S l -semimo dules. The S l - S r -bi- semimo dule E ( g , H ) is endo wed with an in v o lutiv e automorphism transforming its t w o semimodule structures in to eac h other in a w ay compatible with the an tip o de isomorphisms C op ≃ C and S l op ≃ S r . When the algebraic g roup H is unimo dular, the semialgebras S l ( g , H ) and S r ( g , H ) are naturally isomorphic and endo w ed with an in v olutiv e an ti-automorphism. Remark. Let us assume for simplicity that the ﬁeld k has c haracteristic 0 a nd the Harish-Chandra pair ( g , H ) originates fro m an em b edding of aﬃne algebraic groups H ⊂ G . Then the bisemimodule E ( g , H ) can b e inte rpreted geometrically as the Harish-Chandra bimo dule of distributions on G , supp orted on H and regular along H . T ec hnically , the desired v ector space of distributions can b e deﬁned a s the direct image of the righ t Diﬀ H -mo dule of top forms on H under the closed em b edding H − → G , where Diﬀ denotes the rings of diﬀeren t ia l op erato rs [11]. This v ector space has tw o comm ut ing structures of a Harish-Chandra mo dule o v er ( g , H ), one giv en by the action of H b y left shifts and the action of g b y r ig h t inv ar ia n t v ector ﬁelds, t he other in the opp o site wa y; so it can b e considered as an S l - S r -bisemimo dule. The desired map from the ve ctor space E ≃ E ⊗ K R to the space of distributions can b e deﬁned as the unique map f orming a comm utativ e dia gram with the embeddings o f the space o f top fo rms E into b oth v ector spaces and preserving the righ t R - mo dule structures. T o pro v e that this map is an isomorphism, it suﬃces t o consider the ﬁltration of E induced b y the natural ﬁltr ation of the unive rsal env eloping algebra R and the ﬁltratio n of the space of distributions induced by the ﬁltra t ion o f Diﬀ G b y the order of diﬀeren tia l op erator s. When the a lg ebraic group H is unimo dular, 248 one can iden tify the semialgebra S l ≃ S = S r itself with the ab ov e v ector space of distributions b y c ho osing a nonzero biin v ariant top form ω o n H . The semiunit and semim ultiplication in S are then described as follo ws. Giv en a function on C , one has to m ultiply it with ω and take the push-forw ard with resp ect to the closed em b edding H − → G to obtain the corresp onding distribution under the semiunit map. T o describ e the semim ultiplication, denote b y G × H G the quotien t v ariety of the Carthesian pro duct G × G b y the equiv alence relatio n ( g ′ h, g ′′ ) ∼ ( g ′ , hg ′′ ). Then the pull-ba ck of distributions with resp ect to the smo oth map G × G − → G × H G using the relativ e top form ω iden tiﬁes S  C S with the spac e of distributions on G × H G supp o rted in H ⊂ G × H G a nd regular along H . The push-forw ard of distributions with resp ect to the m ultiplication map G × H G − → G pro vides the semim ultiplication in S . (Cf. App endix F.) C.5. Semiin v arian ts and semicontrain v arian ts. C.5.1. Let h ⊂ g b e a Lie algebra with a ﬁnite-dimensional subalgebra; let N b e a g -mo dule. Then there is a natural map ( det( h ) ⊗ k g / h ⊗ k N ) h − → (det( h ) ⊗ k N ) h , where det( V ) is the top exterior p o w er of a ﬁnite-dimensional v ector space V and the sup erindex h denotes t he h -inv ariants. This natura l map is constructed a s follows. T ensoring the action map g ⊗ k N − → N with det( h ) and passing to the h -inv ari- an ts, w e obtain a map (det ( h ) ⊗ k g ⊗ k N ) h − → (det( h ) ⊗ k N ) h . Let us c hec k that the comp osition (det( h ) ⊗ k h ⊗ k N ) h − → (det( h ) ⊗ k g ⊗ k N ) h − → (det( h ) ⊗ k N ) h v anishes. Notice t hat this comp ostion only dep ends on t he h - mo dule structure on N . Let n b e an h -inv ariant elemen t of det( h ) ⊗ k h ⊗ k N ; it can b e also considered as an h -mo dule map ˇ n : (det ( h ) ⊗ k h ) ∨ − → N , where V ∨ denotes the dual v ector space Hom k ( V , k ). Let t denote the trace elemen t of the tensor pro duct (det( h ) ⊗ k h ) ⊗ k (det( h ) ⊗ k h ) ∨ ; then t is an h -inv ar ian t elemen t and one has (id ⊗ ˇ n )( t ) = n . So it remains to c hec k that the image of t under the action map (det( h ) ⊗ k h ) ⊗ k (det( h ) ⊗ k h ) ∨ − → det( h ) ⊗ k (det( h ) ⊗ k h ) ∨ ≃ h ∨ v anishes; this is straightforw a r d. W e hav e constructed a ma p (det ( h ) ⊗ k g ⊗ k N ) h / (det( h ) ⊗ k h ⊗ k N ) h − → (det( h ) ⊗ k N ) h . When N is an injectiv e U ( h )-mo dule, this pro vides the desired map ( det( h ) ⊗ k g / h ⊗ k N ) h − → (det( h ) ⊗ k N ) h . T o construct the latter map in the general case, it suﬃces t o represen t N a s the k ernel of a morphism of U ( h )-injectiv e U ( g )-mo dules (notice that an y injectiv e U ( g )-mo dule is an injectiv e U ( h )-mo dule). Indeed, b o t h the left and the right hand sides of t he desired map preserv e kerne ls. The v ector space of ( g , h ) -se miinvariants N g , h of a g - mo dule N is deﬁned as the cok ernel of the map (det( h ) ⊗ k g / h ⊗ k N ) h − → (det( h ) ⊗ k N ) h that w e hav e obtained. The ( g , h )-semiinv ariants a re a mixture of in v aria n ts along h and coinv aria n ts in t he direction of g relativ e to h . 249 C.5.2. Let P b e another g -mo dule. Then there is a na t ur a l map Hom k (det( h ) , P ) h − → Hom k (det( h ) ⊗ k g / h , P ) h , where the subindex h denotes the h -coinv ariants. This map is constructed as fo llo ws. T ensoring the action map P − → Hom k ( g , P ) with det( h ) ∨ and passing to the h -coinv ariants, we obta in a map Hom k (det( h ) , P ) h − → Hom k (det( h ) ⊗ k g , P ) h . Let us c hec k that the compo sition Hom k (det( h ) , P ) h − → Hom k (det( h ) ⊗ k g , P ) h − → Hom k (det( h ) ⊗ k h , P ) h v anishes. Notice that this comp osition only dep ends on the h -mo dule structure on P . Let p b e an h -inv aria n t map Hom k (det( h ) ⊗ k h , P ) − → k ; it can b e also considered as an h - mo dule map ˇ p : P − → det( h ) ⊗ h . Then the map p factorizes t hr o ugh the map Hom k (det( h ) ⊗ k h , P ) − → Hom k (det( h ) ⊗ k h , det( h ) ⊗ h ) induced b y ˇ p . So it suﬃces to consider the case of a ﬁnite-dimensional h -mo dule P = det( h ) ⊗ h , when the assertion follo ws by duality f rom the result of C.5.1. W e hav e constructed a map from Hom k (det( h ) , P ) h to the ke rnel of the map Hom k (det( h ) ⊗ k g , P ) h − → Hom k (det( h ) ⊗ k h , P ) h . When P is a pro jectiv e U ( h )-mo dule, t his provide s the desired map Hom k (det( h ) , P ) h − → Hom k (det( h ) ⊗ k g / h , P ) h . In the general case, represen t P as the cok ernel of a morphism of U ( h )-pro- jectiv e U ( g )-mo dules and notice tha t b oth the left and the rig h t hand side of the desired map preserv e cok ernels. The vector space of ( g , h ) -semi c ontr ainva ria nts P g , h of a g -mo dule P is deﬁned as the k ernel of the map ( P ⊗ k det( h ) ∨ ) h − → Hom k (det( h ) ⊗ k g / h , P ) h that w e hav e obtained. The ( g , h )-semicontrainv ariants are a mixture of coin v ariants along h and in v ariants in the direction of g relativ e to h . C.5.3. No w let ( g , H ) b e an alg ebraic Harish-Chandra pair. Let N b e a righ t S r ( g , H )-semimo dule, that is a Harish-Chandra mo dule ov er ( g , H ). Then the action map g ⊗ k N − → N is a morphism of Harish-Chandra mo dules. T ensoring it with det( h ) a nd passing to the H -inv arian ts, w e obtain a map (det( h ) ⊗ k g ⊗ k N ) H − → (det( h ) ⊗ k N ) H . By the result of C.5.1 , the comp o sition (det( h ) ⊗ k h ⊗ k N ) H − → (det( h ) ⊗ k g ⊗ k N ) H − → (det( h ) ⊗ k N ) H v anishes. When N is a coﬂat C ( H )-como dule, this provide s a natural map (det( h ) ⊗ k g / h ⊗ k N ) H − → (det( h ) ⊗ k N ) H ; to deﬁne this map in the general case, it suﬃces to represen t N as the k ernel of a morphism of C -coﬂat S r -semimo dules (see Lemma 1 .3.3). The vec tor space of ( g , H ) -semiinvaria n ts N g ,H is deﬁned as the coke rnel of the map (det( h ) ⊗ k g / h ⊗ k N ) H − → (det( h ) ⊗ k N ) H that w e hav e constructed. C.5.4. Let P b e a left S r ( g , H )-semicontramo dule (see C.4.2) . Then the action map P − → Hom k ( g , P ) is a morphism of S r -semicontramo dules. Applying to it the functor Hom k (det( h ) , − ), we get a mo r phism of C ( H )- contramo dules. P assing to the H -coinv ariants, i. e., the maximal quotien t C - contramo dules with the trivial con- traaction, w e obta in a map Ho m k (det( h ) , P ) H − → Hom k (det( h ) ⊗ k g , P ) H . By the result of C.5.2, the comp osition Hom k (det( h ) , P ) H − → Hom k (det( h ) ⊗ k g , P ) H − → 250 Hom k (det( h ) ⊗ k h , P ) H v anishes. Whe n P is a coinjectiv e C -contramo dule, this pro vides a na tural map Ho m k (det( h ) , P ) H − → Hom k (det( h ) ⊗ k g / h , P ) H ; to deﬁne this ma p in the g eneral case, it suﬃces to represen t P as the cok ernel of a morphism of C -coinjectiv e S r -semicontramo dules (see L emma 3.3.3). The vec tor space of ( g , H ) -semic ontr ainvariants P g ,H is deﬁned as the ke rnel of the map Hom k (det( h ) , P ) H − → Hom k (det( h ) ⊗ k g / h , P ) H that w e hav e constructed. C.5.5. Let N b e a righ t S r ( g , H )-semimo dule and M b e a left S l ( g , H )-semimo d- ule; assume tha t either N or M is a coﬂat C ( H )-como dule. Then there is a natura l isomorphism N ♦ S r E ∨ ♦ S r M ≃ ( N ⊗ k M ) g ,H , where N ⊗ k M is considered a s the tensor pro duct of Harish-Chandra mo dules N and M . Indeed, in tro duce an increasing ﬁltration F of the k -algebra R = U ( g ) whose comp onen t F t R , t = 0, 1 , . . . is the linear span of all pro ducts of elemen t s of g where at most t factors do not b elong to h . In particular, w e ha v e F 0 R ≃ K = U ( h ). Set F t S r = C ⊗ K F t R ; then w e ha v e F 0 S r ≃ C , the natural maps F t − 1 S r − → F t S r are injectiv e, their cokerne ls a re coﬂat left a nd righ t C -como dules, S r ≃ lim − → F t S r , and the semim ultiplication map F p S r  C F q S r − → S  C S − → S facto r izes throug h F p + q S r . Moreo v er, the maps F p S r  C F q S r − → F p + q S r are surjectiv e and their k ernels a re coﬂat left and right C -como dules. (Cf. 11.5.) Let N b e a right S r -semimo dule and L b e a left S r -semimo dule suc h that either N or L is a coﬂat C -como dule. Denote by η t : N  C F t S r  C L − → N  C L the map equal to the diﬀerence of the map induced by the semiaction map F t S r  C L − → L and the map induced by the semiaction map N  C F t S r − → N . Let us sho w that the images of η t coincide for t > 1. Let p , q > 1; t hen the map N  C F p S r  C F q S r  C L − → N  C F p + q S r  C L is surjectiv e in view of our assumption on N and L . The comp osition of the ma p N  C F p S r  C F q S r  C L − → N  C F p + q S r  C L with t he map η p + q is equal to the sum of the comp osition of the map N  C F p S r  C F q S r  C L − → N  C F p S ∼  C L induced b y the semiaction map F q S r  C L − → L and the map η p , a nd the comp osition of the map N  C F p S r  C F q S r  C L − → N  C F q S r  C L induced b y the semiaction map N  C F p S r − → N a nd the map η q . So the assertion follows b y induction. Therefore, the semitensor pro duct N ♦ S L is isomorphic to the cokerne l of the map η 1 . On the o ther hand, the map η 0 v anishes. In view of our assumption on N and L , the quotien t space ( N  C F 1 S r  C L ) / ( N  C F 0 S r  C L ) is isomorphic to N  C F 1 S r /F 0 S r  C L . Hence the semitensor pro duct N ♦ S L is isomorphic to the cok ernel of the induced map ¯ η 1 : N  C ( F 1 S r /F 0 S r )  C L − → N  C L . No w when L = E ∨ ♦ S M for a left S l -semimo dule M , the na tural isomorphisms N  C E ∨  C M ≃ ( N ⊗ k E ∨ ⊗ k M ) H ≃ ( E ∨ ⊗ k N ⊗ k M ) H and N  C ( F 1 S r /F 0 S r )  C E ∨  C M ≃ N  C ( C ⊗ k g / h )  C E ∨  C M ≃ ( N ⊗ k g / h ⊗ k E ∨ ⊗ k M ) H ≃ ( E ∨ ⊗ k g / h ⊗ k N ⊗ k M ) H giv en b y the form ulas ˇ ı ⊗ n ⊗ m 7− → n ⊗ ˇ ı ⊗ m and n ⊗ ¯ z ⊗ ˇ ı ⊗ m 7− → ˇ ı ⊗ ¯ z ⊗ n ⊗ m 251 iden tify the map ¯ η 1 with the map whose cok ernel is, b y the deﬁnition, the space o f semiin v arian ts ( N ⊗ k M ) g ,H . C.5.6. Let M b e a left S l ( g , H )-semimo dule and P b e a left S r ( g , H )-semicontra- mo dule; a ssume that either M is an copro jectiv e C ( H )-como dule, or N is a coinjec- tiv e C ( H )-contra mo dule. Then there is a na tural isomorphism SemiHom S r ( E ∨ ♦ S l M , P ) ≃ Hom k ( M , P ) g ,H , where the structure of left S r -semicontramo dule on Hom k ( M , P ) w as in tro duced in C.3.6. The pro of is analogo us to that of C.5.5. 252 Appendix D. T a te Harish-Chandra P airs and T a te Lie Alge bras b y Sergey Arkhip ov and Leonid P ositselsk i In or der to form ulate the comparison theorem relating the functors SemiT or and SemiExt to the semi-inﬁnite ( co) ho mology o f T ate Lie algebras, one has to consider Harish-Chandra pairs ( g , H ) with a T ate Lie algebra g a nd a proalgebraic group H corresp onding to a compact op en subalgebra h ⊂ g . In suc h a situation, the con- struction of a Morita equiv alence from App endix C no lo ng er w orks; instead, there is an isomorphism of “left” and “right” semialgebras corresp o ding to diﬀeren t cen tral c harges. The pro of of t his isomorphism is based on the nonhomogeneous quadratic dualit y theory dev elop ed in Section 11 (see a lso 0.4). Once the isomorphism o f semi- algebras is constructed and the standard semi-inﬁnite (co)homological complexes are in tro duced, the pro of o f the comparison theorem b ecomes prett y straigh tfor- w ard. The equiv alence b etw een the semideriv ed categories o f Har ish-Chandra mo d- ules and Harish-Chandra contramodules with complemen tary (o r rather, shifted) cen tral c harges f o llo ws immediately fr o m the isomorphism of semialgebras. D.1. C ontin uous coact ions. D.1.1. Let k b e a ﬁxed ground ﬁeld. A line ar top olo gy on a v ector space o v er k is a top ology compatible with the v ector space structure f or whic h op en vector subspaces form a base of neigh b orho o ds of zero. In the sequel, by a to p ological v ector space w e will mean a k - v ector space endo w ed with a complete and separated linear top ology . Equiv alen tly , a t o p ological v ector space is a ﬁltered pro jectiv e limit of discrete v ector spaces with its pro jectiv e limit top olog y . Accordingly , the (separated) completion of a vec tor space endo wed with a linear top ology is j ust the pro jectiv e limit of its quotien t spaces by op en v ector subspaces. The category of top ological v ector spaces and contin uous linear maps b etw een them has an exact category structure in whic h a triple of top ological vec tor spaces V ′ − → V − → V ′′ is exact if it is an exact triple of vec tor spaces strongly compatible with the top ologies, i. e., the map V ′ − → V is closed a nd the map V − → V ′′ is op en. An y op en surjectiv e map of top ological v ector spaces is an admissible epimorphism. An y closed injectiv e map from a top o logical vector space a dmitting a countable ba se of neigh b orho o ds of zero is a split admissible monomorphism. A top olo g ical v ector space is called ( lin e arly ) c om p act if it has a base of neigh- b orho o ds of zero consisting of v ector subspaces of ﬁnite co dimension. Equiv alen tly , a top ological vec tor space is compact if it is a pro jectiv e limit of ﬁnite- dimensional discrete v ector spaces. A T ate ve ctor sp ac e is a top ological v ector space admitting a compact op en subspace. Equiv alently , a to p ological v ector space is a T a te vec tor space if it is top olog ically isomorphic to the direct sum of a compact v ector space 253 and a discrete v ector space. The dual T ate v e ctor s p ac e V ∨ to a T ate v ector space V is deﬁned as the space of contin uous linear functions V − → k endow ed with the top ology where annihilators of compact op en subspaces of V form a base o f neigh b or- ho o ds of zero. In particular, the dual T ate v ector spaces to compact vec tor spaces are discrete and vice v ersa; for any T at e ve ctor space V , the natural map V − → ( V ∨ ) ∨ is a top ological isomorphism. D.1.2. The pro jectiv e limit o f a pro jectiv e system of top ological v ector spaces en- do w ed with the top ology o f pro jectiv e limit is a top ological vec tor space. This is called the top olo gic al pr oje ctive l i mit . The inductiv e limit o f a n inductiv e system of top o lo gical v ector spaces can b e en- do w ed with the top o lo gy of inductiv e limit of vec tor spaces with linear top ologies; w e will call the inductiv e limit endo w ed with this top o logy the unc omplete d inductive limit . The c omplete d inductive limit is the (separated) completion of t he uncom- pleted inductiv e limit. F or an y coun table ﬁltered inductiv e system formed b y closed em b eddings of top ological v ector spaces the uncompleted and completed inductiv e limits coincide. Moro ve r, let V α b e a ﬁltered inductiv e system of top olo g ical ve c- tor spaces satisfying the follo wing condition. F or any increasing sequence of indices α 1 6 α 2 6 · · · the uncompleted inductiv e limit of V α i is a direct summand of the uncompleted inductiv e limit o f V α considered as a n ob ject of the category of v ec- tor spaces endow ed with noncomplete linear top ologies. Then the uncompleted and completed inductiv e limits o f V α coincide. D.1.3. W e will consider three op erat ions of tensor pro duct of top o logical v ector spaces [6]. F or an y t w o top ological v ector spaces V and W , denote b y V ⊗ ! W the completion of the t ensor pro duct V ⊗ k W with resp ect to the top olog y with a base of neigh b orho o ds of zero consisting of the ve ctor subspaces V ′ ⊗ W + V ⊗ W ′ , where V ′ ⊂ V and W ′ ⊂ W are op en vec tor subspaces in V and W . F urthermore, denote by V ⊗ ∗ W the completion of V ⊗ k W with resp ect to the to p ology f ormed b y the subspaces of V ⊗ W satisfyin g the following conditions: a v ector subspace T ⊂ V ⊗ W is op en if (i) there exist op en subspaces V ′ ⊂ V , W ′ ⊂ W suc h that V ′ ⊗ W ′ ⊂ T , (ii) for a n y v ector v ∈ V there exists a subspace W ′′ ⊂ W suc h that v ⊗ W ′′ ⊂ T , and (iii) for an y v ector w ∈ W there exists a subspace V ′′ ⊂ V suc h that V ′′ ⊗ w ⊂ T . Finally , denote by V → ⊗ W the completion of V ⊗ k W with resp ect to the top olo g y formed b y the subspaces satisfying the follo wing conditio ns: a v ector subspace T ⊂ V ⊗ k W is op en if (i) there exists an op en subspace W ′ ⊂ W suc h that V ⊗ k W ′ ⊂ T , a nd (ii) for any v ector w ∈ W there exists an op en subspace V ′′ ⊂ V suc h that V ′′ ⊗ w ⊂ T . Set V ← ⊗ W = W → ⊗ V . The to p ological tensor pro ducts ⊗ ! and ⊗ ∗ deﬁne tw o structures of asso ciativ e and commutativ e tensor category on the category of to p ological v ector spaces. The top ological tensor pro duct → ⊗ deﬁnes a structure of a sso ciativ e, but not comm utativ e 254 tensor category on the catego ry of top o logical v ector spaces. F o r any top ological v ector spaces V 1 , . . . , V n and W the vec tor space of con tin uous p o lylinear maps V 1 × · · · × V n − → W is naturally isomorphic to the v ector space of contin uous linear maps V 1 ⊗ ∗ · · · ⊗ ∗ V n − → W . When b o th top ological v ector spaces V and W are compact (discrete), the top olo gical tensor pro duct V ⊗ ∗ W ≃ V → ⊗ W ≃ V ← ⊗ W ≃ V ⊗ ! W is also compact (discrete). The functor ⊗ ! preserv es top olog ical pro jective limits. The functor ⊗ ∗ preserv es (uncompleted o r completed) inductiv e limits of ﬁltered inductiv e systems of op en injections. The top ological t ensor pro duct V → ⊗ W is the top olo g ical pro jectiv e limit of → ⊗ -pro ducts of V w ith discrete quotient spaces of W . The functor ( V , W ) 7− → V → ⊗ W preserv es completed inductiv e limits in its second arg umen t W . The underlying v ector space of the to p ological tensor pro duct V → ⊗ W is determined b y ( t he top olog ical v ector space W and) t he underlying vec tor space o f the top ological vec tor space V . F or T ate v ector spaces V 1 , . . . , V n and a top olog ical vec tor space U , consider the v ector space of con tin uous p olylinear maps Q i V i − → U endo w ed with the top o lo gy with a base of neigh b orho o ds of zero for med by t he subspaces of all p olylinear maps mapping the Carthesian pro duct o f a collection of compact subspaces V ′ i ⊂ V i in to an op en subspace U ′ ⊂ U (the “compact-o p en” top ology). This v ector space is naturally top ologically isomorphic to the top ological tensor pro duct V ∨ 1 ⊗ ! · · · ⊗ ! V ∨ n ⊗ ! W [7]. F or an y top ological v ector spaces U , W a nd T ate v ector space V , the v ector space of con tin uous linear maps V ⊗ ∗ W − → U is naturally isomorphic to the v ector space of con tin uous linear maps W − → V ∨ ⊗ ! U . D.1.4. Let C b e a coalg ebra o v er the ﬁeld k and V be a top olog ical v ector space. A c ontinuous right c o action of C in V is a contin uous linear map V − → V ⊗ ! C , where C is considered as a discrete v ector space, satisfying t he coasso ciativity and counit y equations. Namely , the map V − → V ⊗ ! C should ha v e equal comp ositions with the t w o maps V ⊗ ! C ⇒ V ⊗ ! C ⊗ ! C induced by the map V − → V ⊗ ! C and the comultiplic ation in C , and the comp osition of the map V − → V ⊗ ! C with the map V ⊗ ! C − → V induce d by the counit o f C should b e equal to t he identit y map. Equiv alen tly , a con tin uous right coaction of C in V can b e deﬁned a s a con tin uous linear map V ⊗ ∗ C ∨ − → V , where C ∨ is considered as a compact vec tor space, satisfying the asso ciativity and unit y equations. Continuous left c o actions are deﬁned in the analogo us w ay . A closed subspace W ⊂ V of a top ological v ector space V endo we d with a con- tin uous right coaction of a coalgebra C is said to b e inv arian t with resp ect to the con tin uous coaction ( o r C -inv ariant) if the image of W under the con tin uous coa ctio n map V − → V ⊗ ! C is con tained in the closed subspace W ⊗ ! C ⊂ V ⊗ ! C . It follo ws from the next Lemma that any top ological v ector space with a con tin uous coa ction 255 of a coalgebra C is a ﬁlt ered pro jec tiv e limit of discrete vec tor spaces endo w ed with C -como dule structures. Lemma. F or any top olo gic al ve ctor sp ac e V endo w e d with a c on tinuous c o action V − → V ⊗ ! C of a c o algeb r a C , op e n subsp a c es of V inva riant under the c on tinuous c o action form a b ase of ne ighb orho o ds of zer o in V . Pr o of . Let U ⊂ V b e an op en subspace; then the full preimage U ′ of the op en subspace U ⊗ ! C ⊂ V ⊗ ! C under the con tin uous coa ctio n map V − → V ⊗ ! C is an in v ariant op en subspace in V con tained in U . T o c hec k that U ′ is C -inv arian t, use the fact the functor of ⊗ ! -pro duct preserv es k ernels in the category of top olog ical v ector spaces , and in particular, the ⊗ ! -pro duct with C preserv es the k ernel o f the comp osition V − → V ⊗ ! C − → V / U ⊗ ! C . T o c hec k that U ′ is contained in U , use the counit y equation f or the con tin uous coa ctio n.  The category of t o p ological vec tor spaces endo w ed with a con tin uous coaction of a coalgebra C has an exact category structure suc h tha t a triple of top o logical v ector spaces with con tin uous coactions o f C is exact if and only if it is exact as a triple of top ological ve ctor spaces. If V is a T ate vec tor space with a con tin uous righ t coaction of C , then the dual T ate v ector space V ∨ is endo w ed with a contin uous left coaction of C . Let V b e a to p ological v ector space with a con tin uous righ t coaction of a coalgebra C a nd W b e a t o p ological v ector space with a con tin uous coaction of a coalgebra D . Then all the three t o p ological tensor pro ducts V ⊗ ! W , V ⊗ ∗ W , and V → ⊗ W a re endo w ed with con tin uous right coa ctions of the coalgebra C ⊗ k D . T o construct the con tin uous coactio n on V ⊗ ! W , one uses the natural isomorphism ( V ⊗ ! C ) ⊗ ! ( W ⊗ ! D ) ≃ ( V ⊗ ! W ) ⊗ ! ( C ⊗ k D ). The con tin uous coaction o n V ⊗ ∗ W is deﬁned in terms of the natural contin uous map ( V ⊗ ! C ) ⊗ ∗ ( W ⊗ ! D ) − → ( V ⊗ ∗ W ) ⊗ ! ( C ⊗ k D ), which exists for any top ological v ector spaces V , W and any discrete v ector spaces C , D . The con tin uous coaction on V → ⊗ W is deﬁned in terms of the na tural con tinuous map ( V ⊗ ! C ) → ⊗ ( W ⊗ ! D ) − → ( V → ⊗ W ) ⊗ ! ( C ⊗ k D ). It follows that for a comm utative Hopf algebra C the top olo gical tensor pro ducts V ⊗ ! W , V ⊗ ∗ W , a nd V → ⊗ W of top ological v ector spaces with con tin uous righ t coactions of C are also endo w ed with contin uous righ t coactions of C . Besides, one can transform a con tinuous left coaction of C in V in to a contin uous rig h t coaction using the an tip o de. No w let W , U b e to p ological ve ctor spaces and V b e a T ate v ector space; supp ose that W , U , and V are endo w ed with con tin uous coactions of a comm utative Hopf algebra C . L et f : V ⊗ ∗ W − → U a nd g : W − → V ∨ ⊗ ! U b e contin uous linear maps corresp onding to eac h other under the isomorphism fro m D.1.3; then f preserv es the con tin uous coactions o f C if and only if g do es. 256 D.1.5. A top olog ical Lie algebra g is top o logical v ector space endo w ed with a Lie algebra structure such that the brac k et is a contin uous bilinear map g × g − → g . T op olog ical asso ciative algebras ar e deﬁned in the analogo us w a y . F or example, let V b e a T ate v ector space. Denote b y End ( V ) the asso ciative algebra of contin uous endomorphisms of V endow ed with the compact-op en top ology and b y gl ( V ) the Lie algebra corresp onding to End( V ). Then End( V ) is a t o p ological asso ciat ive algebra and gl ( V ) is a top ological Lie algebra. Let U , V , W b e top ological v ector spaces endow ed with contin uous coactions of a commutativ e Hopf algebra C . Then a contin uous bilinear map V × W − → U is called compatible with the con tin uous coactions o f C if the corresp onding linear map V ⊗ ∗ W − → U preserv es the con tin uous coactions of C . So one can sp eak ab out compatibilit y of contin uous pairings, Lie or asso ciativ e algebra structures, Lie or asso ciativ e actions, etc., with con tin uous coactions of a comm utativ e Hopf algebra. Explicitly , a bilinear map V × W − → U is con tin uous and compatible with the con tin uous coactions of C if and only if t he follow ing condition holds. F or an y C -in- v ariant o p en subspace U ′ ⊂ U and a ny ﬁnite-dimensional subspaces E ⊂ V , F ⊂ W there should exist in v ariant op en subspaces V ′ ⊂ V ′′ ⊂ V , W ′ ⊂ W ′′ ⊂ W suc h that E ⊂ V ′′ , F ⊂ W ′′ , the map V ′′ ⊗ k W ′′ − → U /U ′ factorizes throug h V ′′ /V ′ ⊗ k W ′′ /W ′ , and t he induced map V ′′ /V ′ ⊗ k W ′′ /W ′ − → U /U ′ is a morphism o f C -como dules. D.1.6. F or an y T ate v ector spaces V and W , there is a split exact triple of top olo gical v ector spaces V ⊗ ∗ W − → V → ⊗ W ⊕ W → ⊗ V − → V ⊗ ! W , where the ﬁrst map is the sum of the natural maps V ⊗ ∗ W − → V → ⊗ W , V ⊗ ∗ W − → W → ⊗ V , while the second map is the diﬀerence of the nat ura l maps V → ⊗ W − → V ⊗ ! W , W → ⊗ V − → V ⊗ ! W . Let us take W = V ∨ . Then V ⊗ ! V ∨ is naturally isomorphic t o gl ( V ); the spaces V → ⊗ V ∨ and V ∨ → ⊗ V can b e iden tiﬁed with the subspaces in gl ( V ) formed by the linear op erator s with op en k ernel and compact closure of image, respectiv ely; and V ⊗ ∗ V ∨ is the in tersection of V → ⊗ V ∨ and V ∨ → ⊗ V in gl ( V ). T aking the push-forw ard of the exact triple V ⊗ ∗ V ∨ − → V → ⊗ V ∨ ⊕ V ∨ → ⊗ V − → V ⊗ ! V ∨ with respect to the nat ur a l tra ce map tr : V ⊗ ∗ V ∨ − → k corresp onding to the pairing V × V ∨ − → k , one obtains a n exact triple o f top o logical v ector spaces k − → gl ( V ) ∼ − → gl ( V ). This is also an exact triple of gl ( V )- mo dules, whic h allows to deﬁne a Lie algebra structure on gl ( V ) ∼ making it a cen tral extension of the Lie algebra gl ( V ). The an ti-comm utativity and the Jacobi iden tit y follow fro m the fact that the comm utator of an o p erator with op en k ernel and an op erator with compact closure of image has zero trace. No w assume that a T ate vector space V is endow ed with a contin uous coaction of a commutativ e Hopf algebra C . Then V ⊗ ∗ V ∨ − → V → ⊗ V ∨ ⊕ V ∨ → ⊗ V − → V ⊗ ! V ∨ is an exact triple of top olo gical v ector spaces endo w ed with contin uous coactions of C ; 257 the tra ce map also preserv es the con tin uous coactions. Th us the top ological v ector space gl ( V ) ∼ acquires a contin uous coaction of C . D.1.7. Here is another construction of the Lie algebra gl ( V ) ∼ (see [7]). Consider the quotien t space of the vec tor space V ⊗ k V ∨ ⊕ V ∨ ⊗ k V ⊕ k b y the relation v ⊗ g + g ⊗ v = h g , v i , where h , i denotes the pairing of V ∨ with V . This v ector space is a Lie subalgebra o f the Cliﬀord algebra Cl ( V ⊕ V ∨ ) o f the v ector space V ⊕ V ∨ with the symmetric bilinear form given by the pairing h , i ; the Lie brac k et on this subalgebra is giv en b y t he formu las [ v 1 ⊗ g 1 , v 2 ⊗ g 2 ] = h g 1 , v 2 i v 1 ⊗ g 2 − h g 2 , v 1 i v 2 ⊗ g 1 , [ v ⊗ g , 1] = 0. This Lie a lgebra acts in the v ector space V b y the form ulas ( v ⊗ g )( v ′ ) = h g , v ′ i v , 1( v ) = 0. There is a separated top ology on this Lie algebra with a base of neigh b orho o ds of zero for med by the Lie subalgebras V ⊗ W ′ + V ′ ⊗ V ∨ , where V ′ ⊂ V and W ′ ⊂ V ∨ are op en subspaces such that h W ′ , V ′ i = 0. The completion of this Lie algebra with resp ect to this top ology can b e easily identiﬁe d with the Lie algebra gl ( V ) ∼ deﬁned ab ov e. Hence the Lie br a c k et on gl ( V ) ∼ is con tin uous. In a dditio n, w e need to c hec k that when V is endo w ed with a con tinuous coaction of a comm utativ e Ho pf algebra C , the Lie brac k et is compatible with the con tin uous coaction of C in gl ( V ) ∼ . The latter follo ws from the existence of a w ell-deﬁned comm utator map Hom( X 4 , X 3 , X 1 ; X/X 1 , X 4 /X 1 , X 2 /X 1 ) ⊗ 2 − → gl ( X ) ∼ / ( X ⊗ X ⊥ 3 + X 2 ⊗ X ∨ ) for an y ﬂag of ﬁnite-dimensional v ector spaces X 1 ⊂ X 2 ⊂ X 3 ⊂ X 4 ⊂ X , where the Hom space in the left hand side consists of all maps X 4 − → X/X 1 sending X 3 to X 4 /X 1 and X 1 to X 2 /X 1 , and Y ⊥ ⊂ X ∨ denotes the or t ho gonal complemen t to a v ector subspace Y ⊂ X . D.1.8. A T ate Lie a l g ebr a is a T ate ve ctor space endo w ed with a t o p ological Lie algebra structure. Let g b e a T ate Lie algebra endo w ed with a contin uous coaction of a comm utativ e Hopf algebra C suc h that the Lie algebra structure is compatible with the con tin uous coa ctio n. Then C -inv aria n t compact op en subalgebras form a base of neigh b orho o ds of zero in g . Indeed, c ho ose a C -inv arian t compact op en subspace U ⊂ g ; let h b e the normalizer of U in g , i. e., the subspace of all x ∈ g suc h that [ x, U ] ⊂ U . The n h is a C -inv ariant op en subalgebra in g , since it is the k ernel of the adjoin t action map g − → Hom k ( U, g /U ). Therefore, the in tersection h ∩ U is an C -inv ariant compact op en subalgebra in g contained in U . The canonical cen tral extension g ∼ of a T ate Lie a lg ebra g is deﬁned as the ﬁb ered pro duct of g and gl ( g ) ∼ o v er gl ( g ), where g maps to gl ( g ) b y t he adjoint represen- tation. The ve ctor space g ∼ is endow ed with the top ology of ﬁb ered pro duct; this mak es g ∼ a T ate Lie algebra. The cen tral extension g ∼ − → g splits canonically a nd con tin uously ov er an y compact op en Lie subalgebra h ⊂ g . Indeed, the image of h 258 in gl ( g ) is con tained in the o p en Lie subalgebra gl ( h , g ) ⊂ gl ( g ) of endomorphisms preserving h , and the map from the op en Lie subalgebra of gl ( g ) ∼ constructed as the completion of g ⊗ k h ⊥ + g ∨ ⊗ k h o n to gl ( h , g ) is a top ological isomorphism. The natural con tin uous coaction o f C in g ∼ is constructed as the ﬁb ered pro duct of the coactions in g and gl ( g ) ∼ ; it is clear that the Lie alg ebra structure on g ∼ is compatible with the con tinu ous coaction. If h ⊂ g is a C -inv ariant compact op en subalgebra, then the cano nical splitting h − → g ∼ preserv es the con tin uous coactions. When a T a te ve ctor space V is decomposed into a direct sum V ≃ E ⊕ F of a compact ve ctor space E a nd a discrete vec tor space F , there is a natural section gl ( V ) − → gl ( V ) ∼ of the central extension gl ( V ) ∼ − → gl ( V ); t he image of this section is the completion of V ⊗ F ∨ + F ⊗ V ∨ + V ∨ ⊗ E + E ∨ ⊗ V . Consequen tly , when a T ate Lie algebra g is decomp osed into a direct sum g ≃ h ⊕ b of a compact op en Lie subalgebra h and a discrete ve ctor subspace b , there is a natural section g − → g ∼ of the central extension g ∼ − → g ; this section agrees with the natural splitting h − → g ∼ . D.2. C onst r uction of semialgeb ra. D.2.1. W e will sometimes use Sw eedler’s notation [45] c 7− → c (1) ⊗ c (2) for the com ultiplication map in a coasso ciativ e coalgebra C . The a nalogous notation for coactions of C in a right C -como dule N and a left C -como dule M is n 7− → n (0) ⊗ n (1) and m 7− → m ( − 1) ⊗ m (0) , where n , n (0) ∈ N , m , m (0) ∈ M , and n (1) , m ( − 1) ∈ C . A Lie c o algebr a L is a k -v ector space endo wed with a k -linear map L − → V 2 k L from L to the second exterior p ow er of L denoted b y l 7− → l { 1 } ∧ l { 2 } , whic h should satisfy the dual ve rsion of Jacobi iden tity l { 1 }{ 1 } ∧ l { 1 }{ 2 } ∧ l { 2 } = l { 1 } ∧ l { 2 }{ 1 } ∧ l { 2 }{ 2 } , where l ′ ∧ l ′′ ∧ l ′′′ denotes an elemen t of V 3 k L . A c omo dule M ov er a Lie coalgebra L is a k -ve ctor space endow ed with a k -linear map M − → L ⊗ M denoted by m 7− → m {− 1 } ⊗ m { 0 } satisfying the equation m {− 1 } ∧ m { 0 }{− 1 } ⊗ m { 0 }{ 0 } = m {− 1 }{ 1 } ∧ m {− 1 }{ 2 } ⊗ m { 0 } , where l ′ ∧ l ′′ ⊗ m denotes an elemen t of V 2 k L ⊗ k M . A T ate Harish-Chand r a p air ( g , C ) is a set of data consisting of a T ate Lie algebra g , a comm utativ e Hopf algebra C , a con tin uous coa ction of C in g suc h that the Lie algebra structure on g is compatible with the con tin uous coa ction, a C -inv a rian t compact op en subalgebra h ⊂ g , and a con tin uous pairing ψ : C × h − → k , where C is considered with the discrete top ology . This data should satisfy the follow ing conditions (cf. [9]): (i) The pairing ψ is compatible with the m ultiplication and com ultiplication in C , i. e., the map ˇ ψ : C − → h ∨ corresp onding to ψ is a morphism of Lie coalgebras suc h that ˇ ψ ( c ′ c ′′ ) = ε ( c ′ ) ˇ ψ ( c ′′ ) + ε ( c ′′ ) ˇ ψ ( c ′ ) for c ′ , c ′′ ∈ C . Here the Lie coalgebra structure on C is deﬁned by the form ula c 7− → c (1) ∧ c (2) and the Lie coalgebra structure o n h ∨ is giv en by the f o rm ula h x ∗ , [ x ′ , x ′′ ] i = h x ∗ { 1 } , x ′′ ih x { 2 } , x ′ i − h x ∗ { 1 } , x ′ ih x ∗ { 2 } , x ′′ i for x ∗ ∈ h ∨ , x ′ , x ′′ ∈ h . By ε w e denote t he counit of C . 259 (ii) The pair ing ψ is compat ible with the contin uous coaction of C in h obtained b y restricting the coaction in g and the adjoint coaction of C in it self. The latter is deﬁned by the formula c 7− → c [0] ⊗ c [1] = c (2) ⊗ s ( c (1) ) c (3) , where s denotes the an tip o de map of the Hopf a lgebra C (the square brack ets ar e used to av oid am biguit y o f notatio n) . T he compatibility means that the contin uous linear map C ⊗ ∗ h − → k corresp onding to ψ preserv es the contin uous coactions, or equiv alen tly , the map ˇ ψ is a morphism of C - como dules. (iii) The action of h in g induced b y the contin uous coaction of C in g and the pairing ψ coincides with the adjoin t action o f h in g . Here the former a ctio n is constructed as the pro jective limit of the actions of h in quotient spaces of g by C -inv ariant op en subspaces; for a righ t C -como dule N , the h -mo dule structure on N induced b y the pairing ψ is deﬁned by the form ula xn = − ψ ( n (1) , x ) n (0) for x ∈ h , n ∈ N . Giv en a T ate Harish-Chandra pair ( g , C ), one can construct a T ate Harish-Chandra pair ( g ∼ , C ) with the same Lie subalgebra h , where g ∼ is the canonical cen tral ex- tension of a T ate Lie algebra g . A con tin uous coaction of C in g ∼ and a canonical em b edding of h in to g preserving the con tinuous coactions of C were constructed ab ov e; it remains to c hec k the condition (iii). Here it suﬃces to notice that the adjoin t action of gl ( g ) in gl ( g ) ∼ coincides with the action of gl ( g ) in gl ( g ) ∼ induced b y the action of gl ( g ) in g , hence the adjoint action o f h in gl ( g ) ∼ coincides with the action of h in gl ( g ) ∼ induced b y the coaction of C in gl ( g ) ∼ and the pairing ψ . D.2.2. Let ( g ′ , C ) b e a T ate Harish-Chandra pair suc h that t he T a t e Lie algebra g ′ is a cen t r al extension of a T ate Lie algebra g with the ke rnel iden tiﬁed with k ; a ssume that C coacts trivially on k ⊂ g ′ and the Lie subalgebra h ⊂ g ′ that is a part of the T ate Harish-Chandra pair structure do es not con tain k . Then ( g , C ) is naturally also a T ate Harish-Chandra pair with the induced contin uous coaction o f C in g and the Lie subalgebra h ⊂ g deﬁned a s the image of h in g . In this case, w e will sa y that ( g ′ , C ) − → ( g , C ) is a cen tral extension o f T ate Harish-Chandra pairs with the k ernel k . One example of a cen tral extens ion of T ate Harish-Chandra pairs is the canonical cen tral extension ( g ∼ , C ) − → ( g , C ). Let κ : ( g ′ , C ) − → ( g , C ) b e a cen tral extension of T ate Harish-Chandra pairs with the ke rnel k . Consider the tensor pro duct S r κ ( g , C ) = C ⊗ U ( h ) U κ ( g ), where U ( h ) and U ( g ′ ) denote the univ ersal en v eloping alg ebras of the Lie algebras h and g ′ considered as Lie alg ebras without an y t o p ologies, U κ ( g ) = U ( g ′ ) / (1 U ( g ′ ) − 1 g ′ ) is the mo diﬁcatio n of the univ ersal env eloping algebra of g corresp onding to the cen tr al extension k − → g ′ − → g , a nd 1 U ( g ′ ) and 1 g ′ denote the unit elemen ts of the algebra U ( g ′ ) and the v ector subspace k ⊂ g ′ , resp ectiv ely . The structure of righ t U ( h )-mo dule on C comes from the pairing φ : C ⊗ k U ( h ) − → k corresp o nding to the algebra morphism U ( h ) − → C ∨ induced by the Lie algebra morphism ˇ ˇ ψ : h − → C ∨ , where the m ultiplication on 260 C ∗ is deﬁned by the formula h c ′ ∗ c ′′ ∗ , c i = h c ′ ∗ , c (2) ih c ′′ ∗ , c (1) i for c ′ ∗ , c ′′ ∗ ∈ C ∗ , c ∈ C and t he Lie brac k et is giv en b y the fo rm ula [ c ′ ∗ , c ′′ ∗ ] = c ′ ∗ c ′′ ∗ − c ′′ ∗ c ′ ∗ . W e claim that the vec tor space S r κ ( g , C ) has a na t ural structure of semialgebra ov er the coalgebra C prov ided by the general construction o f 1 0.2.1. The construction of this semialgebra structure b ecomes a little simpler if one a ssumes that (iv) the pair ing φ : C ⊗ k U ( h ) − → k is nondegenerate in C , but this is not necessary . D.2.3. T o construct a righ t C -como dule structure on S r κ ( g , C ), we will hav e to a p- pro ximate this v ector space b y ﬁnite-dimensional spaces. Let V 1 , . . . , V t b e a se- quence o f C - inv ariant compact op en subspaces of g ′ con taining h a nd k suc h that V i + [ V i , V i ] ⊂ V i − 1 . Let N b e a ﬁnite-dimensional right C -como dule. Cho ose a C -in- v ariant compact op en subspace W 1 ⊂ h suc h that the C -como dule N is annihilated b y the action of W 1 obtained b y restricting the action of h induced b y the pa ir ing ψ . F or eac h i = 2, . . . , t c ho o se a C -inv ariant compact op en subspace W i ⊂ h suc h that W i + [ V i , W i ] ⊂ W i − 1 . Denote b y S r κ ( V 1 , . . . , V t ; N ) the quotient space of the vector space N ⊗ k ( k ⊕ V 1 /W 1 ⊕ · · · ⊕ ( V t /W t ) ⊗ t ) by the obvious relatio ns imitating the r ela- tions in the en v eloping algebra U κ ( g ) and its tensor pro duct with N ov er U ( h ). It is easy to see that this quotien t space do es not dep end on the c hoice o f the subspaces W i . In other w ords, denote b y R ( V 1 , . . . , V t ) the subspace U ( h )( k + V 1 + · · · + V t t ) ⊂ U κ ( g ); it is an U ( h )- U ( h )-subbimo dule of U κ ( g ) and a free left U ( h )- mo dule. The tensor pro duct N ⊗ U ( h ) R ( V 1 , . . . , V t ) is naturally isomorphic to S r κ ( V 1 , . . . , V t ; N ). This is an isomorphism o f right U ( h )-mo dules; when N = D is a ﬁnite-dimensional sub coal- gebra o f C , this is also an isomorphism of left C -como dules. Clearly , the inductiv e limit o f S r κ ( V 1 , . . . , V t ; D ) o v er increasing t , V i , a nd ﬁnite-dimensional sub coalgebras D ⊂ C is naturally isomorphic to S r κ ( g , C ). No w the v ector space S r κ ( V 1 , . . . , V t ; N ) has a righ t C - como dule structure induced b y the right C -como dule structure on N ⊗ k ( k ⊕ V 1 /W 1 ⊕ · · · ⊕ ( V t /W t ) ⊗ t ) obt a ined b y taking the tensor pro duct of the C -como dule structures on V i /W i and the r igh t C -como dule structure on D . The inductiv e limit of these C -como dule structures for N = D provid es the desired right C -como dule structure on S r κ ( g , C ). It comm utes with the left C -como dule structure on S r κ ( g , C ) and ag rees with the rig h t U ( h )-mo dule structure, since suc h commutativit y and agreemen t hold o n the lev el of the spaces S r κ ( V 1 , . . . , V t ; D ). F urthermore, b y the (classical) P oincare–Birkhoﬀ–Witt theorem U κ ( g ) is a free left U ( h )-mo dule. If the condition (iv) holds, the construction o f the semialgebra S r κ ( g , C ) is ﬁnished; otherwise, we still ha v e to c hec k that the semiunit map C − → S r κ ( g , C ) and the semim ultiplication map S r κ ( g , C )  C S r κ ( g , C ) − → S r κ ( g , C ) are mo r phisms of right C -como dules. The f ormer is clear, and the latter can b e pro v en in the following w a y . Any ﬁnite- dimensional C -como dule N is a como dule ov er a ﬁnite-dimensional sub coalgebra E ⊂ 261 C . There is a natural isomorphism N ⊗ U ( h ) R ( V 1 , . . . , V t ) ≃ N  C ( E ⊗ U ( h ) R ( V 1 , . . . , V t )). The corresp onding isomorphism S r κ ( V 1 , . . . , V t ; N ) ≃ N  C S r κ ( V 1 , . . . , V t ; E ), which is induced b y the isomorphism N ≃ N  C E , preserv es the rig ht C -como dule structures. All of this is applicable t o the case of N = S r κ ( V ′ 1 , . . . , V ′ t ; D ), where V ′ 1 , . . . , V ′ t is another sequence of subspaces of g ′ satisfying the ab ov e conditions. Now let V ′′ 1 , . . . , V ′′ 2 t ⊂ g ′ b e a sequenc e of subspace s satisfying the ab ov e conditions a nd suc h that V ′ i , V i ⊂ V ′′ t + i . The map C ⊗ U ( h ) U κ ( g ) ⊗ U ( h ) U κ ( g ) − → C ⊗ U ( h ) U κ ( g ) induced b y the m ultiplication map U κ ( g ) ⊗ U ( h ) U κ ( g ) − → U κ ( g ) is the inductiv e limit of the maps D ⊗ U ( h ) R ( V ′ 1 , . . . , V ′ t ) ⊗ U ( h ) R ( V 1 , . . . , V t ) − → D ⊗ U ( h ) R ( V ′′ 1 , . . . , V ′′ 2 t ) ov er increasing t , V i , V ′ i , V ′′ i , and D . The corresp onding map S r κ ( V 1 , . . . , V t ; S r κ ( V ′ 1 , . . . , V ′ t ; D )) − → S r κ ( V ′′ 1 , . . . , V ′′ 2 t ; D ) is induced by the map D ⊗ k ( k ⊕ V ′ 1 /W ′ 1 ⊕ · · · ⊕ ( V ′ t /W ′ t ) ⊗ t ) ⊗ k ( k ⊕ V 1 /W 1 ⊕ · · · ⊕ ( V t /W t ) ⊗ t ) − → D ⊗ k ( k ⊕ V ′′ 1 /W ′′ 1 ⊕ · · · ⊕ ( V ′′ 2 t /W ′′ 2 t ) ⊗ 2 t ), where the sequence s of subspaces W ′ i , W i , W ′′ i satisfy the ab o v e conditions with resp ect to the sequence s of subspaces V ′ i , V i , V ′′ i , and the righ t C -como dules D , D ⊗ k ( k ⊕ V ′ 1 /W ′ 1 ⊕ · · · ⊕ ( V ′ t /W ′ t ) ⊗ t ), D , resp ectiv ely , and the additional condition that W ′ i , W i ⊂ W ′′ t + i . One can easily see that the latter map is a morphism of right C - como dules. D.2.4. The semialgebra S r κ ( g , C ) ov er t he coalgebra C is constructed. Analogously one deﬁnes a semialgebra structure on the tensor pro duct S l κ ( g , C ) = U κ ( g ) ⊗ U ( h ) C . The semialgebras S r κ = S r κ ( g , C ) and S l κ = S l κ ( g , C ) are essen tially opp o site to eac h other (see C.2.7). More precisely , the an tip o de an ti-automorphisms of U ( g ′ ) a nd C induc e a natura l isomorphism of semialgebras S r κ ≃ S l op − κ compatible with the isomorphism o f coalgebras C op ≃ C , where w e denote b y − κ the cen tral extension of T ate Harish-Chandra pairs with the k ernel k that is obtained from the cen tral extension κ by m ultiplying the embedding k − → g ′ with − 1. D.2.5. A dis c r ete mo dule M ov er a to p ological Lie algebra g is g -mo dule suc h that the action map g × M − → M is contin uous with resp ect to the discrete top ology of M . Equiv alently , a g -mo dule M is discrete if the a nnihilat o r of any elemen t of M is an op en Lie subalgebra in g . In particular, if ψ : C × h − → k is a con tin uous pairing b et w een a compact Lie algebra h and a coalgebra C suc h that the map ˇ ψ : C − → h ∨ is a morphism of Lie coalgebras, then the h -mo dule structure induced b y a C - como dule structure b y the for mula o f D.2.1 (iii) is alw ay s discrete. Let κ : ( g ′ , C ) − → ( g , C ) b e a cen tral extension of T ate Harish-Chandra pairs with the k ernel k . Then the catego ry of left semimo dules o v er S l κ ( g , C ) is isomorphic to the category of k - v ector spaces M endo w ed with C -como dule and discrete g ′ -mo dule structures suc h that the induced discrete h -mo dule structures coincide, the action map g / U ⊗ k L − → M is a morphism of C -como dules for a ny ﬁnite-dimensional C - sub- como dule L ⊂ M and an y C -inv aria n t compact o p en subspace U ⊂ g annihilating L , and the unit elemen t of k ⊂ g ′ acts by the identit y in M . The second of these three conditions can b e reform ulated as fo llo ws: for any C -inv arian t compact subspace 262 V ⊂ g ′ , the natural Lie coaction ma p M − → V ∨ ⊗ k M is a morphism of C -como dules. When the assumption (iv) of D .2.2 is satisﬁed, the second conditio n is redundan t. Abusing terminology , w e will call vec tor spaces M endo w ed with suc h a structure Harish-Chandr a mo dules over ( g , C ) with the c entr al char ge κ . Analogously , the category o f righ t semimo dules ov er S r κ ( g , C ) is isomorphic to the category of Harish- Chandra mo dules ov er ( g , C ) with cen tral charge − κ . D.2.6. F or a top ological v ector space V and a v ector space P , denote by V ⊗ b P the tensor pro duct V ⊗ ! P = V ← ⊗ P c onsider e d as a ve ctor sp ac e without any top olo gy , where P is endow ed with the discrete top o logy for the purp ose of making the top ological tensor pro duct. In other w ords, one has V ⊗ b P = lim ← − U V /U ⊗ k P , where the pro jective limit is ta k en ov er all op en subspaces U ⊂ V . F or a top ological v ector space V , denote b y V ∗ , 2 ( V ) t he completion of V 2 k ( V ) with resp ect to the top ology with the base of neighborho o ds of zero formed b y all the subspaces T ⊂ V 2 k ( V ) suc h that there exists an op en subspace V ′ ⊂ V for whic h V 2 k ( V ′ ) ⊂ T and for an y v ector v ∈ V there exists an op en subspace V ′′ ⊂ V for whic h v ∧ V ′′ ⊂ T . F o r a ny top ological v ector spaces V and W , the v ector space o f con tin uous sk ew-symmetric bilinear maps V × V − → W is naturally isomorphic to the v ector space of con tin uous linear maps V 2 , ∗ ( V ) − → W . The space V ∗ , 2 ( V ) is a closed subspace of the space V ⊗ ∗ V ; the sk ew-symme trization map V ⊗ ∗ V − → V ⊗ ∗ V factorizes through V ∗ , 2 ( V ). Let g b e a top ological Lie algebra. A c ontr amo dule ov er g is a v ector space P endo w ed with a linear map g ⊗ b P − → P satisfying the follow ing v ersion of Jacobi equation. Consider the v ector space V ∗ , 2 ( g ) ⊗ b P . There is a natural map V ∗ , 2 ( g ) ⊗ b P − → g ⊗ b P induced b y the brac ket map V ∗ , 2 ( g ) − → g . F urthermore, there is a natural map ( g ⊗ ∗ g ) ⊗ b P − → g ⊗ b ( g ⊗ b P ), whic h is constructed as follows. F o r an y op en subspace U ⊂ g there is a natural surjection ( g ⊗ ∗ g ) ⊗ b P − → ( g /U ⊗ ∗ g ) ⊗ b P and f or an y discrete vec tor space F there is a natural isomorphism ( F ⊗ ∗ g ) ⊗ b P ≃ F ⊗ k ( g ⊗ b P ), so the desired map is obta ined as the pro j ective limit ov er U . Compo sing the map V ∗ , 2 ( g ) ⊗ b P − → ( g ⊗ ∗ g ) ⊗ b P induced b y the em b edding V ∗ , 2 ( g ) − → g ⊗ ∗ g with the map ( g ⊗ ∗ g ) ⊗ b P − → g ⊗ b ( g ⊗ b P ) that w e hav e constructed and with the map g ⊗ b ( g ⊗ b P ) − → g ⊗ b P induced b y the contraaction map g ⊗ b P − → P , w e obtain a second map V ∗ , 2 ( g ) ⊗ b P − → P . Now the contramodule Jacobi equation claims that the t w o maps V ∗ , 2 ⊗ b P − → g ⊗ b P should hav e equal comp ositions with the con traaction map g ⊗ b P − → P . Alternativ ely , the map ( g ⊗ ∗ g ) ⊗ b P − → g ⊗ b ( g ⊗ b P ) can b e constructed a s the comp osition ( g ⊗ ∗ g ) ⊗ b P − → ( g ← ⊗ g ) ⊗ b P ≃ g ← ⊗ g ← ⊗ P ≃ g ⊗ b ( g ⊗ b P ) of the map induced by the nat ural con tin uous map g ⊗ ∗ g − → g ← ⊗ g and the natural isomorphisms whose existence follow s from the f act that the top olo gical tensor product W ← ⊗ V 263 considered as a ve ctor space without any top ology do es not dep end on the top ology of V . The follo wing comparison b etw een the deﬁnitions of a discrete g - mo dule and a g -con tramo dule can b e made: a discrete g - mo dule structure on a v ector space M is giv en b y a con tin uous linear map g ⊗ ∗ M ≃ g → ⊗ M − → M , while a g -con tramo dule structure on a v ector space P is giv en b y a discon tin uous linear map g ⊗ ! P ≃ g ← ⊗ P − → P , where M a nd P ar e endow ed with discrete top olo gies. F or an y top olog ical Lie alg ebra g , the category of g -contramo dules is ab elian (cf. D.5.2). There is a natural exact forgetful functor fr o m the catego ry of g -contramo dules to the catego r y of mo dules o v er the Lie algebra g considered without an y top o lo gy . F or an y discrete g -mo dule M and any v ector space E there is a natural structure of g -contramo dule on the space of linear ma ps Hom k ( M , E ). The con traaction map g ⊗ b Hom k ( M , E ) − → Hom k ( M , E ) is constructed as the pro jective limit o v er all op en subspaces U ⊂ g of the maps g /U ⊗ k Hom k ( M , E ) − → Hom k ( M U , E ) giv en b y the formu la ¯ z ⊗ g 7− → ( m 7− → − g ( ¯ z m )) for ¯ z ∈ g /U , g ∈ Ho m k ( M , E ), and m ∈ M U , where M U ⊂ M denotes the subspace of all elemen ts of M annihilated by U . More generally , for a n y discrete mo dule M o v er a top olog ical Lie algebra g 1 and any contramo dule P ov er a top ological Lie alg ebra g 2 there is a natu- ral ( g 1 ⊕ g 2 )-contramo dule structure on Hom k ( M , P ) with the con traaction map ( g 1 ⊕ g 2 ) ⊗ b Hom k ( M , P ) − → Hom k ( M , P ) deﬁned a s the sum of t w o comm uting con traactions of g 1 and g 2 in Hom k ( M , P ), one of whic h is in tro duced ab ov e and the other one is giv en b y t he comp osition g 2 ⊗ b Hom k ( M , P ) − → Hom k ( M , g 2 ⊗ b P ) − → Hom k ( M , P ) of the natura l map g 2 ⊗ b Hom k ( M , P ) − → Hom k ( M , g 2 ⊗ b P ) and the map Hom k ( M , g 2 ⊗ b P ) − → Hom k ( M , P ) induced by the g 2 -con traaction in P . Hence for any discrete g -mo dule M and any g -contramo dule P there is a natural g -contramo dule structure on Hom k ( M , P ) induced b y the diagonal em b edding of Lie algebras g − → g ⊕ g . D.2.7. When g is a T at e Lie a lg ebra, a g -contramo dule P can b e also deﬁned as a k -v ector space endo w ed with a linear map Hom k ( V ∨ , P ) − → P for ev ery compact op en subspace V ⊂ g . These linear maps should satisfy the follo w- ing tw o conditions: when U ⊂ V ⊂ g a r e compact o p en subspaces , the maps Hom k ( U ∨ , P ) − → P and Hom k ( V ∨ , P ) − → P should form a comm utativ e dia- gram with the map Ho m k ( U ∨ , P ) − → Hom k ( V ∨ , P ) induced b y the natural sur- jection V ∨ − → U ∨ , and for a ny compact op en subspaces V ′ , V ′′ , W ⊂ g such that [ V ′ , V ′′ ] ⊂ W the comp osition Hom k ( V ′′ ∨ ⊗ k V ′ ∨ , P ) − → Hom k ( W ∨ , P ) − → P of the map induced b y the cobrac k et map W ∨ − → V ′′ ∨ ⊗ k V ′ ∨ and the con traaction map Hom k ( W ∨ , P ) − → P should b e equal to the diﬀerence of the iterated con traaction map Hom k ( V ′′ ∨ ⊗ k V ′ ∨ , P ) ≃ Hom k ( V ′ ∨ , Hom k ( V ′′ ∨ , P )) − → Hom k ( V ′ ∨ , P ) − → P and the composition o f the isomorphism Hom k ( V ′′ ∨ ⊗ k V ′ ∨ , P ) ≃ Hom k ( V ′ ∨ ⊗ k 264 V ′′ ∨ , P ) induced by the isomorphism V ′ ∨ ⊗ k V ′′ ∨ ≃ V ′′ ∨ ⊗ k V ′ ∨ with the iterated con traaction map Hom k ( V ′ ∨ ⊗ k V ′′ ∨ , P ) ≃ Hom k ( V ′′ ∨ , Hom k ( V ′ ∨ , P )) − → P . F or a T ate Lie algebra g , a discrete g - mo dule M , and an g -contra mo dule P , the structure of g -contramo dule on Hom k ( M , P ) deﬁned ab ov e is give n b y the formu la π ( g )( m ) = π P ( x ∗ 7→ g ( x ∗ )( m )) − g ( m {− 1 } )( m { 0 } ) for a compact op en subspace V ⊂ g , a linear map g ∈ Hom k ( V ∨ , Hom k ( M , P )), and elemen ts x ∗ ∈ V ∨ , m ∈ M , where m 7− → m {− 1 } ⊗ m { 0 } denotes the map M − → V ∨ ⊗ k M corresponding to the g -action map V × M − → M and π P denotes the g -contraaction map Hom k ( V ∨ , P ) − → P . If ψ : C × h − → k is a con tin uous pairing b et w een a coalgebra C and a compact Lie algebra h suc h that the map ˇ ψ : C − → h ∨ is a morphism of Lie coalgebras, then for an y left C -contramo dule P the induced contraaction of h in P is deﬁned as the comp osition Hom k ( h ∨ , P ) − → Hom k ( C , P ) − → P of the map induced b y the map ˇ ψ and t he C -con traaction map. D.2.8. Let κ : ( g ′ , C ) − → ( g , C ) b e a cen tral extension of T ate Har ish-Chandra pairs with t he k ernel k . Then the category o f left semicon tramo dules ov er the semialgebra S r κ ( g , C ) is isomorphic to t he catego r y of k -v ector spaces P e ndow ed with a left C -contramo dule a nd a g ′ -contramo dule structures suc h that the induced h -contra- mo dule structures coincide, for an y C - inv ariant compact op en subspace V ⊂ g the g -con traaction map Hom k ( V ∨ , P ) − → P is a morphism of C -contramo dules, and the unit elemen t of k ⊂ g ′ acts b y the identit y in P . Here the left C -contramo dule structure on the v ector space Hom k ( M , P ) for a left C -como dule M and a left C -con- tramo dule P is deﬁned b y the formula π ( g )( m ) = π P ( c 7→ g ( s ( m ( − 1) ) c )( m (0) )) for m ∈ M , g ∈ Hom k ( C , Hom k ( M , P )). Indeed, according to 10.2.2, a left S r κ -semicontramo dule structure on P is the same that a left C - contramo dule and a left U κ ( g )-mo dule structures suc h that induced U ( h )-mo dule structures on P coincide and the (semicon tra)action map P − → Hom U ( h ) ( U κ ( g ) , P ) ≃ Cohom C ( S r κ , P ) is a morphism of C - contr a mo dules. The la tter conditio n is equiv alent t o the map P − → Hom U ( h ) ( U ( h ) · V , P ) ≃ Cohom C ( C ⊗ U ( h ) U ( h ) · V , P ) b eing a morphism of C -contramo dules fo r any compact C -inv ariant subspace h ⊕ k ⊂ V ⊂ g ′ , where U ( h ) · V ⊂ U κ ( g ). G iv en this data, o ne can use t he short exact sequences h ⊗ k P − → h ⊗ b P ⊕ V ⊗ k P − → V ⊗ b P to construct the Lie con traaction maps V ⊗ b P − → P . Then the map P − → Hom U ( h ) ( U ( h ) · V , P ) is a morphism of C -contramo dules if and only if the map Hom k ( V ∨ , P ) − → P is a morphism of C -contramo dules. T o che c k this, one can express the ﬁrst condition in terms of the equalit y of t w o appropriate maps Hom k ( C , P ) ⇒ Hom k ( V , P ) and the second condition in t erms of the equality of tw o maps Hom k ( V ∨ ⊗ k C , P ) ⇒ P . These tw o pairs of maps corresp ond to eac h o ther under a natural isomorphism V ∨ ⊗ k C ≃ C ⊗ k V ∨ . In particular, our maps Hom k ( V ∨ , P ) − → P are morphisms o f h -contramo dules, and it fo llows that they deﬁne a g ′ -contramo dule structure. 265 W e will call ve ctor spaces P endo w ed with suc h a structure Harish-Chandr a c ontr a m o dules over ( g , C ) with the c entr al char ge κ . If κ 1 : ( g ′ , C ) − → ( g , C ) and κ 2 : ( g ′′ , C ) − → ( g , C ) are tw o cen tral extensions of T ate Har ish-Chandra pairs with the k ernels k , and M and P are a Harish-Chandra mo dule and a Harish-Chandra con tramo dule o v er ( g , C ) with the central c harges κ 1 and κ 2 , resp ective ly , then the v ector space Hom k ( M , P ) has a natural structure of Harish-Chandra contramo dule with the cen tral c harge κ 2 − κ 1 . Here κ 2 − κ 1 : ( g ′′′ , C ) − → ( g , C ) denotes the Baer diﬀerence of the cen tral extensions κ 2 and κ 1 . This Harish-Chandra con tra- mo dule structure consists of the g ′′′ -contramo dule and C -contramo dule structures on Hom k ( M , P ) deﬁned by the ab ov e rules. D.3. I somorphism of semialgebras. D.3.1. F or any tw o cen tra l extensions of T ate Harish-Chandra pairs κ ′ : ( g ′ , C ) − → ( g , C ) and κ ′′ : ( g ′′ , C ) − → ( g , C ) with the k ernels iden tiﬁed with k w e denote b y κ ′ + κ ′′ their Baer sum, i. e., the cen tral extension of T ate Harish-Chandra pairs ( g ′′′ , C ) − → ( g , C ) with g ′′′ = k er( g ′ ⊕ g ′′ → g ) / im k , where the map g ′ ⊕ g ′′ − → g is the diﬀerence of the maps g ′ − → g and g ′′ − → g , and the map k − → g ′ ⊕ g ′′ is the diﬀerence of the maps k − → g ′ and k − → g ′′ . The canonical cen tral extension ( g ∼ , C ) − → ( g , C ) will b e denoted b y κ 0 . W e claim that for any cen tral extension of T ate Harish-Chandra pairs κ : ( g ′ , C ) − → ( g , C ) with the k ernel k suc h that the condition (iv) of D.2.2 is satisﬁed there is a natural isomorphism S r κ + κ 0 ( g , C ) ≃ S l κ ( g , C ) of semialgebras o v er the coalgebra C . This isomorphism is c haracterized b y the following three prop erties. (a) Consider the increasing ﬁltration F of t he k -alg ebra U κ ( g ) with the compo - nen ts F i U κ ( g ) = ( k + g ′ + · · · + g ′ i ) U ( h ) = U ( h )( k + g ′ + · · · + g ′ i ) and the induced ﬁltration F i S l κ = F i U κ ( g ) ⊗ U ( h ) C o f the semialgebra S l κ = S l κ ( g , C ). Then w e hav e F 0 S l κ ≃ C , S l κ ≃ lim − → F i S l κ , and the semim ultiplication maps F i S l κ  C F j S l κ − → S l κ  C S l κ − → S l κ factorize thro ugh F i + i S l κ . There is an ana logous ﬁltra tion F i S r κ + κ 0 = C ⊗ U ( h ) F i U κ + κ 0 ( g ) of the semialgebra S r κ + κ 0 = S r κ + κ 0 ( g , C ). The desired isomorphism S r κ + κ 0 ≃ S l κ preserv es the ﬁltrations F . (b) The natural maps F i − 1 S l κ − → F i S l κ are injectiv e and t heir cok ernels are coﬂat left and righ t C - como dules, so the asso ciated graded quotien t semialgebra gr F S l κ = L i F i S l κ /F i − 1 S l κ is deﬁned (cf. 1 1.5). The semialgebra gr F S l κ is naturally isomorphic to the tensor pro duct Sym k ( g / h ) ⊗ k C o f the symmetric algebra Sym k ( g / h ) of the k -v ector space g / h and the coalgebra C , endo w ed with the semialgebra structure corresp onding to the left en t wining structure Sym k ( g / h ) ⊗ k C − → C ⊗ k Sym k ( g / h ) for the coalgebra C and the algebra Sym k ( g / h ) (see 10.3). Here the en tw ining map is giv en b y the form ula u ⊗ c 7− → 266 cu ( − 1) ⊗ u (0) , where u 7− → u ( − 1) ⊗ u (0) denotes the C -coaction in Sym k ( g / h ) induced b y the C -coaction in g / h . Analo gously , the semialgebra gr F S r κ + κ 0 is naturally isomorphic to the tensor pro duct C ⊗ k Sym k ( g / h ) endo w ed with the semialgebra structure corresp o nding to the r ig h t ent wining structure C ⊗ k Sym k ( g / h ) − → Sym k ( g / h ) ⊗ k C . Here the en tw ining map is giv en b y the form ula c ⊗ u 7− → u (0) ⊗ cu (1) , where the right coaction u 7− → u (0) ⊗ u (1) is obtained from the ab o v e left coaction u 7− → u ( − 1) ⊗ u (0) b y applying the an tip o de. These left and righ t ent wining maps are in v erse to each other, hence there is a natural isomorphism of semialgebras gr F S l κ ≃ gr F S r κ + κ 0 . This isomorphism can b e obtained b y passing to the asso ciated g raded quotien t semialgebras in the desired isomorphism S l κ ≃ S r κ + κ 0 . (c) Cho o se a section b ′ : g / h − → g ′ of the natural surjection g ′ − → g ′ / ( h ⊕ k ) ≃ g / h . Comp osing b ′ with the surjection g ′ − → g , we obtain a section b of the natural surjection g − → g / h , hence a direct sum decomp osition g ≃ h ⊕ b ( g / h ). So there is the cor r esp o nding section g − → g ∼ of the canonical cen tral extension g ∼ − → g ; denote by ˜ b the comp osition g / h − → g − → g ∼ of the section b and the section g − → g ∼ . The Baer sum o f t he sections b ′ and ˜ b pro vides a section b ′′ : g / h − → g ′′ , where ( g ′′ , C ) − → ( g , C ) denotes the cen tral extension κ + κ 0 . No w the comp osition g / h ⊗ k C − → g ′ ⊗ k C ≃ F 1 U κ ( g ) ⊗ k C − → F 1 S l κ of the ma p induced by the map b ′ , the isomorphism induced by the natural isomorphism g ′ ≃ F 1 U κ ( g ), and t he surjection U κ ( g ) ⊗ k C − → U κ ( g ) ⊗ U ( h ) C pro vides a section of the natural surjection F 1 S l κ − → F 1 S l κ /F 0 S l κ ≃ g / h ⊗ k C . This section is a morphism of rig h t C - como dules. Hence the corresp onding retractio n F 1 S l κ − → F 0 S l κ ≃ C is also a morphism of righ t C -como dules. Ana lo gously , t he comp osition C ⊗ k g / h − → C ⊗ k g ′′ ≃ C ⊗ k F 1 U κ + κ 0 ( g ) − → F 1 S r κ + κ 0 , where t he ﬁrst morphism is induced by the map b ′′ , is a section o f the na t ural surjection F 1 S r κ + κ 0 − → F 1 S r κ + κ 0 /F 0 S r κ + κ 0 ≃ C ⊗ k g / h ; this section is a morphism of left C -como dules. Hence so is the corresp onding retraction F 1 S r κ + κ 0 − → F 0 S r κ + κ 0 ≃ C . The desired isomorphism F 1 S l κ ≃ F 1 S r κ + κ 0 iden tiﬁes t he comp ositions F 1 S l κ − → C − → k and F 1 S r κ + κ 0 − → C − → k o f the retractions F 1 S l κ − → C and F 1 S r κ + κ 0 − → C with the counit map C − → k . This condition holds for all sections b ′ . Theorem. Ther e exists a unique isomorphism of se mialgebr a s S r κ + κ 0 ( g , C ) ≃ S l κ ( g , C ) over C s atisfying the ab ove pr op erties (a-c). Pr o of . Uniqueness is clear, since a morphism from a C - C -bicomo dule to the bicomo d- ule C is determined by its comp osition with the counit map C − → k . The pro of o f existence o ccupies subsections D.3.2 – D.3.7. 267 The next result is obtained b y sp ecializing the semimo dule-semicon tramo dule cor- resp ondence theorem to the case of Harish-Chandra mo dules and contramo dules. Corollary . Ther e is a natur al e quiva l e n c e R Ψ S l κ = L Φ − 1 S r κ + κ 0 b etwe en the s emiderive d c ate go ry of Harish-C handr a mo d ules with the c entr al char ge κ over ( g , C ) and the semiderive d c ate gory of Harish-C handr a c ontr am o dules with the c entr a l char ge κ + κ 0 over ( g , C ) . Her e the semideriv e d c ate gory of Harish-Cha ndr a mo dules is deﬁne d as the quotient c ate gory o f the hom o topy c ate gory o f c omplexes o f Harish -Chandr a mo dules by the thick sub c ate gory of C -c o acyclic c omp l e xes; the semide rive d c ate gory of Harish-Chan d r a c ontr amo dules is ana l o gously deﬁne d as the quotient c ate gory by the thick sub c ate go ry of C -c ontr aacyclic c omp lexes. Pr o of . This fo llo ws from the results of D.2.5 and D.2.8, the ab ov e The orem, and Corollary 6.3.  Remark. The main prop ert y of the equiv alence of semideriv ed categories provided b y the ab o v e Corollary is that it transforms the Harish-Chandra mo dules M that, considered as C -como dules, are the cofree como dules C ⊗ k E cogenerated by a ve ctor space E , in to t he Harish-Chandra con tramo dules P tha t, considered as C -contra- mo dules, are the free con tramo dules Hom k ( C , E ) generated b y t he same v ector space E , and vice v ersa. The similar assertion ho lds for a n y complexes of C -cofree Harish- Chandra mo dules and C -free Harish-Chandra con tramo dules. The a b o v e Coro llary is a w ay to form ulate the classical duality b et w een Harish-Chandra mo dules with the complemen tary cen tral c harges κ and − κ − κ 0 [22, 42]. Of course, t here is no hop e of establishing an anti-e quivalenc e b et w een an y kinds of exotic deriv ed cat- egories of arbitr ary Harish-Chandra mo dules ov er ( g , C ) with the complemen tary cen tral charges, as t he deriv ed categor y of v ector spaces is not an ti-equiv alen t to itself. A t the v ery least, one w ould hav e to imp o se some ﬁniteness conditions on the Harish-Chandra mo dules. The intro duction of con tramo dules allows t o resolv e this problem. Still one can use the functor Φ S to construct a c ontr avariant functor b et w een the semideriv ed catego ries of Harish-Chandra mo dules with t he comple- men tary cen tral charges. Choo se a v ector space U ; for example, U = k . Consider the functor N 7− → Hom k ( N , U ) acting from the semideriv ed category of Harish- Chandra semimo dules ov er ( g , C ) with the central c harge − κ − κ 0 to the semideriv ed category of Harish-Chandra semicon tramo dules ov er ( g , C ) with the central c harge κ + κ 0 . Comp o sing this functor Hom k ( − , U ) with the functor L Φ S r κ + κ 0 , one obta ins a con tra v ariant functor D si ( simo d – S r κ + κ 0 ) − → D si ( S l κ – simo d ). The latter functor trans- forms the Harish-Chandra mo dules that as C -como dules are cofreely cogenerated by a v ector space E in t o t he Harish-Chandra mo dules that as C - como dules are cofreely cogenerated by the v ector space Hom k ( E , U ), and similarly for complexes of C -cofree Harish-Chandra mo dules. One cannot a v oid using the exotic deriv ed categories in 268 this construction, b ecause the functor L Φ S do es not preserv e acyclicit y , in general (see 0 .2.7). D.3.2. The semialgebras S l κ and S r κ + κ 0 endo w ed with the increasing ﬁltra t ions F are left and r ig h t coﬂat nonhomogeneous Koszul semialgebras o v er the coalge- bra C (see 11.5). Indeed, there are natural isomorphisms o f complexes of C - C -bi- como dules Bar • gr (gr F S l κ , C ) ≃ Bar • gr (Sym k ( g / h ) , k ) ⊗ k C and Ba r • gr (gr F S r κ + κ 0 , C ) ≃ C ⊗ k Bar • gr (Sym k ( g / h , k ), and the k -algebra Sym k ( g / h ) is Ko szul. Here the left C -coaction in Bar • gr (Sym k ( g / h ) , k ) ⊗ k C is the tensor pro duct of the C -coaction in Bar • gr (Sym k ( g / h ) , k ) induced by the C - coa ction in g / h and the left C -coactio n in C , while the right C - coaction in Bar • gr (Sym k ( g / h ) , k ) ⊗ k C is induced by the right C -coaction in C . The C - C -bicomo dule structure on C ⊗ k Bar • gr (Sym k ( g / h , k ) is deﬁned in the analogo us w ay (with t he left and righ t sides switch ed). The left and righ t coﬂat Koszul coalgebras D l and D r o v er C quadratic dual to the left a nd right coﬂat Koszul semialgebras gr F S l κ and gr F S r κ + κ 0 are described as follo ws. One has D l ≃ V k ( g / h ) ⊗ k C , where V k ( g / h ) denotes the exterior coa lgebra of the k -v ector space g / h , i. e., the coalg ebra quadrat ic dual t o the symmetric algebra Sym k ( g / h ). The counit of V k ( g / h ) ⊗ k C is the tensor pro duct of the counits of V k ( g / h ) and C , while the com ultiplication in V k ( g / h ) ⊗ k C is constructed as the comp osition V k ( g / h ) ⊗ k C − → V k ( g / h ) ⊗ k V k ( g / h ) ⊗ k C ⊗ k C − → V k ( g / h ) ⊗ k C ⊗ k V k ( g / h ) ⊗ k C o f the map induced b y the comu ltiplications in V k ( g / h ) a nd C and the map induced b y the “p ermutation” map V k ( g / h ) ⊗ k C − → C ⊗ k V k ( g / h ). The latter map is giv en by the formula u ⊗ c 7− → cu [ − 1] ⊗ u [0] for u ∈ V k ( g / h ) and c ∈ C , where u 7− → u [ − 1] ⊗ u [0] denotes the C -coaction in V k ( g / h ) induced by the C -coa ction in g / h . Analogously , one has D r ≃ C ⊗ k V k ( g / h ), where the counit of C ⊗ k V k ( g / h ) is the tensor pro duct o f the counits of V k ( g / h ) and C , while the com ultiplication in C ⊗ k V k ( g / h ) is deﬁned in terms of the “ p erm utation” map C ⊗ k V k ( g / h ) − → V k ( g / h ) ⊗ k C . The latter map is g iven by the form ula c ⊗ u 7− → u [0] ⊗ c u [1] , where the righ t coaction u 7− → u [0] ⊗ u [1] is obtained from the left coaction u 7− → u [ − 1] ⊗ u [0] b y applying the an tip o de. Both coalgebras V k ( g / h ) ⊗ k C a nd C ⊗ k V k ( g / h ) ha v e gradings induced b y the grading of V k ( g / h ). The tw o “p ermutation” maps are inv erse to eac h other, and they provide an isomorphism of graded coalgebras D l ≃ D r . No w recall that we hav e assumed the condition (iv) of D.2.2 . Denote b y · · · ⊂ V 2 C ⊂ V 1 C ⊂ V 0 C = C the decreasing ﬁltration of C orthog o nal to the natural increasing ﬁltration of the univ ersal env eloping algebra U ( h ), tha t is V i C ⊂ C consists of all c ∈ C suc h that φ ( c, x ) = 0 for all x ∈ k + h + · · · + h i − 1 ⊂ U ( h ). Notice that the decreasing ﬁltr a tion V is compatible with b oth the coalgebra and algebra structures on C ; in particular, it is a ﬁltration by ideals with resp ect to the m ultiplication. The subspace V 1 C is the kernel of the counit map C − → k ; the subspace V 2 C is the k ernel of the map C − → h ∨ ⊕ k which is t he sum of the map ˇ ψ and the counit map. 269 Deﬁne decreasing ﬁltra t io ns V on the coa lg ebras D l and D r b y the formulas V i D l ≃ V k ( g / h ) ⊗ k V i C and V i D r ≃ V i C ⊗ k V k ( g / h ); these ﬁltratio ns a re compatible with the coalgebra structures on D l and D r , a nd corr espo nd to eac h other under the isomorphism D l ≃ D r . Set D l = D ≃ D r . The coa lgebra D is cog enerated b y the maps D − → D 0 /V 2 D 0 and D − → D 1 /V 1 D 1 , i. e., the iterated com ultiplication map from D to the direct pro duct of all tensor p ow ers D 0 /V 2 D 0 ⊕ D 1 /V 1 D 1 is injectiv e. Moreo ver, the decreasing ﬁltratio n V on D is cogenerated b y the ﬁltrations on D 0 /V 2 D 0 and D 1 /V 1 D 1 , i. e., the subspaces V i D are the full preimages of the subspaces of the induced ﬁltratio n on t he pro duct of all tensor p ow ers of D 0 /V 2 D 0 ⊕ D 1 /V 1 D 1 under the iterated comultiplication map. D.3.3. Comp osing the equiv alences of categories fr om 11.2.2 and Theorem 1 1 .6, w e obtain a n equiv alence b et w een the category of left (righ t) coﬂat nonhomoge- neous Koszul semialgebras ov er C and the category of left (r igh t) coﬂat Koszul CDG-coalgebras o v er C . Here a CDG -coagebra ( D , d , h ) is called Koszul o v er C if the underlying g raded coalgebra D is Koszul ov er C . Recall that for a left (right) coﬂat nonhomogeneous Koszul semialgebra S ∼ and the corr esp o nding quasi-diﬀeren tial coalgebra D ∼ one has F 1 S ∼ ≃ D ∼ 1 , so t o construct a sp eciﬁc CDG-coalgebra ( D , d, h ) corresp onding to a giv en ﬁltered semialgebra S ∼ one has to c ho o se a linear map δ : F 1 S ∼ − → k suc h that the comp osition of the injection C ≃ F 0 S ∼ − → F 1 S ∼ with δ coincides with the counit map o f C . Cho ose a section b ′ : g / h − → g ′ and construct the related section b ′′ : g / h − → g ′′ ; denote b y δ l b ′ : F 1 S l κ − → k and δ r b ′′ : F 1 S r κ + κ 0 − → k the corresp onding linear functions constructed in (c) of D .3 .1. In order to o btain an isomorphism of left and righ t coﬂat nonhomogeneous Koszul semialgebras S l κ ≃ S r κ + κ 0 , we will construct an isomorphism b et w een the CDG-coalgebras ( D l , d l b ′ , h l b ′ ) and ( D r , d r b ′′ , h r b ′′ ) corresp onding to the ﬁl- tered semialgebras S l κ and S r κ + κ 0 endo w ed with the linear functions δ l b ′ and δ r b ′′ . The isomorphism o f coalgebras D l ≃ D r is already deﬁned; all w e ha v e to do is to chec k that it identiﬁes d l b ′ with d r b ′′ and h l b ′ with h r b ′′ . Besides, w e need to sho w that the iso- morphism S l κ ≃ S r κ + κ 0 so obtained do es not dep end on the c hoice of b ′ . Here it suﬃces to che c k that c hanging the section b ′ to b ′ 1 leads to isomorphisms of CDG-coalgebras (id , a l ) : ( D l , d r b ′ , h r b ′ ) − → ( D l , d r b ′ 1 , h r b ′ 1 ) and (id , a r ) : ( D r , d r b ′′ , h r b ′′ ) − → ( D r , d r b ′′ 1 , h r b ′′ 1 ) with the linear functions a l and a r b eing iden tiﬁed by the isomorphism D l ≃ D r . Since the coalgebra D l = D ≃ D r is cogenerated b y the maps D − → D 0 /V 2 D 0 and D − → D 1 /V 1 D 1 , it suﬃce s to c hec k tha t the comp ositions of d l b ′ and d r b ′′ with these t w o maps coincide in order to show that d l b ′ = d r b ′′ . W e will also see that these comp o sitions f actorize throug h D 1 /V 2 D 1 and D 2 /V 1 D 2 , resp ectiv ely , and the induced map D 1 /V 2 D 1 − → D 0 /V 2 D 0 preserv es the images o f V 1 (actually , eve n maps the whole of D 1 /V 2 D 1 in to V 1 D 0 /V 2 D 1 ), hence it will f o llo w that the diﬀeren tial d l b ′ = d r b ′′ preserv es the decreasing ﬁltration V . Beside s, we will see that the linear 270 function h l b ′ = h r b ′′ annihilates the subspace V 2 D 2 and the linear function a l = a r corresp onding to a change of section b ′ annihilates the subspace V 2 D 1 . D.3.4. Let us intro duce not a tion f or t he comp o nen ts of the comm utator map with resp ect t o the direct sum decomp osition g ′ ≃ h ⊕ b ′ ( g / h ) ⊕ k . As ab ov e, the Lie coalgebra structure on h ∨ is denoted b y x ∗ 7− → x ∗ { 1 } ∧ x ∗ { 2 } . Denote t he Lie coa ctio n of h ∨ in g / h , i. e., t he map g / h − → h ∨ ⊗ k g / h corresp onding to the commutator map h × g / h − → g / h , by u 7− → u {− 1 } ⊗ u { 0 } . These t w o maps do not dep end on the choice of the section b ′ ; the rest of them do. Denote b y u ⊗ x ∗ 7− → u ( x ∗ ) t he map g / h ⊗ k h ∨ − → h ∨ corresp onding to t he pro jec- tion of the commutator map b ′ ( g / h ) × h − → g ′ − → h . D enote b y u ∧ v 7− → { u, v } the map V 2 k ( g / h ) − → g / h corresponding to the comm utator map V 2 k b ′ ( g / h ) − → g / h . Denote by u ∧ v ⊗ x ∗ 7− → ( u, v ) x ∗ the map V 2 k ( g / h ) ⊗ k h ∨ − → k corresp onding to the pro jection of the commutator map V 2 k b ′ ( g / h ) − → g ′ − → h . The ab o v e ﬁve ma ps only dep end on the Lie algebra g with the subalgebra h and the section b : g / h − → g , but the following tw o will dep end essen tially on g ′ and b ′ . Denote by ρ ′ : g / h − → h ∨ the map corresp onding to the pro jection of the commutator map b ′ ( g / h ) × h − → g ′ − → k . D enote b y σ ′ : V 2 k ( g / h ) − → k the map corresponding to the pro jection of the comm utator map V 2 k b ′ ( g / h ) − → g ′ − → k . Denote by ˜ ρ , ˜ σ and ρ ′′ , σ ′′ the analogous maps corresp onding to the cen tral extensions g ∼ − → g and g ′′ − → g with the sections ˜ b and b ′′ . Clearly , we hav e ρ ′′ = ρ ′ + ˜ ρ a nd σ ′′ = σ ′ + ˜ σ . Set b = b ( g / h ) ⊂ g . The comp osition θ of the commutator map in gl ( g ) ∼ with the pro j ection gl ( g ) ∼ − → k corresp onding to the section gl ( g ) − → gl ( g ) ∼ coming from the direct sum decomp osition g ≃ h ⊕ b is written explicitly a s follo ws. F or an y con tin uous linear op erato r A : g − → g denote b y A h → b : h − → b , A b → h : b − → h , etc., its comp o nen ts with resp ect to o ur direct sum decomp osition. Then the co cycle θ is g iv en b y the form ula θ ( A ∧ B ) = tr( A b → h B h → b ) − tr( B b → h A h → b ), where tr denotes the trace of a linear op erator h − → h with an op en k ernel. Using this formula, one can ﬁnd that, in the ab o v e notation, ˜ ρ ( u ) = − u { 0 } ( u {− 1 } ) and ˜ σ ( u ∧ v ) = − ( u, v { 0 } ) v {− 1 } + ( v , u { 0 } ) u {− 1 } . D.3.5. W e hav e D l 0 ≃ C , D l 1 ≃ g / h ⊗ k C , and D l 2 ≃ V 2 k ( g / h ) ⊗ k C . The comp osition of the map d l b ′ : g / h ⊗ k C − → C with the counit map ε : C − → k v anishes, since d l b ′ is a co deriv ation. Let us start with computing the comp o sition of the map d l b ′ with the map ˇ ψ : C − → h ∨ . The class of an elemen t u ⊗ c ∈ g / h ⊗ k C can b e represen ted by the elemen t b ′ ( u ) ⊗ U ( h ) c ∈ F 1 U κ ( g ) ⊗ U ( h ) C ≃ D l 1 ∼ in the quasi-diﬀeren tial coalgebra D l ∼ corresp onding to the ﬁltered semialgebra S l κ . D enote the imag e of b ′ ( u ) ⊗ U ( h ) c under the com ultiplica- tion map D l 1 ∼ − → C ⊗ k D l 1 ∼ b y c 1 ⊗ ( z ⊗ U ( h ) c 2 ), where z ∈ F 1 U κ ( g ). The to tal com ulti- plication of b ′ ( u ) ⊗ U ( h ) c is then equal to c 1 ⊗ ( z ⊗ U ( h ) c 2 ) + ( b ′ ( u ) ⊗ U ( h ) c (1) ) ⊗ c (2) . W e hav e 271 d l b ′ ( u ⊗ c ) = δ l b ′ ( b ′ ( u ) ⊗ U ( h ) c (1) ) c (2) − δ l b ′ ( z ⊗ U ( h ) c 2 ) c 1 = − δ l b ′ ( z ⊗ U ( h ) c 2 ) c 1 . F urthermore, ψ ( d l b ′ ( u ⊗ c ) , x ) = − ψ ( c 1 , x ) δ l b ′ ( z ⊗ U ( h ) c 2 ) = − δ l b ′ ( xb ′ ( u ) ⊗ U ( h ) c ) = − δ l b ′ ([ x, b ′ ( u )] ⊗ U ( h ) c ) − δ l b ′ ( b ′ ( u ) ⊗ U ( h ) xc ) = δ l b ′ ([ b ′ ( u ) , x ] ⊗ U ( h ) c ) = h x, u ( ˇ ψ ( c )) i + h x, ρ ′ ( u ) i ε ( c ) for x ∈ h , since ψ ( x, c 1 ) z ⊗ U ( h ) c 2 = xb ′ ( u ) ⊗ U ( h ) c . So the comp osition of the ma p d l b ′ : g / h ⊗ k C − → C with the map ˇ ψ : C − → h ∨ is equal to the comp osition of the map id ⊗ ( ˇ ψ , ε ) : g / h ⊗ k C − → g / h ⊗ k ( h ∨ ⊕ k ) with the map g / h ⊗ k ( h ∨ ⊕ k ) − → h ∨ giv en by the formula u ⊗ x ∗ + v 7− → u ( x ∗ ) + ρ ′ ( v ). No w let us compute the comp o sition of t he map d l b ′ : V 2 k ( g / h ) ⊗ k C − → g / h ⊗ k C with the map id ⊗ ε : g / h ⊗ k C − → g / h . T he v ector space D l 2 ∼ is the ke rnel of the semim ultiplication map F 1 S l κ  C F 1 S l κ − → F 2 S l κ , whic h can b e identiﬁed with the k ernel of the map F 1 U κ ( g ) ⊗ U ( h ) F 1 U κ ( g ) ⊗ U ( h ) C − → F 2 U κ ( g ) ⊗ U ( h ) C induced b y the m ultiplication map F 1 U κ ( g ) ⊗ U ( h ) F 1 U κ ( g ) − → F 2 U κ ( g ). The class of a n elemen t u ∧ v ⊗ c ∈ V 2 k ( g / h ) ⊗ k C can b e represen ted by the elemen t b ′ ( u ) ⊗ U ( h ) b ′ ( v ) ⊗ U ( h ) c − b ′ ( v ) ⊗ U ( h ) b ′ ( u ) ⊗ U ( h ) c − [ b ′ ( u ) , b ′ ( v )] ⊗ U ( h ) 1 ⊗ U ( h ) c in the latter ke rnel. Denote the image of b ′ ( v ) ⊗ U ( h ) c under the com ultiplication map D l 1 ∼ − → C ⊗ k D l 1 ∼ b y c 1 ⊗ ( z ⊗ U ( h ) c 2 ), where z ∈ F 1 U κ ( g ); then t he image of b ′ ( u ) ⊗ U ( h ) b ′ ( v ) ⊗ U ( h ) c under the comultiplication map D l 2 ∼ − → D l 1 ∼ ⊗ k D l 1 ∼ is equal to ( b ′ ( u ) ⊗ U ( h ) c 1 ) ⊗ ( z ⊗ U ( h ) c 2 ). The ima g e of [ b ′ ( u ) , b ′ ( v )] ⊗ U ( h ) 1 ⊗ U ( h ) c under the same map D l 2 ∼ − → D l 1 ∼ ⊗ k D l 1 ∼ is equal to ([ b ′ ( u ) , b ′ ( v )] ⊗ U ( h ) c (1) ) ⊗ (1 ⊗ U ( h ) c (2) ). W e hav e δ l b ′ ( b ′ ( u ) ⊗ U ( h ) c 1 ) z ⊗ U ( h ) c 2 = 0 and δ l b ′ ( z ⊗ U ( h ) c 2 )(id ⊗ ε ) b ′ ( u ) ⊗ U ( h ) c 1 = ε ( c 1 ) δ l b ′ ( z ⊗ U ( h ) c 2 ) u = δ l b ′ ( b ′ ( v ) ⊗ U ( h ) c ) u = 0, where p ∈ D l 1 denotes the image of an elemen t p ∈ D l 1 ∼ . F urthermore, δ l b ′ ([ b ′ ( u ) , b ′ ( v )] ⊗ U ( h ) c (1) ) 1 ⊗ U ( h ) c (2) = 0. Hence (id ⊗ ε ) d l b ′ ( u ∧ v ⊗ c ) = − δ l b ′ (1 ⊗ U ( h ) c (2) )(id ⊗ ε ) [ b ′ ( u ) , b ′ ( v )] ⊗ U ( h ) c (1) = − (id ⊗ ε )[ b ′ ( u ) , b ′ ( v )] ⊗ U ( h ) c = −{ u, v } ε ( c ). So the comp o sition of t he map d l b ′ : V 2 k ( g / h ) ⊗ k C − → g / h ⊗ k C with the map id ⊗ ε : g / h ⊗ k C − → g / h is equal to the comp o sition of t he map id ⊗ ε : V 2 k ( g / h ) ⊗ k C − → V 2 k ( g / h ) with the map V 2 k ( g / h ) − → g / h give n by the formula u ∧ v 7− → −{ u, v } . Finally , let us compute the linear function h l b ′ : V 2 k ( g / h ) ⊗ k C − → k . W e hav e δ l b ′ ( b ′ ( u ) ⊗ U ( h ) c 1 ) δ l b ′ ( z ⊗ U ( h ) c 2 ) = 0, hence h ( u ∧ v ⊗ c ) = − δ l b ′ ([ b ′ ( u ) , b ′ ( v )] ⊗ U ( h ) c (1) ) δ l b ′ (1 ⊗ U ( h ) c (2) ) = − δ l b ′ ([ b ′ ( u ) , b ′ ( v )] ⊗ U ( h ) c ) = − ( u, v ) ˇ ψ ( c ) − σ ′ ( u ∧ v ) ε ( c ). So the linear function h l b ′ is equal to the comp osition of the map id ⊗ ( ˇ ψ , ε ) : V 2 k ( g / h ) ⊗ k C − → V 2 k ( g / h ) ⊗ k ( h ∨ ⊕ k ) and the linear function V 2 k ( g / h ) ⊗ k ( h ∨ ⊕ k ) − → k giv en by the form ula u 1 ∧ v 1 ⊗ x ∗ + u ∧ v 7− → − ( u 1 , v 1 ) x ∗ − σ ′ ( u ∧ v ) . D.3.6. Analogously , w e hav e D r 0 ≃ C , D r 1 ≃ C ⊗ k g / h , and D r 2 ≃ C ⊗ k V 2 k ( g / h ). The comp osition of the map d r b ′′ : C ⊗ k g / h − → C with t he map ε : C − → k v anishes. The comp osition of the map d r b ′′ : C ⊗ k g / h − → C with the map ˇ ψ : C − → h ∨ is equal to the comp osition of the map ( ˇ ψ , ε ) ⊗ id : C ⊗ k g / h − → ( h ∨ ⊕ k ) ⊗ k g / h and the map ( h ∨ ⊕ 272 k ) ⊗ k g / h − → h ∨ giv en by the form ula x ∗ ⊗ u + v 7− → u ( x ∗ ) + ρ ′′ ( v ). The comp osition of the map d r b ′′ : C ⊗ k V 2 k ( g / h ) − → C ⊗ k g / h with the map ε ⊗ id : C ⊗ k g / h − → g / h is equal to the comp osition o f the map ε ⊗ id : C ⊗ k V 2 k ( g / h ) − → V 2 k ( g / h ) and the map V 2 k ( g / h ) − → g / h giv en by the fo rm ula u ∧ v 7− → −{ u, v } . The linear function h r b ′′ : C ⊗ k V 2 k ( g / h ) − → k is equal to the compo sition of the map ( ˇ ψ , ε ) ⊗ id : C ⊗ k V 2 k ( g / h ) − → ( h ∨ ⊕ k ) ⊗ k V 2 k ( g / h ) and the linear function ( h ∨ ⊕ k ) ⊗ k V 2 k ( g / h ) − → k giv en b y the f o rm ula x ∗ ⊗ u 1 ∧ v 1 + u ∧ v 7− → − ( u 1 , v 1 ) x ∗ − σ ′′ ( u ∧ v ) . The isomorphism g / h ⊗ k C ≃ C ⊗ k g / h forms a commutativ e diagra m with the map g / h ⊗ k C − → g / h ⊗ k ( h ∨ ⊕ k ), the map C ⊗ k g / h − → ( h ∨ ⊕ k ) ⊗ k g / h , and the isomorphism g / h ⊗ k ( h ∨ ⊕ k ) ≃ ( h ∨ ⊕ k ) ⊗ k g / h giv en by the for mula u ⊗ x ∗ + v 7− → x ∗ ⊗ u + v {− 1 } ⊗ v { 0 } + v . Analogously , the isomorphism V 2 k ( g / h ) ⊗ k C ≃ C ⊗ k V 2 k ( g / h ) forms a comm utativ e diagram with t he map V 2 k ( g / h ) ⊗ k C − → V 2 k ( g / h ) ⊗ k ( h ∨ ⊕ k ), the map C ⊗ k V 2 k ( g / h ) − → ( h ∨ ⊕ k ) ⊗ k V 2 k ( g / h ), and the isomorphism V 2 k ( g / h ) ⊗ k ( h ∨ ⊕ k ) ≃ ( h ∨ ⊕ k ) ⊗ k V 2 k ( g / h ) give n by the f o rm ula u 1 ∧ v 1 ⊗ x ∗ + u ∧ v 7− → x ∗ ⊗ u 1 ∧ v 1 + u {− 1 } ⊗ u { 0 } ∧ v + v {− 1 } ⊗ u ∧ v { 0 } + u ∧ v . No w it is straightforw ard to c hec k that the isomorphism D l ≃ D r iden tiﬁes d l b ′ with d r b ′′ mo dulo V 2 D 0 ⊕ V 1 D 1 ⊕ D 2 ⊕ D 3 ⊕ · · · and h l b ′ with h r b ′′ . Indeed, one has u ( x ∗ ) + v { 0 } ( v {− 1 } ) + ρ ′′ ( v ) = u ( x ∗ ) + ρ ′ ( v ) and − ( u 1 , v 1 ) x ∗ − ( u { 0 } , v ) u {− 1 } − ( u, v { 0 } ) v {− 1 } − σ ′′ ( u ∧ v ) = − ( u 1 , v 1 ) x ∗ − σ ′ ( u ∧ v ) . D.3.7. Finally , let b ′ 1 : g / h − → g ′ b e another section of t he surjection g ′ − → g ′ / ( h ⊕ k ) ≃ g / h . The n w e can write b ′ 1 = b + t + t ′ with t : g / h − → h and t ′ : g / h − → k . Analogously , the sections ˜ b 1 : g / h − → g ∼ and b ′′ 1 : g / h − → g ′′ corresp onding to b ′ 1 ha v e the fo rm ˜ b 1 = ˜ b + t + ˜ t and b ′′ 1 = b ′′ + t + t ′′ with t ′′ = t ′ + ˜ t . Denote by τ , τ 1 : gl ( g ) − → gl ( g ) ∼ the sections corresp onding to direct sum de- comp ositions g = h ⊕ b ( g / h ) and g = h ⊕ b 1 ( g / h ) with b 1 = b + t , t : g / h − → h . Then o ne has τ 1 ( A ) − τ ( A ) = tr( tA h → g / h ) f or any A ∈ gl ( g ), where A h → g / h denotes the comp osition h − → g − → g − → g / h o f the endomorphism A with the injection h − → g and the surjection g − → g / h . Using this fo rm ula, one can ﬁnd that ˜ t ( u ) = −h u {− 1 } , t ( u { 0 } ) i , where h , i denotes the natural pairing h ∨ × h − → k . The natural isomorphism (id , a l ) : ( D l , d l b ′ , h l b ′ ) − → ( D l , d l b ′ 1 , h l b ′ 1 ) b et w een the CDG-coalgebras corresp o nding t o t he sections b ′ and b ′ 1 can b e computed easily; the linear function a l : D l 1 − → k is the comp osition o f the map id ⊗ ( ˇ ψ , ε ) : g / h ⊗ k C − → g / h ⊗ k ( h ∨ ⊕ k ) and the linear function g / h ⊗ k ( h ∨ ⊕ k ) − → k giv en b y the form ula u ⊗ x ∗ + v 7− → −h x ∗ , t ( u ) i − t ′ ( v ). Analog ously , the linear function a r in the natur a l isomorphism (id , a r ) : ( D r , d r b ′′ , h r b ′′ ) − → ( D r , d r b ′′ 1 , h r b ′′ 1 ) b etw een the CDG-coalgebras corresp onding to the sections b ′′ and b ′′ 1 is the comp osition of t he map 273 ( ˇ ψ , ε ) ⊗ id : C ⊗ k g / h − → ( h ∨ ⊕ k ) ⊗ k g / h and the linear function ( h ∨ ⊕ k ) ⊗ k g / h − → k giv en b y the f o rm ula x ∗ ⊗ u + v 7− → −h x ∗ , t ( u ) i − t ′′ ( v ). No w it is straightforw a r d to chec k that the isomorphism D l ≃ D r iden tiﬁes a l with a r . Indeed, −h x ∗ , t ( u ) i − h v {− 1 } , t ( v { 0 } ) i − t ′′ ( v ) = −h x ∗ , t ( u ) i − t ′ ( v ). Theorem D.3.1 is prov en.  D.4. Semiin v arian ts and semicontrain v arian ts. D.4.1. Let g b e a T ate Lie algebra, g ∼ − → g b e the canonical cen tral extension, and h ⊂ g b e a compact op en subalgebra; r ecall that the central extension g ∼ − → g splits canonically o v er h . Let N b e a discrete g ∼ -mo dule where the unit elemen t of k ⊂ g ∼ acts by min us the identit y . W e would lik e to construct a natural map ( g / h ⊗ k N ) h − → N h , where the sup erindex h denotes the h -inv a rian ts and the action of h in N is deﬁned in terms of the canonical splitting h − → g ∼ . Cho ose a section b : g / h − → g o f the surjection g − → g / h . The direct sum decom- p osition g ≃ h ⊕ b ( g / h ) leads to a section of the cen tral extension gl ( g ) ∼ − → gl ( g ), and consequen tly to a section of the central extens ion g ∼ − → g . Composing the section b with the latt er section, w e get a section ˜ b : g / h − → g ∼ of the surjection g ∼ − → g ∼ / ( h ⊕ k ) ≃ g / h . Consider the comp osition ( g / h ⊗ k N ) h − → g / h ⊗ k N − → g ∼ ⊗ k N − → N of the natural injection ( g / h ⊗ k N ) h − → h , the map g / h ⊗ k N − → g ∼ ⊗ k N induced b y the section ˜ b : g / h − → g ∼ , and t he g ∼ -action map g ∼ ⊗ k N − → N . Let us c hec k tha t this comp osition do es not dep end on the choice of b and its image lies in the subspace of in v ariants N h ⊂ N , so it provide s t he desired natural map ( g / h ⊗ k N ) h − → N h . Let u ⊗ n b e a formal not a tion for an elemen t o f g / h ⊗ k N . Denote by n 7− → n {− 1 } ⊗ n { 0 } the map N − → h ∨ ⊗ k N correspo nding to the h -action map h × N − → N . Rewriting the iden t ity x ˜ b ( u ) n = ˜ b ( u ) xn + [ x, ˜ b ( u )] n for x ∈ h in the no t a tion of D .3.4, w e obtain the iden tity ( ˜ b ( u ) n ) {− 1 } ⊗ ( ˜ b ( u ) n ) { 0 } = n {− 1 } ⊗ ˜ b ( u ) n { 0 } + u {− 1 } ⊗ ˜ b ( u { 0 } ) n − u ( n {− 1 } ) ⊗ n { 0 } − u { 0 } ( u {− 1 } ) ⊗ n . Now whenev er u ⊗ n is an h - inv ari- an t elemen t o f g / h ⊗ k N one has n {− 1 } ⊗ u ⊗ n { 0 } + u {− 1 } ⊗ u { 0 } ⊗ n = 0, hence ( ˜ b ( u ) n ) {− 1 } ⊗ ( ˜ b ( u ) n ) { 0 } = 0 and ˜ b ( u ) n is an h -inv ar ia n t elemen t of N . Let b 1 : g / h − → h b e another section of the surjection g − → g / h and ˜ b 1 : g / h − → g ∼ b e the corresp onding section of the surjection g ∼ − → g / h . According to D.3 .7, w e ha v e ˜ b 1 = ˜ b + t + ˜ t with a map t = b 1 − b : g / h − → h and the linear function ˜ t : g / h − → k giv en b y the form ula ˜ t ( u ) = −h u {− 1 } , t ( u { 0 } ) i . Let u ⊗ n b e an h -inv ari- an t elemen t of g / h ⊗ N ; t hen the equation n {− 1 } ⊗ u ⊗ n { 0 } + u {− 1 } ⊗ u { 0 } ⊗ n = 0 implies h n {− 1 } , t ( u ) i n { 0 } + h u {− 1 } , t ( u { 0 } ) i n = 0 and t ( u ) n − ˜ t ( u ) n = 0. The cok ernel N g , h of the natural map ( g / h ⊗ k N ) h − → N h that w e hav e constructed is called the space of ( g , h ) -se m iinvariants of a discrete g -mo dule N . The ( g , h )-semi- inv ariants are a mixture o f h -inv ariants and “coinv aria n ts along g / h ” . 274 D.4.2. F or a t op ological Lie algebra h and a n h - contr amo dule P the space of h - co- inv ariants P h is deﬁned as the maximal quotien t contramodule o f P where h acts b y zero, i. e., the coke rnel of t he con traaction map h ⊗ b P − → P . Let g b e a T ate Lie a lgebra with a compact op en subalgebra h . Let P b e a g ∼ -contramo dule where the unit elemen t of k ⊂ g ∼ acts b y the identit y . W e w ould lik e to construct a natural map P h − → Hom k ( g / h , P ) h , where the h -contraaction in Hom k ( g / h , P ) is induced b y the discrete action of h in g / h and the h -contraaction in P as explained in D.2.6 – D.2 .7. As ab o v e, choose a section b : g / h − → g a nd construct the corresp onding section ˜ b : g / h − → g ∼ . Consider the comp osition P − → Hom k ( g ∼ , P ) − → Hom k ( g / h , P ) − → Hom k ( g / h , P ) h of the map P − → Hom k ( g ∼ , P ) corresp onding to the action of g ∼ in P induced b y t he contraaction of g ∼ in P , the map Hom k ( g ∼ , P ) − → Hom k ( g / h , P ) in- duced b y the section ˜ b , and the natural surjection Hom k ( g / h , P ) − → Hom k ( g / h , P ) h . Let us c hec k that this comp osition factorizes through the natural surjection P − → P h and do es not dep end on the c hoice of b , so it deﬁnes the desired map P h − → Hom k ( g / h , P ) h . Let f a linear function h ∨ − → P and π P ( f ) ∈ P b e its image under the con- traaaction map. The image o f π P ( f ) under the comp osition P − → Hom k ( g ∼ , P ) − → Hom k ( g / h , P ) is giv en by the f orm ula u 7− → ˜ b ( u ) π P ( f ) = π P ( x ∗ 7→ ˜ b ( u ) f ( x ∗ )) − ˜ b ( u { 0 } ) f ( u {− 1 } ) + π P ( x ∗ 7→ f ( u ( x ∗ ))) − f ( u { 0 } ( u {− 1 } )) in the notation of D .3.4. This elemen t of Hom k ( g / h , P ) is the image of the elemen t g ∈ Hom k ( h ∨ , Hom k ( g / h , P )) giv en b y the f o rm ula g ( x ∗ )( u ) = ˜ b ( u ) f ( x ∗ ) + f ( u ( x ∗ )) under the con traaction map. If b 1 : g / h − → g is a diﬀeren t section, then ˜ b 1 ( u ) = b 1 ( u ) + t ( u ) − h u {− 1 } , t ( u { 0 } ) i and for a n y p ∈ P the elemen t of Hom k ( g / h , P ) given by the form ula u 7− → t ( u ) p − h u {− 1 } , t ( u { 0 } ) i p is the image of the elemen t g ∈ Hom k ( h ∨ , Hom k ( g / h , P )) given b y the form ula g ( x ∗ )( u ) = h x ∗ , t ( u ) i p under the contraaction map. The kerne l P g , h of the natural map P h − → Hom k ( g / h , P ) h is called the space of ( g , h ) -semic ontr ainvaria nts of a g -contramo dule P . The ( g , h )-semicontrainv aria nts are a mixture of h -coinv arian ts a nd “in v ariants a lo ng g / h ”. Remark. The ab ov e deﬁnitions of ( g , h )-semiinv arian ts and ( g , h )-semicontrainv ari- an ts agree with the deﬁnitions f rom C.5.1 – C.5.2 up to twis ts with a o ne- dimensional v ector space det( h ), essen tially for the follo wing reason. When g is a discrete Lie algebra, the cen tral extension g ∼ − → g ha s a canonical splitting induced by the canonical splitting of the cen tral extens ion gl ( g ) ∼ − → gl ( g ). When g is a T at e Lie algebra and h ⊂ g is a compact op en Lie subalgebra, the cen tral extension g ∼ − → g has a canonical splitting o v er h . When g is a discrete Lie algebra and h ⊂ g is a ﬁnite-dimensional Lie subalgebra, these tw o splittings do not agree ov er h ; instead, they diﬀer b y the mo dular c haracter of the Lie algebra h . 275 D.4.3. Let C b e a coalgebra endo w ed with a coaugmen tation (morphism of coalge- bras) e : k − → C , h b e a compact Lie algebra, and ψ : C × h − → k b e a pairing suc h that the map ˇ ψ : h ∨ − → C is a morphism of coalgebras and ψ annihilates e ( k ). F or an y righ t C -como dule N , the maximal sub como dule of N where the coaction of C is trivial can b e describ ed as the cotensor pro duct N  C k . Here a coa ction of C is called trivial if it is induced b y e ; the v ector space k is endo w ed with the t rivial coaction. There is a natural injectiv e map N  C k − → N h , whic h is an isomorphism provided that the assumption (iv) o f D .2.2 holds. Analogously , for a ny left C -contramo dule P the maximal quotien t C -contramo dule of P with the trivial contraaction can b e describ ed as t he space of cohomomorphisms Cohom C ( k , P ) . There is a natural surjectiv e map P h − → Cohom C ( k , P ) , whic h is also an isomorphism prov ided that the condition (iv) holds. Indeed, it suﬃces to consider the case when P = Hom k ( C , E ) is an induced C - contramo dule; in this case one only has to c hec k t ha t the k ernel of the comp o sition C − → C ⊗ k C − → C ⊗ k h ∨ of the comultiplication map and the map induced b y t he map ˇ ψ coincides with e ( k ). Let C b e a comm utativ e Hopf algebra. Then for an y rig ht C -como dule N and left C -como dule M there is a natural isomorphism N  C M ≃ ( N ⊗ k M )  C k , where the righ t C -como dule structure on N  C M is deﬁned using the antipo de and m ultiplication in C . Analogo usly , for an y left C -como dule M and left C -contramo dule P there is a natural isomorphism Cohom C ( M , P ) ≃ Cohom C ( k , Ho m k ( M , P )). D.4.4. No w let κ : ( g ′ , C ) − → ( g , C ) b e a cen tra l extension of T ate Harish-Chandra pairs with t he ke rnel k satisfying the assumption (iv) of D.2.2 and S l κ = S ≃ S r κ + κ 0 b e the corresp onding semialgebra ov er C . Lemma. (a) L et N b e a Harish-Chandr a mo dule with the c entr al char ge − κ − κ 0 and M b e a Harish-Chandr a mo d ule with the c entr al char ge κ ove r ( g , C ) ; in other wor ds, N is a ri g h t S r κ + κ 0 -semimo dule and M is a left S l κ -semimo dule. Assume that either N or M is a c oﬂat C -c om o dule. Then ther e is a natur al isomorphism N ♦ S M ≃ ( N ⊗ k M ) g , h , wher e the tensor pr o duct N ⊗ k M is a Harish-Chandr a mo dule with the c entr al char ge − κ 0 . (b) L et M b e a Harish-Chandr a mo dule with the c entr al char ge κ and P b e a Harish-Chandr a c ontr amo dule with the c entr al char ge κ + κ 0 over ( g , C ) ; in o ther wor ds, M is a left S l κ -semimo dule and P is a l e ft S r κ + κ 0 -semic o n tr amo d ule. Assume that either M is a c opr oje ctive C -c omo dule, or P is a c oinje ctive C - c ontr amo dule. Then ther e is a natur al isomorphism SemiHom S ( M , P ) ≃ Hom k ( M , P ) g , h , wher e the sp ac e Hom k ( M , P ) is a Harish-Chandr a c ontr amo dule with the c entr al char ge κ 0 . Pr o of . P art ( a ): denote by η i : N  C F i S  C M − → N  C M the map equal to the diﬀerence of the map induced b y the semiaction map F i S  C M − → M and the map induced by the semiaction map N  C F 1 S − → N . The map η 0 v anishes and the 276 quotien t space ( N  C F 1 S  C M ) / ( N  C F 0 S  C M is isomorphic to N  C ( F 1 S /F 0 S )  C M , hence the induced map ¯ η 1 : N  C ( F 1 S /F 0 S )  C M − → N  C M . The cok ernel of the map ¯ η 1 coincides with the semitensor pro duct N ♦ S M for the reasons explained in C.5.5. The cotensor pro duct N  C ( F 1 S /F 0 S )  C M ≃ N  C ( g / h ⊗ k C )  C M ≃ N  C ( g / h ⊗ k M ) is isomorphic t he space o f inv arian ts ( g / h ⊗ k M ⊗ k N ) h in view of the assumption (iv); this isomorphism coincides with the isomorphism N  C ( F 1 S /F 0 S )  C M ≃ ( g / h ⊗ k M ⊗ k N ) h induced b y the isomorphism F 1 S /F 0 S ≃ C ⊗ k g / h . Let us c hec k that this isomorphism iden tiﬁes t he map ¯ η 1 with the map whose cok ernel is, b y the deﬁnition, t he space of semiin v arian ts ( N ⊗ k M ) g , h . Cho ose a section b ′ : g / h − → g ′ and consider the corresp onding section b ′′ : g / h − → g ′′ . There is an isomorphism of righ t C -como dules C ⊕ g / h ⊗ k C ≃ F 1 S l κ giv en b y the formula c ′ 1 + u ′ ⊗ c ′ 7− → 1 ⊗ U ( h ) c ′ 1 + b ′ ( u ′ ) ⊗ U ( h ) c ′ and an analogous isomorphism of left C -como dules C ⊕ C ⊗ k g / h ≃ F 1 S r κ + κ 0 giv en by the formula c ′′ 1 + c ′′ ⊗ u ′′ 7− → c ′′ 1 ⊗ U ( h ) 1 + c ′′ ⊗ U ( h ) b ′′ ( u ′′ ). The induced isomorphism M ⊕ g / h ⊗ k M ≃ F 1 S l κ  C M ≃ F 1 U κ ( g ) ⊗ U ( h ) M is given b y the form ula m 1 + u ⊗ m 7− → 1 ⊗ U ( h ) m 1 + b ′ ( u ) ⊗ U ( h ) m . No w let z = n ⊗ ( c ′ 1 + u ′ ⊗ U ( h ) c ′ ) ⊗ m = n ⊗ ( c ′′ 1 + c ′′ ⊗ U ( h ) u ′′ ) ⊗ m b e an elemen t of N  C F 1 S  C M . Then the correspo nding elemen t of N  C F 1 U κ ( g ) ⊗ U ( h ) M can b e written as ε ( c ′ 1 ) n ⊗ 1 ⊗ U ( h ) m + ε ( c ′ ) n ⊗ b ′ ( u ′ ) ⊗ U ( h ) m , hence the ima g e of z under the map N  C F 1 S  C M − → N  C M induced b y the semiaction map F 1 S  C M − → M is equal to ε ( c ′ 1 ) n ⊗ m + ε ( c ′ ) n ⊗ b ′ ( u ′ ) m . Analogously , the image of z under the map N  C F 1 S  C M − → N  C M induced by the semiaction map N  C F 1 S − → N is equal to ε ( c ′′ 1 ) n ⊗ m − ε ( c ′′ ) b ′′ ( u ′′ ) n ⊗ m . One has ε ( c ′ 1 ) = ε ( c ′′ 1 ) by the condition (c) of D.3.1. Th us η 1 ( z ) = n ⊗ b ′ ( u ) m + b ′′ ( u ) n ⊗ m = ˜ b ( u )( n ⊗ m ), where u = ε ( c ′ ) u ′ = ε ( c ′′ ) u ′′ . P art (a) is prov en; the pro of of part (b) is completely analogous.  D.5. Semi-inﬁnite homology and cohomology. D.5.1. A discr ete right mo d ule N ov er a top ological asso ciative algebra R is a righ t R -mo dule suc h that the action map N × R − → N is contin uous with resp ect to the discrete to p ology of N . Equiv alently , a r ig h t R -mo dule N is discrete if the annihilator of an y elemen t of N is a n o p en righ t ideal in R . Let A and B b e top olog ical asso ciat ive algebras in whic h op en right ideals form bases of neighbor ho o ds o f zero. Then the top ological tensor product A ⊗ ! B has a natural structure o f top o logical asso ciative alg ebra with the same pr o p ert y . The tensor pro duct of a discrete righ t A - mo dule and a discrete righ t B -module is naturally a discrete righ t A ⊗ ! B -mo dule. Let κ : g ′ − → g b e a cen tral extension of top ological Lie a lgebras with the k ernel k . Then t he mo diﬁed en v eloping algebra U κ ( g ) = U ( g ′ ) / (1 U ( g ′ ) − 1 g ′ ) can b e endow ed with the to p ology where right ideals generated b y op en subspaces of g ′ form a base of neigh b orho o ds of zero [7, 6]. D enote the completion o f U κ ( g ) with resp ect to this top ology by U b κ ( g ); this is a top ological a sso ciativ e alg ebra. The category o f discrete 277 g ′ -mo dules where the unit elemen t of k ⊂ g ′ acts by minus the iden t it y is isomorphic to the category of discrete right U b κ ( g )-mo dules. D.5.2. Let R b e a top o logical associat ive algebra where op en right ideals form a base o f neigh b orho o ds of zero. Then for a n y k -v ector space P there is a natural map R ⊗ b ( R ⊗ b P ) − → R ⊗ b P induced by the m ultiplication in R ; it is constructed as the pro jectiv e limit o v er all op en right ideals U ⊂ R of the maps R/U ⊗ k ( R ⊗ b P ) − → R/U ⊗ k P induced by the discrete r igh t action of R in R/ U . A lef t c ontr amo dule ov er R is a ve ctor space P endo w ed with a linear map R ⊗ b P − → P satisfying the following con traasso ciativit y and unit y equations. First, the tw o maps R ⊗ b ( R ⊗ b P ) − → R ⊗ b P , one induced b y the m ultiplication in R and the other induced b y the contraaction map R ⊗ b P − → P , should ha v e equal comp ositions with t he contraaction map. Second, the comp o sition P − → R ⊗ b P − → P of the ma p induced b y the unit of R and t he con traaction map should b e equal to the identit y endomorphism of P . The category of left R -contramo dules is a b elian and there is a natural exact forgetful functor from it to the category of left mo dules o v er the algebra R con- sidered without any top ology (cf. Remark A.3). Notice also the isomorphisms R ⊗ b ( R ⊗ b P ) ≃ R ← ⊗ R ← ⊗ P ≃ ( R ← ⊗ R ) ⊗ b P , demonstrating the similarity of the ab ov e deﬁnition with the deﬁnition of a contramo dule ov er a Lie alg ebra g iv en in D .2.6. The ab o v e natural map R ⊗ b ( R ⊗ b P ) − → R ⊗ b P is induced b y the con tin uous multiplication map R ← ⊗ R − → R , which exists for an y top ological a sso- ciativ e algebra R where op en righ t ideals f o rm a base of neighborho o ds o f zero. Just as for Lie algebras, a structure of a discrete righ t R -mo dule on a v ector space N is giv en b y a contin uous linear map N ⊗ ∗ R ≃ N ← ⊗ R − → N , while a structure of a left R -contramo dule on a v ector space P is giv en b y a discon tinuous linear map R ⊗ ! P ≃ R ← ⊗ P − → P , where N and P are endo w ed with discrete top ologies. F or an y discrete righ t R -mo dule N and an y k -v ector space E , the v ector space Hom k ( N , E ) has a natural structure of left contramodule ov er R . The con traaction map R ⊗ b Hom k ( N , E ) − → Hom k ( N , E ) is constructed as the pro jectiv e limit ov er all o p en righ t ideals U ⊂ R of the maps R/U ⊗ k Hom k ( N , E ) − → Hom k ( N U , E ) giv en by the form ulas ¯ r ⊗ k g 7− → ( n 7− → g ( n ¯ r )) for ¯ r ∈ R/U , g ∈ Hom k ( N , E ), and n ∈ N U , where N U ⊂ N denotes t he subspace o f all elemen ts of N annihilated b y U . More generally , let A and B b e top olo g ical asso ciativ e a lg ebras where op en right ideals form bases o f neigh b orho o ds of zero, N b e a discrete righ t B - mo dule, and P b e a left A - contramo dule. Then the v ector space Hom k ( N , P ) has a natural structure of con tramo dule o v er A ⊗ ! B . The contraaction map ( A ⊗ ! B ) ⊗ b Hom k ( N , E ) − → Hom k ( N , E ) is constructed as the pro j ectiv e limit o ver all op en right ideals U ⊂ B of the comp ositions A ⊗ b ( B /U ⊗ k Hom k ( N , P )) − → A ⊗ b Hom k ( N U , P ) − → Hom k ( N U , A ⊗ b P ) − → Hom k ( N U , P ), where the ﬁrst map is induced b y the right B -action in N and the third map is induced b y the A - contraaction in P . 278 D.5.3. Let κ : g ′ − → g b e a cen tral extens ion of top ological Lie algebras with the k ernel k . Theorem. Assume that the top olo gic al Lie algebr a g ′ has a c ountable b ase of neigh- b orho o ds of zer o c o nsisting of op en Lie sub algebr as. The n the c ate gory of g ′ -c ontr a - mo dules wher e the unit elem e nt of k ⊂ g ′ acts by the ide n tity is isomorphic to the c ate go ry of left c ontr amo dules over the top olo gic al algebr a U b κ ( g ) . Pr o of . It is easy to see that the comp osition g ′ ⊗ b P − → U b κ ( g ) ⊗ b P − → P deﬁnes a g ′ -contramo dule structure on an y left U b κ ( g )-contramo dule P (so, in particular, U b κ ( g ) itself is a g ′ -contramo dule). Let us construct the functor in the opp osite direction. The standard homolo g ical Chev alley complex · · · − → V 2 k g ′ ⊗ k U κ ( g ) − → g ′ ⊗ k U κ ( g ) − → U κ ( g ) − → 0 is acyclic. F or any op en L ie subalgebra h ⊂ g ′ not containing k ⊂ g ′ , the complex · · · − → V 2 k h ⊗ k U κ ( g ) − → h ⊗ k U κ ( g ) − → h U κ ( g ) − → 0 is an acyclic subcomplex of the previous complex. T aking the quotien t complex and passing to the pro jectiv e limit ov er h , we obtain a split exact complex of top ological v ector spaces · · · − → V s , 2 g ′ ⊗ ! U κ ( g ) − → g ′ ⊗ ! U κ ( g ) − → U b κ ( g ) − → 0, where w e denote b y V s ,i g ′ the completion of V i k g ′ with resp ect to the top olog y with a base of neighborho o ds of zero fo rmed b y the subspaces V i k h and the en v eloping algebra U κ ( g ) is considered a s a discrete top olo gical v ector space. Applying the functor ⊗ b P , we obtain an exact sequence of v ector spaces V s , 2 g ′ ⊗ b ( U κ ( g ) ⊗ k P ) − → g ′ ⊗ b ( U κ ( g ) ⊗ k P ) − → U b κ ( g ) ⊗ b P − → 0 for a ny k -v ector space P . No w let P b e a g ′ -contramo dule where the unit elemen t o f k ⊂ g ′ acts b y the iden tit y; then, in particular, P is a g ′ -mo dule and a U κ ( g )-mo dule. It is clear from the ab o v e exact sequence that the comp osition g ′ ⊗ b ( U κ ( g ) ⊗ k P ) − → g ′ ⊗ b P − → P of the map induced b y the U κ ( g )-action map and the g ′ -con traaction ma p facto r izes through U b κ ( g ) ⊗ b P , pro viding the desired contraaction map U b κ ( g ) ⊗ b P − → P . Let us c hec k that this contraaction ma p satisﬁes the con traasso ciativit y equation. An y elemen t z of U b κ ( g ) ⊗ b P can b e presen ted in the fo r m z = P ∞ i =0 u i ⊗ p i with u i ∈ U κ ( g ) and p i ∈ P , where u i → 0 in U b κ ( g ) as i → ∞ and the inﬁnite sum is understo o d as the limit in the top ology of U b κ ( g ) ⊗ ! P . Let us denote the imag e of the elemen t P i u i ⊗ p i under the contraaction map U b κ ( g ) ⊗ b P − → P by P i u i p i ∈ P . In this notation, the U b κ ( g )-con traaction map is deﬁned by the formula P i ( x i 1 x i 2 · · · x i k i ) p i = P i x i 1 ( x i 2 · · · x i k i p i ) for any x i t ∈ g ′ and p i ∈ P suc h that x i 1 → 0 in g ′ as i → ∞ . W e ha v e to show that P i u i P j v ij p ij = P i,j ( u i v ij ) p ij for an y u i , v ij ∈ U κ ( g ) and p ij ∈ P suc h that u i → 0 a s i → ∞ and v ij → 0 a s j → ∞ for any i . Let us ﬁrst c hec k that P i x i P j y ij p ij = P i,j ( x i y ij ) p ij for any x i , y ij ∈ g ′ and p ij ∈ P such tha t x i → 0 in g ′ as i → ∞ and y ij → 0 in g ′ as j → ∞ f o r an y i . Cho o se an in teger j i for eac h i such that { y ij | j > j i } con v erges to zero in g ′ as i + j → ∞ . Then we ha v e P i,j ( x i y ij ) p ij = P j 6 j i x i ( y ij p ij ) + P j >j i y ij ( x i p ij ) + P j >j i [ x i , y ij ] p ij . T o 279 c hec k that P i x i P j >j i y ij p ij = P j >j i y ij ( x i p ij ) + P j >j i [ x i , y ij ] p ij , apply the equation on the con tra a ction map of a con tramo dule ov er a to p ological L ie algebra to the elemen t P j >j i x i ∧ y ij ⊗ p ij of the v ector space V ∗ , 2 ( g ′ ) ⊗ b P . It follows that P i x i P j v ij p ij = P i,j ( x i v ij ) p ij for any x i ∈ g ′ , v ij ∈ U κ ( g ), and p ij ∈ P such that x i → 0 in g ′ as i → ∞ and v ij → 0 in U b κ ( g ) as j → ∞ for any i . Indeed, assuming that v ij = y ij 1 y ij 2 · · · y ij k ij , where y ij t ∈ g ′ and y ij 1 → 0 in g ′ as j → ∞ , w e ha v e P i x i P j ( y ij 1 y ij 2 · · · y ij k ij ) p ij = P i x i P j y ij 1 ( y ij 2 · · · y ij k ij p ij ) = P i,j ( x i y ij 1 )( y ij 2 · · · y ij k ij p ij ) = P i,j ( x i y ij 1 y ij 2 · · · y ij k ij ) p ij . F urthermore, it fo llows that x 1 · · · x s P j v j p j = P j ( x 1 · · · x s v j ) p j for an y x t ∈ g ′ , v j ∈ U κ ( g ), and p j ∈ P suc h that v j → 0 in U b κ ( g ) a s j → ∞ . Now to ch ec k tha t P i u i P j v ij p ij = P i,j ( u i v ij ) p ij , we can assume that u i = x i 1 x i 2 · · · x i k i , where x i t ∈ g ′ and x i 1 → 0 in g ′ as i → ∞ . Then we ha v e P i ( x i 1 x i 2 · · · x i k i ) P j v ij p ij = P i x i 1 ( x i 2 · · · x i k i P j v ij p ij ) = P i x i 1 P j ( x i 2 · · · x i k i v ij p ij ) = P i,j ( x i 1 x i 2 · · · x i k i v ij ) p ij .  Question. Can one construct an isomorphism b etw een the categories of “ g -contra - mo dules with cen tral c harge κ ” and left U b κ ( g )-contramo dules without the coun tabil- it y assumption on the top ology of g ′ ? D.5.4. The following weak er ve rsion of Theorem D.5 .3 holds without the countabilit y assumption. Let κ : g ′ − → g b e a cen tral extension of top o lo gical Lie algebras with the k ernel k ; assume that op en subalgebras from a base of neigh b orho o ds of zero in g ′ . Let B b e a top ological a sso ciativ e algebra where op en righ t ideals form a base of neighborho o ds of zero, N b e a discrete righ t B -mo dule, a nd P b e a g ′ -contramo d- ule where the unit elemen t of k ⊂ g ′ acts by the iden tit y . Then one can deﬁne the con traaction map ( g ′ ⊗ ! B ) ⊗ b Hom k ( N , P ) − → Hom k ( N , P ) as in D.5.2. Consider the iterated con traaction map (( g ′ ⊗ ! B ) ← ⊗ ( g ′ ⊗ ! B )) ⊗ b Hom k ( N , P ) ≃ ( g ′ ⊗ ! B ) ⊗ b (( g ′ ⊗ ! B ) ⊗ b Hom k ( N , P )) − → Hom k ( N , P ). It w as noticed in [6] that a top ological asso ciative algebra A has the prop ert y that op en righ t ideals form a base of neighborho o ds of zero if and only if the multiplication map A ⊗ ∗ A − → A factorizes through A ← ⊗ A . Let K denote the k ernel of the m ultiplication map ( g ′ ⊗ ! B ) ← ⊗ ( g ′ ⊗ ! B ) − → U b κ ( g ) ⊗ ! B . W e claim that the comp osition of the injection K ⊗ b Hom k ( N , P ) − → (( g ′ ⊗ ! B ) ← ⊗ ( g ′ ⊗ ! B )) ⊗ b Hom k ( N , P ) and the iterated con traaction map (( g ′ ⊗ ! B ) ← ⊗ ( g ′ ⊗ ! B )) ⊗ b Hom k ( N , P ) − → Hom k ( N , P ) v anishes. F or an y top olo gical vec tor spaces U , V , X , Y there is a natural map ( U ⊗ ! X ) ← ⊗ ( V ⊗ ! Y ) − → ( U ← ⊗ V ) ⊗ ! ( X ← ⊗ Y ). The comp osition ( g ′ ⊗ ! B ) ← ⊗ ( g ′ ⊗ ! B ) − → ( g ′ ← ⊗ g ′ ) ⊗ ! ( B ← ⊗ B ) − → ( g ′ ← ⊗ g ′ ) ⊗ ! B induces the map (( g ′ ⊗ ! B ) ← ⊗ ( g ′ ⊗ ! B )) ⊗ b Hom k ( N , P ) − → (( g ′ ← ⊗ g ′ ) ⊗ ! B ) ⊗ b Hom k ( N , P ). A contraaction map (( g ′ ← ⊗ g ′ ) ⊗ ! B ) ⊗ b Hom k ( N , P ) − → Hom k ( N , P ) can b e deﬁned in terms of the discrete r ig h t action of B in N and the iterated con tr a action map ( g ′ ← ⊗ g ′ ) ⊗ b P − → P . The iterated con tra a ction map 280 (( g ′ ⊗ ! B ) ← ⊗ ( g ′ ⊗ ! B )) ⊗ b Hom k ( N , P ) − → Hom k ( N , P ) is equal to the comp osition (( g ′ ⊗ ! B ) ← ⊗ ( g ′ ⊗ ! B )) ⊗ b Hom k ( N , P ) − → (( g ′ ← ⊗ g ′ ) ⊗ ! B ) ⊗ b Hom k ( N , P ) − → Hom k ( N , P ) of the ab o v e induced map and con traaction map. Let Q denote the k ernel of the m ultiplication map g ′ ← ⊗ g ′ − → U b κ ( g ). The ima g e of K under the map ( g ′ ⊗ ! B ) ← ⊗ ( g ′ ⊗ ! B ) − → ( g ′ ← ⊗ g ′ ) ⊗ ! B is con tained in Q ⊗ ! B . So it suﬃces to c hec k that t he comp osition of the injection Q ⊗ b P − → ( g ′ ← ⊗ g ′ ) ⊗ b P and the iterated con traaction map ( g ′ ← ⊗ g ′ ) ⊗ b P − → P v anishes. The top o lo gical v ector space Q is the top ological pro jectiv e limit of the kerne ls of m ultiplication maps g ′ / h ⊗ ∗ g ′ − → U κ ( g ) / h U κ ( g ) ov er all o p en subalgebras h ⊂ g ′ not containing k ⊂ g ′ . Since the inte rsection o f h U κ ( g ) and g ′ 2 inside U κ ( g ′ ) is equal to hg ′ , the ke rnel of t he ( no n top olog ical) multiplication map g ′ ⊗ k g ′ − → U κ ( g ′ ) maps surjectiv ely on to the ke rnels we are in terested in. This nontopolo gical k ernel is the image of the map V 2 k ( g ′ ) − → g ′ ⊗ k g ′ giv en b y the form ula x ∧ y 7− → x ⊗ y − y ⊗ x − 1 ⊗ [ x, y ]. The kerne l of the comp osition V 2 k ( g ′ ) − → g ′ ⊗ k g ′ − → g ′ / h ⊗ k g ′ is the subspace V 2 k ( h ) ⊂ V 2 k ( g ′ ). Hence the k ernel of the map g ′ / h ⊗ ∗ g ′ − → U κ ( g ) / h U κ ( g ) is the subspace V 2 k ( g ′ ) / V 2 k ( h ) ⊂ g ′ / h ⊗ ∗ g ′ , em b edded b y the ab o v e formula, endow ed with the induced top olo g y of a closed subspace. One can easily c hec k that this top ology on V 2 k ( g ′ ) / V 2 k ( h ) is the top ology of the quotient space V ∗ , 2 ( g ′ ) / V ∗ , 2 ( h ). Th us the top ological ve ctor space K is isomorphic t o V ∗ , 2 ( g ′ ). D.5.5. The f o llo wing constructions ar e due t o Beilinson and Drinfeld [7]. Let V b e a T ate v ector space and E ⊂ V b e a compact op en sub space. The graded v ector space o f semi-inﬁnite forms V ∞ / 2 E ( V ) = L i V ∞ / 2+ i E ( V ) is deﬁned as the inductiv e limit of the spaces V k ( V /U ) ⊗ k det( E /U ) ∨ o v er a ll compact o p en subspaces U ⊂ E . Here det( X ) denotes the top exterior p o w er of a ﬁnite-dimensional v ector space X and V k ( W ) denotes the direct sum of all exterior p o w ers of a v ector space W ; the gra ding on V k ( V /U ) ⊗ k det( E /U ) ∨ is deﬁned so that V j k ( V /U ) ⊗ k det( E /U ) ∨ is the comp onen t of degree j − dim ( E /U ). The limit is tak en ov er the maps induced b y the nat ura l maps V j k ( V /U ′ ) ⊗ k det( U ′ /U ′′ ) − → V j + m k ( V /U ′′ ), where U ′′ ⊂ U ′ and m = dim( U ′ /U ′′ ). The spaces o f semi-inﬁnite forms corresp onding to diﬀeren t compact op en subspaces E ⊂ V , only diﬀer b y a dimensional shift and a determinantal t wist: if F ⊂ V is another compact op en subspace, then there are natural isomorphisms V ∞ / 2+ i F ( V ) ≃ V ∞ / 2+ i +dim( E ,F ) E ( V ) ⊗ k det( E , F ), where dim( E , F ) = dim( E /E ∩ F ) − dim( F /E ∩ F ) and det( E , F ) = det( E /E ∩ F ) ⊗ k det( F /E ∩ F ) ∨ . Denote by Cl( V ) the algebra of endomorphisms of the vector space V ∞ / 2 E ( V ) endo w ed with the to p ology where annihilato r s of ﬁnite-dimensional subspaces of V ∞ / 2 E ( V ) form a base of neigh b orho o ds of zero. Clearly , the top olog ical asso ciative algebra Cl( V ) do es not depend on the choice o f a compact op en subspace E ⊂ V ; 281 op en left ideals fo rm a base of neigh b orho o ds of zero in Cl( V ) . Denote b y Cl i ( V ) t he closed subspace of homogeneous endomorphisms of degree i in Cl( V ). The Cliﬀord algebra Cl( V ⊕ V ∨ ) a cts naturally in V ∞ / 2 E ( V ), so there is a morphism of asso ciative algebras e : Cl( V ⊕ V ∨ ) − → Cl( V ); in particular, the map e sends V to Cl 1 ( V ) and V ∨ to Cl − 1 ( V ). Let V ! ,i ( V ∨ ) denote the completion of V i k ( V ∨ ) with resp ect to the top ology with the ba se of neigh b orho o ds of zero formed b y the subspaces U ∧ V i − 1 k ( V ∨ ) ⊂ V i k ( V ∨ ), where U ⊂ V ∨ is an op en subspace. The comp osition V i k ( V ∨ ) − → Cl( V ⊕ V ∨ ) − → Cl( V ) can b e extended b y contin uity to a map V ! ,i k ( V ∨ ) − → Cl( V ), whic h we will denote also b y e . The construction of D.1.7 pro vides a morphism of to p ological Lie alg ebras gl ( V ) ∼ − → Cl 0 ( V ). Let g b e a T a te Lie algebra and κ 0 : g ∼ − → g b e its canonical cen tral extension. Consider the top olo gical tens or pro duct g ∼ ⊗ ! Cl( g ) op , where Cl( g ) op denotes the top ological a lg ebra opp o site to Cl( g ); this to p ological tensor pro duct is a bimo dule o v er Cl( g ) op . The unit elemen ts of Cl( g ) op and k ⊂ g ∼ induce em b eddings of g ∼ and Cl( g ) op in to g ∼ ⊗ ! Cl( g ) op . Consider the diﬀerence of the comp o sition g ∼ − → gl ( g ) ∼ − → Cl( g ) ≃ Cl( g ) op − → g ∼ ⊗ ! Cl( g ) op and the em b edding g ∼ − → g ∼ ⊗ ! Cl( g ) op ; this diﬀerence maps k ⊂ g ∼ to zero and so induces a natural map l : g − → g ∼ ⊗ ! Cl 0 ( g ) op . The comp osition of the map l with the em b edding g ∼ ⊗ ! Cl − 1 ( g ) op − → U b κ 0 ( g ) ⊗ ! Cl − 1 ( g ) op is an anti-homomorphism of Lie algebras, i. e., it tr ansforms the comm utators to minus the comm uta tors. Denote b y δ : g ∨ − → V ! , 2 ( g ∨ ) the contin uous linear map g iv en b y the f o rm ula δ ( x ∗ ) = x ∗ { 1 } ∧ x ∗ { 2 } , where h x ∗ , [ x ′ , x ′′ ] i = h x ∗ { 1 } , x ′′ ih x { 2 } , x ′ i − h x ∗ { 1 } , x ′ ih x ∗ { 2 } , x ′′ i for x ∗ ∈ g ∨ , x ′ , x ′′ ∈ g . D eﬁne the map χ : g ⊗ g ∨ − → g ∼ ⊗ ! Cl − 1 ( g ) op b y the formula χ ( x ⊗ x ∗ ) = l ( x ) e ( x ∗ ) op − e ( x ) op e ( δ ( x ∗ )) op = e ( x ∗ ) op l ( x ) − e ( δ ( x ∗ )) op e ( x ) op , where a op denotes the elemen t of Cl( g ) op corresp onding t o an elemen t a ∈ Cl( g ), and extend this map b y con tinuit y to a map g ⊗ ! g ∨ − → g ∼ ⊗ ! Cl − 1 ( g ) op . Identify g ⊗ ! g ∨ with End( g ) and set d = χ (id g ) ∈ g ∼ ⊗ ! Cl − 1 ( g ) op . Denote the image of d under the em b edding g ∼ ⊗ ! Cl − 1 ( g ) op − → U b κ 0 ( g ) ⊗ ! Cl − 1 ( g ) op also b y d . Using the equalit y [ l ( x ) , e ( y ) op ] = − e ([ x, y ]) op , o ne can c hec k that [ d , e ( x ) op ] = l ( x ) and [[ d 2 , e ( x ) op ] , e ( y ) op ] = 0 for all x , y ∈ g , where [ , ] denotes the sup ercomm utator with r esp ect to the gr ading in whic h U b κ 0 ( g ) ⊗ ! Cl i ( g ) op lies in the degree i . It is easy to see that any elemen t of Cl i ( g ) sup ercomm uting with e ( x ) for all x ∈ g is zero when i < 0 ; hence the same applies to elemen ts of U b κ 0 ( g ) ⊗ ! Cl i ( g ) op with i < 0. It follows that d is the unique elemen t of U b κ 0 ( g ) ⊗ ! Cl − 1 ( g ) op satisfying the equation [ d , e ( x ) op ] = l ( x ) and that d 2 = 0 . D.5.6. Let g b e a T ate L ie algebra and E ⊂ g b e a compact o p en vec tor subspace. Let N be a discrete g ∼ -mo dule where the unit elemen t of k ⊂ g ∼ acts by min us the 282 iden tit y . Then N can b e considered as a discrete righ t U b κ 0 ( g )-mo dule and V ∞ / 2 E ( g ) is a discrete rig h t Cl( g ) op -mo dule, so t he tensor pro duct V ∞ / 2 E ( g ) ⊗ k N is a discrete rig h t mo dule ov er U b κ 0 ( g ) ⊗ ! Cl( g ) op . The a ction of the elemen t d ∈ U b κ 0 ( g ) ⊗ ! Cl − 1 ( g ) op deﬁnes a diﬀeren tial d ∞ / 2 of degree − 1 on the gra ded v ector space C E ∞ / 2+ • ( g , N ) with the comp onen ts C E ∞ / 2+ i ( g , N ) = V ∞ / 2+ i E ( g ) ⊗ k N . One has d 2 ∞ / 2 = 0, since d 2 = 0 ; so C E ∞ / 2+ • ( g , N ) b ecomes a complex. This complex is called the semi- inﬁnite homolo g ic al c o mplex and its ho mology is called the se m i-inﬁnite hom olo gy of a T ate Lie algebra g with co eﬃcien ts in a discrete g ∼ -mo dule N . Let P b e a g ∼ -contramo dule where the unit elemen t of k ⊂ g ∼ acts b y the iden- tit y . First assume that g has a coun table base o f neigh b orho o ds of zero. Then P can b e considered as a left U b κ 0 ( g )-contramo dule a nd V ∞ / 2 E ( g ) is a discrete righ t Cl( g ) op -mo dule, so the space of homomorphisms Hom k ( V ∞ / 2 E ( g ) , P ) is a left contra- mo dule ov er U b κ 0 ( g ) ⊗ ! Cl( g ) op . The a ction of the elemen t d ∈ U b κ 0 ( g ) ⊗ ! Cl − 1 ( g ) op deﬁnes a diﬀeren tial d ∞ / 2 of degree 1 on the graded vec tor space C ∞ / 2+ • E ( g , P ) with the comp onen ts C ∞ / 2+ i E ( g , P ) = Hom k ( V ∞ / 2+ i E ( g ) , P ). One has ( d ∞ / 2 ) 2 = 0, since d 2 = 0. Without the countabilit y assumption, the elemen t d ∈ g ∼ ⊗ ! Cl − 1 ( g ) op still acts on the gra ded v ector space C ∞ / 2+ • E ( g , P ) by an op erator d ∞ / 2 of degree 1. By the result of D.5.4, the iden tit y d 2 = 0 in U b κ 0 ( g ) ⊗ ! Cl( g ) op implies the equa- tion ( d ∞ / 2 ) 2 = 0 ; so C ∞ / 2+ • E ( g , P ) b ecomes a complex. This complex is called the semi-inﬁnite c ohomolo gi c al c omp lex and its cohomolog y is called the semi-inﬁnite c ohomo l o gy of a T ate Lie algebra g with co eﬃcien ts in a g ∼ -contramo dule P . D.6. C omparison theorem. D.6.1. The corr esp o ndence h 7− → L = h ∨ pro vides an anti-equiv alence b etw een the categories of compact Lie algebras and Lie coalgebras. The corresp ondence b etw een the action maps h × M − → M and the coaction maps M − → h ∨ ⊗ k M deﬁnes an equiv alence b et w een the categories o f discrete h -mo dules and L - como dules. A Lie coalgebra L is called c oni l p otent if it is a ﬁltered inductive limit of ﬁnite- dimensional Lie coalgebras dual to ﬁnite-dimensional nilp oten t Lie algebras; in o ther w ords, L is conilp ot en t if the dual compact Lie algebra h is pr oni l p otent . A como dule M ov er a Lie coalgebra L is called c onilp otent if it is an inductive limit of ﬁnite- dimensional como dules whic h can b e r epresen ted as iterated extensions of trivial como dules (that is como dules with a zero coaction map); analog ously o ne deﬁnes nilp otent discrete mo dules o v er top o logical Lie alg ebras. A coassociative coalgebra C endo w ed with a coaugmen tation map k − → C is called c onilp otent if for ev ery elemen t c of the coalgebra without counit C / im k there exists a p o sitive integer i suc h that the iterated comultiplication map C / im k − → ( C / im k ) ⊗ i annihilates c . 283 F or a conilp ot en t Lie coalgebra L , the c onilp otent c o enveloping c o algebr a C ( L ) is constructe d as follo ws. Consider the category of ﬁnite-dimensional conilp oten t L -como dules to gether with the fo r g etful functor f r o m it to the category of ﬁnite- dimensional v ector spaces; by a result of [19], this category is equiv alen t (actually , iso- morphic) to the category of ﬁnite-dimensional left como dules o v er a certain uniquely deﬁned coalgebra C ( L ) together with the forgetful functor fro m this categor y to the category of ﬁnite-dimensional v ector spaces. Cle arly , the category of (arbitrary) left C ( L )-como dules is isomorphic to the category of conilp oten t L -como dules. The trivial L -como dule k deﬁnes a coaugmen tation k − → C ( L ); since this is the only irreducible left C ( L )- como dule, the coalgebra C ( L ) is conilp otent. The coa lg ebra C ( L ) is the univ ersal ﬁnal ob ject in the category of conilp oten t coal- gebras C endo w ed with a Lie coa lg ebra morphism C − → L suc h that the comp osition k − → C − → L v anishes. Indeed, there is a morphism of Lie coalgebras C ( L ) − → L , since there is a natura l L -como dule structure on ev ery left C ( L )-como dule, and in particular, on the left como dule C ( L ). Con v ersely , a morphism C − → L with the ab ov e pro p erties deﬁnes a functor assigning to a left C -como dule M a conilp otent L -como dule structure on t he same v ector space M , hence a left C ( L )-como dule struc- ture on M ; this induces a coalgebra morphism C − → C ( L ). Since the category of ﬁnite-dimensional conilp oten t L -como dules is a tensor catego r y with duality , the coalgebra C ( L ) acquires a Hopf a lgebra structure. Let h b e t he compact Lie algebra dual to L ; t hen the pairing φ : C ( L ) × U ( h ) − → k is nondegenerate in C , since the morphism C ( L ) − → L factorizes through the quotient coalgebra of C ( L ) b y the k ernel of φ , so a nonzero ke rnel w ould b e con tradict the univ ersalit y pro p ert y . Let M be a conilp oten t C -como dule; set C = C ( L ) and C + = C / im k . Then the natural surjectiv e morphism fr om the r educed cobar complex M − → C + ⊗ k M − → C + ⊗ k C + ⊗ k M − → · · · c omputing Cotor C ( k , M ) ≃ Ext C ( k , M ) onto t he cohomological Chev alley complex M − → L ⊗ k M − → V 2 k L ⊗ k M − → · · · is a quasi-isomorphism. It suﬃces to c hec k this for a ﬁnite-dimensional Lie coalgebra L dual to a ﬁnite-dimensional nilp otent Lie algebra h ; essen tially , one has show that the fully fa it hf ul functor from the category of nilp o ten t h -mo dules to the category of arbitrar y h -mo dules induces isomorphisms on the Ext spaces. This w ell-kno wn fact can b e pro v en b y induction on the dimension of h using the Serre–Ho c hsc hild sp ectral sequences for b oth t yp es of cohomolog y under consideration. The ke y step is to c hec k that for a Lie sub coalgebra E ⊂ L the C ( L / E ) - como dule C ( L ) is inj ective and the E -como dule C ( E ) is the como dule of L / E -inv a r ia n ts in t he L -como dule C ( L ); it suﬃces to consider the case when L / E is one-dimensional. D.6.2. Let g b e a T ate Lie algebra and h ⊂ g b e a compact o p en Lie subalgebra. Assume that h is pronilp oten t and the dis crete h -mo dule g / h is nilp oten t. Then 284 the conilp otent coalgebra C = C ( h ∨ ) coacts con tin uously in g , making ( g , C ) a T ate Harish-Chandra pair. L et k − → g ′ − → g b e a cen tral extension of T ate Lie algebras endo w ed with a splitting ov er h ; then there are a T at e Harish-Chandra pair ( g ′ , C ) and a cen tral extension of T ate Harish-Chandra pairs κ : ( g ′ , C ) − → ( g , C ) with the k ernel k . D enote b y κ 0 : ( g ∼ , C ) − → ( g , C ) the canonical cen tral extension. Set S l κ = S l κ ( g , C ) ≃ S r κ + κ 0 ( g , C ) = S r κ + κ 0 . Theorem. ( a) L e t N • b e a c om plex of right S r κ + κ 0 -semimo dules an d M • b e a c omplex of le f t S l κ -semimo dules; in other wor ds, N • is a c o mplex of Harish- C handr a mo dules with the c entr al char ge − κ − κ 0 and M • is a c om p lex o f Harish-Chandr a mo d ules with the c entr al c h ar ge κ over ( g , C ) . Th e n the total c omplex of the semi-inﬁnite homolo g ic al bic omple x C h ∞ / 2+ • ( g , N • ⊗ k M • ) c onstructe d by taking inﬁnite dir e ct sums along the diago nals r e pr esents the obje ct SemiT or S l κ ( N • , M • ) in the d e ri v e d c ate gory of k -ve ctor sp ac es. Her e the tensor pr o duct N • ⊗ k M • is a c omple x of Harish-Chandr a mo dules with the c entr al cha r ge − κ 0 . (b) L et M • b e a c omp l e x of left S l κ -semimo dules and P • b e a c omplex of left S r κ + κ 0 -semic o n tr amo d ules; in other wo r ds, M • is a c o mplex of Harish-Chand r a mo d- ules with the c entr al char ge κ and P • is a c omplex of Harish -Chandr a c ontr a mo dules with the c entr al c har ge κ + κ 0 over ( g , C ) . Th en the total c omple x of the sem i- inﬁnite c ohomolo gic al b i c omplex C ∞ / 2+ • h ( g , Hom k ( M • , P • )) c onstructe d by taking in- ﬁnite pr o d ucts along the diagonals r epr esents the obje ct SemiExt S l κ ( M • , P • ) in the derive d c ate gory of k -ve ctor sp ac es. Her e Hom k ( M • , P • ) is a c omplex of Haris h - Chandr a c o ntr amo dules with the c entr al char ge κ 0 . Pr o of . P art ( a ): set S l − κ 0 ≃ S = S r 0 . Consider t he semi-inﬁnite homological complex C h ∞ / 2+ • ( g , S ) of the g ∼ -mo dule S with the discrete g ∼ -mo dule structure originating from the left S l − κ 0 -semimo dule structure. The complex C h ∞ / 2+ • ( g , S ) is a complex o f righ t S r 0 -semimo dules. Let us c hec k that it is a semiﬂat complex natur a lly isomorphic to the semimo dule k in the semideriv ed catego r y of r ig h t S r 0 -semimo dules. Let F i S denote the increasing ﬁltration of the semialgebra S in tro duced in D.3.1. Set F i  V ∞ / 2 h ( g )  = V i k ( g ) ∧ V ∞ / 2 h ( h ). Denote b y F the induced ﬁltratio n of the tensor pro duct C h ∞ / 2+ • ( g , S ) = V ∞ / 2 h ( g ) ⊗ k S ; this is an increasing ﬁltration of the complex of right S r 0 -semimo dules C h ∞ / 2+ • ( g , S ) by sub complexes of r igh t C -como dules. The complex gr F C h ∞ + • ( g , S ) can b e identiﬁed with the to tal complex of the cohomolog ical Chev alley bicomplex ^ k ( h ∨ ) ⊗ k ^ k ( g / h ) ⊗ k Sym k ( g / h ) ⊗ k C of the complex of h ∨ -como dules V k ( g / h ) ⊗ k Sym k ( g / h ) ⊗ k C obtained as the t ensor pro duct of t he Koszul complex V k ( g / h ) ⊗ k Sym k ( g / h ) with the coaction of h ∨ induced 285 b y the coaction in g / h and the left C -como dule C with the induced h ∨ -como dule structure. It follows that the cone of the injection F 0 C h ∞ / 2+ • ( g , S ) − → C h ∞ / 2+ • ( g , S ) is a coacyclic complex of right C -como dules. The complex F 0 C h ∞ / 2+ • ( g , S ) is naturally isomorphic to t he cohomolog ical Chev al- ley complex V k ( h ∨ ) ⊗ k C ; it is a complex of righ t C - como dules b ounded from b elo w and endo w ed with a quasi-isomorphism of complexes of rig h t C -como dules k − → F 0 C h ∞ / 2+ • ( g , S ). Let us c hec k t ha t the right S r 0 -semimo dule structure on H 0 C h ∞ / 2+ • ( g , S ) ≃ k corresp onds to the trivial g - mo dule structure. The unit ele- men t of this homology group can b e represen ted by the cycle λ ⊗ 1 ∈ C h ∞ / 2+0 ( g , S ), where λ denotes the unit elemen t of k ≃ V ∞ / 2+0 h ( h ) ⊂ V ∞ / 2+0 h ( g ) and 1 ∈ C ⊂ S is the unit (coaug men tation) elemen t of C . Then fo r an y z ∈ g one has ( λ ⊗ 1 ) z = λ ⊗ (1 ⊗ U ( h ) z ) = λ ⊗ ( ˜ b ( z (0) ) ⊗ U ( h ) s ( z ( − 1) )) = d ∞ / 2 (( z (0) ∧ λ ) ⊗ s ( z ( − 1) )), where z denotes t he image of z in g / h and ˜ b : g / h − → g ∼ is the section corresp onding to any section b : g / h − → g . The second equation holds, since the elemen ts 1 ⊗ U ( h ) z and ˜ b ( z (0) ) ⊗ U ( h ) s ( z ( − 1) ) of F 1 S ha v e the same images in F 1 S /F 0 S and are b o th annihi- lated b y the left action of h and the map δ l ˜ b = δ r b . T o c heck the third equation, one can use the sup ercomm utation relation [ d , e ( y ) op ] = l ( y ) for y ∈ g . No w let τ > 0 C ∞ / 2+ • ( g , S ) denote the quotien t complex of canonical t r uncatio n of the complex o f righ t S r 0 -semimo dules C h ∞ / 2+ • ( g , S ) concentrated in the nonnegativ e cohomological (nonp ositive homolog ical) degrees; then there are natural morphisms of complexes of right S r 0 -semimo dules k − − → τ > 0 C h ∞ / 2+ • ( g , S ) ← − − C h ∞ / 2+ • ( g , S ) with C -coa cyclic cones. Indee d, recall that any acyclic complex b ounded from b elo w is coacyclic. The em b edding F 0 C h ∞ / 2+ • ( g , S ) − → C h ∞ / 2+ • ( g , S ) has a C -coacyclic cone, as has the comp o sition F 0 C h ∞ / 2+ • ( g , S ) − → C h ∞ / 2+ • ( g , S ) − → τ > 0 C h ∞ / 2+ • ( g , S ), so the cone of the map C h ∞ / 2+ • ( g , S ) − → τ > 0 C h ∞ / 2+ • ( g , S ) is also C -coa cyclic. F or an y complex of left S l − κ 0 -semimo dules K • , the semitensor pro duct C h ∞ / 2+ • ( g , S ) ♦ S K • is naturally isomorphic to the total complex of the bicomplex C h ∞ / 2+ • ( g , K • ), constructed b y taking inﬁnite direct sums a long the diagonals. Consider the increas- ing ﬁltra tion of the tota l complex of C h ∞ / 2+ • ( g , K • ) = V ∞ / 2 h ( g ) ⊗ k K • induced by the ab ov e ﬁltratio n F of V ∞ / 2 h ( g ). The asso ciated graded quotien t complex of this ﬁltration can b e identiﬁed with the total complex of the cohomolo g ical Chev alley bi- complex V k ( h ∨ ) ⊗ k V k ( g / h ) ⊗ k K • of the t ensor pro duct of the gra ded h ∨ -como dule V k ( g / h ) and the complex K • with the h ∨ -como dule structure induced b y the left C -como dule structure. It follows that the complex C h ∞ / 2+ • ( g , S ) ♦ S K • is acyclic 286 whenev er a complex of left S l − κ 0 -semimo dules K • is C -coacyclic, so the complex of righ t S r 0 -semimo dules C h ∞ / 2+ • ( g , S ) is semiﬂat. The tensor pro duct C h ∞ / 2+ • ( g , S ) ⊗ k N • of the complex of Harish-Chandra mo dules C h ∞ / 2+ • ( g , S ) with cen tral charge 0 a nd the complex of Harish-Chandra mo dules N • with cen tral c harge − κ − κ 0 is a complex of Harish-Chandra mo dules with the cen tral c harge − κ − κ 0 . This complex of righ t S r κ + κ 0 -semimo dules is semiﬂat and naturally isomorphic to N • in the semideriv ed category of right S r κ + κ 0 -semimo dules. The latt er is clear, and to ch ec k the former, notice the isomorphisms of L emma D.4.4(a) ( C h ∞ / 2+ • ( g , S ) ⊗ k N • ) ♦ S l κ L • ≃ ( C h ∞ / 2+ • ( g , S ) ⊗ k N • ⊗ k L • ) g , h ≃ C h ∞ / 2+ • ( g , S ) ♦ S r 0 ( N • ⊗ k L • ) for an y complex of left S l κ -semimo dules L • . No w the ob ject SemiT or S l κ ( N • , M • ) is represen ted by the complex ( C h ∞ / 2+ • ( g , S ) ⊗ k N • ) ♦ S l κ M • ≃ C h ∞ / 2+ • ( g , S ) ♦ S r 0 ( N • ⊗ k M • ) ≃ C h ∞ / 2+ • ( g , N • ⊗ k M • ) in the deriv ed category of k -v ector spaces. Another wa y to iden tify SemiT or S l κ ( N • , M • ) with C h ∞ / 2+ • ( g , N • ⊗ k M • ) is to con- sider the semiﬂat complex of left S l κ -semimo dules C h ∞ / 2+ • ( g , S ) ⊗ k M • naturally iso- morphic to M • in the semideriv ed category of left S l κ -semimo dules. T o chec k that these t w o iden tiﬁcatio ns coincide, represen t the images of N • and M • in the semi- deriv ed categories of se mimo dules by arbitrary semiﬂat complexes. The pro of of part (b) is completely analo g ous.  Question. Can one obtain the semi-inﬁnite homology o f arbitrary discrete mo dules o v er a T ate Lie a lgebra with a ﬁxed compact op en subalgebra (rather than only Harish-Chandra mo dules under the nilp ot ency conditions) as a kind of double-sided deriv ed functor of the functor of se miin v ariants on an appropriate exotic deriv ed category o f discrete mo dules? Notice that the cohomology of the Chev alley complex M − → L ⊗ k M − → V 2 k L ⊗ k M − → · · · for a como dule M ov er a Lie coalgebra L is in- deed the rig ht deriv ed functor of the functor of L -inv ariants on the ab elian category of L -como dules, since the category of L -como dules has enough injectiv es and t he coho- mology of the Chev alley complex is a n eﬀaceable cohomological functor. The for mer holds since the category o f discrete mo dules ov er a compact Lie algebra h = L ∨ has exact functors o f ﬁltered inductiv e limits preserv ed b y the forgetful functor to the cat- egory of k - v ector spaces, and t he discrete h -mo dules U ( h ) ⊗ U ( a ) k induced f r om tr ivial mo dules ov er o p en subalgebras a ⊂ h form a set of generators, so the forgetful functor 287 ev en has a righ t adjoin t. T o c hec k t he latter, one can r epresen t co cycles in the coho- mological Chev alley complex b y discrete h -mo dule morphisms into M from the rela- tiv e homological complexes · · · − → V 2 k ( h / a ) ⊗ U ( a ) U h − → h / a ⊗ U ( a ) U ( h ) − → U ( h ), whic h are quotien t complexes o f the Chev alley homological complex of the h - mo dule U ( h ) and ﬁnite discrete h -mo dule resolutions o f the trivial h -mo dule k . F urthermore, the semideriv ed catego ry of discrete mo dules o v er a T ate Lie algebra g , deﬁned as the quotien t catego r y of the homotopy catego r y of discrete g -mo dules b y the thic k sub category of complexes coacyclic as complexes o f discrete h -mo dules, do es not de- p end on the c hoice of an op en compact subalgebra h ⊂ g . This can b e demonstrated b y considering the tensor pr o duct o v er k of the a b ov e relativ e homological complex with a complex of discrete h - mo dules coacyclic o v er a . Notice that in t he ab o v e pro of w e hav e essen tially shown that the semi-inﬁnite homolo gy is a functor on the semideriv ed category of discrete g -mo dules. D.6.3. W e ke ep the assumptions and notation of D.6.2, and also use the notation of Corollary D.3 .1. The following result makes use of t he semimo dule-semicon tramo dule corresp ondence in order to express the semi-inﬁnite homolog y and cohomology in terms of comp ositions of o ne- sided deriv ed functors. Corollary . (a) L et M • b e a c om p lex o f Harish-Chandr a mo dules with the c en- tr al char ge κ and P • b e a c omplex of Harish-Cha ndr a c ontr amo dules with the c entr al char ge κ + κ 0 over ( g , C ) . Then t he semi-inﬁn ite c ohomol o gic al c om- plex C ∞ / 2+ • h ( g , Hom k ( M • , P • )) r epr esents the obje ct Ext S l κ ( M • , L Φ S r κ + κ 0 ( P • )) ≃ Ext S r κ + κ 0 ( R Ψ S l κ ( M • ) , P • ) in the d e ri v e d c ate g ory of k -ve ctor sp ac es. (b) L et M • b e a c omplex of Harish-Chandr a mo dules with the c entr al ch a r ge κ and N • b e a c omplex of Harish-Ch andr a mo dules with the c e n tr al char ge − κ − κ 0 over ( g , C ) . Then the semi-inﬁnite homolo gic al c omplex C h ∞ / 2+ • ( g , N • ⊗ k M • ) r epr e- sents the obje ct Ctrtor S r κ + κ 0 ( N • , R Ψ S l κ ( M • )) ≃ Ctrtor S r − κ ( M • , R Ψ S l − κ − κ 0 ( N • )) in the derive d c ate gory of k -ve c tor sp ac es. Pr o of . This follows from Theorem D.6.2 a nd Corollary 6 .6.  Remark. Set S l 0 = S ≃ S r κ 0 and consider the complex of left S -semimo dules R • = S ⊗ k V h ∞ / 2+ • ( g ). This is a semipro jectiv e complex of semipro jectiv e left S -semi- mo dules isomorphic to the trivial S -semimo dule k in D si ( S – simo d ). Assume that t he pronilp oten t Lie algebra h is inﬁnite-dimensional (cf. 0 .2.7). Then the complex of left S -semic ontramo dules Ψ S ( R • ) is acyclic . Indeed, it suﬃce s to ch ec k that the complex of left C -contramo dules o bta ined b y applying the functor Ψ C to the co- homological Chev alley complex C ⊗ k V k ( h ∨ ) is acyclic; o ne can r educe this prob- lem to the case of an a b elian Lie algebra b y considering the decreasing ﬁltration h ⊃ [ h , h ] ⊃ [ h , [ h , h ]] ⊃ · · · on h and the induced increasing ﬁltrations on h ∨ and C . 288 The complex Ψ S ( R • ) is also a pro jectiv e complex of pro jectiv e left S -semicontra- mo dules (see Remark 6 .5 and subsection 9.2); it can b e though t of as the “pro j ectiv e S -semicontramo dule resolution of a (nonexisten t) one-dimensional left S - semicontra- mo dule placed in t he degree + ∞ ”. F or an y complex of rig ht S -semimo dules N • , the con tratensor pro duct complex N • ⊚ S Ψ S ( R • ) computes the semi-inﬁnite homol- ogy of g with co eﬃcien ts in N • . F or any complex of left S - semicontramo dules P • , the complex of semicon tramo dule homomorphisms Hom S (Ψ S ( R • ) , P • ) computes the semi-inﬁnite cohomology of g with co eﬃcien ts in P • . (Cf. [46, subsection 3.1 1.4].) 289 Appendix E. Groups with Ope n P ro finite Subgroups T o a lo cally compact totally disconnected top ological group G and a commutativ e ring k one asso ciates a family of left and r igh t semipro jectiv e Morita equiv alen t semialgebras S = S k ( G, H ) nu m b ered b y op en proﬁnite subgroups H ⊂ G . As explained in 8.4.5 , Morit a equiv alences of semialgebras do not hav e t o preserv e the semideriv ed categories or the deriv ed functors SemiT or and SemiExt, and this is indeed the case here: SemiT or S and SemiExt S dep end ve ry essen tially on H . F or a complex of smo ot h G -mo dules N • o v er k and a complex of k -ﬂa t smo oth G - mo dules M • o v er k , w e sho w that SemiT or S k ( G,H ) ( N • , M • ) only dep ends on the complex of smo oth G -mo dules N • ⊗ k M • , a nd analogously for SemiExt S k ( G,H ) . When k is a ﬁeld o f zero c haracteristic, one can climb one step higher and assign to a “go o d enough” gro up ob ject G in the category of ind-pro-to p ological spaces with a subgroup ob jec t H b elonging to the categor y of pro-top ological gr o ups a coring ob ject in the tensor catego r y o f represen tations of H × H in pro-v ector spaces o v er k . So there is a functor of cotensor pro duct [2 4] on certain cat ego ries of represen tations of cen tral extensions of G ; it has a double-sided deriv ed functor ProCotor. E.1. Morita equiv alen t semialgebras . E.1.1. In the seque l, all top ological spaces a nd top ological gro ups are presumed to b e lo cally compact and to tally disconnected. F or a top olog ical space X and an ab elian group A , denote by A ( X ) the ab elian group of lo cally constant compactly supp o r ted A -v alued functions o n X . F or any prop er map of t op ological spaces X − → Y , the pull-bac k map A ( Y ) − → A ( X ) is deﬁned. F or an y ´ etale map (lo cal homeomorphism) of top olo g ical spaces X − → Y , the push-forw ard map A ( X ) − → A ( Y ) is deﬁned. F or an y top ological spaces X and Y and an ab elian group A , there is a natur a l isomorphism A ( X × Y ) ≃ A ( X )( Y ). F or an y top ological space X , a comm utat ive ring k , and a k -mo dule A , there is a na tural isomorphism A ( X ) ≃ A ⊗ k k ( X ). F or a top ological space X and an ab elian g r oup A , denote b y A [[ X ]] the ab elian group of ﬁnitely-additiv e compactly supp o rted A -v a lued measures deﬁned on the op en-closed subsets of X . F or an y map of top ological spaces X − → Y , the push- forw ard map A [[ X ]] − → A [[ Y ]] is deﬁned. F or an y compact top ological space X , a comm utativ e ring k , a nd a k -mo dule A , there is a natural isomorphism A [[ X ]] ≃ Ho m k ( k ( X ) , A ). E.1.2. Let H b e a proﬁnite group and k b e a commutativ e ring . Then the mo dule of lo cally constan t functions k ( H ) has a natural structure of coring ov er k where the left and right actions of k coincide. This coring structure, whic h w e denote by C = C k ( H ), is deﬁne d as follo ws. The counit map k ( H ) − → k is the ev alua tion at t he unit elemen t e ∈ H . The com ultiplication map k ( H ) − → k ( H ) ⊗ k k ( H ) is 290 pro vided b y the pull-bac k map k ( H ) − → k ( H × H ) induced by the m ultiplicatio n map H × H − → H together with the iden tiﬁcation k ( H × H ) ≃ k ( H ) ⊗ k k ( H ). Let G b e a top ological group and H ⊂ G b e an o p en pro ﬁnite subgroup. Then the mo dule of lo cally constan t compactly suppo rted functions k ( G ) has a natura l structure of semialgebra ov er the coring C k ( H ). This semialgebra structure, whic h w e denote b y S = S k ( G, H ), is deﬁned as follows. The bicoaction map k ( G ) − → k ( H ) ⊗ k k ( G ) ⊗ k k ( H ) ≃ k ( H × G × H ) is the pull-bac k map induced by the m ultiplication map H × G × H − → G . The semiunit map k ( H ) − → k ( G ) is the push-forward map induced b y the injection H − → G . Denote by G × H G the quotien t space of the Cart hesian square G × G by the equiv alence relatio n ( g ′ h, g ′′ ) ∼ ( g ′ , hg ′′ ) for g ′ , g ′′ ∈ G and h ∈ H . The pull- bac k map k ( G × H G ) − → k ( G × G ) induced b y the na tural surjection G × G − → G × H G iden tiﬁes k ( G × H G ) with the cotensor pro duct k ( G )  C k ( G ) ⊂ k ( G ) ⊗ k k ( G ) ≃ k ( G × G ). The semim ultiplication map k ( G )  C k ( G ) − → k ( G ) is the push-forw ard map induced b y the m ultiplication map G × H G − → G . The in v olutions k ( H ) − → k ( H ) and k ( G ) − → k ( G ) induced b y the in v erse elemen t maps H − → H and G − → G provid e the isomorphism of semialgebras S k ( G, H ) op ≃ S k ( G, H ) compatible with the isomorphism of corings C k ( H ) op ≃ C k ( H ) ov er k (see C.2.7 ). No w let H 1 , H 2 ⊂ G be t w o op en proﬁnite subgroups of G . T hen the k -mo dule k ( G ) = S k ( G, H 1 ) has a natural left S k ( G, H 1 )-semimo dule structure and at the same time k ( G ) = S k ( G, H 2 ) has a natural right S k ( G, H 2 )-semimo dule structure. Ob viously , these t w o semimodule structures comm ute; w e denote this bisemimo dule structure o n k ( G ) b y S k ( G, H 1 , H 2 ). F or any three op en proﬁnite subgroups H 1 , H 2 , H 3 ⊂ G , there is a natura l isomorphism S k ( G, H 1 , H 2 ) ♦ S k ( G,H 2 ) S k ( G, H 2 , H 3 ) ≃ S k ( G, H 2 ) ♦ S k ( G,H 2 ) S k ( G, H 2 ) ≃ S k ( G, H 2 ) ≃ S k ( G, H 1 , H 3 ); t his is a n isomorphism of S k ( G, H 1 )- S k ( G, H 3 )-bisemimo dules. One can chec k that k ( H ) is a pro jectiv e k -mo dule. Clearly , S k ( G, H ) is a copro jec- tiv e left and righ t C k ( H )-como dule. So t he pair ( S k ( G, H 1 , H 2 ) , S k ( G, H 2 , H 1 )) is a left and righ t semipro jectiv e Morita equiv alence b etw een the semialgebras S k ( G, H 1 ) and S k ( G, H 2 ). E.1.3. The semialgebra S k ( G, H ) can b e also obta ined by the construction of 10.2.1. Denote by k [ H ] and k [ G ] the group k -algebras o f the groups H and G considered as gro ups without an y top ology . There is a pair ing φ : C k ( H ) ⊗ k k [ H ] − → k satisfying the conditions of 10 .1 .2, give n by the formula ( c, h ) 7− → c ( h − 1 ) for an y c ∈ k ( H ) and h ∈ H The induced functor ∆ φ : como d – C k ( H ) − → mo d – k [ H ] is f ully faithful; its image is describ ed as f ollo ws. 291 A mo dule M ov er a top ological group G is called smo oth (discrete), if the action map G × M − → M is contin uous with resp ect to the discrete top o logy of M ; equiv- alen tly , M is smo oth if the stabilizer of ev ery its elemen t is an op en subgroup in G . The functor ∆ φ iden tiﬁes the category o f right C k ( H )-como dules with the category of smo oth H -mo dules ov er k . The tensor pro duct C k ( H ) ⊗ k [ H ] k [ G ] is a smo oth G -mo dule with respect to the action of G by righ t multiplic ations, so it b ecomes a semialgebra o v er C k ( H ). This semialgebra can b e iden tiﬁed with S k ( G, H ) by the formula b y the form ula ( c ⊗ g ) 7− → ( g ′ 7− → c ( g ′ g − 1 )), where a lo cally constan t function c : H − → k is presumed t o b e extended to G b y zero. By the result of 10 .2 .2, the category of right S k ( G, H )-semi- mo dules is isomorphic to the category of smo oth G - mo dules o v er k . One also obta ins the following description of the category of left S k ( G, H )-semicon- tramo dules. F or a to p ological group G and a comm utative ring k , a G -c on tr amo dule over k is a k -mo dule P endo w ed with a k -linear map P [[ G ]] − → P satisfying the follo wing conditions. First, t he p oint measure supp orted in the unit elemen t e ∈ G and cor r esp o nding to an elemen t p ∈ P should map to the elemen t p . Second, the comp osition P [[ G × G ]] − → P [[ G ]][[ G ]] − → P of the natural map P [[ G × G ]] − → P [[ G ]][[ G ]] and the iterated con traaction map P [[ G ]][[ G ]] − → P [[ G ]] − → P should b e equal to the comp o sition P [[ G × G ]] − → P [[ G ]] − → P of the push-forward map P [[ G × G ]] − → P [[ G ]] induced b y the m ultiplication map G × G − → G and the con traaction map P [[ G ]] − → P . The images o f the p oint measures under the con traaction map deﬁne the forget- ful functor from the category of G -contramo dules ov er k to the category of (non- top ological) G -mo dules ov er k . The category of left S k ( G, H )-semicontramo dules is isomorphic to the category of G - contramo dules ov er k . E.1.4. F or any smooth G -mo dule M ov er k and an y k - mo dule E t here is a natural G - contramo dule structure on the space of k -linear maps Hom k ( M , E ). The con traaction map Hom k ( M , E )[[ G ]] − → Hom k ( M , E ) is constructed as the pro j ectiv e limit o v er all op en subgroups U ⊂ G of the comp ositions Hom k ( M , E )[[ G ]] − → Hom k ( M , E )[ G/U ] − → Hom k ( M U , E ) o f the maps Hom k ( M , E )[[ G ]] − → Hom k ( M , E )[ G/U ] induced b y the surjections G − → G/U and the maps Hom k ( M , E )[ G/U ] − → Hom k ( M U , E ) induced by the a ctio n maps G/U × M U − → M , where M U denotes t he k -submo dule of U -inv arian ts in M and G/U is the set of all left cosets of G mo dulo U . More generally , let G 1 and G 2 b e to p ological groups. Then for an y smo oth G 1 -mo dule M ov er k and G 2 -contramo dule P o ve r k there is a natural G 1 × G 2 -con- tramo dule structure on Hom k ( M , P ) with the con traaction map Ho m k ( M , P )[[ G 1 × G 2 ]] − → Hom k ( M , P ) deﬁned as either of the comp ositions Hom k ( M , P )[[ G 1 × G 2 ]] − → Hom k ( M , P )[[ G 1 ]][[ G 2 ]] − → Hom k ( M , P )[[ G 2 ]] − → Hom k ( M , P ) or 292 Hom k ( M , P )[[ G 1 × G 2 ]] − → Hom k ( M , P )[[ G 2 ]][[ G 1 ]] − → Hom k ( M , P )[[ G 1 ]] − → Hom k ( M , P ), where the G 1 -con traaction map Hom k ( M , P )[[ G 1 ]] − → Hom k ( M , P ) is deﬁned ab ov e and the G 2 -con traaction map Hom k ( M , P )[[ G 2 ]] − → Hom k ( M , P ) is constructed as the comp osition Hom k ( M , P )[[ G 2 ]] − → Hom k ( M , P [[ G 2 ]]) − → Hom k ( M , P ). Hence for any smo ot h G -mo dule M ov er k and an y G -contramo d- ule P ov er k there is a natural G -contramo dule structure on Hom k ( M , P ) induced b y the diagonal map of to p ological groups G − → G × G . E.2. Semiin v arian ts and semicon train v arian t s. E.2.1. Let G b e a top olo gical group and H ⊂ G b e an op en proﬁnite subgroup. F o r a smo oth H -mo dule M ov er k , let Ind G H M denote the induced G - mo dule k [ G ] ⊗ k [ H ] M . F or an y smo o th G - mo dule N o v er k w e will construct a pair of maps (Ind G H N ) H ⇒ N H , where the sup erindex H denotes the k -submo dule of H - inv arian ts. Namely , the ﬁrst map (Ind G H N ) H − → N H is obtained b y applying the functor o f H -inv aria n ts to the map (Ind G H N ) − → N giv en b y the formula g ⊗ n 7− → g ( n ). The second map (Ind G H N ) H − → N H only dep ends on t he H -mo dule structure on N . T o deﬁne this second map, iden tify the induced represen tation Ind G H N with the k -mo dule of all compactly supp orted functions G − → N t r a nsforming the right action of H in G into the action of H in N ; this identiﬁc ation assigns to a function g 7− → n g the formal linear com bination P g ∈ G/H g ⊗ n g , where G/H denotes the set of all left cosets of G mo dulo H . An H -inv ariant elemen t of Ind G H N is then represen ted b y a compactly suppor t ed function G − → N denoted b y g 7− → n g and satisfying the equations n g h = h − 1 ( n g ) and n hg = n g for g ∈ G , h ∈ H . The second map (Ind G H N ) H − → N H sends a function g 7− → n g to the elemen t P g ∈ H \ G n g ∈ N , where H \ G denotes the set of all righ t cosets o f G mo dulo H . The cokerne l N G,H of this pair of maps (Ind G H N ) H ⇒ N H is called the mo dule of ( G, H ) -semiinv ariants of a smo o th G -mo dule N . The ( G, H )-semiinv arian ts are a mixture of H -inv aria n ts a nd coin v arian ts along G relativ e to H . E.2.2. F or an H -contramo dule Q ov er k , let Coind G H ( Q ) denote the coinduced G -con- tramo dule Hom k [ H ] ( k [ G ] , Q ) ≃ Cohom C k ( H ) ( S k ( G, H ) , Q ). F o r any G -contramo dule P ov er k w e will construct a pair of maps P H ⇒ Coind G H ( P ) H , where the subindex H denotes the k -mo dule of H -coinv ar ia n ts, i. e., the maximal quotien t H - contra- mo dule with the trivial contraaction. Namely , the ﬁrst map P H − → Coind G H ( P ) H is obtained b y applying the functor of H -coinv ar ian ts t o t he semicon traaction map P − → Coind G H ( P ), whic h is given by the fo r mula p 7− → ( g 7− → g ( p )). The second map P H − → Coind G H ( P ) H only dep ends on the H -contramo dule struc- ture on P . It is giv en by the formula p 7− → P g ∈ G/H g ∗ p for p ∈ P , whe re g ∗ p : k [ G ] − → P is the k [ H ]- linear map deﬁned by the rules hg 7− → h ( p ) for h ∈ H 293 and g ′ 7− → 0 for all g ′ ∈ G not b elonging to the right coset H g . Clearly , the inﬁnite sum ov er G /H con v erges elemen t-wise on k [ G ] for an y c hoice of represen tative s of the left cosets; its imag e in the mo dule of coin v ariants do es not dep end on this choice and is determined b y the image of p in P H . The k ernel P G,H of this pa ir of maps P H ⇒ Coind G H ( P ) H is called the mo dule of ( G, H ) -semic ontr ainvarian ts of a G -contramo dule P . The ( G, H )-semicontrainv ari- an ts are a mixture of H - coinv arian ts and inv aria n ts along G relative to H . E.2.3. Denote b y χ : G − → Q ∗ the mo dular c haracter of G , i. e., the c hara cter with whic h G acts by left shifts on the one-dimensional Q -v ector space of Q χ of right in v ariant Q -v a lued measures deﬁned o n the op en compact subsets o f G . Equiv alently , χ ( g ) is equal to the ratio of the num b er of left cosets contained in the double coset H g H to t he n um b er of right cosets. Whenev er the comm utativ e ring k contains Q , there is a natural isomorphism N G,H ≃ ( N ⊗ Q Q χ ) G for an y smo oth G -mo dule N o v er k , where the subindex G denotes t he k -mo dule of G - coinv ariants . Indeed, the comp osition M H − → M − → M H is an isomorphism for any smo oth H -mo dule M ov er k , so in particular there are isomorphisms N H ≃ N H and (Ind G H N ) H − → (Ind G H ) H . These isomorphisms transform the a b o v e pair of maps (Ind G H N ) H ⇒ N H in to the pair of maps (Ind G H N ) H ⇒ N H giv en by the form ulas g ⊗ n 7− → g ( n ) and g ⊗ n 7− → χ − 1 ( g ) n . E.2.4. F or any G -contramo dule P o v er k denote by P G the k -mo dule of G -inv ari- an ts in P deﬁned as the submo dule of all p ∈ P suc h that for an y measure m ∈ k [[ G ]] the image of t he measure pm ∈ P [[ G ]] under the con traaction map P [[ G ]] − → P is equal to the v alue m ( G ) o f m at G . Assuming that t he comm utativ e ring k con tains Q , the comp osition Q H − → Q − → Q H is an isomorphism for any H -contramo dule Q ov er k , as one can show using the action o f the Haar measure o f the proﬁnite group H in the contramo dule Q . One can also use the Haar measure to c hec k that Q H is the maximal sub con tramo dule of Q with the trivial contraaction of H , and it follow s that P G is the maximal sub con tramo dule of P with the trivial con traaction of G , under o ur assumption. Finally , when k ⊃ Q there is a natura l isomorphism P G,H ≃ Hom Q ( Q χ , P ) G . E.2.5. Let N be a left S k ( G, H )-semimo dule and M b e a right S k ( G, H )-semimo d- ule; in other words, N and M are smo oth G -mo dules ov er k . Then there is a natural isomorphism N ♦ S k ( G,H ) M ≃ ( N ⊗ k M ) G,H , where N ⊗ k M is considered as a smo oth G -mo dule ov er k . Here the semitensor pro duct is we ll-deﬁned b y Prop osition 1.2.5( f ). Indeed, there is an ob vious isomorphism N  C k ( H ) M ≃ ( N ⊗ k M ) H . The k -mo dule N  C k ( H ) S k ( G, H )  C k ( H ) M can b e iden tiﬁed with the mo dule of lo cally constan t compactly supp orted functions f : G − → N ⊗ k M satisfying the equations f ( hg ) = hf ( g ) and f ( g h ) = f ( g ) h f or g ∈ G , h ∈ H , where ( g , a ) 7− → g a and ( a, g ) 7− → ag 294 denote t he actions of G in N ⊗ k M induced b y the a ctio ns in N and M , resp ectiv ely . A t the same time, the k -mo dule (Ind G H ( N ⊗ k M )) H can b e iden tiﬁed with the mo dule of lo cally constant compactly supp orted functions f ′ : G − → N ⊗ k M satisfying the equations f ( hg ) = f ( g ) and f ( g h ) = h − 1 f ( g ) h . The formula f ′ ( g ) = g − 1 f ( g ) deﬁnes an isomorphism b etw een these tw o k -mo dules transforming t he pair of maps whose cok ernel is N ♦ S k ( G,H ) M in to the pair o f maps whose cok ernel is ( N ⊗ k M ) G,H . E.2.6. Let M b e a left S k ( G, H )-semimo dule and P b e a left S k ( G, H )-semi- contramo dule. Then there is a natural isomorphism SemiHom S k ( G,H ) ( M , P ) ≃ Hom k ( M , P ) G,H , where Hom k ( M , P ) is considered as a G -contra mo dule ov er k a nd the semihomomorphism mo dule is w ell-deﬁned by Prop osition 3.2.5(j). Indeed, the quotien t mo dules Cohom C k ( H ) ( M , P ) and Hom k ( M , P ) H of the k -mo dule Hom k ( M , P ) coincide. There are tw o comm uting con traactions of G in Hom k ( M , P ) induced b y the smo oth a ction in M and the con traaction in P ; denote these contraaction maps b y π M and π P , and the corresp o nding actions o f G in Hom k ( M , P ) b y ( g , x ) 7− → g M ( x ) and ( g , x ) 7− → g P ( x ). The k -mo dule Cohom C k ( H ) ( S k ( G, H )  C k ( H ) M , P ) can b e iden tiﬁed with the quo- tien t mo dule o f the mo dule of all ﬁnitely-additiv e measures deﬁned o n compact op en subsets of G and taking v alues in Hom k ( M , P ) by the submo dule generated by mea- sures of the form U 7− → π M ( W 7− → µ ( U × W − 1 )) − µ ( { ( g , h ) | g h ∈ U } ) and U 7− → π P ( W 7− → ν ( U × W )) − ν ( { ( h, g ) | hg ∈ U } ), where µ and ν are ﬁnitely- additiv e measures deﬁned on compact op en subsets of G × H and H × G , resp ectiv ely , and taking v a lues in Hom k ( M , P ). Here U denotes a compact op en subset of G and W is an op en-closed subset o f H ; b y W − 1 w e denote the (pre)image of W under the in v erse elemen t map. A t the same time, t he k - mo dule Coind G H (Hom k ( M , P )) H can b e identiﬁed with the quotien t mo dule of the same mo dule of measures on G b y the submo dule generated by measures of the form U 7− → µ ′ ( U × H ) − µ ′ ( { ( g , h ) | g h ∈ U } ) a nd U 7− → π P ( W 2 7− → π M ( W 1 7− → ν ′ (( W 1 ∩ W 2 ) × U ))) − ν ′ ( { ( h, g ) | hg ∈ U } ), where µ ′ and ν ′ are ﬁnitely- additiv e measures deﬁned on compact op en subsets of G × H and H × G , resp ectiv ely , and ta king v alues in Hom k ( M , P ). Here W 1 and W 2 denote op en-closed subsets of H . The f orm ulas m ′ ( U ) = π M ( V 7− → m ( U ∩ V − 1 )) and m ( U ) = π M ( V 7− → m ′ ( U ∩ V )), where V denotes a n op en-closed subset of G , deﬁne a s isomorphism b et w een these t w o quotien t spaces of measures. The pair of maps Cohom C k ( H ) ( M , P ) ⇒ Cohom C k ( H ) ( S k ( G, H )  C k ( H ) M , P ) whose k ernel is SemiHom S k ( G,H ) ( M , P ) is given b y the rules x 7− → P g ∈ G/H g − 1 M ( x ) δ g and x 7− → P g ∈ H \ G g P ( x ) δ g , where y δ g denotes the Hom k ( M , P )- v alued measure de- ﬁned on compact o p en subsets of G that is supp orted in the p oint g ∈ G a nd corresp onds to an elemen t y ∈ Hom k ( M , P ). The pa ir of maps Ho m k ( M , P ) H ⇒ Coind G H (Hom k ( M , P )) H whose k ernel is Hom k ( M , P ) G,H is giv en b y the rules x 7− → 295 P g ∈ G/H xδ g and x 7− → P g ∈ H \ G g P g M ( x ) δ g . The ab o v e isomorphism b et w een tw o quotien t spaces of measures transforms one o f these tw o pairs o f maps into the ot her. E.2.7. Let H 1 ⊂ H 2 b e tw o op en proﬁnite subgroups of a top olo g ical gro up G and N b e a smo ot h G -mo dule ov er k . Then, at least, in the follow ing tw o situatio ns there is a natural isomorphism N G,H 1 ≃ N G,H 2 : when N as a G -mo dule is induced from a smo oth H 1 -mo dule ov er k , and when N as an H 2 -mo dule ov er k is coinduced from a mo dule ov er the t rivial subgroup { e } (i. e., N is a coinduced C k ( H 2 )-como dule). These isomorphisms are constructed as follows. In the ﬁrst case, one shows that the triple semitensor pro duct N ♦ S k ( G,H 1 ) S k ( G, H 1 , H 2 ) ♦ S k ( G,H 2 ) k is a sso ciativ e in the sense that t he conclusion of Prop osition 1 .4.4 applies to it. In the second case, one shows that the triple semitensor pro duct N ♦ S k ( G,H 2 ) S k ( G, H 2 , H 1 ) ♦ S k ( G,H 1 ) k is asso ciativ e in the similar sense. In b o th cases, the argumen t is analog ous to that of Prop osition 1.4 .4. E.2.8. Let H 1 ⊂ H 2 b e t w o op en pro ﬁnite subgroups of G and P b e a G -contra- mo dule ov er k . Then, at least, in the follo wing tw o situations there is a natural isomorphism P G,H 1 ≃ P G,H 2 : when P is coinduced fro m an H 1 -contramo dule ov er k , and when P as an H 2 -contramo dule ov er k is induced from a con tramo dule ov er the triv al subgroup { e } ( i. e., P is an induced C k ( H 2 )-contramo dule). E.3. SemiT or and SemiExt. E.3.1. Assume that the ring k has a ﬁnite w eak homolo gical dimension. Then for an y complexes of smoo th G -mo dules N • and M • o v er k the ob ject SemiT or S k ( G,H ) ( N • , M • ) in the deriv ed category of k -mo dules is deﬁned. F urthermore, whe nev er either N • or M • is a complex of k -ﬂat smo oth G -mo dules ov er k there is a natura l isomorphism SemiT or S k ( G,H ) ( N • , M • ) ≃ SemiT o r S k ( G,H ) ( N • ⊗ k M • , k ) in D ( k – mo d ). Indeed, assume that M • is a complex of ﬂat k -mo dules. If N • is a semiﬂat complex of righ t S k ( G, H )-semimo dules, then so is the tensor pro duct N • ⊗ k M • , since ( N • ⊗ k M • ) ♦ S k ( G,H ) L • ≃ ( N • ⊗ k M • ⊗ k L • ) G,H ≃ N • ♦ S k ( G,H ) ( M • ⊗ k L • ) fo r any complex of smo oth G -mo dules L • o v er k , and the complex of left C k ( H )-como dules M • ⊗ k L • is coacyclic whenev er the complex of left C k ( H )-como dules L • is. It r emains to use the natural isomorphism N • ♦ S k ( G,H ) M • ≃ ( N • ⊗ k M • ) G,H . E.3.2. Let k − → k ′ b e a morphism of comm utativ e rings o f ﬁnite w eak ho mo lo gical dimension. Then for an y complex of smo oth G -mo dules N • o v er k ′ the image of the ob ject SemiT or S k ′ ( G,H ) ( N • , k ′ ) under the restriction of scalars functor D ( k ′ – mo d ) − → D ( k – mo d ) is naturally isomorphic to the ob ject SemiT or S k ( G,H ) ( N • , k ). This follo ws from Corollary 8.3.3(a ). 296 Remark. When G is a discrete group, H = { e } is the trivial subgroup, and N is a G -mo dule o v er k , the cohomology of the ob j ect SemiT or S k ( G, { e } ) ( k , N ) coincides with the discrete group homology H ∗ ( G, N ) and is concen trated in the nonp ositive coho- mological degrees. When G is a proﬁnite gro up, H = G is the whole group, and N is a smo oth (discrete) G -mo dule o v er k , the cohomology o f SemiT or S k ( G,G ) ( k , N ) coincides with the proﬁnite group cohomology H ∗ ( G, N ) and is concen trated in the no nnegativ e cohomological degrees. More generally , when G = G 1 × G 2 is the pro duct of a discrete group G 1 and a pro ﬁnite group G 2 , H = G 2 ⊂ G , k is a ﬁeld, and N = N 1 ⊗ k N 2 is the tensor pro duct of a G 1 -mo dule and a smo oth G 2 -mo dule, the cohomology o f SemiT or S k ( G,H ) ( k , N ) is isomorphic t o the tensor pro duct H ∗ ( G 1 , N 1 ) ⊗ k H ∗ ( G 2 , N 2 ). Applying these o bserv ations to t he case of a ﬁnite g r o up G , one can see that the semi- inﬁnite homolog y of top o logical gro ups SemiT or S k ( G,H ) ( k , N • ) do es dep end essen tially on the c hoice of an o p en proﬁnite subgroup H ⊂ G . E.3.3. Assume that the ring k has a ﬁnite homological dimension. Then fo r an y com- plex o f smo oth G -mo dules M • o v er k and an y complex of G -contra mo dules P • o v er k the ob ject SemiEx t S k ( G,H ) ( M • , P • ) in the derive d category of k -mo dules is deﬁned. F urthermore, whenev er either M • is a complex of pro jectiv e k -mo dules o r P • is a com- plex o f inj ective k -mo dules there is a natural isomorphism SemiExt S k ( G,H ) ( M • , P • ) ≃ SemiExt S k ( G,H ) ( k , Ho m k ( M • , P • )) in D ( k – mo d ). E.3.4. Let k − → k ′ b e a morphism o f comm utativ e rings of ﬁnite homological di- mension. Then f o r a ny complex of G -contramo dules P • o v er k ′ the image of the ob ject SemiExt S k ′ ( G,H ) ( k ′ , P • ) under the restriction of scalars f unctor D ( k ′ – mo d ) − → D ( k – mo d ) is naturally isomorphic to the ob ject SemiExt S k ( G,H ) ( k , P • ). E.4. Remarks on t he Gaitsgory–Kazhdan construct ion. This subsection con- tains some commen ts o n the pap ers [23, 24]. E.4.1. Let G b e a top ological gr oup and k b e a ﬁeld of c haracteristic 0. Then the category of discrete G × G -mo dules ov er k has a t ensor category structure with the tensor pro duct of tw o mo dules K ′ ⊗ G K ′′ deﬁned as t he mo dule of coin v ariants of the action of G in K ′ ⊗ k K ′′ induced by the action o f the second copy of G in K ′ and the action of the ﬁrst cop y of G in K ′′ . The category of discrete G -mo dules o v er k has structures of a left and a righ t mo dule cat ego ry ov er this tensor catego ry; and the f unctor ( N , M ) 7− → N ⊗ G M = ( N ⊗ k M ) G deﬁnes a pairing b etw een these t w o mo dule categor ies taking v alues in the catego ry of k -v ector spaces . A ﬁnitely- additiv e k -v alued measure deﬁned on compact op en subsets of G is called sm o oth if it is equal to the pro duct of a lo cally constan t k - v alued function on G and a (left or right inv aria n t) Ha ar measure on G . The G × G -mo dule of compactly supp o rted smo oth k -v alued measures on G is t he unit ob ject of the ab o v e tensor catego ry . 297 Notice that the ab ov e G × G -mo dule has a natural k -algebra structure, giv en by the con v olution of measures. How ev er, this algebra has no unit and the category of (left or right) mo dules o v er it con tains the category of smo oth G -mo dules ov er k as a prop er full sub category . E.4.2. F or an y category C , denote b y Pro C and Ind C the categories of pro- ob jects and ind- ob jects in C . Let Set ﬁn denote the category o f ﬁnite sets. W e will iden tify the category of compact to p ological spaces with the category Pro Set ﬁn , the category of (discrete) sets with the category Ind Set ﬁn , a nd the category of to p ological spaces with a f ull sub category of the category Ind Pro Set ﬁn formed b y the inductiv e systems of proﬁnite sets a nd their o p en em b eddings. In addition, w e will consider the catego ry Ind Pro Set ﬁn as a f ull sub category in Pro Ind Pro Set ﬁn and the latter category as a f ull sub category in Ind Pro Ind Pro Set ﬁn . Let H b e a g r o up ob ject in Pro I nd Pro Set ﬁn suc h that H can b e represen ted b y a pro jectiv e system of top ological groups in Ind Pro Set ﬁn and op en surjectiv e morphisms b et w een them. A represen tation o f H in k – vect is just a smo oth G -mo dule o v er k , where G is a quotien t group o b ject of H that is a top olog ical group in Ind Pro Set ﬁn . The category Rep k ( H ) of represen tations of H in Pro ( k – vect ), deﬁned in [23], is equiv- alen t to the category of pro- ob jects in the category of represen tations o f H in k – vect . So the category Rep k ( H × H ) has a natural tensor category structure with the unit ob ject giv en b y the pro jectiv e system formed b y the mo dules of smo oth compactly supp orted measures on top ological g roups that are quotien t groups of H and the push-forw ard maps b etw een t he spaces o f measures. The category Rep k ( H ) is a left and a righ t mo dule category o v er this tensor category , and there is a pairing b et w een these left and right mo dule categories taking v alues in Pro ( k – vect ). E.4.3. Let G b e a group ob jec t in Ind Pro Ind Pro Set ﬁn and let H ⊂ G b e a subgroup ob ject whic h b elongs to Pro I nd Pro Set ﬁn . Assume that t he ob ject G is g iven by an inductiv e system of ob jects G α ∈ Pro Ind Pro Set ﬁn and the group ob ject H is given b y a pro jectiv e system of its quotient gro up ob jects H / H i ∈ Ind Pro Set ﬁn satisfying the following conditions. It is conv enien t t o assume t ha t G 0 = H = H 0 is the initial o b ject in the inductiv e system of G α and the ﬁnal ob ject in the pro jectiv e system of H i . As the notation suggests, H i are normal subgroup ob jects of H . The group ob jec t H acts in G α in a w a y compatible with the action of H in G by righ t m ultiplications. The quotien t ob jects G α / H i are top ological spaces in Ind Pro Set ﬁn . The morphisms G α / H i − → G β / H i are closed em b eddings of top ological spaces; the morphisms G α / H i − → G α / H j are principal H i / H j -bundles. Finally , the quotient ob jects G α / H are compact to p ological spaces, i. e., b elong to Pro Set ﬁn ⊂ Ind Pro Set ﬁn . Let G ′ b e a group ob ject in Ind Pro Ind Pro Set ﬁn endo w ed with a cen tral subgroup ob ject iden tiﬁed with the multiplicativ e group k ∗ , whic h is considered as a discrete top ological gro up, that is a g roup ob ject in Ind Set ﬁn ⊂ Pro Ind Pro Set ﬁn . Supp ose 298 that the quotien t group o b ject G ′ /k ∗ is iden tiﬁed with G and the cen tral extension G ′ − → G is endo w ed with a splitting ov er H , i. e., H is a subgroup ob ject in G ′ . Moreo v er, denote by G ′ α of the preimages of G α in G ′ and assume that the mor phisms G ′ α / H i − → G α / H i are principal k ∗ -bundles of top ological spaces in Ind Pro Set ﬁn . E.4.4. One example of suc h a group G ′ is pro vided by the canonical cen tral extension G ∼ of the group G with the k ernel k ∗ , whic h is constructed as follows . F or each α let us c ho ose i suc h that Ad G − 1 α ( H i ) ⊂ H and j suc h that Ad G α ( H j ) ⊂ H i . F or an y to p ological group G denote by µ ( G ) the o ne-dimensional v ector space of left in v ariant ﬁnitely-additiv e k -v a lued Haar measures deﬁned on compact op en subsets of G . An elemen t of the top o logical space G ∼ α / H is a pair consisting o f an elemen t of g ∈ G α / H and an isomorphism µ ( H i / Ad g ( H j )) ≃ µ ( H i / H j ). The to p ology o n G ∼ α / H is deﬁned by the condition that the follo wing set of sections of the k ∗ -torsor G ∼ α / H − → G α / H consists of contin uous maps. Cho ose m such tha t Ad G − 1 α ( H m ) ⊂ H j , a compact op en subset U in the quotien t group H i / H m , and an elemen t a ∈ k ∗ ; for eac h g ∈ G α / H deﬁne the isomorphism µ ( H i / Ad g ( H j )) ≃ µ ( H i / H j ) so that the left- in v ariant measure on H i / Ad g ( H j ) fo r which the measure of the image of U is equal to 1 corresp onds to the left- in v ariant measure on H i / H j for whic h the measure of the image of U is equal to a . The ratio of an y t w o suc h sections is a lo cally constan t function. No w the ob j ect G ∼ α of Pro Ind Pro Set ﬁn is the ﬁb ered pro duct of G ∼ α / H and G α o v er G α / H ; it is easy to deﬁne the group structure on G ∼ (one should ﬁrst c hec k that the construction of G ∼ α / H do es not depend on the c hoice of H i and H j ). E.4.5. Let c ′ : G ′ − → G b e a cen tral extension satisfying the ab ov e conditions. D e- note b y Rep c ′ ( G ) the category of represen tations of G ′ in Pro ( k – v ect ) in whic h the cen tral subgroup k ∗ ⊂ G ′ acts tautologically b y automorphisms prop ortional to the iden tit y , as deﬁned in [23]. Then the forgetful functor Rep c ′ ( G ) − → Rep ( H ) admits a righ t adjoint functor, whic h can b e described as the functor of tensor pro duct ov er H with a certain represen tation of G ′ × H in Pro ( k – vect ). The underlying pro-v ector space of this represen tation, denoted by C c ′ ( G , H ), is the space of “pro-semimeasures on G relativ e to H on the lev el c ′ ”; it is giv en b y t he pro jectiv e system formed by the v ector spaces k c ′ ( G α / H i ) ⊗ k µ ( H / H i ), where the ﬁrst factor is the space of lo cally con- stan t compactly supp orted functions on G ′ α / H i whic h transform b y the tautolog ical c haracter under the a ction of k ∗ . The mo r phism in this pro jec tiv e system corr e- sp onding to a change of α is the pull-bac k map with resp ect to a closed em b edding, while t he morphism corresp onding to a c hange of i is the map of in tegration a long the ﬁb ers of a principal bundle. Hence the represen tation C c ′ ( G , H ) considered as an ob ject of the tensor catego ry Rep ( H × H ) is endo w ed with a structure o f coring (with counit), a nd it fo llo ws from Theorem 7.4.1 that the category Rep c ′ ( G ) is equiv alen t to the category of left como dules o v er C c ′ ( G , H ) in Rep ( H ). 299 No w let c ′′ : G ′′ − → G b e the cen tral extension satisfying t he same conditions that is complemen t ary to c ′ , i. e., the Baer sum c ′ + c ′′ is identiﬁe d with min us the canonical cen tral extension c 0 : G ∼ − → G . Gaitsgory and Ka zhdan noticed that one can extend the rig ht a ction o f H in C c ′ ( G , H ) to an action of G ′′ comm uting with the left action of G ′ , with the cen tr a l subgroup k ∗ of G ′′ acting tautologically . Moreo ver, in [24 ] there is a construction of a natural an ti-isomorphism of corings C c ′ ( G , H ) ≃ C c ′′ ( G , H ) p erm uting the left and righ t actions of G ′ and G ′′ . Th us the category Rep c ′′ ( G ) is equiv alen t to the catego r y of r ig h t como dules o v er C c ′ ( G , H ) in Rep ( H ). E.4.6. So t here is the functor of cotensor pr o duct  C c ′ ( G , H ) : Rep c ′′ ( G ) × Rep c ′ ( G ) − − → Pro ( k – vect ) , whic h is called “semi-in v ariants” in [24]. This functor is neither left, nor rig h t exact in general. One can construct its double-sided derive d functor in the w a y analogous to that of Remark 2.7, at least when the set of indices i is countable. The semi-deriv ed category D si ( Rep c ′ ( G )) is deﬁned as the quotien t category of the homotop y category Hot ( Rep c ′ ( G )) b y t he thic k subcategory o f complexes that are con traacyclic as complexes ov er the ab elian category Rep ( H ). Then Lemma 2.7 allo ws to deﬁne the double-sided deriv ed functor ProCotor C c ′ ( G , H ) : D si ( Rep c ′′ ( G )) × D si ( Rep c ′ ( G )) − − → D ( Pro ( k – v ect )) in terms of coﬂat complexes in Hot ( Rep c ′′ ( G )) and Hot ( Rep c ′ ( G )). The k ey step is to construct f or a n y ob ject of Rep c ′ ( G ) a surjectiv e map onto it from an ob ject of Rep c ′ ( G ) that is ﬂat as a r epresen tation of H . This construction is dual to that of Lemma 1.3.3; it is ba sed on the fact tha t an y mo dule ov er a top olog ical group G induced from the tr ivial mo dule k ov er a compact op en subgroup H ⊂ G is ﬂat with resp ect to the tensor pro duct of discrete G -mo dules o v er k . Question. Can the cotensor pro duct N  C c ′ ( G , H ) M of an ob ject N ∈ Rep c ′′ ( G ) and an ob ject M ∈ Rep c ′ ( G ) b e reco v ered fr o m the tensor pro duct N ⊗ p k M in the categor y of pro-v ector spaces, considered as a represen tat io n of G ∼ with the diagonal action? 300 Appendix F. Algebraic Groupoids with Close d Subgroupoids T o any smo oth aﬃne group oid ( M , H ) one can asso ciate a coring C ( H ) o v er a ring A ( M ) and a natural left and righ t coﬂat Morita a uto equiv alence ( E , E ∨ ) of C ( H ); it has the form C ( H ) ⊗ A ( M ) E ≃ E ≃ E ⊗ A ( M ) C ( H ) a nd E ∨ ⊗ A ( M ) C ( H ) ≃ E ∨ ≃ C ( H ) ⊗ A ( M ) E ∨ , where ( E , E ∨ ) is a certain pair of mutually in v erse in vert- ible A ( M )- mo dules. T o any group oid ( M , G ) con taining ( M , H ) as a close d sub- group oid, one can assign t w o opp osite semialgebras S l ( G, H ) a nd S r ( G, H ) ov er C ( H ) together with a natural left and right semiﬂat Morita equiv alence ( E , E ∨ ) b et w een them formed b y the bisemimo dules S l ( G, H )  C ( H ) E ≃ E ≃ E  C ( H ) S r ( G, H ) and E ∨  C ( H ) S l ( G, H ) ≃ E ∨ ≃ S r ( G, H )  C ( H ) E ∨ . T o obtain these results, w e will hav e to assume the existence of a quotien t v ariet y G /H . In this a pp endix, b y a v ariet y w e mean a smo oth algebraic v ariet y (smo oth sepa- rated sc heme) ov er a ﬁxed gro und ﬁeld k of zero c haracteristic. The structure sheaf of a v ariet y X is denoted by O = O X and the sheaf of diﬀerential top forms by Ω = Ω X ; for an y in v ertible sheaf L ov er X , its tensor p o w ers are denoted b y L n for n ∈ Z . F.1. Cor ing asso ciated to aﬃne group oid. A (smo oth) gr oup oid ( M , G ) is a set of data consisting of tw o v arieties M a nd G , tw o smo oth morphisms s , t : G ⇒ M of sour c e and tar get , a unit morphism e : M − → G , a multiplic ation morphism m : G × M G − → G ( where the ﬁrst factor G in t he ﬁb ered pro duct G × M G maps to M b y the morphism s and the second factor G maps to M b y the morphism t ), and an inverse element morphism i : G − → G . The follo wing equations should b e satisﬁed: ﬁrst, se = id M = te and m ( e × id G ) = id G = m (id G × e ) (unit y); second, tm = tp 1 , sm = sp 2 , and m ( m × id G ) = m (id G × m ), where p 1 and p 2 denote the canonical pro jections of the ﬁb ered pro duct G × M G to the ﬁrst and t he sec ond factors, resp ectiv ely ( asso ciativit y); third, si = t , ti = s , m ( i × id G )∆ t = es , and m (id G × i )∆ s = et , where ∆ s and ∆ t denote the diagonal embeddings of G into the ﬁb ered squares of G ov er M with resp ect to t he morphisms s and t , respective ly (in v erseness ). It fo llo ws from these equations tha t i 2 = id G . A gro up oid ( M , H ) is said t o b e a ﬃne if M and H a re aﬃne v arieties. In the sequel ( M , H ) denotes an aﬃne group oid; its structure morphisms are denoted b y the same letters s , t , e , m , i . Let A = A ( M ) = O ( M ) and C = C ( H ) = O ( H ) b e the rings of f unctions on M and H , resp ectiv ely . The maps o f source and target s , t : H ⇒ M induce t w o maps of rings A ⇒ C , whic h endo w C with t w o structures of A -mo dule; we will consider the A -mo dule structure on C coming f r om the morphism t as a left mo dule structure and the A - mo dule structure on C coming f rom the morphism s as a rig h t mo dule structure. Then there is a natural isomorphism O ( H × M H ) ≃ C ⊗ A C , hence the m ultiplication morphism m : H × M H − → H induces a com ultiplicatio n map 301 C − → C ⊗ A C . Besides, the unit map e : M − → H induces a counit map C − → A . It follows from the asso ciativity a nd unity equations of the group oid ( M , H ) that these com ultiplication and counit maps are morphisms of A - A - bimo dules satisfying the coassociativity and counit y equations; so C is a coring ov er A . Clearly , C is a coﬂat left and right A -mo dule. F.2. Canonical Morita auto equiv alence. Denote b y V = V H the inv ertible sheaf Ω H ⊗ s ∗ (Ω − 1 M ) ⊗ t ∗ (Ω − 1 M ) on H . Let q 1 and q 2 denote the canonical pro jections of the ﬁb ered pro duct H × M H to the ﬁrst and the second factors, resp ectiv ely . Then there are natural isomorphisms q ∗ 1 ( V ) ≃ m ∗ ( V ) ≃ q ∗ 2 ( V ) of inv ertible shea v es on H × M H . Indeed, one has q ∗ 1 (Ω H ) ≃ Ω H × M H ⊗ q ∗ 2 (Ω − 1 H ) ⊗ q ∗ 2 t ∗ (Ω M ) and m ∗ (Ω H ) ≃ Ω H × M H ⊗ q ∗ 2 (Ω − 1 H ) ⊗ q ∗ 2 s ∗ (Ω M ). No w denote b y U the in vertible sheaf e ∗ Ω H ⊗ Ω − 2 M on M . Applying the functors of in v erse image with resp ect to the morphisms e × id H , id H × e : H ⇒ H × M H to the a b o v e isomorphisms, o ne obtains nat ural isomorphisms of in v ertible shea ves t ∗ U ≃ V ≃ s ∗ U . Set E = V ( H ) and E ∨ = V − 1 ( H ). Then E and E ∨ are C -mo dules, and consequen tly A - A -bimo dules. The pull-bac k map V ( H ) − → m ∗ ( V )( H × M H ) with respect to the m ultiplication morphism m together with the isomorphisms m ∗ ( V )( H × M H ) ≃ q ∗ 1 ( V )( H × M H ) ≃ E ⊗ A C and m ∗ ( V )( H × M H ) ≃ q ∗ 2 ( V )( H × M H ) ≃ C ⊗ A E deﬁnes the right and left coactions of C in E . It follo ws f r o m the asso ciativit y equation o f H that these coactions comm ute; so E is a C - C -bicomo dule. Analo gously one deﬁnes a C - C -bimo dule structure on E ∨ . Set E = U ( M ) and E ∨ = U − 1 ( M ); then there are natural isomorphisms of C - como dules C ⊗ A E ≃ E ≃ E ⊗ A C and E ∨ ⊗ A C ≃ E ∨ ≃ C ⊗ A E ∨ . Thes e isomorphisms hav e the prop erty t ha t tw o maps E ≃ C ⊗ A E − → E and E ≃ E ⊗ A C − → E induced b y t he counit map C − → A coincide, and analog ously for E ∨ . Besides, there are obvious isomorphisms E ⊗ A E ∨ ≃ A ≃ E ∨ ⊗ A E . It f ollo ws that E  C E ∨ ≃ ( E ⊗ A C )  C ( C ⊗ A E ∨ ) ≃ E ⊗ A C ⊗ A E ∨ ≃ C , a nd analogo usly E ∨  C E ≃ C . So the pair ( E , E ∨ ) is a left and righ t coﬂat Mor it a equiv alence (see 7.5) b et w een C and itself. Since the bicomo dules E and E ∨ can b e expressed in the ab o v e form in terms of A -mo dules E and E ∨ , it follo ws that there ar e natura l isomorphisms of corings E ⊗ A C ⊗ A E ∨ ≃ C ≃ E ∨ ⊗ A C ⊗ A E . F.3. Dist r ibutions and generalized sect ions. Let X ⊃ Z b e a v ariety with a (smo oth) closed subv ariety and L b e a lo cally constan t sheaf o n X . The sheaf L Z of generalized sections of L , suppo rted in Z and regular a long Z , can b e deﬁned as the image of L with respect to the d - th righ t deriv ed functor of the functor assigning t o an y quasi-coheren t sheaf on X its maximal subsheaf supp orted set-theoretically in Z , where d = dim X − dim Z . The sheaf L Z is a quasi-coheren t sheaf on X supp orted set-theoretically in Z . There is a na t ur a l isomorphism L Z ≃ L ⊗ O X O Z X . The sheaf Ω Z X can b e alterna- tiv ely deﬁned as the direct image of the constan t right mo dule Ω Z o v er the sheaf 302 of diﬀeren tial o p erators Diﬀ Z under the closed em b edding Z − → X (se e [11]); this mak es Ω Z X not only an O X -mo dule, but ev en a D iﬀ X -mo dule. The sheaf Ω Z X is called the sheaf of distributions on X , supp orted in Z and regula r along Z . Let g : Y − → X b e a morphism of v arieties and Z ⊂ X b e a closed subv ar iety . Assume tha t the ﬁb ered pro duct Z × X Y is smo oth and dim Y − dim Z × X Y = dim X − dim Z if Z × X Y is no nempty . Then there is a natur a l isomorphism g ∗ ( L Z ) ≃ ( g ∗ L ) Z × X Y of quasi-coheren t shea v es on Y . In particular, there is a natural pull-back map of the mo dules of glo bal generalized sections g + : L Z ( X ) − → ( g ∗ L ) Z × X Y ( Y ). Let h : W − → X b e a morphism of v arieties and Z ⊂ W b e a closed sub v ariet y suc h that the comp osition Z − → W − → X is also a closed em b edding. Then there is a natural push-forw ard map h ∗ (Ω Z W ) − → Ω Z X of quasi-coheren t shea v es on X . Consequen tly , for a n y lo cally constan t shea v es L ′ on X a nd L ′′ on W endow ed with an isomorphism L ′′ ⊗ Ω − 1 W ≃ h ∗ ( L ′ ⊗ Ω − 1 X ) there is a push-forw ard map h ∗ ( L ′′ Z ) − → L ′ Z . In particular, there is a natura l push-forw a rd map of the mo dules of generalized sections h + : L ′′ Z ( W ) − → L ′ Z ( X ). Let g : Y − → X and h : W − → X b e morphisms of v arieties satisfying the ab ov e conditions with resp ect to a closed sub v ariet y Z ⊂ W . Assume also that the ﬁb ered pro duct W × X Y is smo oth and dim W × X Y + dim X = dim W + dim Y if W × X Y is nonempt y . Set ˜ g = id W × g : W × X Y − → W and ˜ h = h × id Y : W × X Y − → Y . Then there is a natural isomorphism of inv ertible shea v es ˜ g ∗ Ω W ⊗ Ω − 1 W × X Y ≃ ˜ h ∗ ( g ∗ Ω X ⊗ Ω − 1 Y ) on W × X Y , as one can see by decomp osing g and h in to closed embeddings f ollo w ed b y smo oth morphisms. F or an y quasi-coheren t sheaf F o n W support ed set-theoretically in Z there is a nat ura l isomorphism ˜ h ∗ ˜ g ∗ F ≃ g ∗ h ∗ F of quasi-coheren t sheav es on W × X Y . The push-forw ard maps of t he shea v es of distributions with resp ect to the morphisms h and ˜ h are compatible with the pull- bac k isomorphisms with resp ect to g and ˜ g in the obvious sense. The sheaf of generalized sections L Z of a lo cally constan t she af L o n a v ariet y X ⊃ Z is endo w ed with a natural increasing ﬁltration b y coheren t subshea v es F n L Z of generalized sections of order no g r eat er than n . This ﬁltration is preserv ed by all the ab o v e natural isomorphisms and maps. The asso ciated graded sheaf gr F Ω Z X is the direct image under our closed em b edding ι : Z − → X of a sheaf of O Z -mo dules naturally isomorphic to the tensor pro duct Ω Z ⊗ O Z Sym O Z N Z,X , where N Z,X is the normal bundle to Z in X and Sym denotes the symmetric algebra. In particular, there is a natural isomorphism λ 0 : ι ∗ Ω Z − → F 0 Ω Z X . F urthermore, there is a na t ur a l ma p of sheav es of k - v ector spaces λ 1 : ι ∗ Ω Z ⊗ O X T X − → F 1 Ω Z X whic h induces the isomorphism ι ∗ (Ω Z ⊗ O Z N Z,X ) ≃ F 1 Ω Z X /F 0 Ω Z X , where T = T X denotes the tangen t bundle of X . The map λ 1 satisﬁes the equation λ 1 ( f ω ⊗ v ) = λ 1 ( ω ⊗ f v ) = f λ 1 ( ω ⊗ v ) − λ 0 ( v ( f ) ω ) for lo cal sections f ∈ O X , ω ∈ Ω Z , and v ∈ T X , where ( v , f ) 7− → v ( f ) denotes the a ctio n of v ector ﬁelds in functions. 303 F.4. Lie algebroid of a gr oup oid. A Lie algebr oid g o v er a comm utativ e ring A is an A -mo dule endo w ed with a Lie algebra structure and a Lie action of g b y deriv atio ns of A satisfying the equations [ x, ay ] = a [ x, y ] + x ( a ) y and ( ax )( b ) = a ( x ( b )) for a , b ∈ A , x , y ∈ g . The enveloping algebr a U A ( g ) o f a Lie algebroid g ov er A is generated by A a nd g with the relations a · b = ab , a · x = ax , x · a = ax + x ( a ), and x · y − y · x = [ x, y ], where ( u, v ) 7− → u · v denotes the m ultiplication in U A ( g ). The algebra U A ( g ) is endow ed with a natur a l increasing ﬁltra tion F n U A ( g ) deﬁned b y the rules F 0 U A ( g ) = im A , F 1 U A ( g ) = im A + im g , and F n U A ( g ) = F 1 U A ( g ) n for n > 1. When g is a ﬂa t A -mo dule, the asso ciated graded algebra gr F U A ( g ) is isomorphic to the symmetric algebra Sym A ( g ) of the A -mo dule g . Let ( M , G ) b e a group oid with an aﬃne base v ariet y M ; set A = A ( M ) = O ( M ). Then the A -mo dule g = N e ( M ) ,G ( M ) has a natural Lie algebroid structure. T o deﬁne the action of g in A , consider the A - mo dule ( e ∗ T G )( M ) of v ector ﬁelds on e ( M ) tangen t to G . There are natural push-forw ard morphisms s + , t + : ( e ∗ T G )( M ) ⇒ T ( M ). Identify g with the k ernel o f the morphism t + ; then the action of g in A is deﬁned in terms of t he map s + : g − → T ( M ). T o deﬁne the Lie a lg ebra structure on g , w e will em b ed g into a certain mo dule of g eneralized sections on G , supp o rted in e ( M ) and regular along e ( M ). Set K l ( G ) = ( t ∗ Ω − 1 M ⊗ Ω G ) e ( M ) ( G ); this mo dule of generalized sections in en- do w ed with a natural ﬁltra tion F . The O ( G )- mo dule structure o n K l ( G ) induces an A - A -bimo dule structure; as in F.1, w e consider the A -mo dule structure com- ing from the morphism t as a left A -mo dule structure and the A -mo dule structure coming from the morphism s a s a rig h t A -mo dule structure. There is a natura l isomorphism of A - A -bimo dules A ≃ F 0 K l ( G ). D eﬁne a k -linear map of shea v es e ∗ N e ( M ) ,G − → ( t ∗ Ω − 1 M ⊗ Ω G ) e ( M ) lo cally b y the formula v 7− → t ∗ ω − 1 ⊗ λ 1 ( ω ⊗ v ), where v is a lo cal vec tor ﬁeld on e ( M ) tang ent to G suc h that t ∗ ( v ) = 0 and ω is a lo cal non- v anishing top form on M ; it is easy this expression do es not dep end on the choice of ω . P assing to the global sections, we obtain an injective map g − → F 1 K l ( G ) inducing an isomorphism g ≃ F 1 K l ( G ) /F 0 K l ( G ). This injectiv e map and the A - A -bimo dule structure satisfy the compatibility equations a · x = ax and x · a = x ( a ) + ax for x ∈ g ⊂ F 1 K l ( G ) and a ∈ A , where ( a, x ) 7− → ax denotes the action of A in g , while ( a, u ) 7− → a · u and ( u, a ) 7− → u · a denote the left and right actions of A in F 1 K l ( G ). Let us deﬁne a k -algebra structure on K l ( G ). There is a natural isomorphism p ∗ 1 ( t ∗ Ω − 1 M ⊗ Ω G ) ⊗ p ∗ 2 ( t ∗ Ω − 1 M ⊗ Ω G ) ≃ p ∗ 1 t ∗ Ω − 1 M ⊗ Ω G × M G of in v ertible shea v es o n G × M G . The pull-back with resp ect to the closed em b edding G × M G − → G × Spec k G pro vides an isomorphism ( p ∗ 1 t ∗ Ω − 1 M ⊗ Ω G × M G ) ( e × e )( M ) ( G × M G ) ≃ K l ( G ) ⊗ A K l ( G ), and the push-forw ard with r esp ect t o the m ultiplication map G × M G − → G deﬁnes an asso ciativ e m ultiplication K l ( G ) ⊗ A K l ( G ) − → K l ( G ). The asso ciated g raded algebra gr F K l ( G ) is naturally isomorphic t o Sym A F 1 K l ( G ) /F 0 K l ( G ). The f o rm ula ( u, a ) 7− → t + ( s + ( a ) u ) deﬁnes a left action of K l ( G ) in A ; the subspace g in F 1 K l ( G ) 304 is characterize d as the annihilat or of the unit elemen t o f A under this action. Hence g is a Lie subalgebra of K l ( G ); this mak es it a L ie algebra and a Lie algebroid ov er A . It f o llo ws that there is a natural isomorphism U A ( g ) ≃ K l ( G ). Analogously one deﬁnes a n algebra structure on the A - A -bimo dule of generalized sections K r ( G ) = ( s ∗ Ω − 1 M ⊗ Ω G ) e ( M ) ( G ); t hen there is a natural isomorphism of k -alg ebras U A ( g ) op ≃ K r ( G ). F.5. Two Morita equiv alen t semialgebras. Let ( M , H ) − → ( M , G ) b e a mor- phism o f smo ot h group o ids with the same base v ariety M suc h t ha t the gro up oid H is aﬃne and the morphism of v arieties H − → G is a closed embedding. Denote b y h = N e ( M ) ,H ( M ) t he Lie algebroid of the g roup oid H ; then there is a natural injectiv e morphism h − → g of Lie algebroids o v er A ( M ). The for m ula φ l ( u, c ) = t + ( i + ( c ) u ) deﬁnes a pairing φ l : K l ( H ) ⊗ A C − → A b etw een the algebra K l ( H ) and the coring C satisfying the conditions of 10.1 .2 with the left and right sides switc hed. The push-forw ard with resp ect to the closed embedding H − → G deﬁnes an injective morphism o f k - algebras K l ( H ) − → K l ( G ); this is the en v eloping algebra morphism induced b y the morphism of Lie a lgebroids h − → g . Since h and g / h are pro jectiv e A -mo dules, K l ( G ) is a pro jectiv e left a nd righ t K l ( H )-mo dule. Set S l = S l ( G, H ) = K l ( G ) ⊗ K l ( H ) C ; w e will use the construction of 10.2.1 to endo w S l with a structure of semialgebra o v er C . Consider the cotensor pro duct S l  C E ≃ K l ( G ) ⊗ K l ( H ) E . Denote b y p 1 and q 2 the pro jections of the ﬁb ered pro duct G × M H to the ﬁrst and the second factors. There is a natura l isomorphism p ∗ 1 ( t ∗ Ω − 1 M ⊗ Ω G ) ⊗ q ∗ 2 ( V H ) ≃ p ∗ 1 t ∗ Ω − 1 M ⊗ Ω G × M H ⊗ q ∗ 2 s ∗ Ω − 1 M of in v ertible shea v es on G × M H , where V H = Ω H ⊗ s ∗ (Ω − 1 M ) ⊗ t ∗ (Ω − 1 M ) is the inv ertible sheaf on H deﬁned in F .2. The pull-back with resp ect to the closed em b edding G × M H − → G × Spec k H iden tiﬁes the tensor pro duct K l ( G ) ⊗ A E with the mo dule of generalized sections ( p ∗ 1 t ∗ Ω − 1 M ⊗ Ω G × M H ⊗ q ∗ 2 s ∗ Ω − 1 M ) e ( M ) × M H ( G × M H ). The push- forw ard with r esp ect to the multiplication morphism G × M H − → G deﬁnes a natura l map K l ( G ) ⊗ A E − → E , where E = V H G ( G ) is the space of g eneralized sections of the in v ertible sheaf V G = Ω G ⊗ s ∗ (Ω − 1 M ) ⊗ t ∗ (Ω − 1 M ) o n G . It fo llo ws from the asso ciativity equation for the t w o iterated m ultiplication maps G × M H × M H ⇒ G that this map factorizes through K l ( G ) ⊗ K l ( H ) E . The induced map K l ( G ) ⊗ K l ( H ) E − → E is an isomorphism, since the asso ciated graded map with resp ect to t he ﬁltrations F is. Denote b y q 1 and p 2 the pro jections of the ﬁb ered pro duct H × M G to the ﬁrst and second factors. One constructs a natural isomorphism m ∗ ( V G ) ≃ p ∗ 2 ( V G ) of inv ertible shea v es o n H × M G in the same wa y as in F.2. The pull- ba c k with respect to the m ultiplication map m : H × G − → G together with this isomorphism pro vide a left coaction of C in E . Analo gously one deﬁnes a right coaction of C in E ; it follows from the asso ciat ivity equation for H × G × H ⇒ G that these t w o coactions commute. 305 W e ha v e S l  C E ≃ E , th us the isomorphism S l ≃ E  C E ∨ pro vides a left coaction of C in S l comm uting with the natural right coaction of C in S l . Since the natural map E − → E provided b y the push-forw ard with resp ect to the closed em b edding H − → G is a morphism of C - C -bicomo dules, so is the semiunit map C − → S l . It remains to sho w t ha t the semim ulplication map S l  C S l − → S l is a morphism of left C -como dules; here it suﬃces to c hec k that the map S l  C S l  C E − → S l  C E is a morphism of left C - como dules. After w e ha v e done with this v eriﬁcation, the latter map will deﬁne a left S l -semimo dule structure on E . Analogously , deﬁne a pairing φ r : C ⊗ A K r ( H ) − → A b y the formula φ r ( c, u ) = s + ( i + ( c ) u ) and set S r = S r ( G, H ) = C ⊗ K r ( H ) K r ( G ). The same construction make s S r a semialgebra o v er C a nd E a right S r -semimo dule. W e will ha v e to ch ec k that the left S l -semimo dule and the righ t S r -semimo dule structures o n E comm ute. After this is done, w e get an S l - S r -bisemimo dule S l  C E ≃ E ≃ E  C S r , where b oth the maps E ⇒ E induced b y the semiunit maps C − → S l and C − → S r coincide with the push-forw ard map V ( H ) − → V H G ( G ) under the closed em b edding H − → G . The isomorphisms E ∨  C S l ≃ E ∨  C S l  C E  C E ∨ ≃ E ∨  C E  C S r  C E ∨ ≃ S r  C E ∨ deﬁne an S r - S l -bisemimo dule E ∨  C S l = E ∨ ≃ S r  C E ∨ endo w ed with bisemimo dule isomorphisms E ♦ S r E ∨ ≃ ( E  C S r ) ♦ S r ( S r  C E ∨ ) ≃ E  C S r  C E ∨ ≃ S l and E ∨ ♦ S l E ≃ ( E ∨  C S l ) ♦ S l ( S l  C E ) ≃ E ∨  C S l  C E ≃ S r . This pro vides a left and righ t semiﬂat Morita equiv alence ( E , E ∨ ) b et w een the semialgebras S l and S r , and isomorphisms of semialgebras S r ≃ E ∨  C S l  C E and S l ≃ E  C S r  C E ∨ . (See 8.4.5 and 8 .4.1 for the relev ant deﬁnition and construction.) F.6. Compatibility veriﬁcations. In order to che c k that the map S l  C E ≃ S l  C S l  C E − → S l  C E ≃ E is a morphism of left C -como dules, w e will iden tify this map with a certain push-forw ard map o f appropriate mo dules o f generalized sections. Here we will need to assume the existence of a v a riet y of left cosets G/H suc h that G − → G/H is a smo o th surjectiv e morphism and the ﬁb ered square G × G/H G can b e iden tiﬁed with G × M H so that the canonical pro jection maps G × G/H G ⇒ G corresp ond to the pro jection and m ultiplication ma ps p 1 , m : G × M H ⇒ G . Actually , w e are intere sted in the quotient v ariety G × H G of G × M G by the equiv alence relatio n ( g ′ h, g ′′ ) ∼ ( g ′ , hg ′′ ) fo r g ′ , g ′′ ∈ G and h ∈ H ; it can b e constructed as either of the ﬁb ered pro ducts H \ G × M G ≃ G × H G = G × M G/H , where H \ G denotes the v ariety of righ t cosets, H \ G ≃ G/H . Analogously one can construct the quotien t v ariet y G × H G × H G of the triple ﬁb ered pro duct G × M G × M G by the equiv alence relation ( g ′ h 1 , g ′′ h 2 , g ′′′ ) ∼ ( g ′ , h 1 g ′′ , h 2 g ′′′ ) f or g ′ , g ′′ , g ′′′ ∈ G and h 1 , h 2 ∈ H . W e hav e S l  C E ≃ E  C E ∨  C E . Consider the natural map r : G × M H × M G − → G × H G giv en by the f orm ula ( g ′ , h, g ′′ ) 7− → ( g ′ h, g ′′ ) = ( g ′ , hg ′′ ). Let p 1 , q 2 , p 3 denote the pro jections of the triple ﬁb ered pro duct G × M H × M G to the three f actors and n : G × H G − → G denote the multiplication morphism. There are natural 306 isomorphisms p ∗ 1 t ∗ (Ω − 1 M ) ⊗ Ω G × M H × M G ⊗ p ∗ 3 s ∗ (Ω − 1 M ) ≃ p ∗ 1 ( V G ) ⊗ q ∗ 2 (Ω H ) ⊗ p ∗ 3 ( V G ) a nd therefore r ∗ ( n ∗ t ∗ Ω − 1 M ⊗ Ω G × H G ⊗ n ∗ s ∗ Ω − 1 M ) ≃ p ∗ 1 ( V G ) ⊗ q ∗ 2 ( V − 1 H ) ⊗ p ∗ 3 ( V G ) of in v ertible shea v es on G × M H × M G . The pull-back with resp ect to the closed em b edding G × M H × M G − → G × Spec k H × Spec k G provid es an isomorphism ( p ∗ 1 V G ⊗ q ∗ 2 V − 1 H ⊗ p ∗ 3 V G ) H × M H × M H ( G × M H × M G ) ≃ E ⊗ A E ∨ ⊗ A E . The pull-bac k with respect to the smo oth morphism r identiﬁe s the mo dule of generalized sections ( n ∗ t ∗ Ω − 1 M ⊗ Ω G × H G ⊗ n ∗ s ∗ Ω − 1 M ) H × H H ( G × H G ) with the submo dule E  C E ∨  C E ⊂ E ⊗ A E ∨ ⊗ A E , a s one can see b y identifying the tensor pro duct E ⊗ A C ⊗ A E ∨ ⊗ A C ⊗ A E with a mo dule of generalized sections on t he ﬁb ered square of G × M H × M G o v er G × H G . Now our map E  C E ∨  C E − → E is identiﬁed with the push-fo r ward map with resp ect to the multiplic ation morphism n ; to c hec k this, one can ﬁrst iden tify the map K l ( G ) ⊗ A E − → K l ( G ) ⊗ K l ( H ) E ≃ E  C E ∨  C E with a push-forw ard map with resp ect to the morphism G × M G − → G × H G . The desired compatibilit y with the left C -como dule structures now follows from the comm utation of the pull-back and push-forw ard maps o f g eneralized sections. 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