Jacobi-Zariski Exact Sequence for Hochschild Homology and Cyclic (Co)Homology

J A COBI-ZARISKI EXA CT SEQUENCE FOR HOCHSCH ILD HOMOLOGY AND CYCLIC (CO)H OMOLOGY A T ABEY KA YGUN A B S T R AC T . W e prove that fo r an inclusion o f u nital associative but not necessarily co m mutative k - algebras B ⊆ A we have long exact sequences in Hoch schild ho m ology and cyclic (co)h o molog y akin to the Jaco b i-Zariski sequen ce in Andr ´ e-Quillen homology , provided that the qu otient B - module A / B is flat. W e also prove that fo r an arbitrar y r-flat morph ism ϕ : B → A with an H-unital kernel, one can express the W odzick i excision seq u ence and o ur Jacobi-Z ariski sequence in Hochsch ild homolog y and cyclic (co)hom o logy as a single lon g exact sequence. I N T RO D U C T I O N Let k be a ground field. Assum e we have an inclusion of associativ e com mutative unital k - algebras B ⊆ A . Then for any A -bimod ule N , one obtains a long exact sequence i n Andr ´ e-Quillen homology [12, Thm.5.1] · · · → D n + 1 ( A | B ; N ) → D n ( B | k ; N ) → D n ( A | k ; N ) → D n ( A | B ; N ) → · · · which is often referred as the Jacobi-Zariski long exact sequence [8, S ect.3.5]. In this paper we show that th ere are analog o us long exact sequences for ordi nary (co)homo logy , Hochschild homology and cyclic (co)homolog y of k -algebras of the form (written here for Hochschild h o- mology) · · · → H H n + 1 ( A | B ) → H H n ( B | k ) → H H n ( A | k ) → H H n ( A | B ) → · · · for n > 1. W e pro ve the existence under t he condition th at we ha ve an unital associative (not necessarily commutative) algebra A and a unital subalgebra B such that the quot ient B -modul e A / B is flat. In the sequel, such inclus i ons B ⊆ A of unital k -algebras are called r educed-flat ( r- flat in short) extensions. The condi tion of r-flatness is slightl y more restrictive than A bei n g flat ov er B but there are plenty of relev ant e xamples. (See Subsectio n 1.5) There are sim i lar lon g exact sequences in the literature for other cohomology theories of k - algebras. The rele vant sequence we cons ider here is the W odzicki excision sequence [14] for Hochschild homology and cyclic (co)homology (written here for Hochs child homology) · · · → H H n ( I ) → H H n ( B ) → H H n ( A ) → H H n − 1 ( I ) · · · for an epimorphism π : B → A of unit al k -algebras with an H-un i tal kernel I : = ker ( π ) (Subsec- tion 1.3). The W o d zicki excision sequence characterizes homotopy cofiber o f the morphism of dif- ferential graded k -modul es π ∗ : CH ∗ ( B ) → CH ∗ ( A ) induced by π as the sus pended Hochschild 1 2 A T ABEY KA YGUN complex Σ CH ∗ ( I ) of the ideal I . Our Jacobi-Zariski sequence, on the other hand , characterizes the sam e hom otopy cofiber as the relative Hochschild chain complex CH ∗ ( A | B ) (relative ` a la Hochschild [6]) for a monomorphis m B → A of k -algebras. Now , assume ϕ : B → A is an arbitrary morphism of unital k -algebras such that I : = ker ( ϕ ) is H-unital and the quoti ent B -module A / i m ( ϕ ) is flat. Under these conditions, we sh ow i n Theorem 4.3 (written here for Hochschi l d chain complexes) that hom o topy cofiber CH ∗ ( A , B ) of the morphis m ϕ ∗ : CH ∗ ( B ) → CH ∗ ( A ) induced by ϕ fits into a homot o py cofibration sequence of the form Σ CH ∗ > 1 ( I ) → CH ∗ > 1 ( A , B ) → CH ∗ > 1 ( A | B ) which gives us an appropriate lon g exact sequence. As one can im mediately see, we get W odz- icki’ s charac terization of the homoto py cofiber when ϕ is an epim o rphism and our Jacobi-Zariski characterization when ϕ is a monomorphism. Overview. In Section 1 we revie w s ome s tandard constructions and facts that we are going to need in the course of proving ou r main result, mostly in order to establish notation. Th en in Section 2 we prove the existence of the Jacobi-Zariski long-exact sequence for ordinary homology of algebras. In S ection 3 we gradually deve lop th e same result for cohomology under certain restrictions on the dim ens ion of the algebra then we remove tho se restriction s and place them on the coef ficient modules. Finall y in Section 4, we prov e the existence of the Jacobi-Zariski long exact sequence for Hochschi ld homology and cyclic (co)homology of associativ e u nital algebras, and then construct a homotopy cofibration s equence extending both the Jacobi-Zariski sequence and W odzicki e xcision sequence in Hochschil d homolog y and cyclic (co)homology . Standing assumptions and con vent ions. W e use k to denote our grou nd field. W e make no assumption s on the characteristi c of k . All unadorned tensor products ⊗ are assumed to be over k . W e will use L i X i and ∑ i X i to denote respectively the external and t he internal sum of a collection of subsp aces { X i } i ∈ I of a k -vector space V . The k -algebras we consi d er are all unital and associative b ut not necessarily com mutative. W e make no assumptions on t he k -dimensions of these algebras unless otherwise is explicitly stated. W e wi ll use A e to denote the en veloping algebra A ⊗ A o p of an associative unital algebra A . W e will use t he term parity to denote the type of an A -module: wheth er it is a left module or a right mo dule. W e us e the notat i on C ∗ > n to denote th e good tr uncation of C ∗ at degree n , and Σ C ∗ for the suspens ion of a differential Z -graded k -module ( C ∗ , d C ∗ ) . Acknowledgmen ts. W e are grateful to the referee for pointi n g ou t t he reference [2] which al- lowed us to connect H-unital i deals of W odzicki [14] and hom ological epimorph isms of [3]. W e would like to thank the referee also for pointing out the references [5, 1 , 10] which allowed us to better contextualize Conjecture 4.4. W e als o thank Claude Cibil s for point ing an error we made in the publi shed v ersion of this preprint. The current version is corrected by explicitly stating the Jacobi-Zariski sequence works only for p > 1. An errata for the published ve rsion is to appear . J ACOBI-ZARISKI EXA C T SEQUENCE FOR HOCHSCHILD HOMOLOGY AND CYCLIC (CO)HOMOLOGY 3 1. P R E L I M I NA R I E S 1.1. Relative (co)homology . A m onomorphis m f : X → Y of A -modules is called an ( A , B ) - monomorphi sm if f is an monomorphism o f A -m odules, and a split mo nomorphism of B - modules. W e define ( A , B ) -epim o rp h i sms similarly . A sho rt exact sequence of A -modules 0 → X i − → Y p − → Z → 0 is called an ( A , B ) -e xact sequence if i i s an ( A , B ) -mono morphism and p is an ( A , B ) -epimorphism. An A -mod ule P is called an ( A , B ) -projective module if for any ( A , B ) -epimorphism f : X → Y and a morphism of A -mo d u les g : P → Y there exists an A -modul e morph ism g ′ : P → X which satisfies g = f ◦ g ′ P g ′   g   X f / / Y / / 0 ( A , B ) -injectiv e m odules are defined si m ilarly . Also, a module T is called ( A , B ) -flat if for e very ( A , B ) -sho rt exact sequence 0 → X → X ′ → X ′′ → 0 the induced sequence of k -modules 0 → X ⊗ A T → X ′ ⊗ A T → X ′′ ⊗ A T → 0 is e xact. Note that e very A -projective (resp. A -flat or A -inj ective) mo d ule i s also ( A , B ) - projectiv e (resp. ( A , B ) -flat or ( A , B ) -i n jectiv e) for any unital subalgebra B ⊆ A . 1.2. The bar complex. The bar complex of a (unital) associati ve alg ebra A is the graded k - module CB ∗ ( A ) : = M n > 0 A ⊗ n + 2 together with the differentials d CB n : CB n ( A ) → CB n − 1 ( A ) which are defined by d CB n ( a 0 ⊗ · · · ⊗ a n + 1 ) = n ∑ j = 0 ( − 1 ) j ( · · · ⊗ a j a j + 1 ⊗ · · · ) for any n > 1 and a 0 ⊗ · · · ⊗ a n ∈ CB n ( A ) . The complex CB ∗ ( X ; A ; Y ) : = X ⊗ A CB ∗ ( A ) ⊗ A Y is c alled the two-sided (homological) bar complex of a pair ( X , Y ) of A -mo d ules o f oppo s ite parity . The cohomol ogical c ounterpart CB ∗ ( X ; A ; Z ) for a pair of rig ht A -modu l es ( X , Z ) i s defined as CB ∗ ( X ; A ; Z ) : = Hom A ( CB ∗ ( X ; A ; A ) , Z ) The corresponding bar com plex for left modules is defined similarly using CB ∗ ( A ; A ; X ) . Since the two-sided bar complex CB ∗ ( A ) is an A e -projectiv e resol u tion of the A -bim odule A , the homology of t he complex CB ∗ ( X ; A ; Y ) yi elds the T o r-groups T or A ∗ ( X , Y ) for the pair ( X , Y ) . Similarly , t he Ext-groups Ext ∗ A ( X , Z ) for the p air ( X , Z ) come from the coho mological v ariant of the two-sided bar complex CB ∗ ( X ; A ; Z ) . 4 A T ABEY KA YGUN T o be technically correct, since our tensor products ⊗ are taken ov er k , the b ar complexes we defined sho uld be referred as th e r elat i ve bar complexes (relativ e to the base field) denoted by CB ∗ ( X ; A | k ; Y ) or CB ∗ ( X ; A | k ; Z ) . Ho wev er , since k is a semi-simple subalgebra of A , the relative (co)homo logy and the abso lute (co)homology agree (cf. [7, Prop.2.5 ]). See also [4, Thm.1.2]. Now , we define the relativ e two-sided bar comp lexes CB ∗ ( X ; A | B ; Y ) and CB ∗ ( X ; A | B ; Z ) similarly for A -mod u les of the correct parity X , Y and Z where we replace the tensor p ro d u ct ⊗ over k with the tensor product ⊗ B over a unital subalgebra B . Since A ⊗ B A is ( A , B ) - projectiv e by [6, Lem . 2 , pg. 248], we see that for any ri ght A -module X , the modu le X ⊗ B A is a ( A , B ) -projectiv e m odule. Then the relati ve com p l exe s CB ∗ ( X ; A | B ; Y ) and CB ∗ ( X ; A | B ; Z ) yield re spectively the relative torsion groups T o r ( A | B ) ∗ ( X , Y ) and the relative extension groups Ext ∗ ( A | B ) ( X , Z ) . 1.3. H-unital ideals. A not necessarily unital k -algebra I is call ed H-unital if the bar com plex CB ∗ ( I ) of I i s a resolution of I viewed as a I -bimodule. One can imm ediately see that any unital algebra is H-unital. Our definiti on of H-unitality di ff ers from W odzicki’ s o ri g inal definition giv en in [14], b ut is still equi valent. W odzicki defines another complex CB ′ ∗ ( I ) (he uses the notation B ∗ ( I ) ) CB ′ ∗ ( I ) = M n > 0 I ⊗ n + 1 together with the differentials d n ( x 0 ⊗ · · · ⊗ x n ) = n − 1 ∑ j = 0 ( − 1 ) j ( · · · ⊗ x j x j + 1 ⊗ · · · ) CB ′ 0 ( I ) is the di ff erential graded k -module CB ∗ ( I ) suspended once and augmented by I . Th e fact that CB ∗ ( I ) is a resolution o f I is equiv alent to the fa ct that CB ′ ∗ ( I ) is acyc lic. Because k is a field and e very k -module i s free, and therefore flat, the acyclicity of CB ′ ∗ ( I ) is equiva lent to W odzicki’ s definition of H-unitalit y (cf. [14, pg.592].) 1.4. The Hochschild complex. The Hochschild complex of A wit h coef ficients in a A -bimodule M is the graded k -modu l e CH ∗ ( A , M ) : = M n > 0 M ⊗ A ⊗ n together with the differentials (tradition ally denoted by b ) b n : CH n ( A , M ) → CH n − 1 ( A , M ) b n ( m ⊗ a 1 ⊗ · · · a n ) =( ma 1 ⊗ a 2 ⊗ · · · ⊗ a n ) + n − 1 ∑ j = 1 ( − 1 ) j ( m ⊗ · · · ⊗ a j a j + 1 ⊗ · · · ) + ( − 1 ) n ( a n m ⊗ a 1 ⊗ · · · ⊗ a n − 1 ) J ACOBI-ZARISKI EXA C T SEQUENCE FOR HOCHSCHILD HOMOLOGY AND CYCLIC (CO)HOMOLOGY 5 defined for n > 1 and m ⊗ a 1 ⊗ · · · ⊗ a n ∈ CH n ( A , M ) . One can define the relative Hochschild complex CH ∗ ( A | B , M ) sim ilarly where we write the tens or prod ucts o ver B instead of k . For every A -bimodul e M the Hochschild complex CH ∗ ( A , M ) can als o be constructed as CH ∗ ( A , M ) = M ⊗ A e CB ∗ ( A ) where A e = A ⊗ A o p is the en veloping algebra of A . Since the bar complex CB ∗ ( A ) i s a projectiv e resol ution of A viewed as a A -bimodule, the Hochschild ho m ology gro u p s we de- fined using the Hochschild com p lex are T o r A e n ( A , M ) [8, Prop.1.1.13]. One can si m ilarly define the Hochschild cohomology g rou ps H H n ( A , M ) as the extension groups Ext n A e ( A , M ) , which can also be computed as the cohomolog y of the Hochschild cochain complex CH ∗ ( A , M ) : = Hom A e ( CB ∗ ( A ) , M ) [8, Def.1.5.1]. W e will use the notation H H ∗ ( A ) and H H ∗ ( A ) to denot e respectively H H ∗ ( A , A ) and H H ∗ ( A , A ) for a (unital) k -algebra A . 1.5. Flat and r -flat extensions. Assume B ⊆ A i s a unit al sub algebra. This makes A into a B -bim o dule. Now , consider the following sho rt exact sequence of B -modules 0 → B → A → A / B → 0 Since B is flat over it s elf, using the short exact sequence above for e very B -module Y we get an exact sequence of k -modules of the form 0 → T or B 1 ( A , Y ) → T or B 1 ( A / B , Y ) → B ⊗ B Y → A ⊗ B Y → ( A / B ) ⊗ B Y → 0 and an is o morphism o f k -modul es T or B n ( A , Y ) ∼ = T or B n ( A / B , Y ) for every n > 2. Thi s means the flat dimension (som etimes referred as the weak dimension ) of A / B is at most 1 when A i s a flat B -module. On the other hand, when t he qu otient A / B is a flat B -module the B -module A must also be flat. Definition 1.1. An inclusion of unital k -algebras B ⊆ A is call ed a flat e xtension i f A viewed as a B -bimodule is flat. An inclusi o n of unital k -algebras B ⊆ A is called a r-flat e xt ens ion i f the quotient B -bimodu le A / B is B -flat. A morphis m of unital k -algebras ϕ : B → A i s called flat (resp. r-flat) if im ( ϕ ) ⊆ A is a flat (resp. r-flat) extension. One can easily see that e very r-flat extension is a flat extension. Con versely , if A is a flat extension ov er B which is also augmented, i.e. we hav e a unital alg ebra morphism ε : A → B splittin g the inclusion of algebras B → A , then it is also a r -flat extension. In o t her words, for augmented e xtensions flatness and r-flatness are equiv al ent . Example 1.2. The polynomi al algebras B [ x 1 , . . . , x n ] wi th commu t ing indeterminates and the polynomial algebras B { x 1 , . . . , x n } wi t h non-comm u ting i ndeterminates are all r-flat extensions of a unital k -algebra B . In g eneral, if G is a monoid then the group algebra B [ G ] of G over B giv es us a r -flat e xtension B ⊆ B [ G ] . 6 A T ABEY KA YGUN 2. T H E J AC O B I - Z A R I S K I S E Q U E N C E F O R T O R S I O N G R O U P S Pr oposition 2.1. Let I be an H-unital ideal of a unital algebra B . Ass ume X and Y ar e B / I - modules of oppo s ite parity . Then o ne has nat u ral isomorph i sms of the form (2.1) T or B p ( X , Y ) ∼ = T or B / I p ( X , Y ) for e very p > 0 . Pr oof. W e begin by defining an auxiliary differential graded k -modul e I ∗ . W e augment CB ′ ∗ ( I ) with k as follows k 0 ← − I ← I ⊗ 2 ← I ⊗ 3 ← · · · Since I is H-unital, the hom ology is concentrated at degree 0 where H 0 ( I ∗ ) = k and H q ( I ∗ ) = 0 for e very q > 0. Then t h e product X ⊗ I ∗ ⊗ Y is a dif ferential graded k -submodul e of CB ∗ ( X ; B ; Y ) since actio n of I on X and Y are by 0. Now , we define an i n creasing filtratio n of differe ntial graded k -submodules F p ∗ ⊆ F p + 1 ∗ of CB ∗ ( X ; B ; Y ) where for 0 6 p 6 n we let F p n : = ∑ n 0 + ··· + n p = n − p X ⊗ I ⊗ n 0 ⊗ B ⊗ I ⊗ n 1 ⊗ · · · ⊗ B ⊗ I ⊗ n p ⊗ Y and allow n i = 0. Then we define F p n = F p + 1 n for e very p > n , and let F − 1 ∗ : = 0. Note that F 0 ∗ = X ⊗ I ∗ ⊗ Y and S p F p ∗ = CB ∗ ( X ; B ; Y ) . In th e associated spectral sequence of the filtration we get E 1 0 , q = H q ( F 0 ∗ / F − 1 ∗ ) = H q ( X ⊗ I ∗ ⊗ Y ) ∼ = X ⊗ H q ( I ∗ ) ⊗ Y = 0 = E ∞ 0 , q for e very q > 0 since I is H-unital and ⊗ is an exact bifunctor . W e also see that for q < 0 E 0 p , q = F p p + q / F p − 1 p + q = 0 = E ∞ p , q because F p n = F p + 1 n for e very p > n . Moreover , for any p > 0 and for any q > 0 E 0 p , q = F p p + q / F p − 1 p + q ∼ = M n 0 + ··· + n p = q X ⊗ I ⊗ n 0 ⊗ ( B / I ) ⊗ I ⊗ n 1 ⊗ · · · ⊗ ( B / I ) ⊗ I ⊗ n p ⊗ Y Again, sin ce the action of I on X , Y and B / I are by 0, t he qu o tient F p ∗ / F p − 1 ∗ can be written as a product of differential graded modules Σ p ( F p ∗ / F p − 1 ∗ ) ∼ = X ⊗ I ∗ ⊗ ( B / I ) ⊗ · · · ⊗ I ∗ ⊗ ( B / I ) | {z } p -times ⊗ I ∗ ⊗ Y Our spectral sequence collapses because the homology of I ∗ is concentrated at degree 0. So, we get E 1 p , q = 0 = E ∞ p , q for q 6 = 0 and p > 0 E 1 p , 0 = H p ( F p ∗ / F p − 1 ∗ ) = X ⊗ ( B / I ) ⊗ p ⊗ Y which means E 1 ∗ , 0 = CB ∗ ( X ; B / I ; Y ) . Hence we get E 2 p , 0 = E ∞ p , 0 = T or B / I p ( X , Y ) . Our spectral sequence con verges to the hom ology of CB ∗ ( X ; B ; Y ) wh ich is T or B ∗ ( X , Y ) . The result follows.  J ACOBI-ZARISKI EXA C T SEQUENCE FOR HOCHSCHILD HOMOLOGY AND CYCLIC (CO)HOMOLOGY 7 Remark 2.2. An epimorphism of alg ebras ϕ : B → A is called a ho m o l ogical epimorph ism if the i n duced morphism s Ext ∗ A ( X , Y ) → Ext ∗ B ( X , Y ) is an isomorphism for e very finitely gener - ated A -m o dules X and Y [3]. If B i s finite dimens i onal and we confine ourselves with finitely generated modules, then on e has isomorphis m s of the form (2.1) if and o n ly if on e has the same isomorphis m s for Ex t-groups. In other words, an epimorph ism ϕ : B → A between two finite dimensional k -algebras is a homology epimorphism wh en ker ( ϕ ) is H-uni tal because of Proposi- tion 2.1. W e would like t o reiterate here again t h at we make no assumpt i ons on the k -dimens i ons of the algebras we work with , nei t her do we assume that our modules are finitely g enerated. Theor em 2.3. A s sume B is a unital k -algebra and let B ⊆ A be a r-flat e xtensi on. T hen p > 1 we have a long e xact sequence of the form (2.2) · · · → T or ( A | B ) p + 1 ( X , Y ) → T or B p ( X , Y ) → T or A p ( X , Y ) → T or ( A | B ) p ( X , Y ) → · · · for any right A -module X and any left A -module Y wher e the last map T or A 1 ( X , Y ) → T or ( A | B ) 1 ( X , Y ) in the sequence is an epimorphism. Pr oof. W e consider the two-sided bar complex CB ∗ ( X ; A ; Y ) for a right A -modul e X and for a left A -mo d ule Y . W e consider the following in creasing filtration on CB ∗ ( X ; A ; Y ) L p n = ∑ n 0 + ··· + n p = n − p X ⊗ B ⊗ n 0 ⊗ A ⊗ B ⊗ n 1 ⊗ · · · ⊗ A ⊗ B ⊗ n p ⊗ Y for p 6 n where we allow n i = 0. W e take L p ∗ = 0 for p < 0 and let L p n = L p + 1 n for p > n . Note that the filtration d egree comes from th e number o f tensor components wh ich are equal to A . W e also see that L 0 ∗ = CB ∗ ( X ; B ; Y ) and colim p L p ∗ = CB ∗ ( X ; A ; Y ) . Then it i s clear th at L 0 ∗ / L − 1 ∗ = L 0 ∗ = CB ∗ ( X ; B ; Y ) , and for n > p we h a ve L p n / L p − 1 n ∼ = M n 0 + ··· + n p = n − p X ⊗ B ⊗ n 0 ⊗ ( A / B ) ⊗ B ⊗ n 1 ⊗ · · · ⊗ ( A / B ) ⊗ B ⊗ n p ⊗ Y In the derived category , the quotient dif ferential graded k -modul e L p ∗ / L p − 1 ∗ represents an ( p + 2 ) - fold product X ⊗ L B ( A / B ) ⊗ L B · · · ⊗ L B ( A / B ) ⊗ L B Y where ⊗ L B denotes the left deriv ed functor of ⊗ B . W e use the sp ectral s equ ence associ ated with this filtration employing the fact that B ⊆ A is a r-flat extension and we see t hat th e only non -zero groups are o n t he p -axis and on th e q -axis E 1 p , q = H p + q ( L p ∗ / L p − 1 ∗ ) =      T or B q ( X , Y ) if p = 0 X ⊗ B ( A / B ) ⊗ B · · · ⊗ B ( A / B ) | {z } p -times ⊗ B Y if q = 0 8 A T ABEY KA YGUN Now , we observe that the edge q = 0 on the E 1 -page is the norm alized bar complex of X and Y over A relative to B [9, Ch.VIII Thm .6.1 and Ch. IX Sec. 8-9]. Then for E 2 p , q we get E 2 p , q = ( T or B q ( X , Y ) if p = 0 T or ( A | B ) p ( X , Y ) if q = 0 For the subsequent pages in this spectral sequence, t he on l y relev ant dif ferentials are d p p , 0 : T or ( A | B ) p ( X , Y ) → T or B p − 1 ( X , Y ) for p > 2 , and the remaini n g terms for p = 0 , 1 satis fy T or ( A | B ) p ( X , Y ) = E 2 p , 0 = E ∞ p , 0 . Then for p > 2 we ha ve short e xact sequ ences of th e form (2.3) 0 → E ∞ p , 0 → T or ( A | B ) p ( X , Y ) → T or B p − 1 ( X , Y ) → E ∞ 0 , p − 1 → 0 So, eventually i n the E ∞ -page there remains only two classes of non -zero groups: one on the p -axis the other on the q -axis. The induced filtration L ∗ T or A ∗ ( X , Y ) on th e hom ology s atisfies E ∞ p , q = L p T or A p + q ( X , Y ) / L p − 1 T or A p + q ( X , Y ) which can be placed into a d i agram o f the form 0 / / / / L 0 T or A p ( X , Y ) / / / /     L 1 T or A p ( X , Y ) / / / /     · · · / / / / L p T or A p ( X , Y )     E ∞ 0 , p E ∞ 1 , p − 1 E ∞ p , 0 where the top row is a sequence of injections with q uotients giv en by the appropriate groups in E ∞ . Since we only had non-zero groups on the p - and q -ax es, we get short exact sequences of the form (2.4) 0 → E ∞ 0 , p → T or A p ( X , Y ) → E ∞ p , 0 → 0 Splicing (2.3) and (2.4) we g et a lo ng e xact sequence of the form (2.2) for the range p > 1.  Corollary 2.4. Assume ϕ : B → A is a r-flat morphism such that ker ( ϕ ) is an H-unital ideal of B . Then ther e is a long e xact sequence of t h e f o rm (2.2) for any right A -modu le X and any left A -module Y . Pr oof. There is an ov erload of notations here. T or ( A | im ( ϕ )) refers t o the relative T or with im ( ϕ ) is a sub al g ebra of A . On the other hand T or ( A | B ) refers t o relative T or using ϕ : B → A which, by definition, i s constructed using th e image of ϕ . In other words, if A i s viewed as a B -module via a morphism ϕ : B → A then A ⊗ B A : = A ⊗ im ( ϕ ) A , and t herefore T or ( A | im ( ϕ )) and T or ( A | B ) are one and the same. Then since ϕ is r -flat, we hav e a lon g exact sequence similar t o (2.2) of the form · · · → T or ( A | B ) p + 1 ( X , Y ) → T or im ( ϕ ) p ( X , Y ) → T or A p ( X , Y ) → T or ( A | B ) p ( X , Y ) → · · · J ACOBI-ZARISKI EXA C T SEQUENCE FOR HOCHSCHILD HOMOLOGY AND CYCLIC (CO)HOMOLOGY 9 for p > 1. The result fol lows since we have i s omorphism s of the form T or B ∗ ( X , Y ) ∼ = T or im ( ϕ ) ∗ ( X , Y ) by Proposition 2.1.  3. T H E J A C O B I - Z A R I S K I S E Q U E N C E F O R E X T E N S I O N G RO U P S For the time being, we assume A is a unital k -algebra which is also finite dimensional ov er k , and B is an arbitrary unital subalgebra of A . W e wil l work wit h th e category of finitel y g enerated A -modules. In the category of finitely generated A -modules of either parity , every mo d u le is necessarily finite dimensional ove r k . For these categories the k -dualit y functor Hom k ( · , k ) give s us an equiva lence o f cate gories of t h e form Hom k ( · , k ) : A - mod → mod - A where A - mod and mod - A denote the categories of finit ely generated left and right A -modu l es, respectiv ely . Usi ng the adjunction between the functors ⊗ A and Hom k ( · , k ) on e can easily prove the following lemma. Lemma 3.1. Ass u me A is a finite d i mensional k -algebra and let Y be a finitely gener ated A - module. Then Y i s ( A , B ) -flat if and only if the k -dual Hom k ( Y , k ) is an ( A , B ) -i n j ective module. Note that t h e absolute case is co vered by B = k . Pr oof. Ass u me Y is ( A , B ) -flat, that is every ( A , B ) -exact sequence of the form 0 → X → X ′ → X ′′ → 0 induce an exact sequence of k -mo d ules the form 0 → X ⊗ A Y → X ′ ⊗ A Y → X ′′ ⊗ A Y → 0 Since k is a field, and therefore H o m k ( · , k ) is an e xact functor , we get another e xact s equ ence of k -modules of t h e form 0 → Hom k ( X ⊗ A Y , k ) → Hom k ( X ′ ⊗ A Y , k ) → Hom k ( X ′′ ⊗ A Y , k ) → 0 Using the adjunction between ⊗ A and Hom k ( · , k ) we get 0 → Hom A ( X , Hom k ( Y , k )) → Hom A ( X ′ , Hom k ( Y , k )) → Hom A ( X ′′ , Hom k ( Y , k )) → 0 is also e xact. This i s equiv alent to the f act that Hom k ( Y , k ) is ( A , B ) -injective . Con versely , assume Y is ( A , B ) -injectiv e. Since Hom k ( · , k ) is a n equiv alence of finitely generated A - modules exchanging parity , there is a finitely generated A -mod u le T of the opposite parity of Y such that Y ∼ = Hom k ( T , k ) . Moreover , one can e asily see that Hom k ( Y , k ) ∼ = T for t he same module since the modul es we con s ider are all finite dimensional . Assume n ow , without any loss of generality , that Y is a right module, and t herefore T is left module. Then 0 → Hom A ( X , Hom k ( T , k )) → Hom A ( X ′ , Hom k ( T , k )) → Hom A ( X ′′ , Hom k ( T , k )) → 0 10 A T ABEY KA YGUN is exact for every ( A , B ) -exa ct sequence 0 → X → X ′ → X ′′ → 0 o f finitely generated A - modules. Usin g t he adjunction again we see that 0 → Hom k ( X ⊗ A T , k ) → Hom k ( X ′ ⊗ A T , k ) → Hom k ( X ′′ ⊗ A T , k ) → 0 is also ( A , B ) -exact. Since Hom k ( · , k ) is an exact equiv al ence for finit e dimensi o n al k -modul es we see that 0 → X ⊗ A T → X ′ ⊗ A T → X ′′ ⊗ A T → 0 is also ( A , B ) -exac t. Th is is equ ivalent to t he f act that T is ( A , B ) -flat.  Lemma 3.2. If A is a finite dimensional k -algebra and B ⊆ A is a r -flat e xtension then for any finitely ge nerated A -modu les X and Y we have a lon g exact sequence in cohomology o f the form (3.1) · · · → Ex t p ( A | B ) ( X , Y ) → Ex t p A ( X , Y ) → Ext p B ( X , Y ) → Ex t p + 1 ( A | B ) ( X , Y ) → · · · for every p > 1 wher e the first map Ext 1 ( A | B ) ( X , Y ) → Ext 1 A ( X , Y ) in the sequence is a monomor- phism. Pr oof. Ass u me X and Y are A -modu les. Let U and V be a unital k -subalgebras of A such th at V ⊆ U . Fix a pre-dual T of Y , i.e. Hom k ( T , k ) ∼ = Y . Now , observe that if T is left A -modul e then CB ∗ ( U ; U | V ; T ) is a ( U , V ) -projectiv e resol ution of T , and therefore Hom k ( CB ∗ ( U ; U | V ; T ) , k ) is an inj ectiv e resolution of Y ∼ = Hom k ( T , k ) by Lemma 3.1 . Then Ext n ( U | V ) ( X , Y ) ∼ = Ext n ( U | V ) ( X , Hom k ( T , k )) = H n Hom U ( X , Hom k ( CB ∗ ( U ; U | V ; T ) , k )) ∼ = H n Hom k ( CB ∗ ( X ; U | V ; T ) , k ) ∼ = Hom k ( T or ( U | V ) n ( X , T ) , k ) W e consider three cases now: (i) U = A and V = k , (ii) U = B and V = k and finally (iii) U = A and V = B . Then we see that if we apply Hom k ( · , k ) to the l ong exact sequence (2.2) we obtain th e result we w ould like to prove.  Remark 3.3. Note that in t he proof of Lemma 3.1 wh ere we show Hom k ( Y , k ) is ( A , B ) -injective when it is ( A , B ) -flat, we did not use the fact that Y is finit ely generated over a finite dimension al k -algebra A . Indeed, it is true that for an arbitrary A -module over any k -algebra A (regardless of k -dimension) and for an arbitrary unital subalgebra B the module Hom k ( Y , k ) i s still ( A , B ) - injective when Y is ( A , B ) -flat. Then t he resul t of Lemma 3.2 can be extended t o not necessarily finite dimensi onal alg ebras, if inst ead of t h e full category o f finitely generated A -mo d ules we consider the category of A -modules which are fi n ite dimensional over the base field k . The result we proved in Lemm a 3 . 1 in the direction we need is st i ll t rue and in the category of finite dimensional A -modules, the duality functor is still an equiv alence. Th ese are enough to prove an analogue of Lemm a 3.2 in the case A is not necessarily finite dimensi o n al over k . J ACOBI-ZARISKI EXA C T SEQUENCE FOR HOCHSCHILD HOMOLOGY AND CYCLIC (CO)HOMOLOGY 11 Definition 3.4. Assum e A is an arbitrary k -algebra, and we mak e no assumptio n on the k - dimension of A . An A -m o dule M is called an ap p r oximately finite dimension a l A -module i f M is a colim it of all of its A -submodules of finite k -dimension. Theor em 3.5. Let A be an arbitrary unital k -algebra wher e we make no assumpt ion on the k - dimension of A . Let B ⊆ A be a r-flat extension and ass u m e Y is an appr oxim a tely finite dimen- sional A -mod u le. Then for every A -module X we have a long exact sequence of the form (3.1 ) . Pr oof. W e k n ow by Remark 3.3 that we hav e a long exact sequence of the form · · · → Ex t p ( A | B ) ( X , Z ) → Ext p A ( X , Z ) → Ext p B ( X , Z ) → Ext p + 1 ( A | B ) ( X , Z ) → · · · for every submodule Z of Y of finite k -dimension for every p > 1. Then we use the fact Y is the colimit of all of its A -subm odules of finite k -dimension and th at colimit is an exac t functor .  Corollary 3.6. Assume ϕ : B → A is a r-flat morphism of uni tal algebras such that ker ( ϕ ) is an H-unital ideal o f B . Assu me Y is an a ppr oximat ely finite dimensiona l A -mod u le. Then for every A -module X we have a long e xact sequence o f the form (3.1) . 4. J A C O B I - Z A R I S K I S E Q U E N C E F O R H O C H S C H I L D H O M O L O G Y A N D C Y C L I C ( C O ) H O M O L O G Y In this section we assume A is a unital k -algebra and B ⊆ A is a r -flat extension. W e make no assumpt ion on the k -dimension of A . Our aim in this section is to repeat the ar gument we gav e in Section 2 and Section 3 for the Hochschild homol o gy and c yclic (co)homology , and prove appropriate versions of Theorem 2.3 and Theorem 3.5. Even thoug h the filt rations are similar , the associated graded compl exe s wil l be different. So, we need t o check the details carefully . Now we define an increasing filtration on the Hochschild chain complex by letting (4.1) G p n = ∑ n 0 + ··· + n p = n − p M ⊗ B ⊗ n 0 ⊗ A ⊗ B ⊗ n 1 ⊗ · · · ⊗ A ⊗ B ⊗ n p for 0 6 p 6 n . W e let G p n = 0 for p < 0 6 n and G p n = G p + 1 n for all p > n > 0. Here a gain, the filtration degree count s the number of tensor components which are equal to A and observe that G 0 ∗ = CH ∗ ( B , M ) . Moreover , we s ee that coli m p G p ∗ = CH ∗ ( A , M ) . Th e associated graded complex is then gi ven by G p n / G p − 1 n ∼ = M n 0 + ··· + n p = n − p M ⊗ B ⊗ n 0 ⊗ ( A / B ) ⊗ B ⊗ n 1 ⊗ · · · ⊗ ( A / B ) ⊗ B ⊗ n p for any n > p > 1, and at degree 0 we see G 0 ∗ / G − 1 ∗ = G 0 ∗ = CH ∗ ( B , M ) . But recall that A / B is flat ov er B since B ⊆ A i s a r -flat extension. Th en o n the E 1 -page of t h e ass ociated spectral sequence we get only two non-zero groups: one on the p -axis and the other on th e q -axis E 1 p , q = H p + q ( G p ∗ / G p − 1 ∗ ) = ( H H q ( B , M ) if p = 0 f CH p ( A | B , M ) if q = 0 and p > 0 12 A T ABEY KA YGUN where f CH ∗ ( A | B , M ) is the normaliz ed relati ve Hochschild complex [8, 1.1.14]. Thi s leads us to the E 2 -page E 2 p , q = ( H H q ( B , M ) if p = 0 and q > 0 H H p ( A | B , M ) if q = 0 The rest of the ar gument is similar t o t he ar gu m ent we gave in Section 2. Theor em 4.1 . Assume B ⊆ A is a r-flat e xtension of unital , associative but not necessarily com- mutative k -algebr as. Then for any A -bimodul e M we have a long exact sequence of the form · · · → H H p + 1 ( A | B , M ) → H H p ( B , M ) → H H p ( A , M ) → H H p ( A | B , M ) → · · · for p > 1 wher e the last map H H 1 ( A , M ) → H H 1 ( A | B , M ) in the sequence is an epim o rphism. If we assu me M is appr o ximately finit e dim ens ional then we obtain have a long exact sequence in cohomology of the form · · · → H H p ( A | B , M ) → H H p ( A , M ) → H H p ( B , M ) → H H p + 1 ( A | B , M ) → · · · for every p > 1 wher e the first ma p H H 1 ( A | B , M ) → H H 1 ( A , M ) in the sequence is a monomor- phism. Now we are rea dy to e xtend this resul t to cyclic (co)homology . Theor em 4.2. Assume A , B and k ar e as befor e. Then we have t h e following lon g exact s e- quences: · · · → H H p + 1 ( A | B ) → H H p ( B ) → H H p ( A ) → H H p ( A | B ) → · · · (4.2) · · · → H C p + 1 ( A | B ) → HC p ( B ) → H C p ( A ) → H C p ( A | B ) → · · · (4.3) · · · → H C p ( A | B ) → H C p ( A ) → H C p ( B ) → H C p + 1 ( A | B ) → · · · (4.4) for p > 1 . Pr oof. In order to prov e our assertion for cyclic (co)homology , we first need to d erive (4.2) which is a version of the Jacobi-Zarisk i sequence for Hochschil d homology . For this purpose, we will need to write a filtration s imilar to the filtration we defined in (4.1). So, for 0 6 p 6 n + 1 we define G p n = ∑ n 0 + ··· + n p = n + 1 − p B ⊗ n 0 ⊗ A ⊗ B ⊗ n 1 ⊗ · · · ⊗ A ⊗ B ⊗ n p W e let G p n = G p + 1 n for all p > n + 1 and G p n = 0 for p < 0. Observe th at the filtration is com patible with t h e actions of the cyclic groups Z / ( n + 1 ) at e very degree n > 0. The rest of t he proof for obtaining the l ong exact sequence (4.2 ) is si m ilar to argument we used at the beginning of this section for Hochschild homol o gy wit h coef ficients and the proof of Theorem 2.3. The proof for cyclic (co)homology also follows after ou r observation that the filtration G p ∗ is compati b le w i th the actions o f the cyclic groups.  J ACOBI-ZARISKI EXA C T SEQUENCE FOR HOCHSCHILD HOMOLOGY AND CYCLIC (CO)HOMOLOGY 13 Let us use C ∗ > n for the g o od truncation [13, 1.2.7] of a d iffe rential graded k -mod u le C ∗ . Then we ha ve the fol lowing: Theor em 4.3. Let ϕ : B → A be a r-flat morphis m of unital k -algebras such t h at I : = ker ( ϕ ) is H-unital. Let CH ∗ ( A , B ) be the h o motopy cofiber o f the morphis m ϕ ∗ : CH ∗ ( B ) → CH ∗ ( A ) . Then ther e is a homotop y cofibration sequence of the form (4.5) Σ CH ∗ > 1 ( I ) → CH ∗ > 1 ( A , B ) → CH ∗ > 1 ( A | B ) which induces a long exact sequence o f the for m · · · → H H p + 2 ( A | B ) → H H p ( I ) → H H p + 1 ( A , B ) → H H p + 1 ( A | B ) → · · · for p > 1 , and we get an isomorp h i sm of the form H H 1 ( A , B ) ∼ = H H 1 ( A | B ) . A nalogous se- quences e xis t for the cyclic homology and cohomology with no additio n al hypothesis. Pr oof. W e will giv e the proof for the Hochschil d homology . Th e p roo fs for cyclic h omology and cohomology are sim ilar , and therefore, omitted. W e consider all of our Hochschild chain com- plexes as differential graded k -modul es inside (bounded below) deri ved category of k -modul es. This is a triang u lated category . In this cate gory we ha ve two distin guished triangles: o n e com ing from the W odzicki excision sequence [14, Thm.3. 1 ] CH ∗ ( I ) → CH ∗ ( B ) π ∗ − − → CH ∗ ( B / I ) → Σ CH ∗ ( I ) and the other coming from o ur Jacobi-Zariski s equence: CH ∗ > 1 ( B / I ) i ∗ − − → CH ∗ > 1 ( A ) → CH ∗ > 1 ( A | B ) → Σ CH ∗ > 1 ( B / I ) where we use the good truncation of the Hochschil d com plexes because our Jacobi-Zarisk i se- quence works only for the range p > 1. Now , cons ider another homotopy cofibration sequence CH ∗ > 1 ( B ) ϕ ∗ − − → CH ∗ > 1 ( A ) → CH ∗ > 1 ( A , B ) where the last term is defined as the homotopy cofiber of the morphism ϕ ∗ : CH ∗ > 1 ( B ) → CH ∗ > 1 ( A ) . W e construct CH ∗ > 1 ( B ) π ∗ / / CH ∗ > 1 ( B / I ) / /   Σ CH ∗ > 1 ( I ) / / γ ∗   Σ CH ∗ > 1 ( B ) CH ∗ > 1 ( B ) ϕ ∗ / / CH ∗ > 1 ( A ) / / CH ∗ > 1 ( A , B ) / / Σ CH ∗ > 1 ( B ) Now , γ ∗ exists because the (boun ded b elow) d erived category of k -modules is a triangulated cate- gory . W e also ha ve that the middle square is homot opy cartesian [11, Lem.1.4.3]. Thus we get a distingui shed triangle of th e form Σ CH ∗ > 1 ( I ) γ ∗ − − → CH ∗ > 1 ( A , B ) → CH ∗ > 1 ( A | B ) → Σ 2 CH ∗ > 1 ( I ) This follows from Theorem 4.2 since CH ∗ > 1 ( A | B ) is the hom otopy cofiber of the m orphism CH ∗ > 1 ( B / I ) → CH ∗ > 1 ( A ) induced b y the inclusion im ( ϕ ) ⊆ A . This finishes the p roo f.  14 A T ABEY KA YGUN One can easily see that Theorem 4.3 gives us the Jacobi-Zariski s equ ence when ϕ is monomor- phism with an r- flat im age and the W o dzicki excision sequence when ϕ is an epimorphis m with an H-unital kernel, for Hochschild homol ogy and cyclic (co)homology . Conjectur e 4.4. W e belie ve that an appropriate analo g ue o f Theorem 4.2 holds for Hochschild cohomology when p is lar g e enough. Howe ver , due to i ts non-functoriality , to prove such an analogue for Hochschild cohomology is not a straightforward task. If B ⊆ A is a n arbitrary extension, we hav e a homotopy cofibration sequence of the form CH ∗ ( B ) → CH ∗ ( B , A ) → CH ∗ ( B , A / B ) com ing from the short exact sequence of B -bim odules 0 → B → A → A / B → 0. The last term CH ∗ ( B , A / B ) calculates H H ∗ ( B , A / B ) w h ich is Ext ∗ B e ( B , A / B ) . So, we hav e H H n ( B ) ∼ = H H n ( B , A ) for n > 2 when A / B is an injective B e -module. If A is ap- proximately finite dimensional and A / B is B e -flat on top of bein g B e -injective , we can us e Theorem 4.1 where we set M = A to obt ain an analog u e of Theorem 4.2 for Hochschild coho- mology but for p > 2 . There are sim ilar results pointing in th i s direction. See for example [5, Thm.5.3], [1, Thm . 4 . 5 ] and [10]. R E F E R E N C E S [1] C. Cibils. T ensor Ho chschild ho m ology and cohom o logy . In Interactions between ring theo ry an d r epr esenta- tions of algebras (Mu r cia ) , volume 210 of Lectur e Notes in Pur e an d Appl. Math. , pag es 35–5 1. 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Spring e r– V erlag, Berlin, secon d edition, 1 998. [9] S. MacL a ne. Homology , volume 114 o f Die Grundlehren der ma thematischen W issenschaften . A c ademic Press Inc., New Y ork, 19 6 3. [10] S. Michelen a and M.I. Platzeck . Ho chschild co homolo gy of triang u lar matrix algebr as. J. Algebra , 233( 2):502 – 525, 200 0. [11] A. Neema n. T rian g ulated cate gories , volume 148 of An nals of Mathematics S tu dies . Prin ceton University Press, 2001. [12] D. Quillen. On the (co- ) homo logy o f co mmutative rings. In Ap plications of Categorical Algebra (Pr oc. Sympo s. Pur e Math., V ol. XVII, New Y ork, 1968 ) , pag es 65–8 7. Amer . Math. So c., Providen c e , R.I., 1 970. [13] C. A. W eibel. An Intr o duction to Homo logical Algebra , volume 3 8 of Cambridge Stu d ies in Adva nced Mathe- matics . Cambrid ge U n iv ersity Press, Camb ridge, 1994. [14] M. W odzicki. Ex cision in cyclic homolog y and in rational algebraic K – theory . Ann. of Math. , 129(3 ):591– 639, 1989. J ACOBI-ZARISKI EXA C T SEQUENCE FOR HOCHSCHILD HOMOLOGY AND CYCLIC (CO)HOMOLOGY 15 E-mail ad dr ess : kaygun @itu.ed u.tr D E PA RT M E N T O F M A T H E M A T I C S , I S TA N B U L T E C H N I C A L U N I V E R S I T Y , I S T A N B U L , T U R K E Y