Hirzebruch-Riemann-Roch theorem for DG algebras

HIRZEBR UCH-RIEMANN-R OCH THEO REM F OR DG ALGEBRAS D. SHKL Y ARO V De dic ate d to the memory of L. L. V aksman 1. Introduction 1.1. Geometry of DG cat egories. T o motiv ate the sub ject of the present r esearc h, we will b egin b y discussing some applications o f triangulated and diﬀeren tial graded categories in al gebraic geometry . Let X b e a quasi-compact sep arated sc heme. 1 Denote by D qcoh ( X ) the d eriv ed cate- gory of complexes of O X -mo dules with quasi-coherent cohomology and by D perf ( X ) its triangulated sub catego ry of p erfect complexes, i.e. complexes whic h are lo cally quasi- isomorphic to ﬁnite complexes of v ector bu ndles. The catego ry D perf ( X ) has prov ed to b e the basic (co)homolog ical inv ariant of X w hic h someho w enco des all other reasonable in v aria nts. This idea under lies R. Th omason’s researc h on the K -theory of sc hemes [64], M. Kontsevic h’s Homologic al Mirror Sym m etry program [39], and A. B ondal’s and D. Orlo v’s r esearc h on th e der ived categories of smooth sc hemes [9]. 2 When w orking with D perf ( X ), one faces the follo wing problem: ev e n though v ario us in v aria nts of X dep end on this category , it is not clear how to compute some of th em in terms o f D perf ( X ), view ed a s an a bstract triangulated category . On e wa y to get around the problem is due to A . Bo ndal and M. Kaprano v [7]. The point is that the derived categories, as opp osed to abstract triangulated categ ories, can b e “up graded” to diﬀeren tial graded (DG) categories. In p ractice this can b e ac hiev ed by , sa y , passing from D perf ( X ) to the DG category Perf X of left b oun ded injectiv e p erfect complexes. The category D perf ( X ) is then reco v ered as the h omotop y category of P erf X . Man y other inv ariants of X can b e extracte d from Pe rf X as w ell. The simp lest example is the compu tation of th e Hodge cohomology of X in terms of P erf X in the case of a smo oth sc heme. One has (1.1) HH n ( P erf X ) = ⊕ i H i − n (Ω i X ) , where the left-hand side stands f or the n -th Ho c hsc hild homology group of Perf X (see Section 2.3). The c ategory P erf X enco des a lso some geometric prop erties of X . F or example, if X is smo oth then the categ ory Perf X is a p erfect bimo dule ov er itself [41]. The DG categories of the form Perf X turn out to b e equiv alen t to th e DG catego ries of p erfect m o dules o v er certain DG algebras. Namely , according to [10, Section 3.1] (see also [57]) D perf ( X ) is generated b y a sin gle p erfect complex, E . Let A = End Pe rf X ( E ). 1 In what follo ws, everything is considered over a ﬁxed ground ﬁeld. 2 One of their results claims that sc hemes of certain type can b e completely reconstruct ed from their derived catego ries. 1 2 D. SHKL Y AROV Then P erf X is quasi-equiv alen t to the DG catego ry P erf A of p erfect right A -mo dules (see Section 2.2 for the deﬁnition of the latter ca tegory). Of course, there is no canonical generator of D perf ( X ) and , as a resu lt, there is no canonical DG algebra asso ciated with the sc heme. Ho wev er any DG algebra su c h that P erf X is quasi-equiv alen t to P erf A can b e view ed as a replacemen t of the algebra of regular functions in the case of a non-aﬃne sc h eme X . Let u s lo ok at the m ost p opular example - the p ro jectiv e line P 1 . Due to the w ell kno wn result of A. Beilinson [3], the d eriv ed category of coherent s hea ves in this case is equiv alen t to the derived category of ﬁn ite dimensional mo dules o ve r the path alg ebra of the Kr onec ker qu iv er: • ( ( 6 6 • F ollo w ing [65], we will s ay that t wo DG algebras A and B are Morita-equiv alen t if their p erfect categorie s P erf A and P erf B are quasi-equiv alent . I n vie w of the ab o ve d iscus- sion, eac h sc heme giv es r ise to a ﬁ xed Morita-equiv alence class. Th erefore it is reasonable to thin k of an arbitr ary Morita-equiv alence class as repr esen ting some noncommutati v e sc h eme or, b etter yet, a nonc ommutative DG-scheme . An y DG algebra from th e equiv a- lence class should b e view ed as “the” algebra of regular functions on this n oncomm utativ e DG-sc heme, and P erf A pla ys the role of P erf X . The abov e p oin t o f view agrees with the ph ilosoph y o f derive d nonc ommutative algebr aic ge ometry . 3 This su b ject w as initiate d in th e b eginnin g of 90 ’s based on the previous extensiv e stud y of derived c ategories of coheren t sh eav es undertaken by the Mosco w sc ho ol (A. Beilinson, A. Bondal, M. K aprano v, D. O r lo v, A. Rudak o v et al). Later on, it w as greatly enriched by new ideas and examples coming from M. Kont sevic h’s Homological Mirror Symmetry program [38]. A particularly imp ortan t implication of the program is that one can asso ciate certain triangulated categories with symp lectic manifolds whic h should play the s ame imp ortant role in symplectic geometry that the d eriv ed categories of coheren t shea v es pla y in algebraic geometry . F urther imp ortant ideas and results in the ﬁeld are d ue to A. Bondal and M. V an den Bergh, T. Bridgeland, V. Drinf eld, B. Keller, M. Kontsevic h and Y. S oib elman, D. Orlov, R. Rouquier, B. T o en and others. Of course, a “real” deﬁn ition of noncommutativ e DG- sc hemes should include also a description of morp hisms b et w een them. It is cle ar that m orp hisms are giv en b y DG functors b et w een the catego ries of p erfect complexes (a protot yp e is the p ull-bac k functor asso ciated w ith a morphism of sc h emes). The real deﬁnition is more subtle and w e won’t discuss it here referring the r eader to more thorough treatments of the sub ject [20, 35, 62, 63, 65, 66]. Here is an interesting question: Is it p ossible to tell whether a noncomm utativ e DG- sc h eme comes from a usual comm utativ e one? T h ere is a simp le necessary condition: the corresp ondin g DG algebra A sh ou ld be Morita-equiv alen t to its opp osite DG algebra A op (the simplest case w hen this is so is wh en the DG algebras A and A op are isomorphic; lo ok at the Kronec ker quiv er!). Of course, this condition is not suﬃcien t: v arious almost comm u tativ e sc h emes, suc h as orb ifolds, also satisfy it. 3 W e are not sure whether this n ame is commonly accepted or not. HIRZEBRU CH-RIEMANN-R OCH THEOREM F OR DG ALGEBRAS 3 Let A b e a DG algebra. Then , follo wing [41], one can d eﬁne the corresp ond ing non- comm u tativ e DG-sc heme to b e pr op er iﬀ so is A , i.e. P n dim H n ( A ) < ∞ ; smo oth iﬀ so is A , i.e. A is qu asi-isomorphic to a p er f ect A -bim o dule. The ﬁr st pr op erty is cen tral to the present w ork, although we will touc h up on smo oth DG algebras as we ll (see Section 6). 1.2. A categorical v ersion of the Hirzebruch-R iemann-Ro ch t heorem. Let us turn no w to the sub ject of this article, Hi rzebruch-Riemann-Roch (HRR) theorem in the ab o v e noncommutati v e setting. W e will start with v ery general (and ov ersimpliﬁed) catego rical considerations. Fix a groun d ﬁeld, k , and consid er the tensor cat egory of small k -linear DG cate gories, morphisms b eing DG fun ctors. Fix also a homolo gy the ory on th e latter catego ry , i.e. a co v ariant tensor functor H to a tensor cat egory of mo dules o v er a comm utativ e ring 4 K , satisfying the follo win g axioms: (1) H resp ects qu asi-equiv alences. (2) F or any D G algebra A the canonica l em b edding A → Pe rf A in duces an isomorphism H ( A ) ≃ H ( P erf A ) . (3) H ( k ) = K (then, by (2), H ( Perf k ) = K ). Notice that (1) and (2) together imply that H descends to an inv ariant of noncomm u- tativ e DG-sc hemes. A lso, by the v ery d eﬁnition of H , there exists a functorial K ¨ unneth t yp e isomorphism H ( A ) ⊗ K H ( B ) ≃ H ( A ⊗ B ) . Let us add to th is list one more condition: (4) F or an y DG categ ory A there is a functorial isomorph ism ∨ : H ( A ) ≃ H ( A op ) whic h equals ident it y in the case A = k . W e will assume that the ab o v e isomorphisms sat isfy all the natural prop erties and compatibilit y conditions one can imagine 5 . T o describ e what w e und erstand b y an abstract HRR theorem for noncomm utativ e DG- sc h emes, w e need to deﬁne the Chern c haracter map with v alues in the homology theory H . This is a fun ction Ch A H : A → H ( A ), one for eac h DG catego ry A , deﬁned as follo ws. T ake an ob ject N ∈ A and consid er the DG f u nctor T N : k → A that sends the unique ob ject of k to N . Then [11, 36] Ch A H ( N ) = H ( T N )(1 K ) . 4 One can take Z - graded, Z / 2-graded mo du les, mod ules that are complete in some top ology etc. 5 The righ t deﬁnition of a homology theory should b e formulated in terms of t he category of noncom- mutativ e motives [40]. 4 D. SHKL Y AROV Clearly , th e C hern c haracter is functorial: F or an y t wo DG categories A , B and any DG functor F : A → B Ch B H ◦ F = H ( F ) ◦ Ch A H . F rom now on, we will fo cus on pr op er DG categories, i.e. DG categories that corresp ond to prop er noncomm utativ e DG-sc hemes. Let A b e a pr op er DG category . Con s ider the DG functor Hom A : A ⊗ A op → Pe rf k , N ⊗ M 7→ Hom A ( M , N ) . By (3) , it induces a linear map H ( Hom A ) : H ( A ⊗ A op ) → K . One can co mp ose it with the K ¨ unneth isomorph ism to get a K -bilinear pairing h , i A : H ( A ) × H ( A op ) → K . No w we are ready to formulat e th e HRR theorem: F or an y prop er DG cate gory A and an y tw o ob jects N , M ∈ A (1.2) Ch Pe rf k H (Hom A ( M , N )) = h Ch A H ( N ) , Ch A H ( M ) ∨ i A . Indeed, it follo ws from the functorialit y of the isomorphism ∨ that ( H ( T M )(1 K )) ∨ = H ( T M op )(1 K ) where M op stands for M viewed as an ob ject of A op . T hen h Ch A H ( N ) , Ch A H ( M ) ∨ i A = H ( Hom A )  H ( T N )(1 K ) ⊗ ( H ( T M )(1 K )) ∨  = H ( Hom A ) ( H ( T N )(1 K ) ⊗ H ( T M op )(1 K )) = H ( Hom A ) ( H ( T N ⊗ M op )(1 K )) = H ( Hom A ◦ T N ⊗ M op )(1 K ) = H (Hom A ( M , N ))(1 K ) = Ch Pe rf k H (Hom A ( M , N )) . In this very general form, the HRR theorem is almost tauto logical. F or it to b e of an y use, one needs to ﬁn d a w a y to compute the right- hand side of (1.2 ) for a giv en prop er noncomm utativ e DG-sc heme and any p air of p erfect complexes on it. In this work, we solv e this problem in the case K = k , H = HH • , where HH • stands for the Ho c hsc hild homology 6 (see Section 2. 3 for the deﬁn ition). This choice of the h omology theory can b e motiv ated as follo ws. First of all, there is a classical c haracter map from th e Grothendiec k group of a rin g to its Ho c h sc hild h omology - the so called Dennis trace map [43]. Its sh eaﬁﬁed v ersion app eared in [11] in connection with the index theorem for elliptic pairs [58, 59] (the deﬁnition of the Chern c haracter giv en ab ov e mimics the one giv en in [11]). In the algebraic geometric con text, the r elev ance of the Ho c hsc hild homology to the HRR theorem can b e explained as follo ws. There is a ve rsion of the HRR theorem for compact complex m anifolds [48 , 49], in w hic h the Chern class of a coheren t sheaf tak es v alues in the Ho dge cohomolog y ⊕ i H i (Ω i X ) (see also [27]). A new pr o of of this result w as obtained in [44, 45] using an algebraic-diﬀeren tial calculus (see also [14, 50]). Th is latter app roac h emphasizes the imp ortance of viewing the Chern characte r as a map to the Ho c h sc hild h omology HH 0 ( X ) of the space X . The “usual” Chern c haracter is then obtained via the Hochsc hild-Kostant -Rosen b er g isomorphism HH 0 ( X ) ∼ = ⊕ i H i (Ω i X ). Th is 6 The most diﬃcult axioms (1) and (2) in our “deﬁn ition” of the homology theory were prov ed for HH • by B. Keller in [34]. HIRZEBRU CH-RIEMANN-R OCH THEOREM F OR DG ALGEBRAS 5 p oint of view was fu rther develo p ed in [13]. Namely , it w as explained in [13] (see also [15]) h ow to obtain a categorical v ersion of the HRR theorem, similar to the one ab o v e, starting from the cohomol ogy theory smo oth spaces → graded v ector spaces , X 7→ HH • ( X ) (“smo oth spaces” are u ndersto o d in a broad sense: these are usu al sc h emes as we ll as v arious almost comm utativ e on es su c h as orbif olds ). Finally , the trans ition from X to its cat egorical in carnation, P erf X , is based on the fact that HH • ( X ) is isomorphic to the Ho c h sc hild homology of the DG category P erf X , whic h w as pro v ed in [36]. Before we m ov e on to th e description of the main r esults of the pap er, we w ould like to men tion a notational con ven tion we are going to follo w. F ollo w ing [11] (see also [36]), w e will call the Chern c haracter Ch HH with v alues in the Ho c h sc hild homology th e Euler c haracter and use the notation Eu . 1.3. Main results. Let us describ e the main resu lts of this w ork. Fix a ground ﬁeld k and a p rop er DG algebra A o v er k (as w e men tioned earlier, the prop er n ess means P n dim H n ( A ) < ∞ ). The ﬁrst main result is the computation of the Euler class eu ( L ) of an arbitrary p erfect DG A -mo d ule L . Here eu ( L ) stands for the uniqu e elemen t in HH 0 ( A ) that corresp onds to Eu ( L ) ∈ HH 0 ( P erf A ) under the canonical isomorphism HH • ( A ) ≃ HH • ( P erf A ) (see axiom (2) in Section 1.2). The follo wing theorem is pr ov ed in Sectio n 4.1. Theorem 1. L et N = ( L j A [ r j ] , d + α ) b e a twiste d DG A -mo dule and L a homotopy dir e ct summand of N which c orr esp onds to a homotopy idemp otent π : N → N . Then eu ( L ) = ∞ X l =0 ( − 1) l str ( π [ α | . . . | α | {z } l ]) Roughly sp eaking, in this f orm ula π and α are elemen ts of a DG analog of the matrix algebra Mat ( A ), π [ α | . . . | α ] is an elemen t o f the Ho chsc hild chain complex of this DG matrix algebra, and str is an analog of the usual trace map tr : Mat ( A ) → A (see [24, 43]). Note that α is upp er-triangular, so the series te rminates. T o present our next resu lt, w e observe that the p airing HH • ( P erf A ) × HH • (( P erf A ) op ) → HH • ( P erf k ) ≃ k , deﬁned ea rlier in Sectio n 1.2, induces a pairing (1.3) HH • ( P erf A ) × HH • ( P erf A op ) → k . This is due to the existence of a canonica l quasi-equiv alence of DG categories (see (3.6 )): D : Perf A op → ( P erf A ) op , M 7→ DM = Hom Pe rf A op ( M , A ) . In fact, w e “t wist” the exp osition in the main text (Section 3.1) and w ork exclusivel y with the p airing (1.3). The reason is that it can b e deﬁn ed ve ry explicitly without referring to 6 D. SHKL Y AROV its categorical nature. Besides, it indu ces a pairing (1.4) h , i : HH • ( A ) × HH • ( A op ) → k via the canonical isomorph isms HH • ( A ) ≃ HH • ( P erf A ), HH • ( A op ) ≃ HH • ( P erf A op ). Th is latter p airing is describ ed explicitly in our n ext theorem, whic h is obtained b y com bining results of Section 3.2 (see form ulas (3.4), (3.5)) and T heorem 4.6. Theorem 2. L e t a , b b e two elements of HH • ( A ) , HH • ( A op ) , r esp e ctively. Then h a, b i = Z a ∧ b. Her e ∧ : HH • ( A ) × HH • ( A op ) → HH • (End k ( A )) , R : HH • (End k ( A )) → k ar e deﬁne d as fol lows: (1) If P a a 0 [ a 1 | . . . | a l ] (r esp. P b b 0 [ b 1 | . . . | b m ] ) i s a cycle in the H o chschild chain c omplex of A (r esp. A op ) r epr esenting th e homolo gy c lass a (r esp. b ) then a ∧ b = X a,b sh ( L ( a 0 )[ L ( a 1 ) | . . . | L ( a l )] ⊗ R ( b 0 )[ R ( b 1 ) | . . . | R ( b m )]) , wher e L ( a i ) (r esp. R ( b j ) ) stands for the op er ator in A of left (r esp. right) multiplic ation with a i (r e sp. b j ); sh is the wel l known shuﬄe-pr o duct (se e Se ction 2.4). (2) R is what we c al l the F eigin-L osev-Shoikhet tr ac e [22, 51] . It is describ e d e xplicitly in The or em 4.6 (Se ction 4.2). F ur thermore, recall that there s h ould exist a canonical isomorp h ism ∨ : HH • ( A ) ≃ HH • ( A op ) (see axiom (4) in S ection 1.2). In fact, th e isomorphism is easy to describ e explicitly (see Section 3.2 ). By summarizing the ab o v e d iscussion, we get the follo win g v ersion of the noncomm utativ e HRR theorem: Theorem 3. F or any p e rfe ct DG A -mo dules N , M χ ( M , N )(:= χ (Hom Pe rf A ( M , N ))) = Z eu ( N ) ∧ eu ( M ) ∨ . The only thing that needs to b e explained here is wh ere χ (Hom Pe rf A ( M , N )) came fr om. According the categorica l HRR theorem, describ ed in the previous section, the left-hand of the ab o ve equalit y shou ld equal eu (Hom Pe rf A ( M , N )). Ho we v er, the Euler class of a p erfect DG k -mo du le is nothing but its Euler c haracteristic. Th is is a consequence of the follo wing “exp ected” fact, whic h w e prov e in Section 3.1: for any A th e Euler charac ter Eu descends to a c haracter on the Grothendiec k group of the triangulated category Ho ( P erf A ), the h omotop y category of Perf A . Note that the n oncomm u tativ e HRR formula d o esn’t includ e any analog of the T o d d class. Th e T o dd class seems to emerge in the case w hen a noncomm utativ e s p ace, b X , is “close” to a commutati v e on e, X (for example, b X is a deformation quan tization of X ). In su c h cases v arious homology theories of b X can b e id en tiﬁed with certain cohomology HIRZEBRU CH-RIEMANN-R OCH THEOREM F OR DG ALGEBRAS 7 rings asso ciated with X and the T o d d class of X app ears b ecause of this identiﬁca tion. F or some classes of n oncomm utativ e spaces one can try to deﬁne an analog of the T o dd class “by hand” but, in general, a categorical deﬁnition of the T o d d class do esn’t seem to exist. In the main text we do not refer to th e categorica l v ersion of th e HRR theorem to pr o v e Theorem 3. In stead, we deriv e it from a m ore general state ment (Theorem 3.4). Roughly sp eaking, this statemen t sa ys the follo win g: If A and B are t w o pr op er DG al gebras and X is a p erfect A − B -b imo dule then the map HH • ( P erf A ) → HH • ( P erf B ), ind uced by the DG fun ctor − ⊗ A X : P erf A → Perf B , is give n by a “con v olution” with the Euler class of X . Later, in Sectio n 6.1, w e use this result again to prov e the f ollo wing Theorem 4. L et A b e a pr op er smo oth D G algebr a. Then the p airing h , i is non-de gener ate. It is th is application that wa s the original motiv ation for the author to study the Eu ler classes in th e DG setting 7 . It imp lies, in particular, the non commutativ e Ho d ge-to-D e Rham degeneration conjecture [41] for smo oth algebras with the trivial diﬀerenti al and grading. Hop efully , the reader w ill accept all th is as an excuse for “t wisting” the exp osition and not men tioning the categorical HRR in wh at follo ws. In the end of this work w e presen t some “to y” examples of p rop er n oncomm utativ e DG-sc hemes and th e HRR formulas for th em. Namely , in Section 5.1 we discus s w hat w e call directed algebras. Basica lly , these are some quiver alge bras with relations but w e ﬁnd the quive r-free description more con v enien t when it comes to pro ving general facts ab out su c h algebras. Man y commuta tiv e sc hemes, viewed as noncommutat iv e ones, are describ ed by d irected algebras. Namely , this is s o w hen th e scheme p ossesses a strongly exceptional collection [6]. The HRR form ula for s uc h algebras (see (5.3)) is a s p ecial case of Rin gel’s formula [54, Section 2.4]. Section 5.2 is ab out p r op er noncommutat iv e DG-sc hemes “resp onsib le” for orb ifold singularities of the form C n /G , wh ere G is a ﬁ n ite subgroup of S L n ( C ). Na mely , we lo ok at the noncomm utativ e DG-sc heme related to the deriv ed category of complexes of G -equiv ariant coherent shea ves on C n with supp orts at the origin. W e conjecture that th e und erlying DG algebra is the cross-pro duct Λ • C n ⋊ C [ G ] and we derive the HRR formula f or some p er f ect mo d ules o v er this alge bra (see (5.4)). Section 6.2 is devote d to a less str aigh tforward app lication of our results. It has b een conjectured b y Y. S oib elman and K. C ostello that for a Calabi-Y au DG algebra the pairin g HH • ( A ) × HH • ( A op ) → k , w e construct in this pap er, coincides with the one coming from the T op ological Field Theory asso ciated with A by [18, 41]. In Section 6.2 we form ulate this conjecture and v erify it in th e particular case of F rob eniu s algebras. 1.4. Other viewpoints on the noncomm utativ e HRR theorem. In this section, we pro vide a very brief accoun t of other Riemann-Ro ch t yp e theorems in Noncomm u tativ e Geometry we are a ware of. 7 I am grateful t o Y. S oib elman for suggesting to me to “write up” the pro of of this statemen t. 8 D. SHKL Y AROV Let us b egin w ith the afore-mentio ned pr eprint [13] whic h partially ins p ired the presen t w ork. Th e approac h tak en in [13] is based on an alte rnative description of the Ho chsc hild homology of a smo oth prop er space X in terms of the S erre functor S X : D b ( X ) → D b ( X ). Namely , HH • ( X ) ≃ Ext • F un ( S − 1 X , I X ) , where I X is the iden tit y endofunctor of D b ( X ) and the extensions are tak en in a s uitably deﬁned triangulated catego ry of en d ofunctors. In [60] we generalized the ab o v e isomor- phism to the case of an arbitrary smooth p rop er noncomm utativ e DG-sc heme. How ev er, pro ving that the ab o ve d eﬁnition give s r ise to a homology theory on the category of smo oth prop er noncomm utativ e DG-sc hemes (in other words, lifting the ab ov e deﬁnition on the lev el of DG categories) will requir e some eﬀorts [13, App endix B]. Besides, the “tradi- tional” deﬁnition of the Ho c h sc hild homolog y we use in this pap er wo rks for an arb itrary , not n ecessarily smo oth sc h eme. Other analogs of the Riemann-Ro ch theorem we re obtained in [28], [46] in the frame- w ork of Noncommutativ e Algebraic Geometry [1], [55], [56], [69]. T h e exp osition in [46] is closer to our s in that it emphasizes the imp ortance of triangulated categories in connection with Riemann-Ro c h t y p e results. Ou r app roac h an d the ab o ve tw o appr oac hes to the n on- comm u tativ e Riemann -Ro c h theorem are n ot completely unrelated since many interesti ng noncomm utativ e schemes give rise to n on commutativ e DG sc h emes [10, Section 4]. Last, but not least, v arious index theorems ha v e b een pro ved in framew orks of A. Connes’ Noncomm u tative Geometry [17, 61] and Deformation Quant ization [11, 21, 47, 67]. 1.5. Ac kno wledgements. I am indebted to Y. S oib elman for prompting m y in terest in deriv ed noncomm utativ e algebraic geometry and n umerous discussions whic h hav e p la yed a crucial role in sh aping m y understand ing of the sub ject. I am also very grateful to B. Keller and D. Orlo v f or patien tly answ ering m y questions ab out triangulated and DG categories, to K. Costello for numerous in teresting remarks and encouragemen t, and to B. Tsygan for in teresting comment s on noncomm utativ e C h ern c h aracters and the n on commutativ e Ho dge-to-De Rham degeneration conjecture. Finally , I would also like to thank B. Keller, Y u. I. Manin, A. L. Rosen b erg, M. V an den Bergh for sendin g me their commen ts which help ed me to imp ro v e the exp osition. Needless to sa y , I am alone resp onsible for typos and more serious mistak es if there are an y . This w ork w as partially sup p orted b y NSF gran t DMS-0504 048. 2. Hochschild homol ogy of DG algebras and DG ca tegories 2.1. DG algebras, DG categories, and DG functors. Th roughout the paper, w e w ork o v er a ﬁ xed ground ﬁeld k . All ve ctor spaces, alg ebras, linear cat egories are deﬁned o ver k . W e consider unital DG algebras with no restrictions on the Z -grading. If A is a DG algebra A = M n ∈ Z A n , d = d A : A n → A n +1 HIRZEBRU CH-RIEMANN-R OCH THEOREM F OR DG ALGEBRAS 9 then A op will stand for th e opp osite DG algebra, i.e. A op coincides with A as a complex of vec tor spaces and th e pr o duct on A op is give n by a ′ ⊗ a ′′ 7→ ( − 1) | a ′ || a ′′ | a ′′ a ′ (here and further, | n | denotes the degree of a homogeneous element n of a graded v ector space). Mo d A will stand for the DG ca tegory of righ t DG A -modu les. The homotop y catego ry of a DG categ ory A w ill b e denoted by Ho ( A ). Let u s recall the d eﬁnition of the standard triangulated stru cture on Ho ( Mo d A ). The shift functor is d eﬁned in the obvious w a y . The distinguished triangles are deﬁned as follo w s . Let p : L → M b e a degree 0 closed morp hism. The cone Cone ( p ) of the morph ism p is a DG A -mo dule deﬁned by Cone ( p ) =   L [1] ⊕ M ,  d L [1] 0 p d M    (the direct sum is tak en in the category of gr ade d A -mo dules). Th ere are ob vious degree 0 closed m orp hisms q : M → Cone ( p ) and r : Cone ( p ) → L [1]. A triangle in Ho ( Mo d A ) is, by d eﬁ nition, a sequence X → Y → Z → X [1] isomorph ic to a sequen ce of the form L p → M q → Cone ( p ) r → L [1]. Let N be a righ t DG A -mo dule. A degree 0 closed endomorph ism π : N → N w ill b e called a h omotop y idemp oten t if π 2 = π in Ho ( Mo d A ). By a homotop y direct summ and of N we will u nderstand a DG A -mo du le L that s atisﬁes the follo wing prop ert y: there exists a homotopy idemp oten t π : N → N and t w o degree 0 morph isms f : N → L and g : L → N suc h that f g = 1 L , gf = π in Ho ( Mod A ). Fix tw o DG categories A and B and consider th e DG catego ry F un ( A , B ) of DG fu n ctors from A to B [35]. A degree 0 closed morp hism f ∈ Hom F un ( A , B ) ( F , G ) will b e called a we ak homot opy e quivalenc e if for any N ∈ A th e morphism f ( N ) : F ( N ) → G ( N ) is a homotop y equiv alence in Ho ( B ). 2.2. P erfect modules. L et A b e a DG algebra. It can b e viewe d as a full DG sub category of Mod A w ith a single ob ject. Th e em b edding A ֒ → M o d A factors through a smaller full sub catego ry Perf A ⊂ M o d A of p erfect A -mo dules. This sub catego ry is deﬁned as follo w s (see [8 ]). Let u s say that a DG A -mo dule N is ﬁnitely gener ate d fr e e if it is isomorphic to a mo du le of th e f orm K ⊗ A where K is a ﬁnite dimensional graded v ector space (equiv alen tly , it is a ﬁnite direct sum of shifts of A ). W e will sa y that N ∈ M o d A is ﬁnitely gener ate d semi-fr e e if it can b e obtained from a ﬁ nite set of ﬁnitely generated fr ee A -mo dules (equiv alen tly , a ﬁnite set of shifts of A ) b y successive taking the cones of degree 0 closed m orphisms. Finally , a p erfe ct DG A -mo du le is a homotop y dir ect summand of a ﬁn itely generated semi-free DG A -mo du le. Note that this deﬁnition is sligh tly more general than th e one give n in [8]. T he authors of [8 ] require p erfect mo dules to b e semi-free but we don’t. F or example, a complex of v ector spaces is p erfect in our sense iﬀ it has ﬁnite dimensional tot al cohomology and it is p erfect in the sense of [8] if, in addition, it is b ounded ab o ve. The reason w e prefer not to 10 D. SHKL Y AROV restrict ourselv es to semi-free m o dules will b e clear fr om Pr op osition 2.4 b elo w. It suﬃces for us to sta y within the class of homotopically pr o jectiv e mo d u les: N is h omotopically pro jectiv e iﬀ Hom Mo d A ( N , L ) is acyclic whenev er L is acyclic. Ev ery ﬁnitely generated semi-free mo du le N is kn own to b e homotopically pro jectiv e [20, Section 13]. It follo ws that ev ery p erfect mod u le in our sense is homotopica lly pr o jectiv e as w ell. The follo w ing result is well known (and is not hard to pro v e): Prop osition 2.1. The DG c ate gory Perf A is close d under p assing to homotopic al ly e qu iv- alent mo dules, taking shifts and c ones of de gr e e 0 morphisms, and taking homoto py dir e ct summands. Let us list some simple useful facts ab out DG functors b et we en the categ ories of p erfect mo dules. Prop osition 2.2. L et A , B b e DG algebr as and F : M o d A → M o d B a D G functor. The DG functor F pr eserves the sub c ate gories of p erfe ct mo dules iﬀ F ( A ) ∈ Pe rf B . T o pr o ve this p r op osition, observ e that F preserv es homotopy d irect s ummands and cones of degree 0 morph isms. F or t wo DG algebras A, B and a bimo dule X ∈ Mod ( A op ⊗ B ) let us denote by T X the DG functor − ⊗ A X : Mo d A → Mo d B . Here is a straigh tforward consequence of th e last prop osition: Corollary 2.3. Supp ose a bimo dule X ∈ Mo d ( A op ⊗ B ) is p erfe ct as a DG B -mo dule. Then T X pr eserves p erfe ct mo dules. Recall that a DG algebra A is called pr op er if P n dim H n ( A ) < ∞ . Prop osition 2.4. L et A b e a pr op er DG algebr a and B an arbitr ary DG algebr a. Then for any X ∈ P erf ( A op ⊗ B ) the D G functor T X pr eserves p erfe c t mo dules. In view of the abov e corollary , it is enough to sho w that X is p erfect as a DG B - mo dule. Supp ose that X is a homotop y direct s u mmand of a ﬁ nitely generated semi- free DG A op ⊗ B -mo dule Y and Y is obtained from ( A op ⊗ B )[ m 1 ] , . . . , ( A op ⊗ B )[ m l ] b y su ccessiv e taking cones of degree 0 closed morph isms. As a B -mo dule, A op ⊗ B is homotopically equiv alent to the ﬁn itely generated f r ee m o dule H • ( A ) ⊗ B (this is where w e u se the prop ern ess of A and the fact that w e are working o ver a ﬁeld!). Th us, as a B - mo dule, Y is homotopically equiv alent to a ﬁnitely generat ed semi-free mo dule. Th en X , as a B -mo d ule, is a homotop y direct summand of a mo dule that is homotop y equiv alent to a ﬁnitely generated semi-free mo du le. This, together with Pr op osition 2.1, ﬁnishes the pro of. Let u s recall one more result ab out p erfect mo d ules whic h w e will need later on. T he fact that p erfect mo du les are homotopically pro jectiv e implies the follo wing result (cf. [4, Corollary 10 .12.4.4]): HIRZEBRU CH-RIEMANN-R OCH THEOREM F OR DG ALGEBRAS 11 Prop osition 2.5. If P is a p erfe ct right DG A -mo dule then P ⊗ A N is acyclic for every acyclic DG A op -mo dule N . 2.3. Ho c hschild homology . W e b egin by recalling the deﬁnition of the Ho c hsc hild ho- mology group s HH n ( A ), n ∈ Z , of a DG alg ebra A . Let us use the notation sa to den ote an element a ∈ A view ed as an elemen t of the “susp en s ion” sA = A [1]. Thus, | sa | = | a | − 1. Let C • ( A ) = A ⊗ T ( A [1]) = ∞ L n =0 A ⊗ A [1] ⊗ n equipp ed with the induced grading. W e w ill denote elemen ts of A ⊗ A [1] ⊗ n b y a 0 , if n = 0, and a 0 [ a 1 | a 2 | . . . | a n ] otherwise (i.e. a 0 [ a 1 | a 2 | . . . | a n ] = a 0 ⊗ sa 1 ⊗ sa 2 ⊗ . . . ⊗ sa n ). C • ( A ) is equipp ed with the diﬀerenti al b = b 0 + b 1 , w here b 0 and b 1 are t w o an ti-comm uting diﬀeren tials giv en by (2.1) b 0 ( a 0 ) = da 0 , b 1 ( a 0 ) = 0 , and b 0 ( a 0 [ a 1 | a 2 | . . . | a n ]) = da 0 [ a 1 | a 2 | . . . | a n ] − n X i =1 ( − 1) η i − 1 a 0 [ a 1 | a 2 | . . . | da i | . . . | a n ] , b 1 ( a 0 [ a 1 | a 2 | . . . | a n ]) = ( − 1) | a 0 | a 0 a 1 [ a 2 | . . . | a n ] + n − 1 X i =1 ( − 1) η i a 0 [ a 1 | a 2 | . . . | a i a i +1 | . . . | a n ] − ( − 1) η n − 1 ( | a n | +1) a n a 0 [ a 1 | a 2 | . . . | a n − 1 ] for n 6 = 0. Here η i = | a 0 | + | sa 1 | + . . . + | sa i | . C • ( A ) is call ed the Ho chsc hild c hain complex of A . Then HH n ( A ) = H n ( C • ( A )) . Let A b e a (small) DG category . Its Ho c hsc hild c hain complex is deﬁned as follo ws . Fix a n on-negativ e intege r n . W e will denote the set of sequences { X 0 , X 1 , . . . , X n } of ob jects of A b y A n +1 (the ob jects in th e sequence are not r equired to b e diﬀeren t). Fix an elemen t X = { X 0 , X 1 , . . . , X n } ∈ A n +1 and denote by C • ( A , X ) the graded v ector space Hom A ( X n , X 0 ) ⊗ Hom A ( X n − 1 , X n )[1] ⊗ . . . ⊗ Hom A ( X 0 , X 1 )[1]. Equ ip the space C • ( A ) = M n ≥ 0 M X ∈A n +1 C • ( A , X ) with the diﬀeren tial b = b 0 + b 1 where b 0 and b 1 are g iv en b y formulas analogous to (2.1),(2.2). T he complex C • ( A ) is the Ho chsc hild c hain complex of the DG category A and its cohomology HH n ( A ) = H n ( C • ( A )) is the Ho c hschild homology of A . Ob viously , any DG fu n ctor F : A → B b et ween tw o DG catego ries A , B induces a morphism of complexes C ( F ) : C • ( A ) → C • ( B ) and, as a result, a linear map HH ( F ) : HH • ( A ) → HH • ( B ) . Being applied to the em b edd ing A → P erf A , the ab o ve construction yields a morphism of complexes C • ( A ) → C • ( P erf A ). The follo wing result w as pro ved in [34] (see also [35]): 12 D. SHKL Y AROV Theorem 2.6. The morphism C • ( A ) → C • ( P erf A ) is a quasi-isomorphism. Later on, we will need ye t another result p ro v ed in [34] (see Sectio n 3.4 of lo c.cit.): Theorem 2.7. L et A and B b e two DG algebr as and F , G : Perf A → Pe rf B two DG functors. If ther e is a we ak homotopy e quivalenc e F → G then HH ( F ) = HH ( G ) . 2.4. K ¨ unneth isomorphism. Let us recall the construction of the K ¨ unneth isomorp hism M n HH n ( A ) ⊗ HH N − n ( B ) ≃ HH N ( A ⊗ B ) where A, B are tw o DG algebras. Th e form u la b elo w is b orr o wed from [43] (see also [67] where the diﬀerent ial graded case is discussed). Let u s ﬁx a DG algebra A . The ﬁrst in gredien t of the construction is the shuﬄe p ro du ct sh : C • ( A ) ⊗ C • ( A ) → C • ( A ) deﬁned as follo ws. F or t wo elemen ts a ′ 0 [ a ′ 1 | a ′ 2 | . . . | a ′ n ] , a ′′ 0 [ a ′′ 1 | a ′′ 2 | . . . | a ′′ m ] ∈ C • ( A ) the sh uﬄe pro du ct is giv en by the formula: (2.2) sh ( a ′ 0 [ a ′ 1 | a ′ 2 | . . . | a ′ n ] ⊗ a ′′ 0 [ a ′′ 1 | a ′′ 2 | . . . | a ′′ m ]) = ( − 1) ∗ · a ′ 0 a ′′ 0 sh nm [ a ′ 1 | . . . | a ′ n | a ′′ 1 | . . . | a ′′ m ] Here ∗ = | a ′′ 0 | ( | sa ′ 1 | + . . . + | sa ′ n | ) and sh nm [ x 1 | . . . | x n | x n +1 | . . . | x n + m ] = X σ ± [ x σ − 1 (1) | . . . | x σ − 1 ( n ) | x σ − 1 ( n +1) | . . . | x σ − 1 ( n + m ) ] where the s um is tak en o ver all per mutations that d on’t shuﬄe the ﬁr st n an d the last m elemen ts and the sign in front of eac h s ummand is computed according to the follo wing rule: f or t wo homogeneous elemen ts x, y , the tran s p osition [ . . . | x | y | . . . ] → [ . . . | y | x | . . . ] con trib utes ( − 1) | x || y | to th e sign. No w let B b e another DG algebra. Denote by ι A , ι B the natural em b eddings A → A ⊗ B , B → A ⊗ B . They ind uce morp hisms of complexes: C ( ι A ) : C • ( A ) → C • ( A ⊗ B ) , C ( ι B ) : C • ( B ) → C • ( A ⊗ B ) . Theorem 2.8. The c omp osition K of the maps C • ( A ) ⊗ C • ( B ) C ( ι A ) ⊗ C ( ι B ) − − − − − − − − → C • ( A ⊗ B ) ⊗ C • ( A ⊗ B ) sh − − − − → C • ( A ⊗ B ) r e sp e cts the diﬀer entials and induc es a quasi-isomorphism of c omplexes. The morphism K : C • ( A ) ⊗ C • ( B ) → C • ( A ⊗ B ) deﬁned ab ov e admits a generaliz ation to the case of DG categories. Namely , let A and B b e t wo (small) DG catego ries. Fix a set { X 0 , X 1 , . . . , X n } of ob jects of A and a set { Y 0 , Y 1 , . . . , Y m } of ob jects of B . F or tw o elemen ts f n [ f n − 1 | . . . | f 0 ] ∈ Hom A ( X n , X 0 ) ⊗ Hom A ( X n − 1 , X n )[1] ⊗ . . . ⊗ Hom A ( X 0 , X 1 )[1] , g m [ g m − 1 | . . . | g 0 ] ∈ Hom B ( Y m , Y 0 ) ⊗ Hom B ( Y m − 1 , Y m )[1] ⊗ . . . ⊗ Hom B ( Y 0 , Y 1 )[1] HIRZEBRU CH-RIEMANN-R OCH THEOREM F OR DG ALGEBRAS 13 deﬁne K ( f n [ f n − 1 | f n − 2 | . . . | f 0 ] ⊗ g m [ g m − 1 | g m − 2 | . . . | g 0 ]) as ± ( f n ⊗ g m ) sh nm [ f n − 1 | . . . | f 0 | g m − 1 | . . . | g 0 ] , where the sign is computed as b efore and sh nm is d eﬁned by the form ula [ f n − 1 ⊗ 1 Y m | . . . | f 0 ⊗ 1 Y m | 1 X 0 ⊗ g m − 1 | . . . | 1 X 0 ⊗ g 0 ]+ +( − 1) | sf 0 || sg m − 1 | [ f n − 1 ⊗ 1 Y m | . . . | 1 X 1 ⊗ g m − 1 | f 0 ⊗ 1 Y m − 1 | . . . | 1 X 0 ⊗ g 0 ] + . . . Other terms in this sum are obtained from the ﬁrst one b y sh u ﬄing the f -terms with the g -terms according to the follo wing rule: [ . . . | f k ⊗ 1 Y l +1 | 1 X k ⊗ g l | . . . ] → ( − 1) | sf k || sg l | [ . . . | 1 X k +1 ⊗ g l | f k ⊗ 1 Y l | . . . ] Let A and B b e t wo DG algebras. W e ha v e the obvious emb ed ding of DG cat egories P erf A ⊗ Perf B → Perf ( A ⊗ B ) , whic h induces a morphism of complexes C • ( P erf A ⊗ Perf B ) → C • ( P erf ( A ⊗ B )) . Let us denote the comp osition (2.3) C • ( P erf A ) ⊗ C • ( P erf B ) K → C • ( P erf A ⊗ Perf B ) → C • ( P erf ( A ⊗ B )) b y the same letter K . As an immediate corollary of Th eorems 2.6 and 2.8, w e get the follo wing result: Prop osition 2.9. The map K : C • ( P erf A ) ⊗ C • ( P erf B ) → C • ( P erf ( A ⊗ B )) is a quasi- isomorph ism. Indeed, we hav e the co mmutat iv e diagram C • ( P erf A ) ⊗ C • ( P erf B ) / / C • ( P erf ( A ⊗ B )) C • ( A ) ⊗ C • ( B ) O O / / C • ( A ⊗ B ) O O in whic h the vertic al arrows and the arr o w on the b ottom are quasi-isomorphisms. Finally , w e will formulate tw o more results ab out the K ¨ unneth map (2.3). Both resu lts follo w directly from the deﬁnition of K . Prop osition 2.10. L et A , B , and C b e thr e e D G algebr as. The diagr am C • ( P erf A ) ⊗ C • ( P erf B ) ⊗ C • ( P erf C ) K ⊗ 1 / / 1 ⊗ K   C • ( P erf ( A ⊗ B )) ⊗ C • ( P erf C ) K   C • ( P erf A ) ⊗ C • ( P erf ( B ⊗ C )) K / / C • ( P erf ( A ⊗ B ⊗ C )) c ommutes. In other wor ds, K i s asso ciative. 14 D. SHKL Y AROV Prop osition 2.11. L et A, B , C, D b e DG algebr as. L et X ∈ Mo d ( A op ⊗ C ) and Y ∈ Mo d ( B op ⊗ D ) b e bi mo dules satisfying the c onditions of Cor ol lary 2.3. Then the diagr am C • ( P erf A ) ⊗ C • ( P erf B ) K / / C ( T X ) ⊗ C ( T Y )   C • ( P erf ( A ⊗ B )) C ( T X ⊗ k Y )   C • ( P erf C ) ⊗ C • ( P erf D ) K / / C • ( P erf ( C ⊗ D )) c ommutes. 3. Hirzebruch-Riemann-R och t heorem 3.1. Euler c haracter. Let A b e a DG algebra and N a p erfect right DG A -mo dule. Consider the DG fun ctor T N = − ⊗ k N : Perf k → P erf A . T h e Euler class Eu ( N ) ∈ HH 0 ( P erf A ) is deﬁned by the formula (cf. [11],[34]) Eu ( N ) = HH ( T N )(1) . In other w ords, Eu ( N ) is the cla ss of the identit y morphism 1 N in HH 0 ( P erf A ). Let us list some basic prop er ties of th e Eu ler characte r map. The follo w ing statemen t follo ws from Theorem 2. 7: Prop osition 3.1. If N , M ∈ Perf A ar e homoto pic al ly e q u ivalent then Eu ( N ) = Eu ( M ) . In other wor ds, Eu desc ends to obje cts of Ho ( P erf A ) . The follo w ing result means that th e Eu ler class descends to the Gr othend iec k group of the triangulated category Ho ( P erf A ). Prop osition 3.2. F or any N ∈ P erf A one has Eu ( N [1]) = − Eu ( N ) and for any triangle L p → M q → N r → L [1] in Ho ( P erf A ) one has (3.1) Eu ( M ) = Eu ( L ) + Eu ( N ) . Let u s pr o ve th e ﬁr st part. W e ha ve to s h o w that 1 N + 1 N [1] is homologous to 0 in C • ( P erf A ). Denote b y 1 N ,N [1] (resp. 1 N [1] ,N ) the iden tity endomorphism of N view ed as a morphism from N to N [1] (resp. from N [1] to N ). Then b (1 N ,N [1] [1 N [1] ,N ]) = b 1 (1 N ,N [1] [1 N [1] ,N ]) = = − (1 N ,N [1] 1 N [1] ,N + 1 N [1] ,N 1 N ,N [1] ) = − (1 N [1] + 1 N ) Let u s pr o ve the second part. By Pr op osition 3.1, it su ﬃces to pro v e (3.1 ) for N = Cone ( p ). C on s ider the follo win g morphisms: j 1 =  1 L [1] 0  : L [1] → Cone ( p ) , q 1 =  1 L [1] 0  : Cone ( p ) → L [1] , j 2 =  0 1 M  : M → Cone ( p ) , q 2 =  0 1 M  : Cone ( p ) → M . It is easy to see that d ( j 1 ) = j 2 · p, d ( q 1 ) = 0 , d ( j 2 ) = 0 , d ( q 2 ) = − p · q 1 . HIRZEBRU CH-RIEMANN-R OCH THEOREM F OR DG ALGEBRAS 15 (In these formulas, p is viewed as a degree 1 morp hism from L [1] to M .) The follo w ing computation ﬁnishes the pro of: 1 Cone ( p ) − 1 L [1] − 1 M = j 1 q 1 + j 2 q 2 − q 1 j 1 − q 2 j 2 = [ j 1 , q 1 ] + [ j 2 , q 2 ] = b ( j 1 [ q 1 ] + j 2 [ q 2 ]) − b 0 ( j 1 [ q 1 ] + j 2 [ q 2 ]) = b ( j 1 [ q 1 ] + j 2 [ q 2 ]) − ( d ( j 1 )[ q 1 ] − j 2 [ d ( q 2 )]) = b ( j 1 [ q 1 ] + j 2 [ q 2 ]) − ( j 2 p [ q 1 ] + j 2 [ pq 1 ]) = b ( j 1 [ q 1 ] + j 2 [ q 2 ] − j 2 [ p | q 1 ]) . T o formulat e the m ain r esult of this sectio n, we need a p airing HH n ( P erf A ) × HH − n ( P erf A op ) → k , n ∈ Z , where A is a prop er DG algebra. Here is the deﬁn ition. Let us equip A with a left DG A ⊗ A op -mo dule structure as follo ws: ( a ′ ⊗ a ′′ ) a = ( − 1) | a ′′ || a | a ′ aa ′′ . W e will denote the resulting A -bimodu le by ∆. Consider the DG functor: T ∆ : Mo d ( A ⊗ A op ) → Mo d k , N 7→ N ⊗ A ⊗ A op A The follo w ing prop osition is an imm ediately consequen ce of Corollary 2.3. Prop osition 3.3. If A is pr op er then T ∆ induc es a DG functor Pe rf ( A ⊗ A op ) → P erf k . W e can use this to deﬁne a pairing (3.2) h , i : HH n ( P erf A ) × HH − n ( P erf A op ) → k , n ∈ Z via the comp osition of m orphisms of complexes C • ( P erf A ) ⊗ C • ( P erf A op ) K − − − − → C • ( P erf ( A ⊗ A op )) C ( T ∆ ) − − − − → C • ( P erf k ) and the fact that HH n ( P erf k ) ≃ HH n ( k ) is k , if n = 0, and 0 otherwise. Before w e formulat e the m ain r esu lt of this sectio n, let us in tr o duce the follo win g nota- tion. F or a b imo dule X ∈ Perf ( A op ⊗ B ) w e will d enote b y Eu ′ ( X ) the elemen t K − 1 ( Eu ( X )) ∈ M n HH − n ( P erf A op ) ⊗ HH n ( P erf B ) , where K is the K ¨ unneth isomorphism . Theorem 3.4. L et A b e a pr op er DG algebr a, B an arbitr ary DG algebr a, and X any obje ct of Pe rf ( A op ⊗ B ) . If y ∈ HH • ( P erf A ) then HH ( T X )( y ) = h y , Eu ′ ( X ) i . That is, if Eu ′ ( X ) = X n x ′ − n ⊗ x ′′ n ∈ M n HH − n ( P erf A op ) ⊗ HH n ( P erf B ) , then HH ( T X )( y ) = P n h y , x ′ − n i · x ′′ n . 16 D. SHKL Y AROV T o pr o v e this, observe th at T X can b e describ ed as a comp osition of the follo wing DG functors P erf A −⊗ k X − − − − → P erf ( A ⊗ A op ⊗ B ) T ∆ ⊗ k B − − − − → P erf B Th us, HH ( T X ) = HH ( T ∆ ⊗ k B ) ◦ HH ( − ⊗ k X ). It follo ws from the deﬁnition of the K ¨ unneth isomorphism K that the diagram HH • ( P erf A ) 1 ⊗ Eu ( X )   HH ( −⊗ k X ) / / HH • ( P erf ( A ⊗ A op ⊗ B )) HH • ( P erf A ) ⊗ HH 0 ( P erf ( A op ⊗ B )) K 3 3 g g g g g g g g g g g g g g g g g g g g g comm u tes. By conjugating with 1 ⊗ K , w e get the follo wing comm utative diagram: HH • ( P erf A ) 1 ⊗ Eu ′ ( X )   HH ( −⊗ k X ) / / HH • ( P erf ( A ⊗ A op ⊗ B )) HH • ( P erf A ) ⊗ HH • ( P erf A op ) ⊗ HH • ( P erf B ) K ◦ (1 ⊗ K ) 2 2 f f f f f f f f f f f f f f f f f f f f f f f F ur thermore, b y Prop osition 2.11 the diagram HH • ( P erf ( A ⊗ A op ⊗ B )) K − 1   HH ( T ∆ ⊗ k B ) / / HH • ( P erf ( k ⊗ B )) ≃ HH • ( P erf B ) HH • ( P erf ( A ⊗ A op )) ⊗ HH • ( P erf B ) HH ( T ∆ ) ⊗ 1 / / HH • ( P erf k ) ⊗ HH • ( P erf B ) K O O comm u tes. C on j ugating with K ⊗ 1 give s us the follo wing comm utativ e diagram: HH • ( P erf ( A ⊗ A op ⊗ B )) ( K − 1 ⊗ 1) K − 1   HH ( T ∆ ⊗ k B ) / / HH • ( P erf ( k ⊗ B )) ≃ HH • ( P erf B ) HH • ( P erf A ) ⊗ HH • ( P erf A op ) ⊗ HH • ( P erf B ) ( HH ( T ∆ ) K ) ⊗ 1 / / HH • ( P erf k ) ⊗ HH • ( P erf B ) K O O By concatenati ng th e top arr o w s of the form er and the latter diagrams, we get the follo w in g result: HH ( T ∆ ⊗ k B ) ◦ HH ( − ⊗ k X ) = K ◦ (( HH ( T ∆ ) K ) ⊗ 1) ◦ ( K − 1 ⊗ 1) ◦ K − 1 ◦ K ◦ (1 ⊗ K ) ◦ (1 ⊗ Eu ′ ( X )) . By asso ciativit y of th e K¨ unneth isomorphism (Prop osition 2.10), the latter prod uct is nothing b ut K ◦ (( HH ( T ∆ ) K ) ⊗ 1) ◦ (1 ⊗ Eu ′ ( X )) w hic h ﬁnishes th e p r o of. Theorem 3.4 generates sev eral corollaries. The ﬁrst one, the Hir zebr uc h-Riemann-Ro ch t yp e form ula, will b e form u lated and p ro v ed in the next section. Another corolla ry , whic h concerns th e so-ca lled smo oth DG algebras, will be describ ed in Section 6.1. HIRZEBRU CH-RIEMANN-R OCH THEOREM F OR DG ALGEBRAS 17 3.2. Hirzebruc h-Riemann-Ro c h t heorem. Essenti ally , the Hirzebruc h-Riemann-Ro ch theorem is the follo w ing result: Theorem 3.5. L et A b e a pr op er D G algebr a. Then, for any N ∈ Pe rf A , M ∈ Pe rf A op , (3.3) X n ( − 1) n dim H n ( N ⊗ A M ) = h Eu ( N ) , Eu ( M ) i . This theorem is an easy corollary of the resu lts of the pr evious section. Indeed, consid er the DG functors: T N = − ⊗ k N : P erf k → Perf A, T M = − ⊗ A M : Perf A → Pe rf k , T N ⊗ A M = − ⊗ k ( N ⊗ A M ) : P erf k → Pe rf k . Clearly , T N ⊗ A M = T M T N and, b y Theorem 3.4, we get the equalit y Eu ( N ⊗ A M ) = h Eu ( N ) , Eu ( M ) i . What remains is to observe that, for a p erfect DG k -mo dule X , Eu ( X ) = X n ( − 1) n dim H n ( X ) . This latter statemen t is a corollary of Prop ositions 3.1 and 3.2, along with the fact that X is homotop y equiv alen t to H • ( X ). Let us explain ho w one can compute the r igh t-hand side of (3.3). First of all, observ e that, b y T heorem 2.6, the pairin g (3.2) indu ces a pairing on HH • ( A ) × HH • ( A op ). Let us ﬁx tw o cycles X a a 0 [ a 1 | . . . | a l ] ∈ C • ( A ) , X b b 0 [ b 1 | . . . | b m ] ∈ C • ( A op ) ( P indicates that a and b are sums of sev eral terms) and d enote by a (resp. b ) the corre- sp ond ing elemen ts in HH • ( A ) (resp. HH • ( A op )). Let us describ e h a, b i more explicitly . Consider the comp osition of DG fun ctors A ⊗ A op → P erf ( A ⊗ A op ) T ∆ → P erf k, where A ⊗ A op is viewed as a DG category with one ob ject. Clearly , the unique ob ject of A ⊗ A op gets mapp ed und er this comp osition to A ∈ P erf k and an element x ⊗ y ∈ A ⊗ A op , view ed as a morp h ism in th e DG category A ⊗ A op , gets mapp ed to the op erator L ( x ) R ( y ) ∈ En d k ( A ), where L ( x ) : c 7→ xc, R ( y ) : c 7→ ( − 1) | c || y | cy are the op erators of left m ultiplication with x resp. righ t m u ltiplication with y . Since the op erators of left multi plication commute with op erators of righ t multiplicat ion, w e can deﬁne a pro duct a ∧ b = X a,b ± L ( a 0 ) R ( b 0 ) sh lm [ L ( a 1 ) | . . . | L ( a l ) | R ( b 1 ) | . . . | R ( b m )] (3.4) 18 D. SHKL Y AROV on HH • ( A ) × HH • ( A op ) with v alues in HH • (End k ( A )) (the formula for ± and the deﬁn ition of sh lm are the same as in (2.2)). Then h a, b i = Z a ∧ b (3.5) where R is deﬁned as follo ws. L et X b e a p er f ect DG k -mo d ule. Then w e hav e an em b eddin g of DG categ ories 8 End k ( X ) → P erf k whic h send s the uniqu e ob ject of the ﬁrst catego ry to X , view ed as an ob ject of Perf k . Then R is the map fr om HH • (End k ( X )) to HH • ( P erf k ) ≃ k ind u ced by this em b edding. F ur thermore, let us us e the n otation eu ( N ) to denote the element in HH 0 ( A ) corresp ond- ing to Eu ( N ) under the isomorphism HH 0 ( A ) → HH 0 ( P erf A ). W e are ready to rewrite the righ t-hand side of (3.3): h Eu ( N ) , Eu ( M ) i = Z eu ( N ) ∧ eu ( M ) . It tu r ns ou t that there are ve ry explicit form ulas for R and eu which will b e deriv ed in the next section. T o conclude this section, w e will rewrite the Hirzebru c h-Riemann-Ro c h formula in a more conv en tional f orm. Namely , w e will use (3.3) to deriv e a form ula that expresses the Euler form χ ( M , N ) = X n ( − 1) n dim Hom Ho ( P erf A ) ( M , N [ n ]) in terms of the Euler classes of M and N , where M and N are t w o p erfect DG A -mo du les. Consider the follo w ing (con tr av arian t) DG functor (3.6) D : Mo d A → Mo d A op , M 7→ D M = Hom Mo d A ( M , A ) . It is not hard to sh o w that this DG fun ctor preserves p erfect mo du les. Moreo v er, its square is isomorphic to the iden tit y endofunctor of P erf A and, thus, D is a qu asi-equiv alence of the DG categories ( Perf A ) op and Perf A op . The crucial prop erty of this functor is the follo w in g fact: for an y p erfect DG A -mo d ules there is a n atural quasi-isomorphism of complexes N ⊗ A D M ∼ = Hom Pe rf A ( M , N ) . Th us, the formula (3.3) can b e wr itten as follo ws: for an y N , M ∈ P erf A (3.7) χ ( M , N ) = h Eu ( N ) , Eu ( D M ) i = Z eu ( N ) ∧ eu ( DM ) . Finally , w e notice that Eu ( D M ) (and eu ( D M )) can b e expressed in terms of Eu ( M ) (resp. eu ( M )). This is based on th e follo w ing result (see App endix A): Prop osition 3.6. F or any DG algebr a, the f ormula (3.8) ( a 0 [ a 1 | a 2 | . . . | a n ]) ∨ = ( − 1) n + P 1 ≤ i j . Of course, f do esn’t ha v e to b e unique. Let us d en ote the algebra of th is catego ry b y A ( V ): A ( V ) = M s,t ∈ S Hom V ( v s , v t ) . W e will cal l such algebras (as wel l as the underlyin g categories) dir e c te d . Let us denote the ab elian cat egory of ﬁnite d imensional righ t A ( V )-mo dules by mo d A ( V ). The follo wing simple result is v ery w ell k n o wn. Prop osition 5.1. Any mo dule N ∈ mod A ( V ) admits a pr oje ctive r esolution of ﬁnite length. Let us pro v e th is. Fix a map f as ab o ve and denote 1 v f ( i ) simply by 1 i . Denote also the p ro jectiv e mo dules 1 i A ( V ) b y P i . Clearly , dim Hom mod A ( V ) ( P i , P j ) = dim Hom V ( v f ( i ) , v f ( j ) ) . HIRZEBRU CH-RIEMANN-R OCH THEOREM F OR DG ALGEBRAS 25 Th us, b y (5.1) (5.2) dim Hom mod A ( V ) ( P i , P j ) = ( k i = j 0 i > j . Fix N ∈ mo d A ( V ). The canonical m orphism p : n M i =1 Hom mod A ( V ) ( P i , N ) ⊗ k P i → N is su rjectiv e. The k ernel of this morphism satisﬁes the prop ert y Hom mod A ( V ) ( P n , Ker p ) = 0 . T o see this, apply the functor Hom mod A ( V ) ( P n , − ) to the short exact sequence 0 → Ker p → n M i =1 Hom mod A ( V ) ( P i , N ) ⊗ k P i → N → 0 and us e the prop ert y (5.2). T o ﬁnish the p ro of, app ly the same argumen t to Ker p instead of N etc. Observe that HH 0 ( A ( V )) is s p anned by the id emp otent s 1 v s , s ∈ S (or rather their classes in the quotien t A ( V ) / [ A ( V ) , A ( V )]). In terms of these elemen ts, the pairing h , i on HH 0 ( A ( V )) × HH 0 ( A ( V ) op ) is giv en b y h 1 t , 1 ∨ s i = dim Hom V ( v s , v t ) . Let us der ive the Hirzebr u c h-Riemann-Ro c h form ula for ﬁn ite dimensional mo d u les o ver directed algebras. It is w ell kno w n and w as obtained in [54, Section 2.4]. Let us ke ep the n otations from the pro of of Prop osition 6.1. Set d ij := dim Hom V ( v f ( i ) , v f ( j ) ) . Let M , N ∈ mo d A ( V ). As w e sa w ab o v e, M and N admit ﬁnite length resolutions by direct s u ms of the pr o jectiv e mo du les P i . Let us ﬁx t w o suc h r esolutions P ( M ) and P ( N ). W e kno w that eu ( P ( M )), eu ( P ( N )) are linear com binations of 1 i ’s: eu ( P ( M )) = n X i =1 a i · 1 i , eu ( P ( N )) = n X i =1 b i · 1 i . Since 1 j = eu ( P j ), w e ha v e (dim M ) j := Hom mod A ( V ) ( P j , M ) = Hom Ho ( Mod A ( V )) ( P j , P ( M )) = h eu ( P ( M )) , 1 ∨ j i = n X i =1 d j i a i and similarly (dim N ) j = P n i =1 d j i b i . Therefore, X l ( − 1) l dim Ext l mod A ( V ) ( M , N ) = χ ( P ( M ) , P ( N )) = h eu ( P ( N )) , eu ( P ( M )) ∨ i = X i,j b i a j d j i . 26 D. SHKL Y AROV Since a j = P k ( d − 1 ) j k (dim M ) k , b i = P k ( d − 1 ) il (dim N ) l , we get the follo w ing general- ization of Ringel’s form u la: (5.3) X l ( − 1) l dim Ext l mod A ( V ) ( M , N ) = X i,j (dim M ) i ( d − 1 ) ij (dim N ) j . 5.2. Proper noncomm utativ e DG-sc hemes arising from orbifold singularities. In this s ection, we will describ e certain prop er DG algebras 10 whic h arise from qu otien t singularities of the form C n /G , wh ere G is a ﬁnite group. Let V = C n b e a ﬁnite dimen sional complex vecto r sp ace and G a ﬁnite sub group of S L ( V ) ∼ = S L n ( C ). Th en G acts on the p olynomial algebra C [ V ] via ( g f )( x ) = f ( g − 1 x ). The sp ectrum X = V /G of the algebra C [ V ] G of G -inv arian t p olynomials is a singular aﬃne v ariet y . The cen tral prob lem in the stud y of such singular v arieties is to construct their “most economica l” resolutions, whic h are called cr ep ant : a resolution π : Y → X is crepan t, if π preserves the canonical classes 11 , i.e. π ∗ ( ω X ) = ω Y . The der ived Mc k a y corresp ondence [52, 53, 31, 12] is a program aroun d the follo wing conjecture and v arious v ers ions thereof: F or any cr ep ant r esolution Y → X , the b ounde d derive d c ate gory D ( Y ) of c oher ent she aves on Y is e quivalent to the b ounde d derive d c ate gory D G ( V ) of G -e quiv ariant c oher ent she aves on V . In other w ords, all crepan t resolutions of a ﬁxed singularit y are exp ected to b e isomorph ic as noncomm utativ e DG-sc hemes. The conjecture is kno wn to b e true for ﬁnite sub groups of S L (2) [31 ] and S L (3) [12] (see also [5] for a result in higher dimensions). Denote b y D G 0 ( V ) the sub category in D G ( V ) of complexes su pp orted at the origin 0 ∈ V and by D 0 ( Y ) the sub category in D ( Y ) of complexes supp orted at the exceptional ﬁb er π − 1 (0) (in the latter f orm ula 0 stands for th e image of the origin of V u nder th e canonical pro jection V → X ). Then the abov e equiv alence of cate gories should indu ce an equiv alence b et w een D 0 ( Y ) and D G 0 ( V ) [12]. The Ext group s b et ween any t w o ob jects of D G 0 ( V ) v anish in all b ut ﬁnitely m any degrees and, thus, we are dealing with a pr op er non commutativ e DG-sc h eme. This scheme is the main sub ject of the sectio n. F ollo w ing [26, Section 6.2], consider the cross-pro d uct Λ( V , G ) of the exterior algebra Λ V and the group algebra of G . In other w ords, as a v ector space Λ( V , G ) is the tensor pro du ct Λ V ⊗ C [ G ]. The pro d uct of t wo elemen ts is giv en b y ( v ⊗ g )( w ⊗ h ) = ( v ∧ g ( w )) ⊗ g h, v , w ∈ Λ V , g , h ∈ G. Equip Λ( V , G ) with the u nique grading suc h that d eg v = 1 and deg g = 0 for any v ∈ V and g ∈ G . W e will view Λ( V , G ) as a DG algebra w ith th e trivial diﬀerentia l. The follo w ing conjecture is motiv ated b y [26]: 10 All of them are DG algebras with the trivial d iﬀerentia l. 11 A crepant resolutions of X , if ex ists, is a noncompact Calabi-Y au v ariety since th e top-degree form on V is G -in v arian t and therefore the canonical shea ves of X and Y are trivial. HIRZEBRU CH-RIEMANN-R OCH THEOREM F OR DG ALGEBRAS 27 Conjecture. Ther e is an e quivalenc e of triangulate d c ate gories D G 0 ( V ) ∼ = Ho ( P erf Λ( V , G )) . Here is h o w the conjecture migh t b e pro v ed. The catego ry D G 0 ( V ) seems to b e equiv alen t to the categ ory D b ( f .d. C [ V ] ⋊ G ), where C [ V ] ⋊ G is the cross-pro duct of the p olynomial algebra and the group algebra of G and f .d. C [ V ] ⋊ G is the ab elian category of ﬁ n ite dimensional graded C [ V ] ⋊ G -mo dules. Eve ry su c h mo dule admits a ﬁnite ﬁltration by simple C [ V ] ⋊ G -mo d ules. The latter are the C [ V ] ⋊ G -mo d ules obtained fr om simple C [ G ]-mo d ules via “restriction of scalars” C [ V ] ⋊ G → C [ G ] , f ( x ) ⊗ g 7→ f (0) g , f ( x ) ∈ C [ V ] , g ∈ G. Let us denote the simple C [ V ] ⋊ G -mo du le, corr esp ondin g to an irr educible represen ta- tions ρ of G , by S ρ . T h en, us ing the tec hniqu e describ ed in [37], we may conclude th at D b ( f .d. C [ V ] ⋊ G ) is equ iv alen t to the cat egory Ho ( P erf A ) for some A ∞ algebra A with H • ( A ) = Ext • ( ⊕ ρ S ρ , ⊕ ρ S ρ ) , where the sum in the righ t-hand side is tak en o v er irr ed ucible repr esen tations of G . Ac- cording to [26, Sectio n 6.2], the algebra C [ V ] ⋊ G is quadr atic and Koszul, and its Koszul dual is exactly Λ( V , G ). Then, by [37, Sectio n 2.2], the A ∞ algebra A is formal. Finally , w e exp ect that Ext • ( ⊕ ρ S ρ , ⊕ ρ S ρ ) is Morita equiv alent to Λ( V , G ). Whether the conjecture is tru e or not, it is clear that the algebraic triangulated cate- gories of the form Ho ( P erf Λ( V , G )) sh ould pla y a r ole in the study of the qu otien t singu- larities. Let us compute the p airing h , i on HH 0 (Λ( V , G )) × HH 0 (Λ( V , G ) op ). W e start by noticing that, in general, th e space HH 0 (Λ( V , G )) is in ﬁnite dimensional (this is already so in the s im p lest case V = C , G = { 1 } ). Ho wev er, the pairing h , i v anishes on a subs pace of ﬁnite codimens ion (this follo ws from C orollary 4.8). In fact, the pairing is determined b y its restrictio n onto the ﬁnite d im en sional subspace HH 0 ( C [ G ]) × HH 0 ( C [ G ] op ) ⊂ HH 0 (Λ( V , G )) × HH 0 (Λ( V , G ) op ) . (Here w e are using the n atural embedd ing C [ G ] → Λ( V , G ) which induces an embedd ing HH 0 ( C [ G ]) → HH 0 (Λ( V , G )).) F urthermore, it is w ell known that HH 0 ( C [ G ]) is spann ed b y (the homology classes of ) the c h aracters of ir reducible represent ations of G . Let us denote the c haracter of an irreducible represen tation ρ by χ ρ : χ ρ = X g tr ( ρ ( g )) g . Using b asic h armonic analysis on G , it is easy to s h o w that the element π ρ = dim ρ | G | χ ρ is an idemp oten t in Λ( V , G ) (it is nothing but the Euler class of the DG Λ( V , G )-mo du le π ρ · Λ( V , G )). Thus, we hav e to compute h π ρ 1 , π ∨ ρ 2 i = str Λ( V ,G ) ( L ( π ρ 1 ) R ( π ρ 2 )) for tw o irreducible represen tations ρ 1 , ρ 2 . 28 D. SHKL Y AROV Let W b e the space of s ome rep resen tation of G . Then W ⊗ C [ G ] carr ies a natural C [ G ]-bimo dule structure, deﬁ ned as follo ws : g ( w ⊗ h ) k = g ( w ) ⊗ g hk , w ∈ W , g, h, k ∈ G. In particular, the graded comp onents Λ n ( V , G ) = Λ n V ⊗ C [ G ] of the alge bra Λ( V , G ) are C [ G ]-bimo dules and we ha v e str Λ( V ,G ) ( L ( π ρ 1 ) R ( π ρ 2 )) = dim V X n =0 ( − 1) n tr Λ n ( V ,G ) ( L ( π ρ 1 ) R ( π ρ 2 )) = dim V X n =0 ( − 1) n dim( π ρ 1 Λ n ( V , G ) π ρ 2 ) . Therefore, w e will start b y computing d im( π ρ 1 ( W ⊗ C [ G ]) π ρ 2 ) for an arbitrary W . Let us introdu ce a matrix d W of n on -n egativ e integ ers b y the follo wing formula: W ⊗ ρ = M σ d W σρ σ, where ρ and σ run through the set of irreducible r epresen tations of G . Let us denote the represent ation, dual to ρ , by ρ ′ . Then, as a C [ G ]-bimo dule W ⊗ C [ G ] = M ρ ( W ⊗ ρ ) ⊠ ρ ′ = M ρ,σ d W σρ σ ⊠ ρ ′ . Th us, dim( π ρ 1 ( W ⊗ C [ G ]) π ρ 2 ) = dim ρ 1 dim ρ 2 d W ρ 1 ρ 2 , whic h giv es us th e follo wing formula for h π ρ 1 , π ∨ ρ 2 i : (5.4) h π ρ 1 , π ∨ ρ 2 i = dim ρ 1 dim ρ 2 dim V X n =0 ( − 1) n d Λ n V ρ 1 ρ 2 . 6. More on th e p airing h , i 6.1. Smoot h proper DG algebras. Reca ll [41] that a DG algebra is said to b e (ho- mologica lly) smo oth if th er e is a p er f ect r igh t DG A op ⊗ A -mod ule P ( A ) together with a quasi-isomorphism P ( A ) → A of r igh t DG A op ⊗ A -mod ules. T o ha v e an example at hand, ob s erv e that Prop osition 6.1. Any dir e cte d algebr a is smo oth. Indeed, it is clea r that A ( V ) op ⊗ A ( V ) ∼ = A ( V op ⊗ V ). Th erefore, by Prop osition 5.1, an y ﬁnite dimensional A ( V ) op ⊗ A ( V )-mo dule admits a ﬁn ite pro jectiv e r esolution. What re- mains is to apply this to A ( V ) and obs er ve th at any ﬁ n ite complex of pro jectiv e bimo dules o ver an asso ciativ e algebra is a p erfect DG bimo dule in our sense. The aim of this s ection is to pr o ve that the p airin g h , i : HH n ( P erf A ) × HH − n ( P erf A op ) → k , HIRZEBRU CH-RIEMANN-R OCH THEOREM F OR DG ALGEBRAS 29 is n on-degenerate for any prop er s m o oth DG algebra A . The pro of is b ased on the ob- serv ation th at the p airing is in v erse to the Eu ler class Eu ( A ) of the A -bimo d u le A . The author lea rned ab out this id ea from [42 ] 12 . Theorem 6.2. L et A b e a p r op er smo oth DG algebr a. Then the p airing h , i is non- de g ener ate. Indeed, ﬁ x a p erfect resolution P ( A ) p → A in the category of right DG A op ⊗ A -mo dules. Then, for an y righ t p erfect DG A -mod ule X , we ha ve a morph ism 1 ⊗ p : X ⊗ A P ( A ) → X ⊗ A A ≃ X . By Prop osition 2.5, 1 ⊗ p is a quasi-isomorphism. O n the other hand, by Prop osition 2.4, b oth X ⊗ A P ( A ) and X are p erfect and, in particular, homotopically pro j ective . It is w ell kno w n that a qu asi-isomorph ism b et we en t w o homotopically pro jectiv e mo dules is actually a homotop y equiv alence (see, for instance, the pr o of of Lemma 10.12.2. 2 in [4]). Th us, 1 ⊗ p : X ⊗ A P ( A ) → X ⊗ A A ≃ X is a homotop y equiv alence. What we ha v e just p ro v ed is that the quasi-isomorphism P ( A ) p → A give s rise to a w eak h omotop y equiv alence of the DG f u nctors T P ( A ) → Id Pe rf A where Id stands for the iden tit y endofun ctor. Then , as a corollary of Theorem 2.7, w e get th e follo wing result: the linear map HH ( T P ( A ) ) : HH • ( P erf A ) → HH • ( P erf A ) coincides with the ident it y map. On the other hand , b y Theorem 3.4, the m ap HH ( T P ( A ) ) is giv en by the ’con v olution’ with Eu ′ ( P ( A )) , so the conv olution with Eu ′ ( P ( A )) is the identi t y map. This prov es that the left k ernel of the pairing is trivial, i.e. f or any n w e ha ve an em b edding HH n ( P erf A ) → HH − n ( P erf A op ) ∗ . One of the results of [60] sa ys that th e Ho c hschild h omology of an arb itrary pr op er smo oth DG algebra is ﬁn ite d imensional. Thus, to prov e that the r igh t kernel of th e pairing is trivial, it is enough to sho w that dim HH n ( P erf A ) = dim HH − n ( P erf A op ). This can b e done by replacing A b y A op in the ab o ve argument. Let us p oin t out one in teresting corollary of this result 13 : Corollary 6.3. If A is a smo oth pr op er asso ciative algebr a then HH n ( A ) = ( A/ [ A, A ] n = 0 0 otherwise Indeed, the Ho chsc hild homolo gy of such an algebra is concen trated in non -p ositiv e degrees. Thus, by the non-degeneracy of the pairing, the Ho c hsc hild homology group s, sitting in negativ e degrees, ha ve to v anish. This corollary , toget her with Prop osition 6.1, imp lies HH n ( A ( V )) = 0 for an y directed algebra A ( V ) and an y n 6 = 0. This result was obtained b y a diﬀerent metho d in [16]. 12 Although [42] is still in preparation, the argument with the Euler class of the “diagonal ” is already w ell known among the exp erts [30]. 13 If k is p erfect, this result also follo ws from Prop osition 2.5 of [32] and Morita in v ariance of the Ho chschil d homology . 30 D. SHKL Y AROV Another application of the corollary is related to the so-c alled n oncomm utativ e Hod ge- to-de Rham degeneration conjecture. R ou gh ly sp eaking, the conjecture claims that the B -op erator B : HH • ( A ) → HH •− 1 ( A ) (see [23, 67]) v anishes w henev er A is prop er an d smo oth. It w as formulate d, in a stronger form, by M. Kontsevic h and Y. Soib elman [41 ] and p ro v ed, in the partial case of DG algebras concen trated in non-negativ e degrees, by D. Kaledin [30 ]. The ab o v e corollary imp lies th e conjecture in the case of algebras with the trivial diﬀerentia l and grad in g. 6.2. Relation to T op ological Field Theories. This section is dev oted to yet another application of our r esults. Namely , w e will discuss the relev an ce of the pairin g (3.5 ) to the T op ological Field Th eories (TFT’s) constructed in [18, 41]. T o b egin with, let us r ecall that b y a tr ac e on a DG algebra A one understands a (homogeneous) fu nctional τ : A → k suc h that τ ( da ) = 0 , τ ([ a, b ]) = 0 , a, b ∈ A. Let A b e a p rop er DG algebra. Sup p ose the algebra p ossesses a d egree − d trace τ satisfying the follo win g condition: the induced degree − d pairing H • ( A ) × H • ( A ) → k , ( a, b ) 7→ τ ( ab ) is non-degenerate. T hen the pair ( A, τ ) is called a d -dimensional (compact) Calabi-Y au DG al gebra [41] 14 . S ometimes, we will write A instead of ( A, τ ). Observe that the algebra Λ( V , G ), we studied in Section 5.2, carries a natural structure of a dim V -dimensional CY DG algebra. Namely , ﬁ x a n on-zero elemen t ω ∈ Λ dim V V and set [26]: τ ω ( v ⊗ g ) = ( 0 v ∈ Λ n V , n < dim V δ 1 g v = ω . Before w e pr o ceed any fur ther, w e would like to mentio n th at there is a diﬀerent class of CY algebras whose th eory is now b eing activ ely deve lop ed [25]. T hese latter CY alg ebras are noncomm u tativ e analogs of nonc omp act smo oth CY v arieties (a go o d example of suc h an algebra is the cross-pro d uct C [ V ] ⋊ G w e men tioned in Sectio n 5.2). A d -dimensional TFT is d eﬁ ned as follo w s (w e r efer to [18, 41] fo r details). Let M ( n, m ) = S g ≥ 0 M g ( n, m ) denote the mo d u li space of Riemann su r faces with n in- coming and m outgoing b oundaries ( g d enotes the gen us). Fix a graded ve ctor space H • . By deﬁn ition, H • carries a str ucture of a d -dimensional TFT if one has a collect ion of linear maps (6.1) H • ( M ( n, m )) ⊗ H ⊗ n • → H ⊗ m • , n ≥ 1 , m ≥ 0 (here H • in th e left-hand side denotes the singular homology) satisfying the f ollo wing conditions: (1) the maps are compatible with the op eration M ( m, l ) × M ( n, m ) → M ( n, l ) 14 Actually , th e aut h ors of [18, 41] w ork with CY A ∞ algebras and categories. HIRZEBRU CH-RIEMANN-R OCH THEOREM F OR DG ALGEBRAS 31 of gluing of tw o sur f aces along the b ound ary comp onen ts and the operation M ( n, m ) × M ( p, q ) → M ( n + p , m + q ) of taking the disjoin t union of su r faces; (2) elements of H • ( M g ( n, m )) act by operators of d egree d (2 − 2 g − n − m ). One has [18, 41]: F or any d -dimensional CY DG algebr a A the H o chschild homolo gy HH • ( A ) c arries a c anonic al structur e of d -dimensional TFT. 15 One immed iate consequ en ce of this resu lt is that there is a natural degree 0 p airing – let us denote it b y h , i τ – on th e Ho c hschild homology of a d -dimensional CY DG algebra, giv en b y a generator of H 0 ( M 0 (2 , 0)). The follo wing conjecture relates this p airing to the one constru cted in the p resen t w ork 16 : Conjecture. F or any CY DG algebr a A , the p airing h , i τ c oincides with the p airing (3.5), i.e. for any a, b ∈ HH • ( A ) (6.2) h a, b i τ = h a, b ∨ i , wher e ∨ is the isomorphism HH • ( A ) → HH • ( A op ) deﬁne d by (3.8). W e n ote that this conjecture, together with T heorem 6.2, w ould imply th e follo w ing result co njectured in [41, Section 11.6 ]: Corollary . F or any smo oth CY D G algebr a A , the p airing h , i τ is non-de gener ate. T o pr esen t a piece of evidence in fav or of the conjecture, let us prov e it in the case of an asso ciativ e Calabi-Y au algebra, when the grading and the d iﬀeren tial are b oth trivial. Observe that s u c h a CY alge bra is nothing b ut a symmetric F rob eniu s algebra [38]. T o compute the left-hand side of (6.2), we will use an explicit description of the ac- tion (6.1) b ased on graph s [41, Section 11.6]. In th e language of [41], the p airing h , i τ corresp onds to the follo wing graph: ✫✪ ✬✩ r r in 1 in 2 Let us ﬁx a sym metric F r ob enius alg ebra A = ( A, τ ). Since A is ﬁnite d imensional and the b ilinear form τ ( ab ) is n on-degenerate, there exists a u nique elemen t Φ = X k φ ′ k ⊗ φ ′′ k ⊗ φ ′′′ k ∈ A ⊗ A ⊗ A satisfying the prop ert y (6.3) τ ( abc ) = X k τ ( aφ ′ k ) τ ( bφ ′′ k ) τ ( cφ ′′′ k ) 15 In fact, a much stronger result is obtained in [18, 41], namely , that th e action (6.1) exists on t he level of complexes that comput e the singular homology of the mo duli sp aces and the Ho chsc h ild homology of the algebra. 16 This conjecture was suggested to the author by Y. Soib elman and K . Costello. 32 D. SHKL Y AROV for ev ery a, b, c ∈ A . Notice th at Φ is cyclically symmetric b ecause τ ( ab ) is symmetric. According to [41], h a, b i τ can b e compu ted by means of th e ab o v e graph as follo ws: attac h a to the v ertex mark ed in 1 and b to the v ertex marked in 2 ; attac h t w o copies of th e tensor Φ to the r emaining t wo v ertices; con tru ct all the tensors along all four edges of the graph, using the pairing a × b 7→ τ ( ab ) . Here is the result: h a, b i τ = X k ,l τ ( aφ ′ k ) τ ( bφ ′ l ) τ ( φ ′′ k φ ′′ l ) τ ( φ ′′′ k φ ′′′ l ) . By (6.3), the latter form ula can b e simpliﬁed as follo ws: h a, b i τ = X k τ ( aφ ′ k ) τ ( bφ ′′ k φ ′′′ k ) . T o simplify th e form u la further, consider the u nique sym metric eleme nt γ = X i γ ′ i ⊗ γ ′′ i ∈ A ⊗ A satisfying the prop ert y (6.4) a = X i γ ′ i τ ( γ ′′ i a ) = X i τ ( aγ ′ i ) γ ′′ i for every a ∈ A . Th en it is easy to see that Φ = P i,j γ ′ i ⊗ γ ′ j γ ′′ i ⊗ γ ′′ j . Indeed, X i,j τ ( aγ ′ i ) τ ( bγ ′ j γ ′′ i ) τ ( cγ ′′ j ) = X i τ ( aγ ′ i ) τ ( b X j γ ′ j τ ( cγ ′′ j ) γ ′′ i ) = X i τ ( aγ ′ i ) τ ( bcγ ′′ i ) = τ ( a X i γ ′ i τ ( bcγ ′′ i )) = τ ( abc ) . Th us, h a, b i τ = X k τ ( aφ ′ k ) τ ( bφ ′′ k φ ′′′ k ) = X i,j τ ( aγ ′ i ) τ ( bγ ′ j γ ′′ i γ ′′ j ) = X j τ ( bγ ′ j X i τ ( aγ ′ i ) γ ′′ i γ ′′ j ) = X j τ ( bγ ′ j aγ ′′ j ) . Since γ is symmetric, w e arriv e at the follo wing formula h a, b i τ = X i τ ( aγ ′ i bγ ′′ i ) . By Corollary 4.7, w e ha v e h a, b ∨ i = tr A ( L ( a ) R ( b )). Thus, for a symm etric F rob enius algebra, the ab ov e conjecture b oils down to the follo wing identi t y: X i τ ( aγ ′ i bγ ′′ i ) = tr A ( L ( a ) R ( b )) , a, b ∈ A. T o pr o ve it, w e observe that und er th e canonical isomorphism End k ( A ) ∼ = A ⊗ A ∗ the op erators L ( a ), R ( b ) get mapp ed to the ele ment s X i aγ ′ i ⊗ τ ( γ ′′ i · − ) , X j γ ′ j b ⊗ τ ( γ ′′ j · − ) , HIRZEBRU CH-RIEMANN-R OCH THEOREM F OR DG ALGEBRAS 33 resp ectiv ely (this f ollo ws from the d eﬁnition (6.4) of γ ). Therefore, tr A ( L ( a ) R ( b )) = X i,j τ ( γ ′′ j aγ ′ i ) τ ( γ ′′ i γ ′ j b ) = X i τ ( γ ′′ i γ ′ j X j τ ( γ ′′ j aγ ′ i ) b ) = X i τ ( γ ′′ i aγ ′ i b ) = X i τ ( aγ ′ i bγ ′′ i ) , whic h ﬁnishes the pro of. The same pro of sh ould work for graded CY DG alge bras with the trivial diﬀeren tial. Appendix A. Proof of Pr opos ition 3.6 Clearly , the morp hism (3.8) is in v ertible. W e ha v e to sho w th at it comm utes with the diﬀerent ials. It is obvio us that ∨ resp ects the ﬁ rst diﬀeren tial b 0 as its deﬁnition do esn’t in v olv e multiplica tion. Let u s sh o w by a direct computation that ∨ comm u tes with the second diﬀerent ial b 1 . Let us denote the m ultiplication in A op b y ∗ . T o simplify computations, we will also use the notatio ns ξ i = | a 0 | + | sa n | + | s a n − 1 | + . . . + | s a i +1 | and f ( a 1 , a 2 , . . . , a n ) = P 1 ≤ i

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