THE MUKAI PAIRING AND INTEGRAL TRANSFORMS IN HOCHSCHILD HOMOLOGY

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1 MOSCOW MATHEMATICAL JOURNAL Volume 10, Number 3, July September 2010, Pages THE MUKAI PAIRING AND INTEGRAL TRANSFORMS IN HOCHSCHILD HOMOLOG AJA C. RAMADOSS Abstract. Let be a separated smooth proper scheme over a field of characteristic 0. Following Shklyarov, we construct a (non-degenerate pairing on the Hochschild homology of perf(, and hence, on the Hochschild homology of. On the other hand the Hochschild homology of also has the Mukai pairing (see papers by Căldăraru. If is Calabi au, this pairing arises from the action of the class of a genus 0 Riemann-surface with two incoming closed boundaries and no outgoing boundary in H 0(M 0(2, 0 on the algebra of closed states of a version of the B-Model on. We show that these pairings almost coincide. This is done via a different view of the construction of integral transforms in Hochschild homology that originally appeared in Căldăraru s work. This is used to prove that the more natural construction of integral transforms in Hochschild homology by Shklyarov coincides with that of Căldăraru. These results give rise to a Hirzebruch Riemann Roch theorem for the sheafification of the Dennis trace map Math. Subj. Class. 19L10, 14F05, 19D55, 14C40. Key words and phrases. Hochschild homology, integral transforms, Mukai pairing, Dennis trace map, Hirzebruch Riemann Roch. Introduction Throughout this paper, the term scheme will mean separated scheme. We will always work over a field K of characeristic 0. Let be a smooth proper scheme. Let perf( denote the DG-category of left bounded injective perfect complexes of O -modules. There is a natural isomorphism of Hochschild homologies (see [8], [17] for instance HH ( HH (perf(. (1 If is any smooth proper scheme, an object Φ perf( can be thought of as the kernel of an integral transform from perf( to perf( (Section 8 of [19]. This is a morphism from perf( to perf( in the homotopy category Ho(dg-cat of dg-categories modulo quasiequivalences. We will abuse notation and denote this Received December 14, 2008; in revised form April 20, c 2010 Independent University of Moscow

2 630 A. RAMADOSS by Φ as well. It follows that Φ induces a map Φ : HH (perf( HH (perf( and hence, by (1, a map Φ nat : HH ( HH (. One also has (see [17] a Künneth quasiisomorphism K : HH (perf( HH (perf( HH (perf(. Since is smooth, the diagonal : is a local complete intersection. Hence, O := R O is a perfect complex on (see [19, Section 8]. We will abuse notation and denote O thought of as the kernel of an integral transform from to Spec K by. One then has a pairing given by the composite map HH (perf( HH (perf( K HH (perf( HH (perf(k = K. Denote this pairing by, Shk. The pairing, Shk was first constructed by D. Shklyarov [17] in the DG-algebra setting. On the other hand, the work of A. Căldăraru [3] constructs the following: A non-degenerate Mukai pairing, M : HH ( HH ( K. For each Φ Perf ( an integral transform Φ cal : HH ( HH (. If is Calabi au, it has been argued implicitly by Căldăraru [5] that, M is precisely the pairing on HH ( arising from the action (on HH ( of the class of a genus 0 Riemann surface with two incoming closed boundaries and no outgoing boundary in H 0 (M 0 (2, 0. Let : HH ( HH ( be the involution whose image under the Hochschild Kostant Rosenberg isomorphism is the involution on Hodge cohomology that acts on the direct summand H q (, Ω p by multiplication by ( 1 p. The natural pairing and the Mukai pairing. The main result of this paper is as follows. Theorem 1. Let be a smooth, proper scheme. Let a, b HH (. Then, b, a M = a, b Shk. If is a smooth proper quasicompact scheme, the category perf( is quasiequivalent to perf(a for some DG-algebra A (see [11], [19]. In this case, the pairing, Shk on HH ( is the pairing on HH (A described in [17]. On the other hand, the Mukai pairing, M has been explicitly computed at the level of Hodge cohomology in [15]. In an implicit form, this computation appeared earlier in [13]. Theorem 1 therefore enables us to relate the familiar Riemann Roch Hirzebruch theorem for a proper scheme over K to the more abstract noncommutative Riemann Roch theorem in [17].

3 THE MUKAI PAIRING AND INTEGRAL TRANSFORMS IN HOCHSCHILD HOMOLOG631 Further, if is Calabi au, so is A. In this case Theorem 1 is very similar to Conjecture 6.2 in [17] for proper homologically smooth Calabi au DG-algebras A such that perf(a is quasiequivalent to perf( for some smooth proper quasicompact scheme. We make a remark about this in Section 2.3. It would also be very interesting to understand to what extent the construction of, M and, Shk carry over to more general contexts. Both pairings are defined for the Hochschild homology of a proper, homologically smooth DG-algebra. The construction of, M would follow [3] using a Serre duality theorem in this context that is due to D. Shklyarov [18]. Another interesting context is that of deformation quantization algebroids developed by M. Kashiwara and P. Schapira [7]. Given the existence of a Serre duality theorem in this setting, it seems as though a pairing analogous to, M can be constructed in following [3]. Regarding, Shk, the author does not know whether derived categories that arise in this theory arise from proper, homologically smooth DG-algebras as in the algebraic geometry setting (though one may expect this to be the case. Integral transforms in Hochschild homology. Let us outline how Theorem 1 is proven. It was stated and proven in [17] that if Φ perf(, then Φ nat is simply convolution with the Chern character of Φ with respect to the pairing, Shk. Besides [17], the reader may refer to Theorems 4 and 5 in this paper for the precise statement. We construct a map Φ : HH ( HH ( that is almost convolution with the Chern character of Φ with respect to the Mukai pairing. We then proceed to prove that Φ has all the good properties one expects of an integral transform in Hochschild homology (Propositions 1, 2 and 3 of this paper. We recall that the integral transform from perf( to perf( arising nat out of the element O of perf( is the identity. It follows that O = id. Proposition 2, which says that O = id as well, is then used to prove Theorem 1. The fact that Φ has all the good properties one expects of an integral transform in Hochschild homology is also exploited to prove the following theorem. Theorem 2. Let, be smooth proper schemes and let Φ perf(. Then, Φ nat = Φ = Φ cal. In other words,the good constructions of integral transforms in Hochschild homology coincide. A Hirzebruch Riemann Roch for the sheafification of the Dennis trace map. We now mention another consequence of Theorems 1 and 2. Recall that we have an isomorphism of higher K groups K i ( K i (perf(. For any DG-category C, let Z 0 (C denote the category such that Obj(Z 0 (C = Obj(C and Hom Z 0 (C(M, N = Z 0 (Hom C (M, N M, N Obj(C.

4 632 A. RAMADOSS Here, Z 0 (C is the space of 0-cocycles for any cochain complex C. If Z 0 (C is exact, one has a Dennis trace map (see [9]. This therefore, yields us a map Ch i : K i (C HH i (C Ch i : K i ( HH i (perf( HH i (. This map is the sheafification of the Dennis trace map constructed in [2]. Let I HKR : HH ( p,q Hp (, Ω q denote the Hochschild Kostant Rosenberg isomorphism. Let ch i : K i ( j H j i (, Ω j denote I HKR Ch i. It was proven in [4, Theorem 4.5] that ch 0 is the usual Chern character. We have the following generalization of the Hirzebruch Riemann Roch theorem. Theorem 3. Let f : be a smooth proper morphism between smooth proper schemes and. Let Z be a smooth quasicompact scheme. Then, for any α K i ( Z. (f id (ch i (απ td(t = ch i ((f id (απ td(t Layout of this paper. Section 1 reviews some basic facts from D. Shklyarov s work [17]. Section 2.1 recalls A. Căldăraru s construction of the Mukai pairing [3] and related results. In Section 2.2, we give an alternate construction of Φ : HH ( HH ( for any Φ perf(. We prove Theorem 1 and Theorem 2 in Section 2.2. Section 2.3 contains some remarks about what Theorem 1 means when is Calabi au. Section 2.4 proves Theorem 3. Acknowledgements. I am grateful to Prof. Kevin Costello, Prof. Madhav Nori and Prof. Boris Tsygan for some very useful discussions. I also thank the referee for his/her very useful comments and suggestions. 1. The Natural Pairing on the Hochschild Homology of Schemes This section primarily recalls material from D. Shklyarov s work [17]. The term DG algebra in this section shall refer to a proper homologically smooth DGalgebra unless explicitly stated otherwise Preliminary recollections. Recall that a DG-algebra A is proper if H n (A is finite-dimensional for each n and H n (A = 0 for almost all n, and is homologically smooth if it is quasiisomorphic to a perfect A op A-module. Here, A op denotes the opposite algebra of A. The term A-module shall refer to a right A-module. Recall that a A-module is said to be semi-free if it is obtained from a finite set of free A-modules after taking finitely many cones of degree 0 closed morphisms. A

5 THE MUKAI PAIRING AND INTEGRAL TRANSFORMS IN HOCHSCHILD HOMOLOG633 perfect A-module is a direct summand of a semi-free A-module. Let perf(a denote the DG-category of perfect A-modules. We recall the following facts from [17]. Fact 1: If A is a DG-algebra, the natural embedding of the category with a unique object whose morphisms are given by A into perf(a induces an isomorphism HH (A HH (perf(a (2 Fact 2: If A and B are DG-algebras and Φ is a perfect A op B-module, then Φ gives a (DG functor Φ therefore induces a map Φ : perf(a perf(b, M M A Φ. Φ nat : HH (perf(a HH (perf(b. Fact 3: Let A be a proper, homologically smooth DG-algebra. Let denote A treated as a a perfect A op A-module in the natural way. Then, by Fact 2, we have a DG functor : perf(a A op perf(k. Further, there is a isomorphism K : HH (perf(a HH (perf(a op HH (perf(a A op. The map nat K : HH (perf(a HH (perf(a op HH (perf(k = K therefore gives rise to a pairing, Shk : HH (A HH (A op K. For any exact K-linear category C, let K 0 (C denote the Grothendieck group of C. Recall from [17] that there is a Chern character Ch: K 0 (perf(a HH 0 (perf(a HH 0 (A. Let A be a proper, homologically smooth DG-algebra and let B be an arbitrary DG-algebra. We abuse notation and denote the composite map HH (A HH (A op HH (B, Shk id HH (B by, Shk itself. Identify HH (A op B with HH (A op HH (B via the inverse of the Kunneth isomorphism. If Φ perf(a op B, the following theorem from [17] (Theorem 3.4 of [17] says that Φ nat is just convolution with Ch(Φ. Theorem 4. for any x HH (A. Φ nat (x = x, Ch(Φ Shk Note that Theorem 4 implies that Φ nat D(perf(A op B. depends only on the image of Φ in

6 634 A. RAMADOSS 1.2. The natural pairing on the Hochschild homology of schemes. In this subsection, whenever f : is a morphism of schemes, f,f etc shall denote the corresponding derived functors. Let be a quasicompact scheme over K. In this case, the (unbounded derived category D qcoh ( of quasicoherent O -modules on admits at least compact generator E (see [19]. This is a perfect complex of O -modules. We recall the following facts. Fact 1: For each compact generator E of D qcoh ( there one can choose a (proper if and only if is proper DG-algebra A(E such that perf(a(e is quasiequivalent to perf( (see [11], [19]. Fact 2: Recall that if E is a compact generator of D qcoh ( and if F is a compact generator of D qcoh ( then E F is a compact generator of D qcoh ( K. Fact 3: The A(E can be chosen so that A(E F = A(E A(F whenever E and F are as in Fact 2 above. Fact 4: If E is a compact generator of D qcoh (, so is the dual perfect complex E. One can choose A(E to be A(E op. Hence, perf(a(e is quasiequivalent to perf(a(e op. From the quasiequivalences perf(a(e perf( and perf(a(e op perf(, we obtain isomorphisms i: HH ( HH (A(E, j: HH ( HH (A(E op. For proper let, Shk be the pairing on HH ( such that a, b Shk = i(a, j(b Shk for all a, b HH (. Note that the RHS of the above equation has been defined in the previous subsection. We identify HH ( with HH ( HH ( via the inverse of the Kunneth isomorphism. Recall from [19] that an element Φ of perf( gives rise to an integral transform Φ from perf( to perf(. This is a morphism in Ho(dg-cat, the category of DG-categories modulo quasiequivalences. The functor from D(perf( to D(perf( induced by Φ is the functor E π (π E L Φ. Φ induces a map from HH (perf( to HH (perf( and hence, a map from HH ( to HH ( which we shall denote by Φ nat. We now state the following consequence of Theorem 4. Like Theorem 4, Theorem 5 implies that Φ nat depends only on the image of Φ in D(perf(. Theorem 5. For any Φ in perf(, for all x HH (. Φ nat (x = x, Ch(Φ Shk HH ( Sketch of proof of Theorem 5. Theorem 5 is a direct consequence of Theorem 4 and the work of B. Toën [19]. Given two DG-categories C and D, [19] constructs a DGcategory RHom(C, D. The objects of this category are DG-functors from C to

7 THE MUKAI PAIRING AND INTEGRAL TRANSFORMS IN HOCHSCHILD HOMOLOG635 D. Suppose that C is cofibrant in the model category of DG-categories (model structure as in [19]. Then, given two DG-functors F, G: C D, the complex Hom RHom(C,D (F, G is quasiisomorphic to the cochain complex whose r cochains are natural transformations of graded functors F G[r] (see [1]. Remark. It would be an interesting exercise to compare RHom(C, D with the DG-category of A -functors Fun (C, D constructed in [2]. Let and be quasicompact schemes over K. Let E and F be compact generators of D qcoh ( and D qcoh (. One obtains identifications and η : perf(a(e perf(, η : perf(a(f perf( η : perf(a(e op A(F perf( (see Section 8.3 of [19]. Further, it was shown in [19] that there is an identification β : perf(a(e op A(F RHom(perf(A(E, perf(a(f, Φ M M A Φ in Ho(dg-cat. Similarly, there is an identification γ : perf( RHom(perf(, perf( in Ho(dg-cat. If Φ is in perf(, γ(φ is the integral transform Φ from perf( to perf( that we described before stating Theorem 5. It was shown in Section 8.3 of [19] that the following diagram commutes in Ho(dg-cat: perf( η 1 perf(a(e op A(F γ RHom(perf(, perf( η 1 ( η RHom(perf(A(E, perf(a(f. β Theorem 5 is then a direct consequence of Theorem 4 and the above commutative diagram. Remark. Instead of choosing a compact generator E of D qcoh ( and using the DG-algebra A(E to define, Shk on HH (, we could make do with any DGalgebra A such that perf(a is quasiequivalent to perf(. 2. The Mukai Pairing 2.1. Some recollections. Let be a smooth proper scheme. Let S denote the shifted line bundle on tensoring with which yields the Serre duality functor on the bounded derived category D b ( of coherent O -modules (the reader may recall that S = Ω n [n]. If f : is a morphism of schemes, f,f etc shall denote the corresponding derived functors in this section. Let :

8 636 A. RAMADOSS denote the diagonal embedding. Let! denote the left adjoint of. Let O denote O. Recall from [3] that there is an isomorphism HH ( RHom (! O, O. Since! O S 1, tensoring with π 2 S yields an isomorphism D: RHom(! O, O RHom( O, S. Definition. The Mukai pairing, M on HH ( is the pairing v w tr (D(v w, where tr denotes the Serre duality trace on. The same pairing was constructed in the DG-algebra setup in [18]. Recall that the Hochschild Kostant Rosenberg map I HKR induces an isomorphism HH i ( H j i (, Ω j, i which we shall also denote by I HKR. Let denote the linear functional on p,q Hp (, Ω q that coincides with the Serre duality trace on Hn (, Ω n and vanishes on other direct summands. Let denote the involution on p,q Hp (, Ω q that acts on the summand H p (, Ω q by ( 1p. The following result (implicitly in [13] and explicitly in [15] computes, M at the level of Hodge cohomology. Theorem 6. For a, b HH (, a, b M = I HKR (a I HKR (btd(t Integral transforms in Hochschild homology. Any Φ perf( yields an integral transform Φ: perf( perf( as described in Section 1.2. Note that if Ψ perf( Z, the image of the kernel of the integral transform Ψ Φ in D(perf( Z is precisely π Z (π Φ L π Z Ψ. A priori, there is more than one construction of the corresponding integral transform Φ : HH ( HH ( such that (a (Ψ Φ = Ψ Φ ; (b The following diagram commutes: Objects(D(perf( Ch Φ Objects(D(perf( Ch HH 0 ( Φ HH0 (. For example, Φ nat is seen to satisfy these properties without much difficulty. Indeed, by Section 8 of [19], we may replace perf(, perf(, perf(z, perf( and perf( Z by perf(a, perf(b, perf(c, perf(a op B and perf(b op C respectively for some proper homologically smooth DG-algebras A, B and C. Then, Φ perf(a op B, Ψ perf(b op C and Ψ Φ = Φ B Ψ perf(a op C. Since

9 THE MUKAI PAIRING AND INTEGRAL TRANSFORMS IN HOCHSCHILD HOMOLOG637 Φ nat, Ψnat and Ψ Φ nat are induced by the DG-functors ( A Φ: perf( perf(, ( B Ψ: perf( perf(z and ( A (Φ B Ψ: perf( perf(z respectively, (a and (b are immediate for Φ nat. Another construction of Φ was given by A. Căldăraru in [3]. Broadly speaking, one views HH ( as an ext of endofunctors of D b (, Ext(S 1, id. This can be done rigorously as in [5]. For this construction, view Φ as an element in D(. Let Φ := RHom(Φ, O L π S. By Section 5.2 of [3], the integral transform π (Φ L π : D( D( is a left adjoint of π (Φ π : D( D(. For notational brevity, we will denote these transforms by Φ and Φ respectively. Then, if α Ext(S 1, id, Φ (α is the following composite, where the unlabeled arrows are the natural adjunctions: S 1 Φ S 1 Φ Φ S 1 = S Φ S 1 id α id id id Φ S Φ S 1 id. Theorem 6 enables us to give yet another construction of Φ. This construction of Φ is motivated by Theorem 4,and plays a key role in relating the Mukai pairing to the natural pairing constructed in Section 1. In the rest of this section, the identification of HH ( with HH ( HH ( will be via the inverse of the Kunneth isomorphism. Recall that if Φ perf(, the Chern character Ch(Φ HH ( HH ( HH ( may be viewed as a K-linear map from K to HH ( HH (. Let W: HH ( HH ( be the unique involution such that W(I 1 HKR (α = ( 1p α for any α H p (, Ω q. Construction. We define Φ : HH ( HH ( to be the composite id Ch(Φ HH ( HH ( 2 HH ( W id id HH ( 2 HH (, M id HH (. Proposition 1. If Φ perf( and Ψ perf( Z then (Ψ Φ = Ψ Φ. Proof. We shall denote p,q Hp (, Ω q by H (. Recall (Theorem 4.5 in [4] that for any smooth scheme Z, I HKR Ch = ch, the right hand side being the familiar Chern character map from D(perf(Z to H (Z. Let a HH (. Note that HH 0 ( i HH i( HH i (. Hence, Ch(Φ = α λ(i β λ(i i λ(i I i

10 638 A. RAMADOSS for some index sets I i and α λ(i HH i ( and β λ(i HH i (. By Theorem 6 and the construction of Φ, I HKR (Φ (a = I HKR (ai HKR (α λ(i td(t I HKR (β λ(i. (3 i Now suppose that λ(i I i Ch(Ψ = j µ(j J j γ µ(j δ µ(j for some index sets J j and γ µ(j HH j ( and δ µ(j HH j (Z. Then, by (3, I HKR (Ψ = i,j Φ (a I HKR (δ µ(j λ(i I i, µ(j J j I HKR (ai HKR (α λ(i td(t I HKR (β λ(i I HKR (γ µ(j td(t. Recall that Ψ Φ = π Z (π Z Ψ π Φ The desired proposition will follow from (3 if we can show that ch(ψ Φ = I HKR (β λ(i I HKR (γ µ(j td(t α λ(i δ µ(j. i,j λ(i I i, µ(j J j (4 Recall that after identifying H ( with H ( H (, π gets identified with id. Also, π is identified with the map a 1 a from ( to H H (. With this in mind, (4 can be rewritten as ch(ψ Φ = π Z (ch(π (Φ ch(π ZΨ π td(t. This follows directly from the Riemann Roch Hirzebruch theorem applied to the map π Z : Z Z. Let O = O be treated as the kernel of an integral transform from to. Proposition 2. O = id. Proof. Since O is the kernel of the identity, O O = O. By Proposition 1, O is an idempotent endomorphism of HH (. To prove that it is the identity, it suffices to show that it is surjective. nat For this, note that O = id. By Theorem 4, Ch(O = e i,k f i,k, i where the e i,k form a basis of HH i ( and the f i,k form a basis of HH i ( such that f i,k, e i,l Shk = δ k,l. The δ on the right hand side of the above equation is the Kronecker delta. k

11 THE MUKAI PAIRING AND INTEGRAL TRANSFORMS IN HOCHSCHILD HOMOLOG639 Let W be the involution on HH ( which we defined earlier before constructing Φ. For any p, q Z, recall that W(I 1 HKR (α = ( 1p α for any α H p (, Ω q. It follows that if x HH i (, then O (x = k W(x, e i,k M f i,k. Recall from [3] that the pairing, M is non-degenerate. Moreover, W is an involution on HH (. Since the e i,k form a basis of HH i (, there exist elements x k in HH i ( such that W(x l, e i,k M = δ kl. Clearly, O (x k = f i,k. This proves that O is surjective, as was desired. We are now ready to prove Theorem 1. Proof of Theorem 1. This follows almost immediately from the fact that O nat = id: HH ( HH (. Since O = id, O nat f i,k, e i,l Shk = δ k,l. On the other hand since O = id by Proposition 2, W(f i,k, e i,l M = δ k,l. It follows from the K bi-linearity of the pairings that (a, b a, b Shk, (a, b W(a, b M = a, b Shk = W(a, b M. (5 Let denote the involution on HH ( such that I 1 HKR (α = ( 1 q I 1 HKR (α for any α H p (, Ω q. Note that for any α Hp (, Ω q and β H2k p (, Ω 2k q, ( 1 p α β = ( 1 q β α. It follows that I HKR (W(a I HKR (b = I HKR (b I HKR (a in H (. Recall that I HKR (W(a = I HKR (a by the definition of W. Hence, Theorem 6 may be rewritten to say that a, b M = I HKR (bi HKR (a td(t. By (5, a, b Shk = W(a, b M = I HKR (W(a I HKR (btd(t = I HKR (ai HKR (btd(t = I HKR (ai HKR ([b ] td(t = b, a M. This proves Theorem 1.

12 640 A. RAMADOSS Let Φ! := RHom(Φ, O L π S D(. Recall from Section 5.2 of [3] that the integral transform π (Φ! L π : D( D( is right adjoint to the transform π (Φ L π : D( D(. For notational brevity, we denote these transforms by Φ! and Φ respectively. We also have the following proposition, which shows that Φ is a good candidate for the integral transform on Hochschild homology defined by Φ. Proposition 3. If x HH ( and y HH (, then Φ (x, y M = x, Φ! (y M. Proof. The notation used in this proof is as in the proof of Proposition 1. Assume that after identifying HH ( with HH ( HH ( (via the inverse of the Kunneth map, Ch(Φ = α λ(i β λ(i i λ(i I i for some index sets I i and α λ(i HH i ( and β λ(i HH i (. Then, by Theorem 6 and (3, Φ (x, y M = i λ(i I i I HKR (xi HKR (α λ(i td(t I HKR (β λ(i I HKR (ytd(t. Note that Ch(Φ! = i λ(i I i ( 1 i W(β λ(i [W(α λ(i. Ch(S ]. The ( 1 i comes from the fact that the composite HH ( HH ( K HH ( K 1 HH ( HH ( is the signed map swapping factors. It follows from Theorem 6 and (3 that x, Φ! (y M = i = i λ(i I i ( 1 i λ(i I i I HKR (β λ(i I HKR (ytd(t I HKR (I HKR (yβ λ(i td(t I HKR (x I HKR (α λ(i ch(s td(t I HKR (x I HKR (α λ(i ch(s td(t. Now, if n is the dimension of, ch(s = ( 1 n ch(ω n. Also, td(t ch(ω n = td(t (see [4]. It follows that I HKR (x I HKR (α λ(i ch(s td(t = ( 1 n (I HKR (xi HKR (α λ(i td(t.

13 THE MUKAI PAIRING AND INTEGRAL TRANSFORMS IN HOCHSCHILD HOMOLOG641 Hence, I HKR (x I HKR (α λ(i ch(s td(t = I HKR (xi HKR (α λ(i td(t. This proves the desired proposition. Note that Proposition 1 and Proposition 3 parallel Theorems 5.3 and 7.3 respectively in [3]. However, since we use the Riemann Roch theorem for (proper projections to prove Proposition 1, the construction of Φ by itself does not amount to a self-contained construction of integral transforms in Hochschild homology at this stage. However, it helps prove Theorem 1, which in turn leads to Theorem 2, showing that all three constructions of integral transforms in Hochschild homology coincide. In particular, it tells us that the integral transform constructed by A. Căldăraru [3] coincides with the more natural construction of the integral transform constructed by D. Shklyarov [17]. Let Φ perf(. Denote the integral transform Φ : HH ( HH ( constructed by A. Căldăraru [3] and described briefly earlier in this section by Φ cal. Proof of Theorem 2. That Φ = Φ nat is an immediate consequence of Theorem 1 and Theorem 5. We therefore need to show that Φ = Φ cal. For this, we will follow D. Shklyarov and imitate the proof of Theorem 4 (Theorem 3.4 in [17]. Step 1: Recall that if Φ perf( and Φ perf(, Φ Φ perf(. We then have integral transforms in Hochschild homology Φ : HH ( HH (, Φ : HH ( HH (, (Φ Φ : HH ( HH (. Identify HH ( and HH ( with HH ( HH ( and HH ( HH ( respectively via the inverse of the relevant Künneth isomorphisms. It follows from the construction of Φ that (Φ Φ = Φ Φ. Similarly, we have integral transforms in Hochschild homology Φ cal : HH ( HH (, Φ cal : HH ( HH (, (Φ Φ cal : HH ( HH (. It can be verified without much difficulty (see [12], Lemma 2.1 for instance that (Φ Φ cal = Φ cal Φ cal. Step 2: Note that Φ perf( may also be thought of as the kernel of an integral transform from Spec K to. We will denote Φ thought of in this manner by Φ pt. Let denote O thought of as the kernel of an integral transform from to Spec K. Also identify HH ( with HH ( HH (Spec K via the map y y 1. Then Φ = (O Φ pt,

14 642 A. RAMADOSS whence Φ Φ cal = = cal Now, by Proposition 2, (O Φ pt (O Φ pt cal O cal = = cal = O (O (O cal = id. (Φ pt (1, (Φ pt cal (1. Also, (Φ pt cal (1 = Ch(Φ by Definition 6.1 in [3] and Theorem 4.5 in [4]. (Φ pt (1 = Ch(Φ by the construction of (Φ pt. Thus, we need to show that = cal : HH ( HH (Spec K = K. With the above identification of HH (Spec K with K, for any x HH (Spec K, x = x, 1 M. Let! denote RHom(, O L S. If α HH (, by Proposition 3. By Theorem 7.3 in [3], (α, 1 M = α,! (1 M cal (α, 1 M = α,!cal (1 M. Now,!cal (1 = Ch(! by Definition 6.1 in [3] and Theorem 4.5 in [4].! (1 = Ch(! by the construction of!. This yields the desired theorem When is Calabi au. In such a situation, D b ( can be be thought of as the category of open states of the B-Model on (see [5]. The corresponding algebra of closed states is the Hochschild cohomology HH (perf( HH (. As is Calabi au, there is an identification HH ( HH (. The Mukai pairing constructed by A. Căldăraru in [3] on HH ( then gives a pairing on HH (. Moreover, for any E D b (, there are natural maps ι E : Hom D b ((E, E HH (, ι E : HH ( Hom D b ((E, E as constructed in [5]. Let E, F D b (. Let f Hom Db ((E, E and let g Hom Db ((F, F. Then, ϕ g ϕ f gives a degree 0 endomorphism of the graded vector space RHom Db ((E, F. Denote this endomorphism by T f,g. The Cardy condition (see [3, Theorem 7.9] tells us that ( 1 i Tr(T f,g HomD b ( (E,F[i] = ι E (f, ι F (g M. i Z For a more explicit/classical version of this statement, the reader may refer to [16]. The Cardy condition verifies that this data gives a topological quantum field theory. Of course, the Mukai pairing in this case is the pairing obtained by the action of the class of a genus 0 Riemann surface with two incoming closed boundaries and no outgoing boundary in H 0 (M 0 (2, 0 on HH (, the action coming from the fact

15 THE MUKAI PAIRING AND INTEGRAL TRANSFORMS IN HOCHSCHILD HOMOLOG643 that HH ( with Mukai pairing is a good algebra of closed states as verified by the Cardy condition. On the other hand, [6] gives the category of open states of the B-Model on as an A enrichment of D b (. The closed TCFT one associates with this category has homology HH ( HH (. This is also equipped with a pairing coming out of the action of the class of a genus 0 Riemann surface with two incoming closed boundaries and no outgoing boundary in H 0 (M 0 (2, 0 on the homology of the closed TCFT one constructs in [6] from the B-Model. Whether these pairings coincide is however, not clear currently. Theorem 1 is similar to Conjecture 6.2 in [17] for Calabi au algebras A such that perf(a is quasiequivalent to perf( for some quasicompact smooth scheme. Proof of Theorem 3. The sheafification of the Dennis trace map. Let us briefly recall how the sheafification of the Dennis trace map is constructed. The material we are recalling is from [9], [2], [10] and [14]. Let be a smooth quasicompact scheme. As in Section 1.2, choose a compact generator E of D qcoh ( and a DG-algebra A(E such that perf(a(e is quasiequivalent to perf(. Let Z 0 (perf(a(e be the exact category whose objects are those of perf(a(e such that Hom Z 0 (perf(a(e(m, N = Z 0 (Hom perf(a(e (M, N. As pointed out by B. Keller [10], using the Waldhausen structure of Z 0 (perf(a(e, we can construct a Dennis trace map Dtr: K i ( K i (Z 0 (perf(a(e HH i,mcc (Z 0 (perf(a(e i 0. Here, HH i,mcc is the Hochschild homology constructed by R. McCarthy in [14]. As Keller further points out in [10], there is a natural transformation HH i,mcc (Z 0 (perf(a(e HH i (Z 0 (perf(a(e. Further, we also have a natural transformation HH i (Z 0 (perf(a(e HH i (perf(a(e. The obvious compositions then give us a map Ch i : K i ( K i (Z 0 (perf(a(e HH i (perf(a(e HH i (. Let be a smooth quasicompact scheme. Let F and A(F be as in Section 1.2. Let Ψ perf(a(f op A(E. The following proposition, analogous to Theorem 7.1 of [3], says that the sheafification of the Dennis trace map is functorial. Proposition 4. The following diagram commutes. K i (Z 0 (perf(a(e Ch i HH i (perf(a(e Ψ Ψ nat K i (Z 0 (perf(a(f Ch i HH i (perf(a(f.

16 644 A. RAMADOSS Proof. This proposition will follow easily once we verify that Ψ: Z 0 (perf(a(e Z 0 (perf(a(f preserves cofibrations and weak equivalences. By [9], the weak equivalences in Z 0 (perf(a for any DG-algebra A are quasiisomorphisms. The cofibrations in Z 0 (perf(a are morphisms of A-modules that admit retractions as morphisms of graded A-modules. That Ψ preserves cofibrations follows without difficulty from the fact that Ψ: perf(a(e perf(a(f is a DG-functor. That Ψ preserves weak equivalences follows from the fact that perfect modules are homotopically projective (see Proposition 2.5 of [17]. Proof of Theorem 3. We warn the reader that in the proof that follows, and denote proper smooth quasicompact schemes. Step 1: Let Φ perf(.the first step is to note that even though Z is not necessarily proper, the kernel Φ O Z perf( Z Z induces an integral transform from perf( Z to perf( Z. This follows from the fact that if E and F are compact generators of D coh ( and D coh (Z respectively, the compact generator E F := π E π Z F of D coh( Z is mapped by the integral transform with kernel Φ O Z to the perfect complex π (Φ L π E F. Also, after identifying HH ( Z and HH ( Z with HH ( HH (Z and HH ( HH (Z respectively via the inverse of the relevant Kunneth isomorphisms, (Φ O Z nat = Φ nat id: HH ( HH (Z HH ( HH (Z. (6 This follows from the facts that O Z nat = id and from Proposition 2.11 of [17]. Step 2: By the Proposition 4, the following diagram commutes. K i (perf( Z (Φ O Z Ch i HH i ( Z (Φ O Z nat K i (perf( Z Ch i HH i ( Z. (7 After identifying HH ( Z and HH ( Z with HH ( HH (Z and HH ( HH (Z respectively via the inverse of the relevant Kunneth isomorphisms,we have the following commutative diagram by (7 and (6: K i (perf( Z (Φ O Z K i (perf( Z p+q=i Ch i HH p ( HH q (Z Φ nat id p+q=i Ch i HH p ( HH q (Z. (8 Now, it follows from Theorem 1 and Theorem 3 that Φ (8 and Proposition 3, = Φ nat. Hence, by (, M id HH (Z(f y Ch i (α = (, M id HH (Z(y (id f Ch i (α (9

17 THE MUKAI PAIRING AND INTEGRAL TRANSFORMS IN HOCHSCHILD HOMOLOG645 for any α K i (Z, y HH (. By Theorem 4, (9 can be rewritten to say that I HKR (f (y ch i (αtd(t = I HKR (y ch i ((f id αtd(t as elements of H (Z. The desired theorem now follows from the facts that f commutes with I HKR (see Theorem 7 of [13] and commutes with the involution. References [1] A. I. Bondal, M. Larsen, and V. A. Lunts, Grothendieck ring of pretriangulated categories, Int. Math. Res. Not. (2004, no. 29, MR [2] P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan, Chern character for twisted complexes, Geometry and dynamics of groups and spaces, Progr. Math., vol. 265, Birkhäuser, Basel, 2008, pp MR [3] A. Căldăraru, The Mukai pairing, I: the Hochschild structure, Preprint ariv:math/ [math.ag]. [4] A. Căldăraru, The Mukai pairing. II. The Hochschild Kostant Rosenberg isomorphism, Adv. Math. 194 (2005, no. 1, MR [5] A. Căldăraru and S. Willerton, The Mukai pairing, I: a categorical approach, Preprint ariv: [math.ag]. [6] K. Costello, Topological conformal field theories and Calabi au categories, Adv. Math. 210 (2007, no. 1, MR [7] M. Kashiwara and P. Schapira, Deformation quantization modules, Preprint ariv: [math.ag]. [8] B. Keller, On the cyclic homology of ringed spaces and schemes, Doc. Math. 3 (1998, MR [9] B. Keller, On differential graded categories, Preprint ariv:math/ [math.kt]. [10] B. Keller, On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999, no. 1, MR [11] M. Kontsevich and. Soibelman, Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I, Preprint ariv:math/ [math.ra]. [12] E. Macri, P. Stellari, and S. Mehrotra, Infinitesimal derived Torelli theorem for K3 surfaces, Preprint ariv: [math.ag]. [13] N. Markarian, Poincaré Birkhoff Witt isomorphism, Hochschild homology and Riemann Roch theorem, Max Planck Institute preprint MPIM [14] R. McCarthy, The cyclic homology of an exact category, J. Pure Appl. Algebra 93 (1994, no. 3, MR [15] A. C. Ramadoss, The relative Riemann Roch theorem from Hochschild homology, New ork J. Math. 14 (2008, MR [16] A. C. Ramadoss, A generalized Hirzebruch Riemann Roch theorem, C. R. Math. Acad. Sci. Paris 347 (2009, no. 5 6, MR [17] D. Shklyarov, Hirzebruch Riemann Roch theorem for DG algebras, Preprint ariv: [math.kt]. [18] D. Shklyarov, On Serre duality for compact homologically smooth DG algebras, Preprint ariv:math/ [math.ra]. [19] B. Toën, The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007, no. 3, MR Department of Mathematics, University of Oklahoma, Norman, OK address: aramadoss@math.ou.edu