The Mukai pairing and integral transforms in Hochschild homology

The Muk ai pairing and in tegral transfor ms in Ho c hsc hild homo logy . Aja y C. Ramadoss Octob er 31, 2018 Abstract Let X b e a smo oth prop er scheme o ver a field of characteristic 0. F ollowing Shklyaro v [10] , w e construct a (non-degenera te) pairing on the Hochschild homology o f p erf ( X ), and hence, on the Ho chschild homology o f X . On the o ther hand the Ho chschild homology of X also has the Muk ai pairing (see [1]). If X is Calabi-Y au, this pairing a rises from the action of the class of a genus 0 Riemann-surface with tw o in- coming closed b oundaries a nd no outgoing b oundary in H 0 ( M 0 (2 , 0)) on the alg e br a of closed sta tes of a version of the B-Mo del on X . W e show that these pair ings ”almost” coincide. This is done via a dif- ferent view of the co ns truction of in tegr al transforms in Ho chsc hild homology that originally app ear ed in Ca ldararu’s work [1]. This is used to pr ov e tha t the mo r e ”natural” constructio n of integral tr a ns- forms in Ho chschild homolo gy by Shklyarov [10] coincides with that o f Caldarar u [1]. These results give ris e to a Hirzebruch Riema nn-Ro ch theorem for the shea fification of the Dennis trac e map. In tro du c tion. Let X b e a smo oth pr op er sc heme o v er a fi eld K of characte r istic 0. Let p erf ( X ) denote the DG-category of left b ound ed p erfect injectiv e complexes of O X -mo dules . Th ere is a natural isomorphism of Ho c hschild homologies (see [5] for instance) HH • ( X ) ≃ HH • (p erf ( X )) . (1) If Y is an y smo oth prop er scheme, an ob ject Φ ∈ p erf ( X × Y ) can b e though t of as the kernel of an inte gral transform f r om p erf ( X ) to p erf ( Y ) (Section 8 of [11]). Th is is a morphism from p erf ( X ) to p erf ( Y ) in the 1 homotop y category Ho(dg-cat) of dg-categories mo du lo quasi-equiv alences. W e will abuse notation and denote this b y Φ as well. It follo ws that Φ induces a map Φ ∗ : HH • (p erf ( X )) → HH • (p erf ( Y )) and h ence, b y (1) , a map Φ nat ∗ : HH • ( X ) → HH • ( Y ) . One also has (see [10] ) a Kunneth quasiisomorphism K : HH • (p erf ( X )) ⊗ HH • (p erf ( Y )) → HH • (p erf ( X × Y )) . Since X is smo oth, the diagonal ∆ : X → X × X is a lo cal complete in tersection. Hence, O ∆ := R∆ ∗ O X is a p erfect complex on X × X (see [11], Section 8). W e will abu s e notation and d enote O ∆ though t of as the k ernel of an int egral transform fr om X × X to S p ec K by ∆. One then h as a pairing giv en by the comp osit e m ap HH • (p erf ( X )) ⊗ HH • (p erf ( X )) K   y HH • (p erf ( X × X )) ∆ ∗ − − − − → HH • (p erf ( K )) = K . Denote this pairing by h , i Shk . On the other hand, the w ork of A. Caldararu [1] constructs the follo wing: • A non-degenerate Muk ai p airing h , i M : HH • ( X ) ⊗ HH • ( X ) → K . • F or eac h Φ ∈ P erf ( X × Y ) an ”inte gral transform” Φ cal ∗ : HH • ( X ) → HH • ( Y ) . If X is Calabi-Y au, it has b een argued implicitly by Caldararu [3] that h , i M is precisely th e pairing on HH • ( X ) arising fr om the action (on HH • ( X )) of the class of a genus 0 Riemann-sur face with t wo incoming closed b ound aries and no outgoing b o u ndary in H 0 ( M 0 (2 , 0)). Let ∨ : HH • ( X ) → HH • ( X ) b e the wh ose image under the Ho c hschild-Kostan t-Rosen b erg isomorphism is the in volutio n on Ho dge cohomology that acts on the d irect summand H q ( X, Ω p X ) by m ultiplication by( − 1) p . 2 The ”natural” pairing and the Muk ai pairing. The main result of this note is as follo ws. Theorem 1. L et a, b ∈ HH • ( X ) . Then, h b ∨ , a i M = h a, b i Shk . If X is a smo oth prop er qu asi-compact sc h eme, the category p erf ( X ) is quasi-equiv alent to p erf ( A ) f or some DG-algebra A (see [6],[11]). In this case, the pairing h , i Shk on HH • ( X ) is the pairing on HH • ( A ) describ ed in [10]. On the other hand , the Mu k ai pairing h , i M has b ee n explicitly computed at the lev el of Ho dge cohomology in [8]. In an implicit f orm, this computation app eared earlier in [7]. Th eorem 1 therefore, en ab les u s to relate the familiar Riemann-Ro ch-Hirzebruc h theorem for a prop er sc heme o v er K to the more abstract ”noncomm utativ e” Riemann-Ro ch theorem in [10]. F ur ther, if X is Calabi-Y au, so is A . In this case Theorem 1 is v ery sim- ilar to Conjecture 6.2 in [10] for prop er homologica lly smo oth C alabi-Y au DG-alge br as A su c h that p erf ( A ) is quasi-equiv alent to p erf ( X ) for some smo oth prop er quasi-compact sc heme X . W e mak e a remark ab out this in Section 2.3. In tegral transforms in Ho c hsc hild homology . Let us outline ho w T heorem 1 is pro v en. It was stated and prov en in [10] that if Φ ∈ p erf ( X × Y ), then Φ nat ∗ is simply con v olution with the Ch er n c haracter of Φ with resp ect to th e pairing h , i Shk . Besides [10], the reader ma y refer to Theorems 4 and 5 in this pap er for the precise statemen t. W e construct a map Φ muk ∗ : HH • ( X ) → HH • ( Y ) th at is ”almost” conv olution with the Chern c haracter of Φ with r esp ect to the Mu k ai p airing. W e then pro ceed to pr o v e that Φ muk ∗ has all the ”goo d prop e r ties” one exp e cts of an in tegral transform in Ho chsc h ild homology (Pr op ositions 1 ,2 and 3 of this pap er). W e r ecall that the in tegral transform f rom p e rf ( X ) to p erf ( X ) arising out of the elemen t O ∆ of p erf ( X × X ) is the ident ity . It f ollo ws that O ∆ nat ∗ = id. Pr op osition 2, wh ic h says th at O ∆ muk ∗ = id as w ell, is then used to pro ve Th eorem 1. 3 The fact that Φ muk ∗ has all the ”go o d p r op erties” one exp ec ts of an in tegral transform in Ho c hsc h ild homology is also exploited to pro ve the follo wing theorem. Theorem 2. Φ nat ∗ = Φ muk ∗ = Φ c al ∗ . In other words,the ”go o d constructions” of in tegral transforms in Ho chsc hild homology coincide. A Hirzebruch- R iemann-Ro ch for the sheafification of the D en- nis trace map. W e no w men tion another consequen ce of T heorems 1 and 2. Recall that w e ha ve an isomorph ism of higher K group s K i ( X ) ≃ K i (p erf ( X )) . F or any DG-category C , let Z 0 ( C ) denote the category su c h that Ob j(Z 0 ( C )) = Ob j( C ) and Hom Z 0 ( C ) ( M , N ) = Z 0 (Hom C ( M , N )) ∀ M , N ∈ Ob j( C ) . Here, Z 0 (C) is the space of 0-cocycles for an y co chain complex C. If Z 0 ( C ) is exact, one has a Dennis trace map Ch i : K i ( C ) → HH i ( C ) (see [12]). T h is therefore, yields us a map Ch i : K i ( X ) → HH i (p erf ( X )) ≃ HH i ( X ) . This map is the ”sh eafification of the Dennis trace map” constructed in [13]. Let I H K R : HH • ( X ) → ⊕ p,q H p ( X, Ω q X ) denote the Ho chsc hild-Kostan t- Rosen b e rg isomorp hism. Let c h i : K i ( X ) → ⊕ j H j − i ( X, Ω j X ) denote I H K R ◦ Ch i . It w as pro ven in [2] (Theorem 4.5) that c h 0 is the u sual Chern c haracter. W e ha ve th e follo wing generalizat ion of the Hirzebr uc h Riemann-Ro c h theorem. 4 Theorem 3. L et f : X → Y b e a smo oth pr op er morphism b etwe en pr op er schemes X and Y . L et Z b e a sm o oth quasi-c omp act sep ar ate d sc heme. Then, ( f × id ) ∗ ( ch i ( α ) π ∗ X td ( T X )) = ch i (( f × id ) ∗ ( α )) π ∗ Y td ( T Y ) for any α ∈ K i ( X × Z ) . La y out of this note. Section 1 reviews some basic facts from D. Shklyaro v’s wo r k [10]. S ection 2.1 recalls A. Caldararu’s constru ction of the Muk ai pairing [1] and r elated results. In Section 2.2, we giv e an alternate construction of Φ ∗ : HH • ( X ) → HH • ( Y ) for an y Φ ∈ p erf ( X × Y ). W e pro ve Theorem 1 and Theorem 2 in Section 2.2. Section 2.3 cont ains s ome remarks ab out what Theorem 1 means when X is Calabi-Y au. S ection 2.4 pro v es T heorem 3. Ac knowled gemen t s. I am grateful to Prof. Kevin Costello, Prof. Madh av Nori and Prof. Boris Tsygan for some v ery u s eful discussions. 1 The ”natural p airing” on the Ho c hsc hild homol- ogy of sc hemes. This section primarily recalls m aterial from D. Shkly aro v’s work [10]. The term ”DG algebra” in this section shall refer to a p rop er homologically smo oth DG-algebra unless explicitly s tated otherwise. 1.1 Preliminary recollections. Recall that a DG-algebra A is prop er if P n dim H n ( A ) < ∞ and is h omo- logica lly smo oth if it is quasi isomorphic to a p erfect A op ⊗ A -mo d ule. Here, A op denotes the opp osite algebra of A . T he term ” A -mo dule” shall r efer to a righ t A -mo d ule. 5 Recall that a A -mo d ule is said to b e semi-free if it is obtained from a fi nite set of f ree A -mo d ules after taking finitely man y cones of degree 0 closed mor- phisms . A p erfect A -mo d ule is a direct sum m and of a semi-free A -mo dule. Let p erf ( A ) den ote the DG-cate gory of p erfect A -mod ules. W e recall the follo wing facts fr om [10]. F act 1: If A is a DG-alg ebr a, the natural em b edding of the category with a unique ob ject whose m orphisms are giv en by A into p erf ( A ) indu ces an isomorphism HH • ( A ) ≃ HH • (p erf ( A )) (2) F act 2: If A and B are DG-algebras and Φ is a p erfect A op ⊗ B -mo dule, then Φ giv es a (DG) functor Φ ∗ : p erf ( A ) → p erf ( B ) M M ⊗ A Φ . Φ ∗ therefore induces a map Φ nat ∗ : HH • (p erf ( A )) → HH • (p erf ( B )) . F act 3: Let ∆ denote A treated as a a p erfect A op ⊗ A -mo du le in the natural w a y . T hen, by F act 2, we ha ve a DG fun ctor ∆ ∗ : p erf ( A ⊗ A op ) → p erf ( K ). F ur ther, there is a isomorph ism K : HH • (p erf ( A )) ⊗ HH • (p erf ( A op )) → HH • (p erf ( A ⊗ A op )) . The map ∆ nat ∗ ◦ K : HH • (p erf ( A )) ⊗ HH • (p erf ( A op )) → HH • (p erf ( K )) = K therefore giv es rise to a pairing h , i Shk : HH • ( A ) ⊗ HH • ( A op ) → K . F or any exact K -linear category C , let K 0 ( C ) denote the Grothendiec k group of C . Recall from [10] that there is a Chern c haracter Ch : K 0 (p erf ( A )) → HH 0 (p erf ( A )) ≃ HH 0 ( A ) . Let A and B b e DG-alge b ras. W e abuse notation and denote th e comp osite map HH • ( A ) ⊗ HH • ( A op ) ⊗ HH • ( B ) h , i Shk ⊗ id − − − − − − − → HH • ( B ) b y h , i Shk itself. I d en tify HH • ( A op ⊗ B ) with HH • ( A op ) ⊗ HH • ( B ) via the in verse of the Kunneth isomorphism. If Φ ∈ p erf ( A op ⊗ B ), th e follo wing theorem fr om [10] (Th eorem 3.4 of [10]) s a ys that Φ nat ∗ is just ”conv olution with Ch(Φ)”. 6 Theorem 4. Φ nat ∗ ( x ) = h x, Ch (Φ) i Shk for any x ∈ HH • ( A ) . Note that T heorem 4 implies that Φ nat ∗ dep end s only on the image of Φ in D(p erf ( A op ⊗ B )). 1.2 The natural pairing on t he Ho c hsc hild homology of sc hemes. In this su bsection, whenev er f : X → Y is a morphism of sc hemes, f ∗ , f ∗ etc shall denote the corresp onding deriv ed functors. Let X b e a quasicompact separated sc heme ov er K . In this case, the (unb ou n ded) derived category D q coh ( X ) of qu asi-coheren t O X -mo dules on X admits at least compact gen- erator E (see [11]). T his is a p erf ect complex of O X -mo dules. W e recall the follo wing facts. F act 1: F or eac h compact generator E of D q coh ( X ) there one can c ho ose a (prop er if and only if X is prop e r ) DG-algebra A ( E ) su c h that p erf ( A ( E )) is quasi-equiv alen t to p e r f ( X ) (see [6],[11 ]). F act 2: Recall that if E is a compact generator of D q coh ( X ) and if F is a compact generator of D q coh ( Y ) then E ⊠ F is a compact generator of D q coh ( X × K Y ). F act 3: T he A ( E ) can b e c hosen so that A ( E ⊠ F ) = A ( E ) ⊗ A ( F ) whenev er E and F are as in F act 2 ab o ve. F act 4 : If E is a compact generator of D q coh ( X ), so is the du al p erfect complex E ∨ . One can c ho ose A ( E ∨ ) to b e A ( E ) op . Hence, p erf ( A ( E )) is quasi-equiv alent to p erf ( A ( E ) op ). F rom the quasi-equiv alences p erf ( A ( E )) ≃ p erf ( X ) and p erf ( A ( E ) op ) ≃ p erf ( X ), we obtain isomorph isms i : HH • ( X ) ≃ HH • ( A ( E )) j : HH • ( X ) ≃ HH • ( A ( E ) op ) . 7 F or X prop er let h , i Shk b e the pairing on HH • ( X ) suc h that h a, b i Shk = h i ( a ) , j ( b ) i Shk for all a, b ∈ HH • ( X ). Note that the RHS of the ab o v e equation has b een defined in the previous su bsection. W e iden tify HH • ( X × Y ) with HH • ( X ) ⊗ HH • ( Y ) via the inv ers e of the Kun neth isomorphism. Recall from [11] that an elemen t Φ of p erf ( X × Y ) give s rise to an inte gral transform Φ from p erf ( X ) to p erf ( Y ). T his is a morphism in Ho(dg-cat), the catego ry of DG-cate gories mo d ulo quasi-equiv alences. The fun ctor f rom D(p erf ( X )) to D(p erf ( Y )) induced by Φ is the functor E 7→ π Y ∗ ( π ∗ X E ⊗ L Φ) . Φ induces a map from HH • (p erf ( X )) to HH • (p erf ( Y )) and hence, a map from HH • ( X ) to HH • ( Y ) wh ic h we shall d en ote by Φ nat ∗ . W e n o w state the follo wing consequence of Theorem 4. Lik e Theorem 4, Theorem 5 implies that Φ nat ∗ dep end s only on the image of Φ in D(p er f ( X × Y )). Theorem 5. F or any Φ in p erf ( X × Y ) , Φ nat ∗ ( x ) = h x, Ch (Φ) i Shk ∈ H H • ( Y ) for al l x ∈ HH • ( X ) . Sk etc h of pro of of Theorem 5. Theorem 5 is a d irect consequence of Theorem 4 an d the work of B. T o en [11]. Giv en tw o DG-categories C and D , [11] constructs a DG-category RHom( C , D ). Let X and Y b e quasi compact separated sc hemes o v er K . Let E and F b e compact generators of D q coh ( X ) and D q coh ( Y ). Recall that in [11] it w as s h o wn that there is an identi fi cation β : p erf ( A ( E ) op ⊗ A ( F )) → RHom(p erf ( A ( E ) ) , p erf ( A ( F ))) Φ M 7→ M ⊗ A Φ in Ho(dg-cat). S imilarly , th ere is an id entificatio n γ : p erf ( X × Y ) → RHom(p erf ( X ) , p erf ( Y )) in Ho(dg-cat). If Φ is in p erf ( X × Y ), γ (Φ) is the in tegral transform Φ from p erf ( X ) to p erf ( Y ) that we describ ed b efore stating Theorem 5. W e abuse n otatio n an d use η to den ote the quasi-equiv alences p erf ( A ( E )) ≃ p erf ( X ),p erf ( A ( E ) op ⊗ A ( F )) ≃ p erf ( X × Y ) and 8 RHom(p erf ( A ( E )) , p erf ( A ( F ))) ≃ RHom(p erf ( X ) , p erf ( Y )) d escrib ed in [11]. It was shown in Section 8 of [11] that the follo wing diagram comm utes in Ho(dg-cat). p erf ( X × Y ) η − 1 − − − − → p erf ( A ( E ) op ⊗ A ( F ))   y γ β   y RHom(p erf ( X ) , p erf ( Y )) η − 1 − − − − → RHom(p erf ( A ( E )) , p erf ( A ( F ))) Theorem 5 is th en a direct consequence of Theorem 4 and the ab o v e com- m utativ e diagram. Remark. I n stead of c ho osing a compact generator E of D q coh ( X ) and using the DG-algebra A ( E ) to define h , i Shk on HH • ( X ), we could make d o w ith an y DG-algebra A such that p erf ( A ) is quasi-equiv alen t to p erf ( X ). 2 The Mu k ai pairing. 2.1 Some recollections. Let X b e a smo oth pr op er sc heme. Let S X denote the shifted line bun dle on X tensoring with whic h yields the Serre dualit y fu nctor on the b ound ed deriv ed catego ry D b ( X ) of coherent O X -mo dules. If f : X → Y is a mor- phism of sc h emes, f ∗ , f ∗ etc shall denote the corresp onding deriv ed functors in this section. Let ∆ : X → X × X denote the diagonal embedd ing. Let ∆ ! denote the left adjoint of ∆ ∗ . Let O ∆ denote ∆ ∗ O X . Recall fr om [1] that there is an isomorphism HH • ( X ) ≃ RHom X × X (∆ ! O X , O ∆ ) . Since ∆ ! O X ≃ ∆ ∗ S − 1 X , tensorin g with π ∗ 2 S X yields an isomorphism D : RHom(∆ ! O X , ∆ ∗ O X ) → RHom(∆ ∗ O X , ∆ ∗ S X ) . Definition. The Muk ai p airing h , i M on HH • ( X ) is th e pairing v ⊗ w tr X × X ( D ( v ) ◦ w ) 9 where tr X × X denotes the Serre dualit y trace on X × X . The same pairing w as constru cted in the DG-algebra setup in [9]. Recall that the Hochsc hild-Kostan t-Rosen b erg map I H K R induces an iso- morphism HH i ( X ) ≃ ⊕ i H j − i ( X, Ω j X ) whic h we shall also denote by I H K R . Le t R X denote the linear f unctional on ⊕ p,q H p ( X, Ω q X ) that coincides with the Serr e dualit y trace on H n ( X, Ω n X ) and v anish es on other direct summands . Let ∗ denote the in vo lution on ⊕ p,q H p ( X, Ω q X ) that acts on th e summand H p ( X, Ω q X ) by ( − 1) p . The fol- lo wing result (implicitly in [7] and explicitly in [8]) computes h , i M at the lev el of Ho d ge cohomology . Theorem 6. F or a, b ∈ H H • ( X ) , h a, b i M = Z X I H K R ( a ) ∗ I H K R ( b ) td ( T X ) . 2.2 In tegral tr ansforms in Ho chsc hild homology . An y Φ ∈ p erf ( X × Y ) yields an inte gral transform Φ : p erf ( X ) → p erf ( Y ) as describ ed in Section 1.2. Note that if Ψ ∈ p erf ( Y × Z ),the image of the k ernel of the in tegral transform Ψ ◦ Φ in D(p erf ( X × Z )) is pr ecisely π X Z ∗ ( π ∗ X Y Φ ⊗ L π ∗ Y Z Ψ). A p riori, there is more than one construction of the corresp ondin g integ ral transform Φ ∗ : HH • ( X ) → HH • ( Y ) su c h that a. (Ψ ◦ Φ ) ∗ = Ψ ∗ ◦ Φ ∗ . b. The follo wing diagram comm utes. D(p erf ( X )) Φ − − − − → D(p erf ( Y ))   y Ch Ch   y HH 0 ( X ) Φ ∗ − − − − → HH 0 ( Y ) F or example, Φ nat ∗ is seen to satisfy th ese p rop erties without muc h difficult y . Another construction of Φ ∗ w as giv en b y A. Caldararu in [1]. Broadly sp eaking, one views HH • ( X ) as an ”ext of functors”, E xt( S − 1 X , id). Th is can b e done rigorously as in [3]. Let Φ ∨ b e a left adjoint of Φ. T h en, 10 if α ∈ Ext( S − 1 X , id), Φ ∗ ( α ) is the follo wing comp osite where the u nlab eled arro ws are adjunctions. S − 1 Y   y Φ ◦ Φ ∨ ◦ S − 1 Y =   y Φ ◦ S − 1 X ◦ S X ◦ Φ ∨ ◦ S − 1 Y id ◦ α ◦ id ◦ id ◦ id − − − − − − − − → Φ ◦ S X ◦ Φ ∨ ◦ S − 1 Y − − − − → id Y Theorem 6 en ables us to giv e y et another construction of Φ ∗ . T his construc- tion of Φ ∗ is motiv ated b y Theorem 4,and pla ys a k ey role in relating th e Muk ai pairing to the natur al pairing constru cted in Section 1. In the rest of th is sectio n , th e id entificatio n of HH • ( X × Y ) with HH • ( X ) ⊗ HH • ( Y ) will b e via the in verse of the Ku n neth isomorph ism. Recall that if Φ ∈ p erf ( X × Y ), the Ch ern c h aracter Ch(Φ) ∈ HH • ( X × Y ) ≃ HH • ( X ) ⊗ HH • ( Y ) ma y b e viewe d as a K -linear map from K to HH • ( X ) ⊗ HH • ( Y ). Let W : HH • ( X ) → HH • ( X ) b e the unique inv olution corresp onding via I H K R to the inv olution ∗ on ⊕ p,q H p ( X, Ω q X ) ment ioned in the pr evious sub- section. Construction. W e define Φ muk ∗ : HH • ( X ) → HH • ( Y ) to b e the comp osite HH • ( X ) id ⊗ Ch(Φ)   y HH • ( X ) ⊗ 2 ⊗ HH • ( Y ) W ⊗ id ⊗ id − − − − − − → HH • ( X ) ⊗ 2 ⊗ HH • ( Y ) h , i M ⊗ id − − − − − − → HH • ( Y ) Prop osition 1. If Φ ∈ p erf ( X × Y ) and Ψ ∈ p erf ( Y × Z ) then (Ψ ◦ Φ) muk ∗ = Ψ muk ∗ ◦ Φ muk ∗ . Pr o of. W e shall denote ⊕ p, q H p ( X, Ω q X ) b y H • ( X ). Recall ( Th eorem 4.5 in [2]) that for any smo oth sc heme Z , I H K R ◦ C h = c h , the righ t h and side b eing the familiar Chern c h aracter map fr om D(p erf ( Z )) to H • ( Z ). Let a ∈ HH • ( X ). Note that HH 0 ( X × Y ) ≃ ⊕ i HH i ( X ) ⊗ HH − i ( Y ). Hence, Ch(Φ) = X i X λ ( i ) ∈ I i α λ ( i ) ⊗ β λ ( i ) 11 for some in dex sets I i and α λ ( i ) ∈ HH i ( X ) and β λ ( i ) ∈ HH − i ( Y ). By Theo- rem 6 and the constr u ction of Φ muk ∗ , I H K R (Φ muk ∗ ( a )) = X i X λ ( i ) ∈ I i ( Z X I H K R ( a )I H K R ( α λ ( i ) )td( T X ))I H K R ( β λ ( i ) ) . (3) No w sup p ose that Ch(Ψ) = X j X µ ( j ) ∈ J j γ µ ( j ) ⊗ δ µ ( j ) for some in dex sets J j and γ µ ( j ) ∈ HH j ( Y ) and δ µ ( j ) ∈ HH − j ( Z ). Then , by (3), I H K R (Ψ muk ∗ ◦ Φ muk ∗ ( a )) = X i,j X λ ( i ) ∈ I i ,µ ( j ) ∈ J j I H K R ( δ µ ( j ) )( Z X I H K R ( a )I H K R ( α λ ( i ) )td( T X ))( Z Y I H K R ( β λ ( i ) )I H K R ( γ µ ( j ) )td( T Y )) . Recall that Ψ ◦ Φ = π X Z ∗ ( π ∗ Y Z Ψ ⊗ π ∗ X Y Φ) The desired pr op osition will follo w from (3) if w e can s ho w that c h(Ψ ◦ Φ) = X i,j X λ ( i ) ∈ I i ,µ ( j ) ∈ J j ( Z Y I H K R ( β λ ( i ) )I H K R ( γ µ ( j ) )td( T Y )) α λ ( i ) ⊗ δ µ ( j ) . (4) Recall that after iden tifying H • ( X × Y ) with H • ( X ) ⊗ H • ( Y ), π Y ∗ gets iden tified with R X ⊗ id. Also, π ∗ Y is identified with the map a 1 ⊗ a from H • ( Y ) to H • ( X × Y ). With this in m in d, (4) can b e r ewritten as, c h(Ψ ◦ Φ ) = π X Z ∗ (c h( π ∗ X Y (Φ)) . c h ( π ∗ Y Z Ψ) .π ∗ Y td( T Y )) . This follo ws directly fr om the Riemann-Ro c h-Hirzebru ch theorem applied to the map π X Z : X × Y × Z → X × Z . Let O ∆ = ∆ ∗ O X b e treate d as the k ern el of an in tegral transform f rom X to X . Then, Prop osition 2. O ∆ muk ∗ = i d . 12 Pr o of. Sin ce O ∆ is the kernel of the identit y , O ∆ ◦ O ∆ = O ∆ . By Prop osi- tion 1, O ∆ muk ∗ is an idemp otent end omorphism of HH • ( X ). T o pro v e that it is the iden tit y , it su ffices to sh o w that it is surjectiv e. F or th is, note that O ∆ nat ∗ = id. By theorem 4, Ch( O ∆ ) = X i X k e i,k ⊗ f i,k where the e i,k form a basis of HH i ( X ) and the f i,k form a basis of HH − i ( X ) suc h that h f i,k , e i,l i Shk = δ k ,l . The δ on the righ t h an d side of the ab o ve equation is the Kr onec k er d elta. Let W b e the inv olution on HH • ( X ) wh ic h we defined earlier b efore con- structing Φ muk ∗ . It follo ws that if x ∈ HH i ( X ), then O ∆ muk ∗ ( x ) = X k h W( x ) , e − i,k i M f − i,k . Recall fr om [1] that the pairing h , i M is non-degenerate. Moreo v er, W is an inv olution on HH • ( X ). Since the e − i,k form a basis of HH − i ( X ), th er e exist elemen ts x k in HH i ( X ) suc h that h W( x l ) , e − i,k i M = δ k l . Clearly , O ∆ muk ∗ ( x k ) = f − i,k . This prov es th at O ∆ muk ∗ is s u rjectiv e, as w as d esired. W e are n ow ready to prov e Theorem 1. Pro of of Theorem 1. Pr o of. Th is follo ws almost immediately from the fact that O ∆ muk ∗ = O ∆ nat ∗ = id : HH • ( X ) → HH • ( X ). S ince O ∆ nat ∗ = id , h f − i,k , e − i,l i Shk = δ k ,l . On the other hand since O ∆ muk ∗ = id by Pr op osition 2, h W( f − i,k ) , e − i,l i M = δ k ,l . 13 It follo ws fr om the K bi-linearit y of the p airings ( a, b ) h a, b i Shk ( a, b ) h W( a ) , b i M that h a, b i Shk = h W ( a ) , b i M . (5) Recall th at ∨ denotes the inv olution on HH • ( X ) corresp ond ing via I H K R to the in vol u tion on H • ( X ) w hic h acts on the direct summand H q ( X, Ω p X ) b y m ultiplication b y ( − 1) p . No w, I H K R ( a ) ∗ ∪ I H K R ( b ) = I H K R ( b ) ∪ I H K R ( a ∨ ) in H • ( X ). Hence, Th eorem 6 may b e rewritten to sa y that h a, b i M = Z X I H K R ( b )I H K R ( a ∨ )td( T X ) . By (5), h a, b i Shk = Z X I H K R ( a )I H K R ( b )td( T X ) = h b ∨ , a i M . This prov es T heorem 1. Recall f rom [1] that the in tegral tr ansform from D(p erf ( Y )) to D(p erf ( X )) due to R H om (Φ , O X × Y ) ⊗ L π ∗ X S X is the right adjoin t of that fr om D(p erf ( X )) to D(p erf ( Y )) du e to Φ. Let Φ ! denote R H om (Φ , O X × Y ) ⊗ L π ∗ X S X . W e also ha ve the follo win g pr op osition, w h ic h shows that Φ muk ∗ is a ”go o d can- didate” for the in tegral trans form on Ho chsc hild homology d efined by Φ. Prop osition 3. If x ∈ HH • ( X ) and y ∈ HH • ( Y ) , then h Φ muk ∗ ( x ) , y i M = h x, Φ ! muk ∗ ( y ) i M . Pr o of. Th e notation used in this pro of is as in the pr o of of Prop osition 1. Assume that after ident ifyin g HH • ( X × Y ) with HH • ( X ) ⊗ HH • ( Y ) (via the in verse of the K unneth map), Ch(Φ) = X i X λ ( i ) ∈ I i α λ ( i ) ⊗ β λ ( i ) for some index sets I i and α λ ( i ) ∈ HH i ( X ) and β λ ( i ) ∈ HH − i ( Y ). Then, b y Theorem 6 and (3), h Φ muk ∗ ( x ) , y i M = 14 X i X λ ( i ) ∈ I i ( Z X I H K R ( x )I H K R ( α λ ( i ) )td( T X ))( Z Y I H K R ( β λ ( i ) ) ∗ I H K R ( y )td( T Y )) . Note th at Ch(Φ ! ) = P i P λ ( i ) ∈ I i ( − 1) i W( β λ ( i ) ) ⊗ [W( α λ ( i ) ) . Ch( S X )]. Th e ( − 1) i comes from the fact that the comp osite HH • ( X ) ⊗ HH • ( Y ) K − − − − → HH • ( X × Y ) K − 1 − − − − → HH • ( Y ) ⊗ HH • ( X ) is the signed map sw app ing factors. It follo ws from Th eorem 6 and (3) that h x, Φ ! muk ∗ ( y ) i M = X i X λ ( i ) ∈ I i ( − 1) i ( Z Y I H K R (I H K R ( y ) β λ ( i ) ) ∗ td( T Y ))( Z X I H K R ( x ) ∗ I H K R ( α λ ( i ) ) ∗ c h( S X )td( T X )) = X i X λ ( i ) ∈ I i ( Z Y I H K R ( β λ ( i ) ) ∗ I H K R ( y )td( T Y ))( Z X I H K R ( x ) ∗ I H K R ( α λ ( i ) ) ∗ c h( S X )td( T X )) . No w, if n is the d imension of X , c h ( S X ) = ( − 1) n c h(Ω n X ). Also, td ( T X )c h(Ω n X ) = td( T X ) ∗ (see [2]). It follo ws that I H K R ( x ) ∗ I H K R ( α λ ( i ) ) ∗ c h( S X )td( T X )) = ( − 1) n (I H K R ( x )I H K R ( α λ ( i ) )td( T X )) ∗ . Hence, Z X I H K R ( x ) ∗ I H K R ( α λ ( i ) ) ∗ c h( S X )td( T X ) = Z X I H K R ( x )I H K R ( α λ ( i ) )td( T X ) This prov es th e desired p rop osition. Note that Prop osition 1 and Prop osition 3 p arallel Theorems 5.3 and 7.3 resp ectiv ely in [1]. Ho wev er, since w e use the Riemann-Ro ch theorem for (prop er) pr o jections to p ro ve Prop osition 1, the constru ction of Φ muk ∗ b y it- self d o es not amount to a s elf-conta ined construction of integ ral transforms in Hochsc hild homology at this stage. Ho w ever, it helps pro ve T heorem 1, whic h in turn leads to Theorem 2, sh o wing th at all three constructions of in tegral transform s in Ho c hsc h ild homology coincide. In particular, it tells us that the integral transform constructed by A. Caldararu [1] coincides with the more ”natural” construction of th e integ r al transform constructed b y D. Sh k lyaro v [10]. Let Φ ∈ p erf ( X × Y ). Denote th e integral transform Φ ∗ : HH • ( X ) → HH • ( Y ) constructed by A. Caldararu [1] and describ ed br iefly earlier in th is section by Φ cal ∗ . 15 Pro of of Theorem 2. Pr o of. Th at Φ muk ∗ = Φ nat ∗ is an immediate consequence of Theorem 1 and Theorem 5. W e th er efore need to sh o w that Φ muk ∗ = Φ cal ∗ . F or this, we w ill follo w D. S hkly arov and imitate th e pro of of Th eorem 4 (Theorem 3.4 in [10]) in [10]). Step 1: Recall that if Φ ∈ p erf ( X × Y ) and Φ ′ ∈ p erf ( X ′ × Y ′ ) , Φ ⊠ Φ ′ ∈ p erf ( X × X ′ × Y × Y ′ ). W e then h a v e in tegral transforms in Hochsc hild homology Φ muk ∗ : HH • ( X ) → HH • ( Y ) , Φ ′ muk ∗ : HH • ( X ′ ) → HH • ( Y ′ ) (Φ ⊠ Φ ′ ) muk ∗ : HH • ( X × X ′ ) → HH • ( Y × Y ′ ) . Iden tify HH • ( X × X ′ ) and HH • ( Y × Y ′ ) with HH • ( X ) ⊗ HH • ( X ′ ) and HH • ( Y ) ⊗ HH • ( Y ′ ) resp ectiv ely via the in v erse of the relev an t Kunneth isomorphisms. It follo ws from the co n struction of Φ muk ∗ that (Φ ⊠ Φ ′ ) muk ∗ = Φ muk ∗ ⊗ Φ ′ muk ∗ . Similarly , w e ha ve integral tran s forms in Ho c hschild h omology Φ cal ∗ : HH • ( X ) → HH • ( Y ) , Φ ′ cal ∗ : HH • ( X ′ ) → HH • ( Y ′ ) (Φ ⊠ Φ ′ ) cal ∗ : HH • ( X × X ′ ) → HH • ( Y × Y ′ ) . It can b e v erified without muc h difficult y (see [16], Lemma 2.1 for instance) that (Φ ⊠ Φ ′ ) cal ∗ = Φ cal ∗ ⊗ Φ ′ cal ∗ . Step 2: Note that Φ ∈ p erf ( X × Y ) ma y also b e though t of as the k ern el of an inte gral transf orm from Sp ec K to X × Y . W e will d enote Φ thought of in th is manner by Φ pt → X × Y . L et ∆ denote O ∆ though t of as the ke rn el of an integ ral transform fr om X × X to S p ec K . Also identi fy HH • ( X ) with HH • ( X ) ⊗ HH • (Sp ec K ) via the map y y ⊗ 1. Then, Φ = ∆ ◦ ( O ∆ ⊠ Φ pt → X × Y ) = ⇒ Φ muk ∗ = ∆ muk ∗ ◦ ( O ∆ ⊠ Φ pt → X × Y ) muk ∗ = ∆ muk ∗ ◦ ( O ∆ muk ∗ ⊗ (Φ pt → X × Y ) muk ∗ (1)) Φ cal ∗ = ∆ cal ∗ ◦ ( O ∆ ⊠ Φ pt → X × Y ) cal ∗ = ∆ cal ∗ ◦ ( O ∆ cal ∗ ⊗ (Φ pt → X × Y ) cal ∗ (1)) No w, by Pr op osition 2, O ∆ cal ∗ = O ∆ muk ∗ = id . 16 Also, (Φ pt → X × Y ) cal ∗ (1) = Ch(Φ) by Definition 6.1 in [1] and Theorem 4.5 in [2]. (Φ pt → X × Y ) muk ∗ (1) = Ch(Φ) b y the construction of (Φ pt → X × Y ) muk ∗ . W e therefore , need to sho w that ∆ muk ∗ = ∆ cal ∗ : HH • ( X × X ) → HH • (Sp ec K ) = K . With th e ab o ve iden tification of HH • (Sp ec K ) with K , for an y x ∈ HH • (Sp ec K ), x = h x, 1 i M . Let ∆ ! denote R H om (∆ , O X × X ) ⊗ L S X × X . If α ∈ HH • ( X × X ), h ∆ muk ∗ ( α ) , 1 i M = h α, ∆ ! muk ∗ (1) i M b y Prop osition 3. By Theorem 7.3 in [1], h ∆ cal ∗ ( α ) , 1 i M = h α, ∆ ! cal ∗ (1) i M . No w, ∆ ! cal ∗ (1) = C h(∆ ! ) by Definition 6.1 in [1] and Theorem 4.5 in [2]. ∆ ! muk ∗ (1) = Ch(∆ ! ) by the construction of ∆ ! muk ∗ . This yields the desired theorem. 2.3 When X is C alabi-Y au. In s u c h a situ ation, D b ( X ) can b e b e thought of as th e category of op en states of the B-Mod el on X (see [3]). The corresp ond ing algebra of closed states is th e Ho chsc hild cohomology HH • (p erf ( X )) ≃ HH • ( X ). As X is Calabi-Y au, there is an id en tification HH • ( X ) ≃ HH • ( X ) . The Muk ai pairing constructed by A. Caldararu in [1] on HH • ( X ) then giv es a pairing on HH • ( X ). Moreo v er, for an y E ∈ D b ( X ), there are natural maps ι E : Hom D b ( X ) ( E , E ) → HH • ( X ) ι E : HH • ( X ) → Hom D b ( X ) ( E , E ) as constructed in [3]. Th e Cardy condition v erifi es that th is data giv es a top ological quantum field theory . O f course, the Muk ai p airing in this case is the pairing obtained b y the action of the class of a gen us 0 Riemann- surface with tw o in coming closed b oundaries and no outgoing b oundary in H 0 ( M 0 (2 , 0)) on HH • ( X ), the action coming from the fact that HH • ( X ) 17 with Muk ai pairing is a ”goo d ” algebra of closed states as ve r ified by the Cardy condition. On the other hand, [4] giv es the category of op en states of th e B-Mod el on X as an A ∞ enric hment of D b ( X ). The closed TCFT one asso ciates with this category has homology HH • ( X ) ≃ HH • ( X ) . This is also equipp ed with a pairing coming out of the action of the class of a gen us 0 Riemann -su rface with t wo incoming closed b ound aries and no outgoing b ound ary in H 0 ( M 0 (2 , 0)) on the homology of the closed TCFT one constructs in [4] from the B-Model. Whether these pairings coincide is ho wev er, not clear cu rrent ly . Theorem 1 is similar Conjecture 6.2 in [10] for C alabi-Y au algebras A such that p erf ( A ) is quasi-equiv alen t to p erf ( X ) for some q u asi-compact sepa- rated smo oth sc heme X . 2.4 Pro of of Theorem 3. The sheafification of the Dennis trace map. Let us br iefly recall ho w the sheafification of the Denn is trace map is constru cted. The material we are recalling is f rom [12],[13],[14 ] and [15]. Let X b e a sm o oth qu asicompact separated s cheme. As in Section 1.2, choose a compact generator E of D q coh ( X ) and a DG-algebra A ( E ) s uc h that p erf ( A ( E )) is qu asiequiv alen t to p erf ( X ). Let Z 0 (p erf( A ( E ))) b e the exact category whose ob jects are those of p erf ( A ( E )) suc h that Hom Z 0 (perf( A ( E ) )) ( M , N ) = Z 0 (Hom perf ( A ( E )) ( M , N )) . As p oin ted ou t b y B. Keller in [14], usin g the W aldhausen stru ctur e of Z 0 (p erf( A ( E ))), w e can constr u ct a Dennis trace map Dtr : K i ( X ) ≃ K i (Z 0 (p erf( A ( E )))) → HH i, McC (Z 0 (p erf( A ( E )))) ∀ i ≥ 0 . Here, HH i, McC is the Ho c hschild h omology co n structed b y R. McCarth y in [15]. As K eller further p oints out in [14], there is a natural transformation HH i, McC (Z 0 (p erf( A ( E )))) → HH i (Z 0 (p erf( A ( E )))) . 18 F ur ther, we also ha ve a natural tran s formation HH i (Z 0 (p erf( A ( E ) ))) → HH i (p erf ( A ( E ))) . The ob vious comp ositions then giv e u s a map Ch i : K i ( X ) ≃ K i (Z 0 (p erf( A ( E ) ))) → HH i (p erf ( A ( E ))) ≃ HH i ( X ) . Let Y b e a smo oth quasicompact separated sc heme. Let F and A ( F ) b e as in Section 1.2. Let Ψ ∈ p erf ( A ( F ) op ⊗ A ( E )). The follo wing prop osition, analogous to Th eorem 7.1 of [1], s ays that the sheafification of the Dennis trace map is ”functorial”. Prop osition 4. The fol lowing diagr am c ommutes. K i ( Z 0 ( p erf ( A ( E )))) Ψ ∗ − − − − → K i ( Z 0 ( p erf ( A ( F ))))   y Ch i Ch i   y HH i ( p erf ( A ( E ) )) Ψ nat ∗ − − − − → HH i ( p erf ( A ( F ) )) Pr o of. Th is prop osition will f ollo w easily once we ve r if y that Ψ : Z 0 (p erf( A ( E ))) → Z 0 (p erf( A ( F ))) pr eserv es cofibrations and w eak equiv alences. By [12], the weak equiv alences in Z 0 (p erf( A )) for an y DG- algebra A are qu asiisomorphisms. Th e cofibr ations in Z 0 (p erf( A )) are mor- phisms of A -mo d ules that ad m it retractions as m orp hisms of graded A - mo dules. Th at Ψ preserves cofibr ations follo ws w ithout difficult y from the fact that Ψ : p erf ( A ( E )) → p erf ( A ( F )) is a DG-functor. That Ψ pr eserv es w eak equiv alences follo ws from the fact that p erfect mo dules are homotopi- cally pro jectiv e (see Pr op osition 2.5 of [10]). Pro of of T heorem 3. W e w arn the r eader that in the pr o of that follo ws, X and Y denote pr op er s m o oth quasicompact separated sc hemes. Pr o of. Step 1: Let Φ ∈ p erf ( X × Y ).The first step is to note that eve n though Z is n ot necessarily prop er, the k ern el Φ ⊠ O ∆ Z ∈ p erf ( X × Z × Y × Z ) induces an inte gral transform from p erf ( X × Z ) to p erf ( Y × Z ). Th is fol- lo ws from the fact that if E and F are compact generators of D coh ( X ) and D coh ( Z ) resp ectiv ely , the compact generator E ⊠ F := π ∗ X E ⊗ π ∗ Z F of D coh ( X × Z ) is mapp ed b y the in tegral transform with ke r n el Φ ⊠ O ∆ Z to the p erf ect complex π Y ∗ (Φ ⊗ L π ∗ X E ) ⊠ F . 19 Also, after identifying HH • ( X × Z ) and HH • ( Y × Z ) with HH • ( X ) ⊗ HH • ( Z ) and HH • ( Y ) ⊗ HH • ( Z ) resp ectiv ely via the inv erse of the relev ant K unneth isomorphisms, (Φ ⊠ O ∆ Z ) nat ∗ = Φ nat ∗ ⊗ id : HH • ( X ) ⊗ HH • ( Z ) → HH • ( Y ) ⊗ HH • ( Z ) . (6) This follo ws from the facts that O ∆ Z nat ∗ = id and f rom Prop osition 2.11 of [10]. Step 2: By the Prop osition 4, the follo win g diagram commutes. K i (p erf ( X × Z )) (Φ ⊠ O ∆ Z ) ∗ − − − − − − − → K i (p erf ( Y × Z ))   y Ch i Ch i   y HH i ( X × Z ) (Φ ⊠ O ∆ Z ) nat ∗ − − − − − − − − → HH i ( Y × Z ) (7) After ident ifyin g HH • ( X × Z ) and HH • ( Y × Z ) with HH • ( X ) ⊗ HH • ( Z ) and HH • ( Y ) ⊗ HH • ( Z ) resp ectiv ely via the inv erse of the relev ant K unneth isomorphisms,we hav e th e follo wing commutativ e diagram b y (7) and (6). K i (p erf ( X × Z )) (Φ ⊠ O ∆ Z ) ∗ − − − − − − − → K i (p erf ( Y × Z ))   y Ch i Ch i   y ⊕ p + q = i HH p ( X ) ⊗ HH q ( Z ) Φ nat ∗ ⊗ id − − − − − → ⊕ p + q = i HH p ( Y ) ⊗ HH q ( Z ) (8) No w, it follo w s from Th eorem 1 and Theorem 3 that Φ muk ∗ = Φ nat ∗ . Hence, b y (8) and Prop osition 3, ( h , i M ⊗ id HH • ( Z ) )( f ∗ y ⊗ Ch i ( α )) = ( h , i M ⊗ id HH • ( Z ) )( y ⊗ (id × f ) ∗ Ch i ( α )) (9) for any α ∈ K i ( Z × X ), y ∈ HH • ( Y ). By Th eorem 4, (9) can b e rewritten to sa y that Z X I H K R ( f ∗ ( y )) ∗ c h i ( α )td( T X ) = Z Y I H K R ( y ) ∗ c h i (( f × id) ∗ α )td( T Y ) as elemen ts of H • ( Z ). Th e d esired theorem no w follo ws from the facts that f ∗ comm utes w ith I H K R (see T heorem 7 of [7]) and comm u tes with the in vo lu tion ∗ . 20 References [1] Caldararu , A., The Muk ai pairing, I: the Ho c hschild structur e. Arxiv preprint math.A G/030807 9 . [2] Caldararu , A., The Muk ai pairing, I I: the Ho chsc hild-Kostan t- Rosen b erg isomorphism. Adv ances in Mathematics 194(2 005), No. 1, 34-66 . [3] Caldararu , A., Willerton, S., The Muk ai pairing, I: a categ orical ap- proac h. Arxiv pr eprint arXiv:0707 .2052 . [4] Costello, K., T op ological conformal field theories and Calabi-Y au cate- gories. 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