A proof of a conjecture of Shklyarov
aa r X i v : . [ m a t h . K T ] S e p A PROOF OF A CONJECTURE OF SHKLYAROV
MICHAEL K. BROWN AND MARK E. WALKER
Abstract.
We prove a conjecture of Shklyarov concerning the relationship between K. Saito’shigher residue pairing and a certain pairing on the periodic cyclic homology of matrix factorizationcategories. Along the way, we give new proofs of a result of Shklyarov ([Shk16, Corollary 2]) andPolishchuk-Vaintrob’s Hirzebruch-Riemann-Roch formula for matrix factorizations ([PV12, Theorem4.1.4(i)]).
Contents
1. Introduction 11.1. Background on Shklyarov’s conjecture 21.2. Outline of the proof of Conjecture 1.4 32. Generalities on Hochschild homology for curved dg-categories 42.1. Hochschild homology of curved dg-categories 42.2. The K¨unneth map for Hochschild homology of cdga’s 52.3. Functoriality of HH II using the shuffle product 62.4. The K¨unneth map for Hochschild homology of cdg-categories 72.5. Naturality of the K¨unneth map 83. Hochschild homology of matrix factorization categories 83.1. Matrix factorizations 93.2. The HKR map 103.3. Relationship between the HKR map and the map I f (0) 133.4. Compatibility of the HKR map with taking duals 143.5. Multiplicativity of the HKR map 164. Proof of Shklyarov’s conjecture 184.1. Computing HH ( mf m ( Q, Introduction
Let Q = C [ x , . . . , x n ], and let m denote the maximal ideal ( x , . . . , x n ) ⊆ Q . Fix f ∈ m , andassume the only singular point of the associated morphism f : Spec( Q ) → A C is m . Let mf ( Q, f ) MB gratefully acknowledges support from the National Science Foundation (award DMS-1502553), and MW grate-fully acknowledges support from the Simons Foundation (grant denote the differential Z / f ; see Section 3.1 for thedefinition of mf ( Q, f ). Shklyarov proves in [Shk16, Theorem 1] that a certain pairing on the periodiccyclic homology of mf ( Q, f ) coincides, up to a constant factor c f (which possibly depends on f ),with K. Saito’s higher residue pairing, via the Hochschild-Kostant-Rosenberg (HKR) isomorphism.Shklyarov conjectures in [Shk16, Conjecture 3] that c f = ( − n ( n +1)2 . The main goal of this paper isto prove this conjecture.We begin by discussing Shklyarov’s conjecture in more detail.1.1. Background on Shklyarov’s conjecture.
Let HN ( mf ( Q, f )) denote the negative cyclic com-plex of mf ( Q, f ), and let HN ∗ ( mf ( Q, f )) denote its homology. See, for instance, [BW19a, Section 3]for the definition of the negative cyclic complex of a dg-category. The dg-category mf ( Q, f ) is proper ,i.e. each cohomology group of the ( Z / C -vector space. As with any such dg-category, there is a canonical pairing of Z / C -vector spaces K mf : HN ∗ ( mf ( Q, f )) × HN ∗ ( mf ( Q, f )) → C [[ u ]] , where u is an even degree variable. The pairing K mf is defined exactly as in [Shk16, page 184], butwith periodic cyclic homology HP ∗ replaced with HN ∗ and C (( u )) replaced with C [[ u ]]. We note that K mf is C [[ u ]] -sesquilinear ; that is, for any α, β ∈ HN ∗ ( mf ( Q, f )) and g ∈ C [[ u ]], we have K mf ( g ( u ) · α, β ) = g ( u ) K mf ( α, β ) = K mf ( α, g ( − u ) · β ) . It follows from work of Segal [Seg13, Corollary 3.4] and Polishchuk-Positselski [PP12, Section 4.8]that there is a quasi-isomorphism(1.1) I f : HN ( mf ( Q, f )) ≃ −→ (Ω • Q/ C [[ u ]] , ud − df ) , which generalizes the classical Hochschild-Kostant-Rosenberg (HKR) theorem. The target of I f iscalled the twisted de Rham complex , and it is a Z / mQ/ C to have(homological) degree m and u to have degree −
2. (Since the twisted de Rham complex is Z / m has degree − m and u has degree 2. Note that the map ud has degree − df has degree 1, but since this is regarded as a Z / I f : HN n ( mf ( Q, f )) ∼ = −→ H (0) f , where H (0) f := H n (Ω • Q/ C [[ u ]] , ud − df ) = Ω nQ/ C [[ u ]]( ud − df ) · Ω n − Q/ C [[ u ]] . In [Sai83], K. Saito equips the C [[ u ]]-module H (0) f with a pairing K f : H (0) f × H (0) f → C [[ u ]]known as the higher residue pairing . Shklyarov has proven the following result concerning the rela-tionship between the canonical pairing and the higher residue pairing under the HKR isomorphism: Theorem 1.2 ([Shk16], Theorem 1) . For each polynomial f as above, there is a constant c f ∈ C (possibly depending on f ) such that the diagram (1.3) HN n ( mf ( Q, f )) × I f × I f ∼ = / / c f · u n · K mf & & ◆◆◆◆◆◆◆◆◆◆◆◆◆ (cid:16) H (0) f (cid:17) × K f z z ✈✈✈✈✈✈✈✈✈ C [[ u ]] commutes. Moreover, Shklyarov makes the following prediction:
PROOF OF A CONJECTURE OF SHKLYAROV 3
Conjecture 1.4 ([Shk16], Conjecture 3) . For any f , c f = ( − n ( n +1) / . Outline of the proof of Conjecture 1.4.
The constant c f can be determined from a related,but simpler, pairing on HH ∗ ( mf ( Q, f )), the Hochschild homology of mf ( Q, f ). We recall that, forany dg-category C , there is a short exact sequence(1.5) 0 → HN ( C ) · u −→ HN ( C ) → HH ( C ) → HN ∗ ( mf ( Q, f )) and HH ∗ ( mf ( Q, f )) are con-centrated in degree n (mod 2). The long exact sequence in homology induced by (1.5) thereforeinduces an isomorphism(1.6) HN ∗ ( mf ( Q, f )) /u · HN ∗ ( mf ( Q, f )) ∼ = −→ HH ∗ ( mf ( Q, f )) . The pairing K mf determines a well-defined pairing modulo u , which we write, via (1.6), as η mf : HH ∗ ( mf ( Q, f )) × HH ∗ ( mf ( Q, f )) → C . The isomorphism I f is C [[ u ]]-linear and, upon setting u = 0, it induces an isomorphism I f (0) : HH n ( mf ( Q, f )) ∼ = −→ H n (Ω • Q/ C , − df ) . The higher residue pairing K f has the form K f ω + X j ≥ ω j u j , ω ′ + X j ≥ ω ′ j u j = h ω, ω ′ i res u n + higher order terms , where h ω, ω ′ i res is the classical residue pairing determined by the partial derivatives of f . It is definedalgebraically as h g · dx · · · dx n , h · dx · · · dx n i res = res " gh · dx · · · dx n∂f∂x , . . . , ∂f∂x n , where the right-hand side is Grothendieck’s residue symbol.Thus, upon dividing the maps in diagram (1.3) by u n and setting u = 0, we obtain the commutativetriangle(1.7) HH n ( mf ( Q, f )) × HH n ( mf ( Q, f )) I f (0) × I f (0) ∼ = / / c f η mf ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ Ω nQ/ C df ∧ Ω n − Q/ C × Ω nQ/ C df ∧ Ω n − Q/ C h− , −i res x x qqqqqqqqqq C . Since I f (0) is an isomorphism, and the residue pairing is non-zero, the value of c f is uniquely deter-mined by the commutativity of (1.7). Note that a result closely related to the commutativity of (1.7)was also proven by Polischuk-Vaintrob ([PV12, Corollary 4.1.3]).In this paper, we re-establish the commutativity of diagram (1.7) using techniques that differ fromthose used by Shklyarov. Our method results in an explicit calculation of c f : Theorem 1.8.
Shklyarov’s Conjecture holds: that is, for any f as above, c f = ( − n ( n +1) / . In fact, we prove the commutativity of diagram (1.7), and Theorem 1.8, in the case where Q is anessentially smooth algebra over a characteristic 0 field k , m is a k -rational maximal ideal, and f ∈ m is such that m is the only singularity of the morphism f : Spec( Q ) → A k . The special case k = C , Q = C [ x , . . . , x n ], and m = ( x , . . . , x n ) yields Shklyarov’s conjecture. MICHAEL K. BROWN AND MARK E. WALKER
The general outline of our proof is summarized by the diagram(1.9) HH n ( mf ( Q, f )) × HH n ( mf ( Q, f )) id × Ψ (cid:15) (cid:15) I f (0) × I f (0) ∼ = / / H n (Ω • Q/k , − df ) × H n (Ω • Q/k , − df ) id × ( − n (cid:15) (cid:15) HH n ( mf ( Q, f )) × HH n ( mf ( Q, − f )) I f (0) × I − f (0) ∼ = / / ⋆ (cid:15) (cid:15) H n (Ω • Q/k , − df ) × H n (Ω • Q/k , df ) ∧ (cid:15) (cid:15) HH n ( mf m ( Q m , ( − n ( n +1) / trace * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ ε / / H n R Γ m (Ω • Q m /k ) res u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ k. The map Ψ is induced by taking Q -linear duals; ⋆ is induced by a K¨unneth map followed by the tensorproduct of matrix factorizations; trace is defined in Section 4; res is Grothendieck’s residue map; ∧ is induced by exterior multiplication of differential forms, using that the complexes (Ω • Q/k , ± f ) aresupported on { m } ; and the map ε is an HKR-type map. We prove:(1) the diagram commutes (Lemma 3.11, Lemma 3.14, Corollary 3.26, and Theorem 4.36),(2) the composition along the left side of this diagram is the canonical pairing η mf (Lemma 4.23),and(3) the composition along the right side of this diagram is the residue pairing h− , −i res (Proposi-tion 4.34).Finally, in Section 5, we use some of our results to give a new proof Polishchuk-Vaintrob’s Hirzebruch-Riemann-Roch theorem for matrix factorizations ([PV12, Theorem 4.1.4(i)]). Acknowledgements.
We thank Srikanth Iyengar for helpful remarks concerning the proof of Lemma4.2. 2.
Generalities on Hochschild homology for curved dg-categories
We review some background on Hochschild homology of curved dg-categories and establish somenew results concerning pairings of such. Throughout this section, k is a field, and “graded” meansΓ-graded for Γ ∈ { Z , Z / } . We will eventually focus on the case Γ = Z / Hochschild homology of curved dg-categories.
We refer the reader to [BW19a, Section 2.1]for the definition of a curved differential Γ-graded category (henceforth referred to as a cdg-category).Recall that a cdg-category with just one object is a curved differential Γ-graded algebra (cdga).For a cdg-category C whose objects form a set, define HH ( C ) ♮ to be the Γ-graded k -vector spacegiven by the direct sum totalization of the Z − Γ-bicomplex which, in Z -degree n , is the Γ-graded k -vector space M X ,...,X n ∈C Hom( X , X ) ⊗ k Σ Hom( X , X ) ⊗ k · · · ⊗ k Σ Hom( X n , X n − ) ⊗ k Σ Hom( X , X n ) . When C is essentially small , so that the isomorphism classes of objects in the Γ-graded categoryunderlying C form a set (see [PP12, Section 2.6]), we define HH ( C ) ♮ by first replacing C with a fullsubcategory consisting of a single object from each isomorphism class. From now on, we will tacitlyassume all of our cdg-categories are essentially small. Given α i ∈ Hom( X i +1 , X i ) for i = 0 , . . . , n (with X n +1 = X ), we write α [ α | · · · | α n ] for the element α ⊗ sα ⊗ · · · ⊗ sα n of HH ( C ) ♮ .The Hochschild complex of C , denoted HH ( C ), is the above graded k -vector space equipped withthe differential b := b + b + b , where b , b , b are defined as in [BW19a, Section 3.1]. Roughly, b is the classical Hochschild differential induced by the composition law in C , b is induced by thedifferentials of C , and b is induced by the curvature elements of C . When C has just one object with PROOF OF A CONJECTURE OF SHKLYAROV 5 trivial curvature, then C is a dga, and the maps b and b are the classical ones (and b = 0 in thiscase).We will also need “Hochschild homology of the second kind”, as introduced by Polishchuk-Positselskiin [PP12]. Define HH II ( C ) ♮ to be the Γ-graded k -vector space given as the direct product totalizationof the above bicomplex. Equivalently, HH II ( C ) ♮ is the completion of HH ( C ) ♮ under the topologydetermined by the evident filtration. Since b is continuous for this topology, it induces a differentialon HH II ( C ) ♮ , which we also write as b , and we write HH II ( C ) for the resulting chain complex.2.2. The K¨unneth map for Hochschild homology of cdga’s.
For a cdga A = ( A, d A , h A ), wehave HH ( A ) ♮ = A ⊗ k T (Σ A ) , where, for any graded k -vector space V , T ( V ) = L n ≥ V ⊗ n . Recall that T ( V ) is a commuative k -algebra under the shuffle product :( v ⊗ · · · ⊗ v p ) • ( v p +1 ⊗ · · · ⊗ v p + q ) = X σ ± v σ (1) ⊗ · · · ⊗ v σ ( p + q ) , where σ ranges over all ( p, q )-shuffles. The sign is given by the usual rule for permuting homogeneouselements in a product.Since A is also an algebra, HH ( A ) ♮ has an algebra structure, whose multiplication rule will bewritten as − ⋆ − : HH ( A ) ♮ ⊗ k HH ( A ) ♮ → HH ( A ) ♮ . It is given explicitly as x [ a | · · · | a p ] ⋆ y [ a p +1 | · · · | a p + q ] = X σ ± xy [ a σ (1 ) | · · · | a σ ( p + q ) ] . Note that the canonical inclusion T (Σ A ) ֒ → HH ( A ) ♮ lands in the center of HH ( A ) ♮ for the ⋆ multi-plication.If B = ( B, d B , h B ) is another cdga, the tensor product of A and B is defined to be A ⊗ k B = ( A ⊗ k B, d A ⊗ ⊗ d B , h A ⊗ ⊗ h B ) . We define the
K¨unneth map − ˜ ⋆ − : HH ( A ) ♮ ⊗ k HH ( B ) ♮ → HH ( A ⊗ k B ) ♮ to be the composition of the tensor product of the maps induced by the canonical inclusions HH ( A ) ♮ ֒ → HH ( A ⊗ B ) ♮ and HH ( B ) ♮ ֒ → HH ( A ⊗ B ) ♮ with the ⋆ product for A ⊗ k B . The ⋆ product on HH ( A ) ♮ can be recovered from the K¨unneth map by setting B = A : the ⋆ product coincides with the compo-sition HH ( A ) ♮ ⊗ k HH ( A ) ♮ ˜ ⋆ −→ HH ( A ⊗ k A ) ♮ µ ∗ −→ HH ( A ) ♮ , where µ ∗ : ( A ⊗ k A ) ⊗ k T (Σ( A ⊗ A )) → A ⊗ k T (Σ A )is induced by the multiplication map µ : A ⊗ A → A .It is important to note that, for an algebra A , the ⋆ product does not, in general, make HH ( A )into a dga, since b is not a derivation for the ⋆ multiplication unless A is commutative. But, b is aderivation for the K¨unneth map; see Lemma 2.6.The ⋆ product does behave well with respect to b . In detail, recall that the tensor algebra functor T ( − ) sends Γ-graded complexes of k -vector spaces to differential Γ-graded algebras under the shuffleproduct. Let d T denote the differential on T (Σ A ) induced from the differential Σ d on Σ A . Then( T (Σ A ) , • , d T ) is a dga, where • is the shuffle product. By examining the explicit formula for b , wesee that b = d A ⊗ ⊗ d T . In other words, ( HH ( A ) ♮ , ⋆, b ) is a dga, and it is given as a tensor product of dga’s:( HH ( A ) ♮ , ⋆, b ) = ( A, · , d A ) ⊗ ( T (Σ A ) , • , d T ) , MICHAEL K. BROWN AND MARK E. WALKER where · is the multiplication rule for A .If z is an element of A of even degree, then we have1[ z ] ⋆ a [ a | · · · | a n ] = X i ( − | a | + | a | + ··· + | a i |− i a [ a | · · · | a i | z | a i +1 | · · · | a n ] . In particular, the component b of the differential in HH ( A ) is given by(2.1) b = 1[ h ] ⋆ − . Since 1[ h ] is a central element of ( A ⊗ T (Σ A ) , ⋆ ) of odd degree, it follows that(2.2) b ( − ) ⋆ − = b ( − ⋆ − ) = ± − ⋆b ( − ) . The ⋆ product extends to HH II since it is continuous for the topology on HH whose completiongives HH II .2.3. Functoriality of HH II using the shuffle product. We recall that a morphism A = ( A, d A , h A ) →B = ( B, d B , h B ) of cdga’s is given by a pair φ = ( ρ, β ), with ρ : A → B a morphism of Γ-graded k -algebras and β ∈ B a degree one element, such that • ρ ( d ( a )) − d ′ ( ρ ( a )) = [ β, ρ ( a )] for all a ∈ A , and • ρ ( h ) = h ′ + d ′ ( β ) + β .Such a morphism is called strict if β = 0.A strict morphism φ induces maps φ ∗ : HH ( A ) → HH ( B ) and φ ∗ : HH II ( A ) → HH II ( B )given by φ ∗ ( a [ a | . . . | a n ]) = ρ ( a )[ ρ ( a ) | · · · | ρ ( a n )] . A non-strict morphism φ does not, in general, induce a map on Hochschild homology, but it doesinduce a map φ ∗ : HH II ( A ) → HH II ( B )given by sending a [ a | . . . | a n ] to(2.3) X i ,...,i n ≥ ( − i + ··· + i n ρ ( a )[ β | · · · | β | {z } i copies | ρ ( a ) | β | · · · | β | {z } i copies | ρ ( a ) | · · · | ρ ( a n ) | β | · · · | β | {z } i n copies ] . We next show how φ ∗ may also be defined using the ⋆ product. Suppose b ∈ B is a degree 1element, and let exp(1[ b ]) denote the degree 0, central element of the algebra ( HH II ( B ) ♮ , ⋆ ) given byevaluating the power series for the exponential function at 1[ b ]:exp(1[ b ]) = 1 + 1[ b ] + 12! (1[ b ] ⋆ b ]) + 13! (1[ b ] ⋆ b ] ⋆ b ]) + · · · = 1 + 1[ b ] + 1[ b | b ] + 1[ b | b | b ] + · · · . The signs are correct, since s ( b ) ∈ T (Σ B ) has even degree. We have:exp(1[ b ]) ⋆ ( b [ b | . . . | b n ]) = ( b [ b | . . . | b n ]) ⋆ exp(1[ b ])= X i ,...,i n ≥ b [ b | · · · | b | {z } i copies | b | b | · · · | b | {z } i copies | b | · · · | b n | b | · · · | b | {z } i n copies ] . By comparing formulas, we see that(2.4) φ ∗ = exp(1[ − β ]) ⋆ ρ ∗ . That is, φ ∗ ( a [ a | . . . | a n ]) = exp(1[ − β ]) ⋆ ρ ( a )[ ρ ( a ) | · · · | ρ ( a n )] = ρ ( a )[ ρ ( a ) | · · · | ρ ( a n )] ⋆ exp(1[ − β ]) . PROOF OF A CONJECTURE OF SHKLYAROV 7
The K¨unneth map for Hochschild homology of cdg-categories.
For a pair of cdg-categories C and D , we write C ⊗ k D for the cdg-category whose objects are ordered pairs ( C, D ) with C ∈ C and D ∈ D and such thatHom(( C, D ) , ( C ′ , D ′ )) = Hom C ( C, C ′ ) ⊗ k Hom D ( D, D ′ ) , with differentials given in the standard way for a tensor product. The composition rules are theevident ones, and the curvature elements are defined by h ( C,D ) = h C ⊗ id D + id C ⊗ h D . Note that, if A = ( A, d A , h A ) and B = ( B, d B , h B ) are cdga’s, then this construction specializes tothe construction given above: A ⊗ k B = ( A ⊗ k B, d A ⊗ id B + id A ⊗ d B , h A ⊗ id B + id A ⊗ h B ) . We define the
K¨unneth map for the cdg-categories C and D to be the map − ˜ ⋆ − : HH ( C ) ♮ ⊗ k HH ( D ) ♮ → HH ( C ⊗ k D ) ♮ given by c [ c | · · · | c m ]˜ ⋆d [ d | · · · | d n ] = X σ ± c ⊗ d [ e σ (1) | · · · | e σ ( m + n ) ] , where σ ranges over all ( m, n )-shuffles, and e i := ( c i ⊗ id , if 1 ≤ i ≤ m , andid ⊗ d i − m , if m + 1 ≤ i ≤ m + n .This map extends to HH II ( − ) ♮ : − ˜ ⋆ − : HH II ( C ) ♮ ⊗ k HH II ( D ) ♮ → HH II ( C ⊗ D ) ♮ . Remark . There does not seem to be an analogue of the ⋆ product for a general cdg-category. Theissue is that, in general, there is no “diagonal map” C ⊗ k C → C . Lemma 2.6.
For any two cdg-categories C and D , the diagram HH ( C ) ♮ ⊗ k HH ( D ) ♮b i ⊗ id + id ⊗ b i (cid:15) (cid:15) − ˜ ⋆ − / / HH ( C ⊗ k D ) ♮b i (cid:15) (cid:15) HH II ( C ) ♮ ⊗ k HH ( D ) ♮ − ˜ ⋆ − / / HH ( C ⊗ k D ) ♮ commutes for i = 0 , , and , and similarly for HH II ( − ) ♮ . In particular, − ˜ ⋆ − : HH ( C ) ⊗ k HH ( D ) → HH ( C ⊗ k D ) and − ˜ ⋆ − : HH II ( C ) ⊗ k HH II ( D ) → HH II ( C ⊗ k D ) are chain maps.Proof. This follows from the definitions by a routine check. (cid:3)
MICHAEL K. BROWN AND MARK E. WALKER
Naturality of the K¨unneth map.
We recall that a morphism
A → B of cdg-categories is apair φ = ( F, β ), where F : A → B is a morphism of categories enriched in Γ-graded k -vector spaces,and β is an assignment to each object X of A a degree 1 element β X ∈ End B ( F ( X )). The pair ( F, β )is required to satisfy: • For all
X, Y ∈ Ob( A ) and f ∈ Hom A ( X, Y ), F ( δ ( f )) = δ ( F ( f )) + β Y ◦ F ( f ) − ( − | f | F ( f ) ◦ β X , where δ is the differential on Hom A ( X, Y ); and • for all X ∈ Ob( A ), F ( h X ) = h F ( X ) + δ ( β X ) + β X .φ is called strict if β X = 0 for all X . Lemma 2.7.
Suppose A , A ′ , B , B ′ are curved differential Γ -graded categories, and φ = ( F, β ) :
A → B , φ ′ = ( F ′ , β ′ ) : A ′ → B ′ are morphisms of such. Then (1) φ ⊗ φ ′ := ( F ⊗ F ′ , β ⊗ ⊗ β ′ ) is a morphism from A ⊗ k A ′ to B ⊗ k B ′ , and, if φ and φ ′ are strict morphisms, then so is φ ⊗ φ ′ ; (2) the diagram HH II ( A ) ⊗ k HH II ( A ′ ) ˜ ⋆ (cid:15) (cid:15) ( φ ) ∗ ⊗ ( φ ′ ) ∗ / / HH II ( B ) ⊗ k HH II ( B ′ ) ˜ ⋆ (cid:15) (cid:15) HH II ( A ⊗ k A ′ ) ( φ ⊗ φ ′ ) ∗ / / HH II ( B ⊗ k B ′ ) commutes; and (3) if φ and φ ′ are strict morphisms, the corresponding diagram involving ordinary Hochschildhomology commutes.Proof. The proof of (1) is a routine check, and (3) is an immediate consequence of (2). For (2), tosimplify the notation, we assume the cdg-categories involved are cdg-algebras; the proof of the generalclaim is notationally more complicated but essentially the same. Write φ = ( ρ, β ), φ ′ = ( ρ ′ , β ′ ), sothat, by (2.4), φ ∗ = exp(1[ − β ]) ⋆ ρ ∗ and φ ′∗ = exp(1[ − β ′ ]) ⋆ ρ ′∗ . Let ι : HH II ( A ) ֒ → HH II ( A ⊗ k A ′ ) and ι ′ : HH II ( A ′ ) ֒ → HH II ( A ⊗ k A ′ ) be the canonicalinclusions. We haveexp(1[ − β ])˜ ⋆ exp(1[ − β ′ ]) = exp( ι (1[ − β ])) ⋆ exp( ι ′ (1[ − β ′ ])) = exp(1[ − β ⊗ − ⊗ β ′ ]);the second equation holds since ι (1[ − β ]) and ι ′ (1[ − β ′ ]) commute. Therefore, for elements α ∈ HH II ( A ) and α ′ ∈ HH II ( A ′ ), using also the associativity of ⋆ , we get( φ ) ∗ ( α )˜ ⋆ ( φ ′ ) ∗ ( α ′ ) = (exp(1[ − β ]) ⋆ ρ ( α ))˜ ⋆ (exp(1[ − β ′ ]) ⋆ ρ ′ ( α ′ ))= (exp(1[ − β ])˜ ⋆ exp(1[ − β ′ ])) ⋆ ( ρ ( α )˜ ⋆ρ ′ ( α ′ ))= exp(1[ − β ⊗ − ⊗ β ′ ]) ⋆ ( ρ ⊗ ρ ′ )( α ˜ ⋆α ′ )= ( φ ⊗ φ ′ ) ∗ ( α ˜ ⋆α ′ ) . (cid:3) Hochschild homology of matrix factorization categories
Let k be a field, and let Q be an essentially smooth k -algebra. Fix f ∈ Q . PROOF OF A CONJECTURE OF SHKLYAROV 9
Matrix factorizations.
The dg-category mf ( Q, f ) of matrix factorizations of f over Q is de-fined as follows: • Objects are pairs (
P, δ P ), where P is a finitely generated Z / Q -module,and δ P is an odd degree endomorphism of P such that δ P = f id P . • Hom mf ( Q,f ) (( P, δ P ) , ( P ′ , δ P ′ )) is the Z / Q ( P, P ′ ) with differential ∂ given by ∂ ( α ) = δ P ′ α − ( − | α | αδ P for α homogeneous. From now on, we will omit the subscript on Hom mf ( Q,f ) ( − , − ).We emphasize that f is allowed to be 0. The homotopy category of mf ( Q, f ), denoted [ mf ( Q, f )], is the Q -linear category with the same objects as mf ( Q, f ) and morphisms given by Hom [ mf ( Q,f )] ( − , − ) := H Hom( − , − ).Let X, Y ∈ mf ( Q, f ), and let α , α ∈ Hom(
X, Y ) be cocycles. We recall that α , α are homotopic if there is an odd degree Q -linear map h : X → Y such that hd X + d Y h = α − α . This is just the usual notion of a homotopy between morphisms of a Z / X ∈ mf ( Q, f ) is contractible if id X isnull-homotopic. Morphisms in mf ( Q, f ) that are cocycles are homotopic if and only if they are equalin [ mf ( Q, f )].
Definition 3.1.
Given X ∈ mf ( Q, f ), the support of X is the setsupp( X ) = { p ∈ Spec( Q ) | X p is not a contractible object of mf ( Q p , f ) } . For a closed subset Z of Spec( Q ), let mf Z ( Q, f ) denote the full dg-subcategory of mf ( Q, f ) consistingof those X with supp( X ) ⊆ Z .We record the following: Proposition 3.2.
Let X ∈ mf ( Q, f ) . (1) When f = 0 , supp( X ) is the set of points at which the Z / -complex X is not exact. Therefore,when f = 0 , the notion of support defined above agrees with the usual notion of support for a Z / -graded complex. (2) We have supp( X ) ⊆ Spec(
Q/f ) . When f is a non-zero-divisor, supp( X ) ⊆ Sing(
Q/f ) .Proof. (1) This is [BMTW17, Lemma 2.3]. (2) It is easy to check that any matrix factorization of aunit is contractible. Suppose f is a non-zero-divisor. By [Orl03, Theorem 3.9], the homotopy category[ mf ( Q, f )] is equivalent to the singularity category of
Q/f , and the singularity category is trivialwhen
Q/f is regular. (cid:3)
Remark . If f is a non-zero-divisor, so that the morphism of schemes f : Spec( Q ) → A k is flat,then Spec( Q/f ) ∩ Sing( f ) = Sing( Q/f ) , where Sing( f ) denotes the set of points of Spec( Q ) at which the morphism f : Spec( Q ) → A k is notsmooth.Let R be another essentially smooth k -algebra, and let g ∈ R . Given X ∈ mf ( Q, f ) and Y ∈ mf ( R, g ), we form the tensor product X ⊗ Y ∈ mf ( Q ⊗ k R, f ⊗ ⊗ g )by adapting the notion of tensor product of Z / mf ( Q, f ) ⊗ k mf ( R, g ) → mf ( Q ⊗ k R, f ⊗ ⊗ g ) . If Z and W are closed subsets of Spec( Q ) and Spec( R ), respectively, one has an induced functor mf Z ( Q, f ) ⊗ k mf W ( R, g ) → mf Z × W ( Q ⊗ k R, f ⊗ ⊗ g ) . If Q = R , composing with multiplication in Q gives a functor mf Z ( Q, f ) ⊗ k mf W ( Q, g ) → mf Z ∩ W ( Q, f + g ) . We also have a duality functor D which determines an isomorphism of dg-categories D : mf ( Q, f ) op ∼ = −→ mf ( Q, − f ) . The functor D sends an object P = ( P, δ P ) of mf ( Q, f ) to the object P ∗ = ( P ∗ , − δ ∗ P ) of mf ( Q, − f ),and it sends an element α of Hom( P , P ) op = Hom( P , P ) to the element α ∗ of Hom( P ∗ , P ∗ ). Notethat α ∗ ( γ ) = ( − | α || γ | γ ◦ α . If X ∈ mf Z ( X, f ) op for some closed Z ⊆ Spec( Q ), then D ( X ) ∈ mf Z ( X, − f ). If X, Y ∈ mf ( Q, f ), there is a canonical isomorphismHom(
X, Y ) ∼ = D ( X ) ⊗ Y. In particular, if X ∈ mf Z ( Q, f ) and Y ∈ mf W ( Q, f ), we have(3.4) Hom(
X, Y ) ∈ mf Z ∩ W ( Q, . The HKR map.
Assume for the rest of Section 3 that char( k ) = 0. Given a Z -graded com-plex ( C • , d ) of k -vector spaces, its Z / -folding is the Z / L i ∈ Z C i (resp. L i ∈ Z C i +1 ) and whose differential is given by d .Let Ω • Q/k denote the Z / Q -algebra given by the Z / Q/k . That is, Ω even
Q/k = L j Ω jQ/k , and Ω odd Q/k = L j Ω j +1 Q/k . We write (Ω • Q/k , − df ) for the Z / Q -modules with underlying graded Q -module Ω • Q/k and with differential givenby left multiplication by − df ∈ Ω Q/k .Let Z be a closed subset of Spec( Q/f ). The goal of the rest of this section is to study, for eachtriple (
Q, f, Z ), a Hochschild-Kostant-Rosenberg (HKR)-type map(3.5) ε Q,f,Z : HH ( mf Z ( Q, f )) → R Γ Z (Ω • Q/k , − df ) . Here, R Γ Z is the right adjoint of the inclusion functor D Z Z / ( Q ) ⊆ D Z / ( Q ), where D Z / ( Q ) denotesthe derived category of Z / Q -modules, and D Z Z / ( Q ) ⊆ D Z / ( Q ) the subcategory spanned bycomplexes with support contained in Z . It will be convenient for us to use the following ˇCech modelfor R Γ Z . Choose g , . . . , g m ∈ Q such that Z = V ( g , . . . , g m ), and let C = C ( g , . . . , g m ) = O j ( Q → Q [1 /g i ])be the ( Z / C ⊗ Q M models R Γ Z ( M )for any M ∈ D Z / ( Q ); i.e., the functor C ⊗ Q − : D Z / ( Q ) → D Z Z / ( Q )is right adjoint to the inclusion. From now on, given g , . . . , g m ∈ Q such that V ( g , . . . , g m ) = Z , wewill tacitly identify R Γ Z ( M ) with C ⊗ Q M . Note that, for any Z / M of Q -modulesthat is supported in Z , the natural morphism of complexes(3.6) C ⊗ Q M → M given by the tensor product of the augmentation map C → Q with id M is a quasi-isomorphism.HKR maps for matrix factorization categories have been widely studied. Segal gives such an HKRmap, involving Hochschild homology of the second kind and without a support condition, in [Seg13,Corollary 3.4]; Efimov generalizes Segal’s result to the non-affine setting in [Efi17, Proposition 3.21];and Preygel gives a map just as in (3.5) (but also in the not-necessarily-affine setting), and proves it isa quasi-isomorphism, in [Pre11, Theorem 8.2.6(iv))]. But [Pre11] doesn’t contain a concrete formulafor where the HKR map (3.5) sends an element of the bar complex computing HH ( mf Z ( Q, f )), andwe will need such a formula later on. So, we develop our own version of (3.5).
PROOF OF A CONJECTURE OF SHKLYAROV 11
Quasi-matrix factorizations.
Define a curved dg-category qmf ( Q, f ), the category of quasi-matrix factorizations , in the following way. • Objects (
P, δ P ) are defined in the same way as those of mf ( Q, f ), except we remove therequirement that δ P is given by multiplication by f . • Morphisms are defined in the same way as in mf ( Q, f ). • The curvature element of End qmf ( Q,f ) ( P, δ P ) is δ P − f . mf ( Q, f ) is precisely the full subcategory of qmf ( Q, f ) spanned by objects with trivial curvature.Let qmf ( Q, f ) denote the full subcategory of qmf ( Q, f ) spanned by those objects (
P, δ P ) suchthat δ P = 0. Note that the curvature element of an object in qmf ( Q, f ) is − f . The pair ( Q, qmf ( Q, f ) , and its endomorphisms form the curved differential Z / Q, , − f ). That is, we have inclusions mf ( Q, f ) ֒ → qmf ( Q, f ) ← ֓ qmf ( Q, f ) ← ֓ ( Q, , − f ) . These functors are all pseudo-equivalences , in the language of [PP12, Section 1.5], and so, by [PP12,Lemma A, page 5319], the induced maps HH II ( mf ( Q, f )) → HH II ( qmf ( Q, f )) ← HH II ( qmf ( Q, f ) ) ← HH II ( Q, , − f )are all quasi-isomorphisms.A key point is that there is a (non-strict) cdg-functor( F, β ) : qmf ( Q, f ) → qmf ( Q, f ) given by F ( P, δ P ) = ( P,
0) and β ( P,δ P ) = δ P . The induced map( F, β ) ∗ : HH II ( qmf ( Q, f )) → HH II ( qmf ( Q, f ) )sends α [ α | · · · | α n ], where α i ∈ Hom(( P i +1 , δ i +1 ) , ( P i , δ i )), to X i ,...,i n ≥ ( − i + ··· + i n α [ i z }| { δ | · · · | δ | α | i z }| { δ | · · · | δ | · · · | α n | i n z }| { δ | · · · | δ ] . The supertrace.
Given a Z / Q -module P , define the su-pertrace map str : End Q ( P ) → Q as the composition End Q ( P ) ∼ = P ∗ ⊗ Q P γ ⊗ p γ ( p ) −−−−−−−→ Q for homogeneous elements γ , p . Equivalently, for α ∈ End Q ( P ) we havestr( α ) = ( tr( α : P → P ) − tr( α : P → P ) , if α has degree 0, and0 , if α has degree 1.Here, tr is the classical trace of an endomorphism of a projective module. We extend str to a mapEnd Ω • Q/k ( P ⊗ Q Ω • Q/k ) ∼ = End Q ( P ) ⊗ Q Ω • Q/k str ⊗ id −−−−→ Ω • Q/k , which we also write as str.3.2.3. The HKR map without supports.
Definition 3.7. A connection on an object ( P, δ P ) ∈ qmf ( Q, f ) is a k -linear map ∇ : P → Ω Q/k ⊗ Q P of odd degree such that ∇ ( qp ) = dq ⊗ p + q ∇ ( p ), i.e. a superconnection , in the language of [Qui85].Notice that the definition does not involve δ P . Choose a connection ∇ P on each object ( P, ∈ qmf ( Q, f ) ; we stipulate that the connectionchosen for Q ∈ qmf ( Q, f ) is the canonical one given by the de Rham differential, d : Q → Ω Q/k .Define ε : HH II ( qmf ( Q, f ) ) ♮ −→ Ω • Q/k by ε ( α [ α | · · · | α m ]) = 1 m ! str( α α ′ · · · α ′ m ) , where, for α : ( P , → ( P , α ′ = ∇ P ◦ α − ( − | α | α ◦ ∇ P . By [BW19a, Theorem 5.18], ε gives a chain map HH II ( qmf ( Q, f ) ) −→ (Ω • Q/k , − df ) . Then the composition ε Q : HH II ( Q, , − f ) ≃ −→ HH II ( qmf ( Q, f ) ) ε −→ (Ω • Q/k , − df ) , where the first map is induced by inclusion, is given by the classical HKR map ε Q ( q [ q | · · · | q n ]) = q dq · · · dq n n ! ∈ Ω nQ/k . In particular, ε is a quasi-isomorphism. ( F, β ) ∗ is also a quasi-isomorphism, since qmf ( Q, f ) ≃ −→ qmf ( Q, f ) ( F,β ) −−−→ qmf ( Q, f ) is the identity.We define the HKR map ε Q,f : HH ( mf ( Q, f )) → (Ω • Q/k , − df )to be the composition HH ( mf ( Q, f )) can −−→ HH II ( mf ( Q, f )) ≃ −→ HH II ( qmf ( Q, f )) ( F,β ) ∗ −−−−→ HH II ( qmf ( Q, f ) ) ε −→ (Ω • Q/k , − df ) , where “can” denotes the canonical map. A more explicit formula for ε Q,f is given as follows. Givenobjects ( P , δ ) , . . . , ( P n , δ n ) of mf ( Q, f ) and maps P α ←− P α ←− · · · α n − ←−−− P n α n ←−− P , set ∇ i = ∇ P i . Then ε Q,f ( α [ α | . . . | α n ]) = X i ,...,i n ≥ ( − i + ··· + i n ( n + i + · · · + i n )! str (cid:0) α ( δ ′ ) i α ′ · · · ( δ ′ n ) i n − α ′ n ( δ ′ ) i n (cid:1) , where, just as above, α ′ j = ∇ j ◦ α j − ( − | α j | α j ◦ ∇ j +1 (with ∇ n +1 = ∇ ) , and δ ′ j = [ ∇ j , δ i ] = ∇ j ◦ δ j + δ j ◦ ∇ j . Note that the sum in this formula is finite, since Ω jQ/k = 0 for j > dim( Q ). PROOF OF A CONJECTURE OF SHKLYAROV 13
Summarizing, we have a commutative diagram(3.8) HH ( mf ( Q, f )) ε Q,f ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ / / HH II ( mf ( Q, f )) ≃ / / HH II ( qmf ( Q, f )) ≃ ( F,β ) ∗ (cid:15) (cid:15) HH II ( qmf ( Q, f ) ) ε ≃ (cid:15) (cid:15) HH II ( Q, , − f ) ≃ o o ε Q ≃ u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ (Ω Q/k , − df ) . Notice that this implies ε Q,f is independent, up to natural isomorphism in the derived category, ofthe choices of connections. In particular, the map on homology induced by ε Q,f is independent ofsuch choices.We include the following result, although it will not be needed in this paper:
Proposition 3.9.
If the only critical value of f : Spec( Q ) → A is 0, ε Q,f is a quasi-isomorphism.Proof.
By [PP12, Section 4.8, Corollary A], the canonical map HH ( mf ( Q, f )) → HH II ( mf ( Q, f ))is a quasi-isomorphism. The statement therefore follows from the commutativity of diagram (3.8). (cid:3)
The HKR map with supports.
We now define the HKR map for a general closed subset Z ofSpec( Q ). Composing ε Q,f with the natural map induced by the inclusion mf Z ( Q, f ) ⊆ mf ( Q, f )gives a map(3.10) HH ( mf Z ( Q, f )) → (Ω • Q/k , − df ) . By Proposition 3.2 (1) and (3.4), if
X, Y ∈ mf Z ( Q, f ), Hom(
X, Y ) is a complex of Q -modules whosesupport is contained in Z . (When f is a non-zero-divisor, this complex is in fact supported on Z ∩ Sing(
Q/f ).) It follows that each row of the bicomplex used to define HH ( mf Z ( Q, f )) is sup-ported on Z . Since HH ( mf Z ( Q, f ) is the direct sum totalization of this bicomplex, we have that HH ( mf Z ( Q, f )) is supported on Z . Adjointness thus gives a canonical isomorphism ε Q,f,Z : HH ( mf Z ( Q, f )) → R Γ Z (Ω • Q/k , − df )in D ( Q ). In other words, ε Q,f,Z is represented in D ( Q ) by the diagram HH ( mf Z ( Q, f )) R Γ Z HH ( mf Z ( Q, f )) ≃ ( . ) o o ( . ) / / R Γ Z (Ω • Q/k , − df ) . We will sometimes refer to ε Q,f,Z as just ε , if no confusion can arise.3.3. Relationship between the HKR map and the map I f (0) . When Q = C [ x , . . . , x n ] and m = ( x , . . . , x n ) is the only singular point of the map f : A n C → A C , Shklyarov defines in [Shk16,Section 4.1] an isomorphism I f (0) : HH ∗ ( mf ( Q, f )) ∼ = −→ H ∗ (Ω • Q/k , − df )as follows. Let A f be the endomorphism dga of the following matrix factorization ( P, δ P ) whichrepresents the residue field Q/ m in the singularity category of Q/f : choose polynomials y , . . . , y n ∈ Q so that f = P i x i y i , let P be the Z / Q on generators e , . . . , e n , anddefine a differential on P given by δ P = X i x i e ∗ i + y i e i . Here, e ∗ i is the Q -linear derivation of P determined by e ∗ i ( e j ) = δ ij . By a theorem of Dyckerhoff([Dyc11, Theorem 5.2 (3)]), the inclusion ι : A f ֒ → mf ( Q, f ) is a Morita equivalence. Since Hochschild homology is Morita invariant, the induced map ι ∗ : HH ∗ ( A f ) ∼ = −→ HH ∗ ( mf ( Q, f ))is an isomorphism.From now on, we identify P with Q ⊗ C Λ, where Λ = Λ C ( e , . . . , e n ), and A f with Q ⊗ C End C (Λ).Shklyarov defines a quasi-isomorphism α : HH ( A f ) ≃ −→ (Ω • Q/k , − df )as the composition HH ( A f ) exp( − δ P ]) −−−−−−−−→ HH II ( A f ) ε ′ −→ (Ω • Q/k , − df ) , where ε ′ ( q ⊗ α [ q ⊗ α | · · · | q n ⊗ α n ]) = ( − P i odd | α i | n ! str( α · · · α n ) q dq · · · dq n . Finally, I f (0) is the composition HH ∗ ( mf ( Q, f )) ι − ∗ −−→ HH ∗ ( A f ) α −→ H ∗ (Ω • Q/k , − df ) . Lemma 3.11.
The map ε ′ coincides with the map ε Q,f restricted to HH (End( P )) for the choice ofconnection ∇ P defined as ∇ P ( q ⊗ α ) = dq ⊗ α . Thus, I f (0) = ε Q,f .Proof.
We have ε Q,f ( q ⊗ α [ q ⊗ α | · · · | q n ⊗ α n ]) = 1 n ! str(( q ⊗ α )( dq ⊗ α ) · · · ( dq n ⊗ α n ))= ( − P i i | α i | n ! str( α · · · α n ) q dq · · · dq n = ( − P i odd | α i | n ! str( α · · · α n ) q dq · · · dq n . (cid:3) Compatibility of the HKR map with taking duals.
Shklyarov proves in [Shk14, Proposition3.2] that, for any differential Z / A , there is a canonical isomorphism of complexes(3.12) Φ : HH ( A ) ∼ = −→ HH ( A op )given by a [ a | · · · | a n ] ( − n + P ≤ i The diagram HH ( mf Z ( Q, f )) Ψ (cid:15) (cid:15) ε Q,f,Z / / R Γ Z (Ω • Q/k , − df ) γ (cid:15) (cid:15) HH ( mf Z ( Q, − f )) ε Q, − f,Z / / R Γ Z (Ω • Q/k , df ) PROOF OF A CONJECTURE OF SHKLYAROV 15 commutes in D ( Q ) , where γ is R Γ Z applied to the map whose restriction to Ω jQ/k is multiplication by ( − j for all j .Proof. The map ε Q,f,Z factors as HH ( mf Z ( Q, f )) → R Γ Z HH ( mf ( Q, f )) ε Q,f −−−→ (Ω • Q/k , − df ) , where the first map is the canonical one. ε Q, − f,Z factors similarly. Since the diagram HH ( mf Z ( Q, f )) Ψ (cid:15) (cid:15) / / R Γ Z HH ( mf ( Q, f )) R Γ Z (Ψ) (cid:15) (cid:15) HH ( mf Z ( Q, − f )) / / R Γ Z HH ( mf ( Q, − f ))evidently commutes, we may assume Z = Spec( Q ).Recall from (3.8) that ε Q,f fits into a commutative diagram(3.15) HH ( mf ( Q, f )) θ / / ε Q,f ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ HH II ( qmf ( Q, f ) ) ε ≃ (cid:15) (cid:15) HH II ( Q, , − f ) can ≃ o o ε Q ≃ u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ (Ω • Q/k , − df ) , where θ ( α [ α | · · · | α n ]) = X i ,...,i n ≥ ( − i + ··· + i n α [ δ i | α | δ i | · · · | α n | δ i n ] . Here, δ i stands for i z }| { δ | · · · | δ .The map Ψ extends to a mapΨ : HH II ( qmf ( Q, f ) ) → HH II ( qmf ( Q, − f ) )using the same formula, and this map in turn restricts to a mapΨ : HH II ( Q, , − f ) → HH II ( Q, , f )given by Ψ( q [ q | · · · | q n ]) = ( − n + ( n ) q [ q n | · · · | q ] . We claim that the diagram(3.16) HH ( mf ( Q, f )) θ / / Ψ (cid:15) (cid:15) HH II ( qmf ( Q, f ) ) Ψ (cid:15) (cid:15) HH II ( Q, , − f ) can ≃ o o Ψ (cid:15) (cid:15) HH ( mf ( Q, − f )) θ / / HH II ( qmf ( Q, − f ) ) HH II ( Q, , f ) can ≃ o o commutes. This is evident for the right square. As for the left, the element α [ α | · · · | α n ] is mappedvia Ψ ◦ θ to X i ,...,i n ≥ ( − I ( − n + I + P ≤ i Let ( Q, f, Z ) and ( R, g, W ) be triples consisting of anessentially smooth k -algebra, an element of the algebra, and a closed subset of the spectrum of thealgebra. The tensor product of matrix factorizations (Section 3.1), along with the K¨unneth map forHochschild homology of dg-categories (Section 2.4), gives a pairing(3.17) − ˜ ⋆ − : HH ( mf Z ( Q, f )) ⊗ k HH ( mf W ( R, g )) → HH ( mf Z × W ( Q ⊗ k R, f ⊗ ⊗ g )) . Write f + g for the element f ⊗ ⊗ g ∈ Q ⊗ k R . Multiplication in Ω • Q ⊗ k R/k defines a pairingof complexes of Q ⊗ k R -modules − ∧ − : (Ω • Q/k , − df ) ⊗ k (Ω • R/k , − dg ) → (Ω • Q ⊗ k R/k , − df − dg ) . We compose this with the canonical maps R Γ Z (Ω • Q/k , − df ) → (Ω • Q/k , − df ) and R Γ W (Ω • R/k , − dg ) → (Ω • R/k , − dg ) to obtain the map R Γ Z (Ω • Q/k , − df ) ⊗ k R Γ W (Ω • R/k , − dg ) → (Ω • Q ⊗ k R/k , − df − dg ) . The source of this map is supported on the closed subset Z × W of Spec( Q ⊗ k R ) = Spec( Q ) × k Spec( R ).Thus, by adjointness, we obtain a pairing(3.18) − ∧− : R Γ Z (Ω • Q/k , − df ) ⊗ Q R Γ W (Ω • R/k , − dg ) → R Γ Z × W (Ω • Q ⊗ k R/k , − df − dg ) . A key fact is that the pairings (3.17) and (3.18) are compatible via the HKR maps: Proposition 3.19. The diagram HH ( mf Z ( Q, f )) ⊗ k HH ( mf W ( R, g )) ε Q,f,Z ⊗ ε R,f,W / / − ˜ ⋆ − (cid:15) (cid:15) R Γ Z (Ω • Q/k , − df ) ⊗ k R Γ W (Ω • R/k , − dg ) ∧ (cid:15) (cid:15) HH ( mf Z × W ( Q ⊗ k R, f + g )) ε Q ⊗ kR,f + g,Z × W / / R Γ Z × W (Ω • Q ⊗ k R/k , − df − dg ) in D ( Q ⊗ k R ) commutes.Proof. It is enough to show the diagrams(3.20) HH ( mf Z ( Q, f )) ⊗ k HH ( mf W ( R, g )) / / − ˜ ⋆ − (cid:15) (cid:15) R Γ Z HH ( mf ( Q, f )) ⊗ k R Γ W HH ( mf ( R, g )) − ˜ ⋆ − (cid:15) (cid:15) HH ( mf Z × W ( Q ⊗ k R, f + g )) / / R Γ Z × W HH ( mf ( Q ⊗ k R, f + g )) PROOF OF A CONJECTURE OF SHKLYAROV 17 and(3.21) R Γ Z HH ( mf ( Q, f )) ⊗ k R Γ W HH ( mf ( R, g )) − ˜ ⋆ − (cid:15) (cid:15) / / − ˜ ⋆ − (cid:15) (cid:15) R Γ Z (Ω • Q/k , − df ) ⊗ k R Γ W (Ω • R/k , − dg ) ∧ (cid:15) (cid:15) R Γ Z × W HH ( mf ( Q ⊗ k R, f + g )) / / R Γ Z × W (Ω • Q ⊗ k R/k , − df − dg )commute. Here, the right-most vertical map in (3.20) (which coincides with the left-most vertical mapin (3.21)) is defined in a manner similar to the map (3.18), and the horizontal maps in (3.20) are thecanonical ones. The commutativity of (3.20) is clear. As for (3.21), it suffices to show the diagram HH ( mf ( Q, f )) ⊗ k HH ( mf ( R, g )) − ˜ ⋆ − (cid:15) (cid:15) ε Q,f ⊗ ε R,g / / − ˜ ⋆ − (cid:15) (cid:15) (Ω • Q/k , − df ) ⊗ k (Ω • R/k , − dg ) ∧ (cid:15) (cid:15) HH ( mf ( Q ⊗ k R, f + g )) ε Q ⊗ kR,f + g / / (Ω • Q ⊗ k R/k , − df − dg )in D ( Q ⊗ k R ) commutes. Factoring the HKR maps as in diagram (3.8), it suffices to show the squares(3.22) HH ( mf ( Q, f )) ⊗ k HH ( mf ( R, g )) − ˜ ⋆ − (cid:15) (cid:15) / / HH II ( qmf ( Q, f )) ⊗ k HH II ( qmf ( R, g )) − ˜ ⋆ − (cid:15) (cid:15) HH ( mf ( Q ⊗ k R, f + g )) / / HH II ( qmf ( Q ⊗ k R, f + g ))and(3.23) HH II ( qmf ( Q, f )) ⊗ k HH II ( qmf ( R, g )) ε ⊗ ε / / − ˜ ⋆ − (cid:15) (cid:15) (Ω • Q/k , − df ) ⊗ k (Ω • R/k , − dg ) ∧ (cid:15) (cid:15) HH II ( qmf ( Q ⊗ k R, f + g )) ε / / (Ω • Q ⊗ k R/k , − df − dg )commute. It follows immediately from Lemma 2.7 that (3.22) commutes. The square(3.24) HH II ( Q, − f ) ⊗ k HH II ( R, − g ) ≃ / / − ˜ ⋆ − (cid:15) (cid:15) HH II ( qmf ( Q, f )) ⊗ k HH II ( qmf ( R, g )) − ˜ ⋆ − (cid:15) (cid:15) HH II ( Q ⊗ k R, − f − g ) ≃ / / HH II ( qmf ( Q ⊗ k R, f + g ))evidently commutes, and concatenating this diagram with (3.23) gives a commutative diagram. Itfollows that (3.23) commutes. (cid:3) For an essentially smooth k -algebra Q , any element f ∈ Q , and any pair of closed subsets Z and W of Spec( Q ), there is a pairing(3.25) HH ( mf Z ( Q, f )) × HH ( mf W ( Q, − f )) ⋆ −→ HH ( mf Z ∩ W ( Q, HH ( mf Z ( Q, f )) × HH ( mf W ( Q, − f )) ˜ ⋆ −→ HH ( mf Z × W ( Q ⊗ k Q, f ⊗ − ⊗ f ))with the map HH ( mf Z × W ( Q ⊗ k Q, f ⊗ − ⊗ f )) → HH ( mf Z ∩ W ( Q, Q ⊗ Q → Q . The previous result, along with the functoriality ofthe HKR map, yields: Corollary 3.26. The diagram HH ( mf Z ( Q, f )) ⊗ k HH ( mf W ( Q, − f )) ε Q,f,Z ⊗ ε Q, − f,Z / / (cid:15) (cid:15) R Γ Z (Ω • Q/k , − df ) ⊗ k R Γ W (Ω • Q/k , df ) ∧ (cid:15) (cid:15) HH ( mf Z ∩ W ( Q, ε Q, ,Z ∩ W / / R Γ Z ∩ W Ω • Q/k in D ( Q ⊗ k Q ) commutes. We will be especially interested in the case where Z ∩ W = { m } .4. Proof of Shklyarov’s conjecture Throughout this section, we assume • k is a field, • Q is a regular k -algebra, and • m is a k -rational maximal ideal of Q ; i.e. the canonical map k → Q/ m is an isomorphism.Let us review our progress on the proof of Conjecture 1.4. Recall from the introduction that, toprove the conjecture, it suffices to show that diagram (1.9) commutes, the composition along the leftside of this diagram computes the pairing η mf , and the composition along the right side computes theresidue pairing. So far, we have shown the two interior squares of (1.9) commute: this follows fromLemma 3.11, Lemma 3.14, and Corollary 3.26. In this section, we show the left side of the diagramgives the canonical pairing η mf (Lemma 4.23), the right side of the diagram gives the residue pairing(Proposition 4.34), and the bottom triangle commutes (Theorem 4.36).4.1. Computing HH ( mf m ( Q, . We carry out a calculation of the Hochschild homology of thedg-category mf m ( Q, 0) that we will use repeatedly throughout the rest of the paper. Let n denotethe Krull dimension of Q m . We recall that a sequence x , . . . , x n ∈ m is called a system of parameters if x , . . . , x n generate an m -primary ideal, and a system of parameters is called regular if the elementsgenerate m .Fix a regular system of parameters x , . . . , x n for Q m , and set K = Kos Q m ( x , . . . , x n ) ∈ mf m ( Q m , Z / x i ’s. Explicitly, K is the differential Z / Q m generated by e , . . . , e n with d K ( e i ) = x i . The differ-ential Z / Q m -algebra E := End mf m ( Q m , ( K ) is generated by odd degree elements e , . . . , e n , e ∗ , . . . , e ∗ n satisfying e i = 0 = ( e ∗ i ) , [ e i , e j ] = 0 = [ e ∗ i , e ∗ j ], and [ e i , e ∗ j ] = δ ij ; and the differential d E is determined by the equations d E ( e i ) = x i and d E ( e ∗ i ) = 0. Let Λ be the dg- k -subalgebra of E generated by the e ∗ i . So, Λ is an exterior algebra over k on n generators, with trivial differential.The inclusion Λ ⊆ E is a quasi-isomorphism of differential Z / k -algebras. Since Λ is gradedcommutative, HH ∗ (Λ) is a k -algebra under the shuffle product, and, by a standard calculation, thereis an isomorphism(4.1) Λ ⊗ k k [ y , . . . , y n ] ∼ = −→ HH ∗ (Λ) , of k -algebras, where e ∗ i ⊗ e ∗ i [], and 1 ⊗ y i e ∗ i ]. Here, and throughout the paper, we use thenotation α [] to denote an element of a Hochschild complex of the form α [ α | · · · | α n ] with n = 0. Lemma 4.2. The canonical morphisms (4.3) E ֒ → mf m ( Q m , and (4.4) mf m ( Q, → mf m ( Q m , of dg-categories are Morita equivalences. In particular, we have canonical quasi-isomorphisms (4.5) HH (Λ) ≃ −→ HH ( mf m ( Q m , ≃ ←− HH ( mf m ( Q, . PROOF OF A CONJECTURE OF SHKLYAROV 19 Proof. To prove (4.3) is a Morita equivalence, we prove the thick closure of K in the homotopycategory [ mf m ( Q m , mf m ( Q m , D denote the derived category of all Z / Q m -modules whose homology groups are finite dimensional over k . Since Q m isregular, it follows from [BMTW17, Proposition 3.4] that the canonical functor[ mf m ( Q m , → D is an equivalence. It therefore suffices to show Thick( K ) = D ; in fact, we need only show every objectin D with free components is in Thick( K ).Let X be an object of D with free components. We may assume that X is minimal , i.e. that k ⊗ Q m X is a direct sum of copies of k and Σ k . The isomorphism K ∼ = −→ k in D induces an isomorphism K ⊗ Q m X ∼ = −→ k ⊗ Q m X, and therefore K ⊗ Q m X ∈ Thick( k ). It thus suffices to prove X ∈ Thick( K ⊗ Q m X ). Since K ⊗ Q m X ∼ = Kos Q m ( x ) ⊗ Q m · · · ⊗ Q m Kos Q m ( x n ) ⊗ Q m X, it suffices to show that, for every Y ∈ D whose components are free Q m -modules, and every x ∈ m \{ } , Y ∈ Thick( Y /xY ). Using induction and the exact sequence0 → Y /x n − Y x −→ Y /x n Y → Y /xY → , we get Y /x n Y ∈ Thick( Y /xY ) for all n . Observing that End D ( Y )[1 /x ] = 0, choose n ≫ x n on Y determines the zero map in D . The distinguished triangle Y x n −−→ Y → Y /x n → Σ Y in D therefore splits, implying that Y is a summand of Y /x n . Thus, Y ∈ Thick( Y /x n ) ⊆ Thick( Y /xY ).As for (4.4), the functor [ mf m ( Q, → [ mf m ( Q m , [ mf m ( Q, ( X, Y )is supported in { m } for any X, Y . It follows that the induced map(4.6) [ mf m ( Q, idem → [ mf m ( Q m , idem on idempotent completions is fully faithful, so we need only show (4.6) is essentially surjective. Bythe above argument, it suffices to show K is in the essential image of (4.6). Choose a Q -free resolution F of k ; F m is homotopy equivalent to the Koszul complex on the x i ’s, and so the Z / F m is isomorphic to K in [ mf m ( Q m , (cid:3) Remark . Let b Q denote the m -adic completion of Q . Letting b Q play the role of Q in Lemma 4.2implies that the inclusion End mf m ( b Q, ( K ⊗ Q m b Q ) ֒ → mf m ( b Q, mf m ( Q, → mf m ( b Q, The trace map. We define an even degree maptrace : HH ∗ ( mf m ( Q, → k of Z / k -vector spaces, with k concentrated in even degree, as follows. Let Perf Z / ( k ) denotethe dg-category of Z / k -vector spaceshaving finite dimensional homology. There is a dg-functor mf m ( Q, → Perf Z / ( k ) induced byrestriction of scalars along the structural map k → Q that induces a map u : HH ∗ ( mf m ( Q, → HH ∗ (Perf Z / ( k )) , and there is a canonical isomorphism v : k ∼ = −→ HH ∗ (Perf Z / ( k )) given by a a []. Here, k is considered as a Z / a is regarded as an endomorphism of this complex. We definetrace := v − u. In the rest of this subsection, we establish several technical properties of the trace map that we willneed later on.Given an object ( P, δ P ) ∈ mf m ( Q, P ) → HH ( mf m ( Q, α α [] and hence an induced map(4.8) H ∗ (End( P )) → HH ∗ ( mf m ( Q, . Proposition 4.9. If ( P, δ P ) ∈ mf m ( Q, , and α is an even degree endomorphism of P , the compo-sition H (End( P )) ( . ) −−−→ HH ( mf m ( Q, trace −−−→ k sends α to the supertrace of the endomorphism of H ∗ ( P ) induced by α : trace( α []) = str( H ∗ ( α ) : H ∗ ( P ) → H ∗ ( P ))= tr( H ( α ) : H ( P ) → H ( P )) − tr( H ( α ) : H ( P ) → H ( P )) . In particular, trace(id P []) = dim k H ( P ) − dim k H ( P ) . Proof. Let Vect Z /2 ( k ) denote the subcategory of Perf Z / ( k ) spanned by finite-dimensional Z / Z /2 ( k ) ֒ → Perf Z / ( k ) in-duces a quasi-isomorphism on Hochschild homology. Composing the map End( H ∗ ( P )) → HH ∗ (Vect Z / ( k ))given by α α [] with the canonical map H ∗ (End( P )) → End( H ∗ ( P )) gives a map(4.10) H ∗ (End( P )) → HH ∗ (Vect Z / ( k )) . We first show that the square(4.11) H ∗ (End( P )) ( . ) (cid:15) (cid:15) ( . ) / / HH ∗ ( mf m ( Q, u (cid:15) (cid:15) HH ∗ (Vect Z /2 ( k )) ∼ = / / HH ∗ (Perf Z / ( k ))commutes. Let β be an even degree cycle in End( P ), and let H ∗ ( β ) denote the induced endomorphismof H ∗ ( P ). We must show the cycles β [] and H ∗ ( β )[] coincide in HH ∗ (Perf Z / ( k )). To see this, chooseeven degree k -linear chain maps ι : H ∗ ( P ) → P , π : P → H ∗ ( P )such that • π ◦ ι = id H ∗ ( P ) , and • ι ◦ π is homotopic to id P via a ( Z / h , i.e. ι ◦ π − id P = δ P ◦ h + h ◦ δ P . Applying the Hochschild differential b to π [ β ◦ ι ] ∈ Hom( P, H ∗ ( P )) ⊗ Hom( H ∗ ( P ) , P ) ⊆ HH (Perf Z / ( k )) , we get b ( π [ β ◦ ι ]) = ( b + b )( π [ β ◦ ι ]) = b ( π [ β ◦ ι ]) = ( π ◦ β ◦ ι )[] − ( ι ◦ π ◦ β )[] = H ∗ ( β )[] − ( ι ◦ π ◦ β )[] . Next, observe that ( b + b )(( h ◦ β )[]) = b (( h ◦ β )[]) = ( ι ◦ π ◦ β − β )[] . It follows that diagram (4.11) commutes. PROOF OF A CONJECTURE OF SHKLYAROV 21 The isomorphism v : k ∼ = −→ HH ∗ (Perf Z / ( k ))factors as k ∼ = −→ HH ∗ ( k ) ∼ = −→ HH ∗ (Vect Z /2 ( k )) ∼ = −→ HH ∗ (Perf Z / ( k )) , where each map is the evident canonical one. There is a chain map HH (Vect Z / ( k )) → HH ( k )given by the generalized trace map described in [Seg13, Section 2.3.1] and an evident isomorphism HH ∗ ( k ) ∼ = −→ k . It follows from [Seg13, Lemma 2.12] that composing these maps gives the inverse of k ∼ = −→ HH ∗ ( k ) ∼ = −→ HH ∗ (Vect Z /2 ( k )) . As discussed in [Seg13, Page 872], the generalized trace sends a class of the form α [] to str( α )[]. Thestatement now follows from the commutativity of (4.11). (cid:3) Remark . If Z and W are closed subsets of Sing( Q/f ) that satisfy Z ∩ W = { m } , then, from(3.25), we obtain the pairing HH ∗ ( mf Z ( Q, f )) × HH ∗ ( mf W ( Q, − f )) ⋆ −→ HH ∗ ( mf m ( Q, . By Proposition 4.9, given X ∈ mf Z ( Q, f ) and Y ∈ mf W ( Q, − f ), the composition H ∗ (End( X )) × H ∗ (End( Y )) → HH ∗ ( mf Z ( Q, f )) × HH ∗ ( mf W ( Q, − f )) ⋆ −→ HH ∗ ( mf m ( Q, trace −−−→ k sends a pair of endomorphisms ( α, β ) to tr( H ( α ⊗ β )) − tr( H ( α ⊗ β )). In particular, it sends (id X , id Y )to θ ( X, Y ) := dim k H ( X ⊗ Y ) − dim k H ( X ⊗ Y ) . Recall from Subsection 4.1 the folded Koszul complex K and the exterior algebra Λ ⊆ End mf m ( Q m , ( K ).Denote by η : Λ → k the augmentation map that sends e ∗ i to 0. Proposition 4.13. The composition (4.14) HH ∗ (Λ) ( . ) −−−→ HH ∗ ( mf m ( Q m , trace −−−→ k coincides with (4.15) HH ∗ (Λ) HH ∗ ( η ) −−−−−→ HH ∗ ( k ) ∼ = −→ k, where the second map in (4.15) is the canonical isomorphism. In particular, if α [ α | . . . | α n ] is acycle in HH (Λ) , where n > , the map (4.14) sends α [ α | . . . | α n ] to 0.Proof. If C is a Z -graded complex, denote its Z / C ). Similarly, given a differential Z -graded category C , define a differential Z / C ) with the same objects as C andmorphism complexes given by taking the Z / C . In this proof,we use the notation HH Z ( − ) (resp. HH Z / ( − )) to denote the Hochschild complex of a differential Z -graded (resp. Z / C is a differential Z -graded category,(4.16) Fold( HH Z ∗ ( C )) = HH Z / ∗ (Fold( C )) . Let Perf m ( Q ) denote the dg-category of perfect complexes of Q -modules with support in { m } ,and let Perf Z ( k ) denote the differential Z -graded category of complexes of (not necessarily finitedimensional) k -vector spaces with finite dimensional total homology. As in the Z / e v : k ∼ = −→ HH Z ∗ (Perf Z ( k )) , where k is concentrated in degree 0, given by a a [].Let e K denote the Z -graded Koszul complex on the regular system of parameters x , . . . , x n for Q m chosen in Subsection 4.1, so that the Z / e K is K . Similarly, denote by e Λ the subalgebra (with trivial differential) of End( e K ), defined in the same way as Λ, so that the Z / e Λ is Λ.Notice that every α i appearing in our cycle α [ α | . . . | α n ] can be considered as an element of e Λ.We consider the composition(4.17) HH Z ∗ ( e Λ) → HH Z ∗ (End( e K )) → HH Z ∗ (Perf m ( Q )) → HH Z ∗ (Perf Z ( k )) ( e v ) − −−−→ k of maps of Z -graded k -vector spaces. We claim (4.17) coincides with the composition(4.18) HH Z ∗ ( e Λ) → HH Z ∗ ( k ) ∼ = −→ k, where the first map is induced by the augmentation map e Λ → k . We need only check this in degree0. HH Z ( e Λ) is a 1-dimensional k -vector space generated by id K []. The map (4.18) sends id K [] to 1,and, by (the Z -graded version of) Lemma 4.9, the map (4.17) does as well.Applying Fold( − ) to (4.17), and using (4.16), we arrive at a composition HH Z / ∗ (Λ) → HH Z / ∗ (Fold(Perf m ( Q ))) → k of maps of Z / k -vector spaces, which may be augmented to a commutativediagram(4.19) HH Z / ∗ (Λ) / / ( . ) ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ HH Z / ∗ (Fold(Perf m ( Q ))) (cid:15) (cid:15) / / kHH Z / ∗ ( mf m ( Q, . trace ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ On the other hand, applying Fold( − ) to (4.18), and once again applying (4.16), we get the map(4.15). (cid:3) Lemma 4.20. Suppose Q and Q ′ are regular k -algebras, and m ⊆ Q , m ′ ⊆ Q ′ are k -rational maximalideals. Let g : Q → Q ′ be a k -algebra map such that g − ( m ′ ) = m , the induced map Q m → Q ′ m ′ is flat,and g ( m ) Q ′ m ′ = m ′ Q ′ m ′ . Then g induces a quasi-isomorphism g ∗ : HH ( mf m ( Q m , ≃ −→ HH ( mf m ′ ( Q ′ m ′ , , and trace Q ′ m ′ ◦ g ∗ = trace Q m . Proof. Let b Q (resp. c Q ′ ) denote the m -adic (resp. m ′ -adic) completion of Q (resp. Q ′ ). The assump-tions on g imply it induces an isomorphism b Q ∼ = −→ c Q ′ . The first assertion follows since the canonicalmaps HH ∗ ( mf m ( Q m , → HH ∗ ( mf m ( b Q, HH ∗ ( mf m ′ ( Q ′ m ′ , → HH ∗ ( mf m ( c Q ′ , n = dim( Q m ), choose a regular system of parameters x , . . . , x n of Q m , and construct the exterior algebra Λ using this system of parameters, as in Subsection 4.1. Thehypotheses ensure that g ( x ) , . . . , g ( x n ) form a regular system of parameters for Q ′ m ′ , and we let Λ ′ be the associated exterior algebra. We have a commutative diagram HH ∗ (Λ) ∼ = (cid:15) (cid:15) ∼ = / / HH ∗ (Λ ′ ) ∼ = (cid:15) (cid:15) HH ∗ ( mf m ( Q m , / / HH ∗ ( mf m ′ ( Q ′ m ′ , , PROOF OF A CONJECTURE OF SHKLYAROV 23 where the vertical isomorphisms are as in Lemma 4.2. By Proposition 4.13, it now suffices to observethat the composition HH ∗ (Λ) ∼ = −→ HH ∗ (Λ ′ ) → k, where the second map is induced by the augmentation Λ ′ → k , coincides with the map induced bythe augmentation Λ → k . (cid:3) Lemma 4.21. Suppose Q , Q ′ are essentially smooth k -algebras and m ′ ⊆ Q ′ , m ′′ ⊆ Q ′′ are k -rationalmaximal ideals. Set Q = Q ′ ⊗ k Q ′′ and m = m ′ ⊗ k Q ′′ + Q ′ ⊗ k m ′′ . Then Q is an essentially smooth k -algebra, m is a k -rational maximal ideal of Q , and the diagram HH ∗ ( mf m ′ ( Q ′ m ′ , ⊗ k HH ∗ ( mf m ′′ ( Q ′′ m ′′ , ˜ ⋆ / / trace ⊗ trace (cid:15) (cid:15) HH ∗ ( mf m ( Q m , trace (cid:15) (cid:15) k ⊗ k k ∼ = / / k commutes.Proof. The first two assertions are standard facts. As for the final one, let n ′ and n ′′ denote thedimensions of Q ′ m ′ and Q ′′ m ′′ , resp. Choose regular systems of parameters x , . . . , x n ′ and y , . . . , y n ′′ of Q ′ m ′ and Q ′′ m ′′ , resp., so that x , . . . , x n ′ , y , . . . , y n ′′ form a regular system of parameters of Q m .As in the proof of Lemma 4.20, let Λ , Λ ′ , and Λ ′′ be exterior algebras associated to these systems ofparameters, as constructed in Subsection 4.1. By Lemma 2.7, we have a commutative square HH ∗ (Λ ′ ) ⊗ k HH ∗ (Λ ′′ ) ∼ = / / ˜ ⋆ (cid:15) (cid:15) HH ∗ ( mf m ′ ( Q ′ m ′ , ⊗ k HH ∗ ( mf m ′′ ( Q ′′ m ′′ , ˜ ⋆ (cid:15) (cid:15) HH ∗ (Λ) ∼ = / / HH ∗ ( mf m ( Q m , , where the horizontal isomorphisms are as in Lemma 4.2. By Proposition 4.13, it now suffices toobserve that the composition Λ ∼ = −→ Λ ′ ⊗ k Λ ′′ → k, where the second map is the tensor product of the augmentations, coincides with the augmentationΛ → k . (cid:3) The canonical pairing on Hochschild homology. A k -linear differential Z / C is called proper if, for all pairs of objects ( X, Y ), dim k H i Hom C ( X, Y ) < ∞ for i = 0 , Definition 4.22. For a proper differential Z / C , the canonical pairing for Hochschildhomology is the map η C ( − , − ) : HH ∗ ( C ) ⊗ k HH ∗ ( C ) → k given by the composition HH ∗ ( C ) ⊗ k HH ∗ ( C ) id ⊗ Φ −−−→ HH ∗ ( C ) ⊗ k HH ∗ ( C op ) ˜ ⋆ −→ HH ∗ ( C ⊗ k C op ) HH (( X,Y ) Hom C ( Y,X )) −−−−−−−−−−−−−−−−→ HH ∗ (Perf Z / ( k )) ∼ = ←− k where Φ is the map defined in (3.12).When Sing( Q/f ) = { m } , mf ( Q, f ) is proper, so we have the canonical pairing η mf : HH ∗ ( mf ( Q, f )) ⊗ k HH ∗ ( mf ( Q, f )) → k. Lemma 4.23. When Sing( Q/f ) = { m } , η mf coincides with the pairing given by the composition HH ∗ ( mf ( Q, f )) ⊗ k HH ∗ ( mf ( Q, f )) id ⊗ Ψ −−−−→ HH ∗ ( mf ( Q, f )) ⊗ k HH ∗ ( mf ( Q, − f )) ⋆ −→ HH ∗ ( mf m ( Q, trace −−−→ k, where Ψ is defined in (3.13) .Proof. By Lemma 2.7, there is a commutative square HH ∗ ( mf ( Q, f )) ⊗ k HH ∗ ( mf ( Q, f ) op ) HH (id) ⊗ HH ( D ) / / − ˜ ⋆ − (cid:15) (cid:15) HH ∗ ( mf ( Q, f )) ⊗ k HH ∗ ( mf ( Q, − f )) − ˜ ⋆ − (cid:15) (cid:15) HH ∗ ( mf ( Q, f ) ⊗ k mf ( Q, f ) op ) HH (id ⊗ D ) / / HH ∗ ( mf ( Q, f ) ⊗ k mf ( Q, − f )) , where D is the dg-functor defined in Subsection 3.4. Therefore, it suffices to show the composition HH ∗ ( mf ( Q, f ) ⊗ k mf ( Q, f ) op ) ⊗ D −−−→ HH ∗ ( mf ( Q, f ) ⊗ k mf ( Q, − f )) can −−→ HH ∗ ( mf m ( Q, Forget −−−−→ HH ∗ (Perf Z / ( k ))coincides with the map induced by the dg-functor mf ( Q, f ) ⊗ k mf ( Q, f ) op → Perf Z / ( k )given by ( X, Y ) Hom mf ( Y, X ), and this is clear. (cid:3) The residue map. Assume that Q is an essentially smooth k -algebra and m is a k -rationalmaximal ideal of Q . Let n be the Krull dimension of Q m . In this subsection, we recall the definitionof Grothendieck’s residue map res G : H n m (Ω nQ m /k ) → k and some of its properties. Recall from Subsection 3.2 that for any system of parameters x , . . . , x n of Q m , we have a canonical isomorphism(4.24) H n m (Ω nQ m /k ) ∼ = H n ( C ( x , . . . , x n ) ⊗ Q m Ω nQ m /k ) . We will temporarily use Z -gradings and index things cohomologically, using superscripts. In particular,Ω • Q m /k is a graded Q m -module with Ω jQ m /k declared to have cohomological degree j .We introduce some notation that will be convenient when computing with the augmented ˇCechcomplex. First form the exterior algebra over Q m [1 /x , . . . , /x n ] on (cohomological) degree 1 gener-ators α , . . . α n , and make it a complex with differential given as left multiplication by the degree 1element P i α i . We identify C ( x , . . . , x n ) as the subcomplex whose degree j component is M i < ···
Every element of E ( x , . . . , x n ) ⊗ Q m Ω nQ m /k is a sum of terms of the form1 x a · · · x a n n ⊗ ω with a i ≥ ω ∈ Ω nQ m /k , and this element corresponds to(4.26) α · · · α n x a · · · x a n n ⊗ ω ∈ H n ( C ( x , . . . , x n ) ⊗ Q m Ω nQ m /k )under the isomorphism (4.25). Definition 4.27. Given a system of parameters x , . . . , x n for Q m , integers a i ≥ ≤ i ≤ n ,and an n -form ω ∈ Ω nQ m /k , the generalized fraction (cid:20) ωx a , . . . , x a n n (cid:21) ∈ H n m (Ω nQ m /k )is the class corresponding to the element in (4.26) under the canonical isomorphism (4.24).To define Grothendieck’s residue map, we now assume x , . . . , x n is a regular system of parameters.Since m is k -rational, the m -adic completion b Q of Q is isomorphic to the ring of formal power series k [[ x , . . . , x n ]], and a basis for E ( x , . . . , x n ) as a k -vector space is given by the set { x a ··· x ann | a i ≥ } .We also have that Ω nQ m /k is a free Q m -module of rank one spanned by dx · · · dx n . It follows that theset (cid:26)(cid:20) dx · · · dx n x a , · · · , x a n n (cid:21) | a i ≥ (cid:27) is a k -basis of H n m (Ω nQ m /k ). Definition 4.28. Grothendieck’s residue map res G : H n m (Ω nQ/k ) → k is the unique k -linear map suchthat, if x , . . . , x n is a regular system of parameters of Q m , then(4.29) res G (cid:20) dx · · · dx n x a , · · · , x a n n (cid:21) = ( a i = 1 for all i , and0 otherwise.See [KCD08, Theorem 5.2] for a proof that this definition is independent of the choice of x , . . . , x n .We now revert to the Z / • Q m /k as a Z / Q m -module with Ω jQ m /k located in degree j (mod 2), and we use subscriptsto indicate degrees. Definition 4.30. The residue map for the Z / Q m -module Ω • Q m /k is the mapres = res Q, m : H n R Γ m (Ω • Q m /k ) → k, defined as the composition H n R Γ m (Ω • Q m /k ) ։ H n R Γ m (Σ − n Ω nQ m /k ) ∼ = H n m (Ω nQ m /k ) res G −−−→ k, where the first map is induced by the canonical projection Ω • Q m /k ։ Σ − n Ω nQ m /k .We will need the following two properties of the residue map: Lemma 4.31. Suppose Q and Q ′ are essentially smooth k -algebras and m ⊆ Q , m ′ ⊆ Q ′ are k -rational maximal ideals. Let g : Q → Q ′ be a k -algebra map such that g − ( m ′ ) = m , the induced map Q m → Q ′ m ′ is flat, and g ( m ) Q ′ m ′ = m ′ Q ′ m ′ . Then Q m and Q ′ m ′ have the same Krull dimension, say n ; g induces an isomorphism g ∗ : H n R Γ m (Ω • Q m /k ) ∼ = −→ H n R Γ m ′ (Ω • Q ′ m ′ /k ) of k -vector spaces; and we have res Q ′ , m ′ ◦ g ∗ = res Q, m . Proof. Let x , . . . , x n be a regular system of parameters for Q m , and set x ′ i = g ( x i ). The assumptionson g give that x ′ , . . . , x ′ n form a regular system of parameters for Q ′ m ′ , and hence the induced mapon completions is an isomorphism. The first two assertions follow.The map E ( x , . . . , x n ) ⊗ Q m Ω nQ m /k → E ( x ′ , . . . , x ′ n ) ⊗ Q ′ m ′ Ω nQ ′ m ′ /k induced by g sends α ··· α n x a ··· x ann ⊗ dx · · · dx n to the expression obtained by substituting x ′ i for x i , and thus g ∗ (cid:20) dx · · · dx n x a , . . . , x a n n (cid:21) = (cid:20) dx ′ · · · dx ′ n ( x ′ ) a , . . . , ( x ′ n ) a n (cid:21) . The equation res Q ′ , m ′ ◦ g ∗ = res Q, m follows from (4.29). (cid:3) Lemma 4.32. Let ( Q ′ , m ′ ) , ( Q ′′ , m ′′ ) , and ( Q, m ) = ( Q ′ ⊗ k Q ′′ , m ′ ⊗ k Q ′′ + Q ′ ⊗ k m ′′ ) be as in Lemma4.21. Set m = dim( Q ′ ) and n = dim( Q ′′ ) . The diagram H m R Γ m (Ω • Q ′ m ′ /k ) ⊗ k H n R Γ m ′′ (Ω • Q ′′ m ′′ /k ) ∧ / / res Q ′ , m ′ ⊗ res Q ′′ , m ′′ (cid:15) (cid:15) H m +2 n R Γ m (Ω • Q m /k ) res Q, m (cid:15) (cid:15) k ⊗ k k ∼ = / / k commutes up to the sign ( − mn .Proof. It suffices to prove the analogous diagram given by replacing Ω • Q ′ m ′ /k and Ω • Q ′′ m ′′ /k with Ω mQ ′ m ′ /k and Ω nQ ′′ m ′′ /k commutes. Let x ′ , . . . , x ′ m and x ′′ , . . . , x ′′ n be regular systems of parameters for Q ′ m ′ and Q ′′ m ′′ . Then, upon identifying x ′ i and x ′′ j with the elements x ′ i ⊗ ⊗ x ′′ i of Q m , the sequence x ′ , . . . , x ′ m , x ′′ , . . . , x ′′ n forms a regular system of parameters for Q m . We use these three regular systemsof a parameters to identify H m R Γ m ′ (Ω mQ ′ m ′ /k ) with H m ( C ( x ′ , . . . , x ′ m ) ⊗ Q ′ m ′ Ω mQ ′ m ′ /k ) and similarlyfor Q ′′ and Q . Under these identifications, the map labelled ∧ in the diagram sends α ′ · · · α ′ m x ′ · · · x ′ m ⊗ dx ′ · · · dx ′ m ⊗ α ′′ · · · α ′′ n x ′′ · · · x ′′ n ⊗ dx ′′ · · · dx ′′ n to ( − mn α ′ · · · α ′ m α ′′ · · · α ′′ n x ′ · · · x ′ m x ′′ · · · x ′′ m ⊗ dx ′ · · · dx ′ m dx ′′ · · · dx ′′ n , with the sign arising since the dx ′ i ’s and α ′′ j ’s have odd degree. The result now follows from Definition4.27 and (4.29). (cid:3) The residue pairing. We assume Q , k and m are as in Subsection 4.4. All gradings in thissection are Z / f ∈ Q , and assume Sing( f : Spec( Q ) → A k ) = { m } . Then the canonicalmap (Ω • Q/k , − df ) → (Ω • Q m /k , − df )is a quasi-isomorphism, and the only non-zero homology module isΩ nQ/k df ∧ Ω n − Q/k ∼ = Ω nQ m /k df ∧ Ω n − Q m /k , located in degree n := dim( Q m ). Choose a regular system of parameters x , . . . , x n ∈ m Q m . Then dx , . . . , dx n forms a Q m -basis for Ω Q m /k , and we write ∂ , . . . , ∂ n ∈ Der k ( Q m ) = Hom Q m (Ω Q m /k , Q m )for the associated dual basis. Set f i = ∂ i ( f ). The sequence f , . . . , f n forms a system of parametersfor Q m . For example, when Q m = k [ x , . . . , x n ] ( x ,...,x n ) , we have ∂ i = ∂/∂x i , so that f i = ∂f /∂x i . PROOF OF A CONJECTURE OF SHKLYAROV 27 Definition 4.33. With the notation of the previous paragraph, the residue pairing is the map h− , −i res : Ω nQ/k df ∧ Ω n − Q/k × Ω nQ/k df ∧ Ω n − Q/k → k that sends a pair ( gdx · · · dx n , hdx · · · dx n ) to res G h ghdx ··· dx n f ,...,f n i . Proposition 4.34. The residue pairing coincides with the composition Ω nQ/k df ∧ Ω n − Q/k × Ω nQ/k df ∧ Ω n − Q/k = H n (Ω • Q/k , − df ) × H n (Ω • Q/k , − df ) ∼ = −→ H n (Ω • Q m /k , − df ) × H n (Ω • Q m /k , − df ) id × ( − n −−−−−−→ H n (Ω • Q m /k , − df ) × H n (Ω • Q m /k , df ) ∼ = ←− H n R Γ m (Ω • Q m /k , − df ) × H n (Ω • Q m /k , df ) K¨unneth −−−−−→ H n ( R Γ m (Ω • Q m /k , − df ) ⊗ Q m (Ω • Q m /k , df )) ∧ −→ H n R Γ m (Ω • Q m /k , res −−→ k. In particular, it is well-defined and independent of the choice of regular system of parameters.Proof. We need a formula for the inverse of the canonical isomorphism(4.35) H n R Γ m (Ω • Q m /k , − df ) ∼ = −→ H n (Ω • Q m /k , − df ) . Since the isomorphism is Q m -linear, we just need to know where the inverse sends dx ∧ · · · ∧ dx n .Note that C ( x , . . . , x n ) ⊗ Q m Ω • Q m /k is a graded-commutative Q m -algebra (but not a dga), and thedifferential is left multiplication by P i α i − f i dx i . Observe that the element ω := ( − f α + dx )( − f α + dx ) · · · ( − f n α n + dx n )= ( − n f · · · f n ( α − f dx )( α − f dx ) · · · ( α n − f n dx n ) ∈ C ⊗ Q m (Ω • Q m /k , − df )is a cocycle, and it maps to dx ∧ · · · ∧ dx n ∈ H n (Ω • Q m /k , − df ) via (4.35). Therefore, the compositionΩ nQ/k df ∧ Ω n − Q/k × Ω nQ/k df ∧ Ω n − Q/k ∼ = −→ H n (Ω • Q m /k , − df ) × H n (Ω • Q m /k , − df ) id × ( − n −−−−−−→ H n (Ω • Q m /k , − df ) × H n (Ω • Q m /k , df ) ∼ = ←− H n ( C ⊗ Q m (Ω • Q m /k , − df )) × H n (Ω • Q m /k , df ) K¨unneth −−−−−→ H n ( C ⊗ Q m (Ω • Q m /k , − df ) ⊗ Q m (Ω • Q m /k , df ))sends ( gdx · · · dx n , hdx · · · dx n ) to g Y i ( − f i α i + dx i ) ⊗ ( − n hdx ∧ · · · ∧ dx n . Under the composition H n ( C ⊗ Q m (Ω • Q m /k , − df ) ⊗ Q m (Ω • Q m /k , df )) ∧ −→ H n ( C ⊗ Q m (Ω • Q m /k , ∼ = −→ E ⊗ Q m Ω nQ m /k , this element maps to ghf · · · f n ⊗ dx ∧ · · · ∧ dx n , which is sent to res G h ghdx ··· dx n f ,...,f n i ∈ k by the residue map. (cid:3) Relating the trace and residue maps. Our goal in this subsection is to prove the followingtheorem: Theorem 4.36. Let k be a field of characteristic , Q an essentially smooth k -algebra, and m a k -rational maximal ideal of Q . Then the diagram HH ( mf m ( Q m , ( − n ( n +1)2 trace & & ◆◆◆◆◆◆◆◆◆◆◆◆◆ ε / / H n R Γ m (Ω • Q m /k ) res x x qqqqqqqqqqq k commutes, where n = dim( Q m ) . Our strategy for proving this theorem is to reduce it to the very special case when Q = k [ x ] and m = ( x ) and then to prove it in that case via an explicit calculation. Lemma 4.37. Given a pair ( Q, m ) and ( Q ′ , m ′ ) satisfying the hypotheses of Theorem 4.36, supposethere is a k -algebra map g : Q → Q ′ such that g − ( m ′ ) = m , the induced map Q m → Q ′ m ′ is flat, and m Q ′ m ′ = m ′ Q ′ m ′ . Then (1) Theorem 4.36 holds for ( Q, m ) if and only if it holds for ( Q ′ , m ′ ) . (2) Theorem 4.36 holds provided it holds in the special case where Q = k [ t , . . . , t n ] and m =( t , . . . , t n ) .Proof. (1) follows from Lemmas 4.20 and 4.31 and the naturality of the HKR map ε . As for (2),for ( Q, m ) as in Theorem 4.36, applying (1) to the map g : Q → Q m allows us to reduce to thecase when Q is local. In this case, let x , . . . , x n be a regular system of parameters for Q , define g : k [ t , . . . , t n ] → Q to be the k -algebra map sending t i to x i , and apply (1) to g . (cid:3) Lemma 4.38. Suppose Q ′ , Q ′′ are essentially smooth k -algebras, and m ′ ⊆ Q ′ , m ′′ ⊆ Q ′′ are k -rationalmaximal ideals. Let Q = Q ′ ⊗ k Q ′′ and m = m ′ ⊗ k Q ′′ + Q ′ ⊗ k m ′′ . If Theorem 4.36 holds for each of ( Q ′ , m ′ ) and ( Q ′′ , m ′′ ) , then it also holds for ( Q, m ). In particular, the Theorem holds in general if itholds for the special case Q = k [ x ] , m = ( x ) .Proof. For brevity, let HH ′ = HH ( mf m ′ ( Q ′ m ′ , HH ′′ = HH ( mf m ′′ ( Q ′′ m ′′ , HH = HH ( mf m ( Q m , R Γ ′ = H dim( Q ′ m ′ ) R Γ m ′ (Ω • Q ′ m ′ /k ), etc. We consider the diagram k ⊗ k k ∼ = (cid:15) (cid:15) = / / k ⊗ k k ∼ = (cid:15) (cid:15) HH ′ ⊗ k HH ′′ ˜ ⋆ (cid:15) (cid:15) g g ◆◆◆◆◆◆◆◆◆◆◆ ε ′ ⊗ ε ′′ / / R Γ ′ ⊗ k R Γ ′′ qqqqqqqqqq ∧ (cid:15) (cid:15) HH ε / / w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ R Γ & & ◆◆◆◆◆◆◆◆◆◆◆◆◆ k = / / k, where the diagonal maps are the appropriate trace or residue maps. The left and right trapezoidscommute by Lemmas 4.21 and 4.32, the middle square commutes by Proposition 3.19, the top trapezoidcommutes by assumption, and the outer square obviously commutes. It follows from (4.1) and Lemma PROOF OF A CONJECTURE OF SHKLYAROV 29 HH ′ ⊗ k HH ′′ ˜ ⋆ −→ HH is an isomorphism. A diagram chase now shows that the bottomtrapezoid commutes, which gives the first assertion. The second assertion is an immediate consequenceof the first assertion and Lemma 4.37. (cid:3) Proof of Theorem 4.36. By Lemma 4.38, we need only showres ◦ ε = − tracein the case where Q = k [ x ] and m = ( x ). Let K be the Koszul complex on x , considered as adifferential Z / E = End mf ( x ) ( k [ x ] ( x ) , ( K ( x ) ). Recall fromSection 4.1 that E is the differential Z / Q -algebra generated by odd degree elements e, e ∗ satisfying the relations e = 0 = ( e ∗ ) and [ e, e ∗ ] = 1, and the differential d E is given by d E ( e ) = x and d E ( e ∗ ) = 0. By Lemma 4.2, we have an isomorphism k [ y ] ∼ = −→ HH ( mf ( x ) ( k [ x ] ( x ) , , where y id K [ e ∗ ] ∈ HH ( E ) ⊆ HH ( mf ( x ) ( k [ x ] ( x ) , , and, more generally, y j j ! id K [ j z }| { e ∗ | · · · | e ∗ ] , for j ≥ H R Γ ( x ) (Ω • k [ x ] ( x ) /k ) with k [ x ] ( x ) [ x − ] k [ x ] ( x ) · α ⊗ k [ x ] ( x ) Ω k [ x ] ( x ) /k , where | α | = 1. Theorem4.36 follows from the calculations(1) res( αx ⊗ dx ) = 1,(2) res( αx i ⊗ dx ) = 0 for all i > y ) = 1,(4) trace( y j ) = 0 for all j ≥ 1, and(5) ε ( y j ) = − j !( αx j +1 ⊗ dx ) for all j ≥ ε is induced by the diagram(4.39) k [ y ] ∼ = −→ HH ( E ) ∼ = ←− H R Γ ( x ) HH ( E ) R Γ ( x ) ε ′ −−−−−→ H R Γ ( x ) (Ω • k [ x ] ( x ) /k ) , where ε ′ denotes the composition HH ( E ) (id ,d K ) ∗ −−−−−→ HH II ( E ) ε −→ Ω • k [ x ] ( x ) /k . Here, E is the same as E , but with trivial differential, (id , d K ) is a morphism E → E of curved dga’s(with trivial curvature), and ε is as defined in 3.2.3.We need to calculate the inverse of the isomorphism H R Γ ( x ) HH ( E ) ∼ = −→ HH ( E ) occuring in (4.39).As usual, we make the identification R Γ ( x ) HH ( E ) = HH ( E ) ⊕ HH ( E )[1 /x ] · α. The differential on the right is ∂ := b + α , where α denotes left multiplication by α ; note that α = 0.So, for a class γ + γ ′ α , we have ∂ ( γ + αγ ′ ) = b ( γ ) − b ( γ ′ ) α + γα. With this notation, the quasi-isomorphism R Γ ( x ) HH ( E ) ≃ −→ HH ( E ) is given by setting α = 0.For j ≥ 0, we define y ( j ) = 1 j ! y j = id K [ j terms z }| { e ∗ | e ∗ | · · · | e ∗ ] and ω j = e [ j terms z }| { e ∗ | e ∗ | · · · | e ∗ ] ∈ HH ( E )[1 /x ] . Then, for j ≥ 0, we have b ( ω j ) = xy ( j ) − y ( j − , where y ( − := 0, from which we get b (cid:18) x ω j + 1 x ω j − + · · · + 1 x j +1 ω (cid:19) = y ( j ) . It follows that, for each j ≥ 0, the class y ( j ) + α (cid:18) x ω j + 1 x ω j − + · · · x j +1 ω (cid:19) is a cycle in R Γ ( x ) HH ( E ) that maps to y ( j ) ∈ HH ( E ) under the canonical map R Γ ( x ) ( HH ( E )) → HH ( E ). We conclude that the inverse of H R Γ ( x ) HH ( E ) ∼ = −→ HH ( E ) = k [ y ]maps y j to the class of η j := y j + j ! α (cid:18) x ω j + 1 x ω j − + · · · + 1 x j +1 ω (cid:19) for each j ≥ 0, and hence ε ( y j ) = R Γ ( x ) ε ′ ( η j ) . Recall that ε ′ sends θ [ θ | · · · | θ n ] ∈ HH ( E ) to X ( − j + ··· + j n n + J )! str( θ ( d ′ K ) j θ ′ · · · θ ′ n ( d ′ K ) j n ) , where the derivatives are computed relative to any specified flat connection on K . Using the Levi-Civita connection associated to the basis { , e } of K , we get e ′ = 0, ( e ∗ ) ′ = 0 and hence d ′ K = − e ∗ dx .It follows that ε ′ ( ω j ) = 0 for j ≥ ε ′ ( ω ) = str( e ) + str( ee ∗ dx ) ,ε ′ ( y ( j ) ) = 0 for j ≥ 1, and ε ′ ( y (0) ) = str(id K ) + str( e ∗ dx ) . It is easy to see that str( ee ∗ ) = − 1, str( e ∗ ) = 0, str( e ) = 0, and str(id K ) = 0, so that ε ′ ( ω ) = − dx , ε ′ ( ω j ) = 0 for all j ≥ 0, and ε ′ ( y j ) = 0 for all j . We obtain ε ( y j ) = R Γ ( x ) ε ′ ( η j ) = − j !( αx j +1 ⊗ dx )for all j ≥ 0, as needed. (cid:3) Proof of the conjecture. Let Q = C [ x , . . . , x n ] and f ∈ m = ( x , . . . , x n ) ⊆ Q , and assume m is the only singular point of the morphism f : Spec( Q ) → A . As discussed in the introduction, aresult of Shklyarov ([Shk16, Corollary 2]) states that there is a commutative diagram(4.40) HH n ( mf ( Q, f )) × I f (0) × I f (0) ∼ = / / c f η mf & & ▲▲▲▲▲▲▲▲▲▲▲▲▲ ( Ω nQ/k df ∧ Ω n − Q/k ) × h− , −i res { { ✈✈✈✈✈✈✈✈✈ C , for some constant c f which possibly depends on f . PROOF OF A CONJECTURE OF SHKLYAROV 31 Theorem 4.41. Let k be a field of characteristic , Q an essentially smooth k -algebra, m a k -rational maximal ideal, and f an element of m such that m is the only singularity of the morphism f : Spec( Q ) → A k . Then the diagram HH n ( mf ( Q, f )) × ε × ε ∼ = / / ( − n ( n +1) / η mf & & ▲▲▲▲▲▲▲▲▲▲▲▲▲ ( Ω nQ/k df ∧ Ω n − Q/k ) × h− , −i res { { ✈✈✈✈✈✈✈✈✈ k commutes.Proof. Consider the diagram(4.42) HH n ( mf ( Q, f )) × HH n ( mf ( Q, f )) id × Ψ (cid:15) (cid:15) ε × ε / / H n (Ω • Q , − df ) × H n (Ω • Q , − df ) id × ( − n (cid:15) (cid:15) HH n ( mf ( Q, f )) × HH n ( mf ( Q, − f )) ε × ε / / ⋆ (cid:15) (cid:15) H n (Ω • Q , − df ) × H n (Ω • Q , df ) ∧ (cid:15) (cid:15) HH n ( mf m ( Q m , ( − n ( n +1) / trace * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ ε / / H n R Γ m (Ω • Q m ) res u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ k. The top square commutes by Lemma 3.14, the square in the middle commutes by Corollary 3.26, andthe triangle at the bottom commutes by Theorem 4.36. By Lemma 4.23, the map HH n ( mf ( Q, f )) × HH n ( mf ( Q, f )) → k obtained by composing the maps along the left edge of (4.42) is ( − n ( n +1) / η mf . By Proposition4.34, the map Ω nQ/k df ∧ Ω n − Q/k ! × = H n (Ω • Q , − df ) × → k obtained by composing the maps along the right edge of (4.42) is h− , −i res . (cid:3) Corollary 4.43. Conjecture 1.4 holds. That is, for f ∈ m = ( x , . . . , x n ) ⊆ Q = C [ x , . . . , x n ] suchthat m is the only singularity of the morphism f : Spec( Q ) → A k , the unique constant c f that makesdiagram (1.3) commute is ( − n ( n +1) / , as predicted by Shklyarov.Proof. Under these assumptions, ε = I f (0) by Lemma 3.11. Theorem 4.41 thus implies that thevalue c f = ( − n ( n +1) / causes the diagram (4.40) to commute. As discussed in the introduction, thisuniquely determines the value of c f , and the unique constant c f which makes diagram (4.40) commuteis the same as that which makes diagram (1.3) commute. (cid:3) Recovering Polishchuk-Vaintrob’s Hirzebruch-Riemann-Roch formula for matrixfactorizations Assume k , Q , m , and f are as in the statement of Theorem 4.41. We recall that, given objects X, Y ∈ mf ( Q, f ), the Euler pairing applied to the pair ( X, Y ) is given by χ ( X, Y ) = dim k H Hom( X, Y ) − dim k H Hom( X, Y ) . In this final section, we give a new proof of a theorem due to Polishchuk-Vaintrob that relates theEuler pairing to the residue pairing via the Chern character map.The following is an immediate consequence of the commutativity of diagram (4.42) in the proof ofTheorem 4.41: Corollary 5.1. Let k , Q , m , and f be as in the statement of Theorem 4.41, and assume n = dim( Q m ) is even. Then the triangle HH ( mf ( Q, f )) ⊗ k HH ( mf ( Q, − f )) ε ⊗ k ε / / ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ Ω nQ/k df ∧ Ω n − Q/k ⊗ k Ω nQ/k df ∧ Ω n − Q/k h− , −i res x x qqqqqqqqqqq k commutes, where the left diagonal map is ( − n ( n +1) / trace ◦ ( − ⋆ − ) , and ε denotes the compositionof the HKR map and the isomorphism H n (Ω • Q/k , ± df ) ∼ = −→ Ω nQ/k df ∧ Ω n − Q/k . Let X ∈ mf ( Q, f ). We recall that the Chern character of Xch ( X ) ∈ HH ( mf ( Q, f )is the class represented by id X [] ∈ End( X ) ⊆ HH ( mf ( Q, f )) . Assume now that n is even. The isomorphism ε : HH ( mf ( Q, f )) ∼ = −→ Ω nQ/k df ∧ Ω n − Q/k sends ch ( X ) to the class 1 n ! str(( δ ′ X ) n ) , where δ ′ X = [ ∇ , δ X ] for any choice of connection ∇ on X . Abusing notation, we also denote thiselement of Ω nQ/k df ∧ Ω n − Q/k as ch ( X ).For example, if the components of X are free, then, upon choosing bases, we may represent δ X as a pair of square matrices ( A, B ) satisfying AB = f I r = BA . Using the Levi-Cevita connectionassociated to this choice of basis, we have(5.2) ch ( X ) = 2 n ! tr( n factors z }| { dAdB · · · dAdB ) . Recall from Remark 4.12 that, for X ∈ mf ( Q, f ) and Y ∈ mf ( Q, − f ), θ ( X, Y ) is given bydim k H ( X ⊗ Y ) − dim k H ( X ⊗ Y ) , and we have(5.3) θ ( X, Y ) = trace( ch ( X ) ⋆ ch ( Y )) . Corollary 5.4. Under the assumptions of Corollary 5.1, (1) If X ∈ mf ( Q, f ) and Y ∈ mf ( Q, − f ) , θ ( X, Y ) = ( − n ) h ch ( X ) , ch ( Y ) i res . (2) If X, Y ∈ mf ( Q, f ) , χ ( X, Y ) = ( − n ) h ch ( X ) , ch ( Y ) i res . Remark . Corollary 5.4 (2) is Polishchuk-Vaintrob’s Hirzebruch-Riemann-Roch formula for matrixfactorizations ([PV12, Theorem 4.1.4(i)]). Proof. (1) is immediate from Corollary 5.1 and (5.3). We now prove (2). Without loss of generality,we may assume Q is local, so that the underlying Z / Q -modules of X and Y are free. Givena matrix factorization ( P, δ P ) ∈ mf ( Q, f ) written in terms of its Z / δ : P → P , δ : P → P ) , PROOF OF A CONJECTURE OF SHKLYAROV 33 we define a matrix factorization N ( P, δ P ) ∈ mf ( Q, − f ) with components( δ : P → P , − δ : P → P ) . We have h ch ( X ) , ch ( N ( Y )) i res = ( − n ) θ ( X, N ( Y )) = χ ( X, Y ) . The first equality follows from (1), and the second equality follows from [BW19b, Corollary 8.5] and[BMTW17, Proposition 3.18]; note that ( − n ) = ( − n , since n is even, and also that the notation χ in [BMTW17, Proposition 3.18] has a different meaning than it does here. 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Department of Mathematics, University of Wisconsin-Madison, WI 53706-1388, USA E-mail address : [email protected] Department of Mathematics, University of Nebraska-Lincoln, NE 68588-0130, USA E-mail address ::