A Chern-Weil formula for the Chern character of a perfect curved module
aa r X i v : . [ m a t h . K T ] S e p A CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECTCURVED MODULE
MICHAEL K. BROWN AND MARK E. WALKER
Abstract.
Let k be a field of characteristic 0 and A a curved k -algebra. We obtain a Chern-Weil-type formula for the Chern character of a perfect A -module taking values in HN II ( A ), the negativecyclic homology of the second kind associated to A , when A satisfies a certain smoothness condition. Contents
1. Introduction 12. Curved differential graded categories 52.1. Basic definitions 62.2. Modules over cdg categories 93. Mixed Hochschild complexes and the HKR theorem 113.1. Mixed Hochschild complexes 113.2. Negative cyclic homology of cdg categories 173.3. The HKR theorem 184. Chern character maps for curved differential graded categories 194.1. The Chern character map for dg categories 194.2. The Chern character map of the second kind 204.3. The Chern character map of the second kind for cdg modules over a cdga 215. The main theorem 225.1. Connections, curvature, and the trace map 235.2. Statement of the main theorem 245.3. Key technical result 255.4. Proof of the main theorem 276. Examples 28Appendix A. Hochschild homology and cohomology via twisting cochains 33Appendix B. Proof of Theorem 5.19 37References 441.
Introduction
Our goal is to obtain a Chern-Weil-type formula for the Chern character map associated to certaincurved algebras. Let us explain what we mean.In its most classical form, the Chern character map is a ring homomorphism from the Grothendieckgroup of complex vector bundles on a finite CW complex X to the even part of its rational singularcohomology: ch : KU ( X ) → M i H i ( X, Q ) . MB and MW gratefully acknowledge support from the National Science Foundation (NSF award DMS-1502553) andthe Simons Foundation (grant
When X is a smooth manifold, by extending coefficients from Q to C and composing with the isomor-phism to the de Rham cohomology of X with complex coefficients, one obtains the de Rham Cherncharacter map ch dR : KU ( X ) → M i H idR ( X ; C ) . For a smooth complex vector bundle V , Chern-Weil theory provides an explicit formula for ch dR ( V ) inthe following way. Let Ω • ( X ) denote the complexified de Rham complex of X , and choose a C -linearconnection ∇ : Γ( V ) → Γ( V ) ⊗ Ω ( X ) Ω ( X )on V . Let R := ∇ : Γ( V ) → Γ( V ) ⊗ Ω ( X ) Ω ( X )denote the associated curvature. Regarding R as an element of End Ω ( X ) (Γ V ) ⊗ Ω ( X ) Ω ( X ), one has ch dR ( V ) = tr( e R ) = X i tr( R i ) i ! ∈ M i H idR ( X ; C ) , where tr = tr End
Ω0( X ) (Γ V ) ⊗ id : End Ω ( X ) (Γ V ) ⊗ Ω ( X ) Ω • ( X ) → Ω • ( X )is the trace map.The de Rham Chern character map and Chern-Weil formula have purely algebraic analogues. Forthe rest of this introduction, let k be a field of characteristic 0 and X a smooth variety over k .Grothendieck defines a very general Chern character map ch : K ( X ) → CH ∗ ( X ) ⊗ Q taking values in the rationalized Chow groups of X (in fact, this map induces an isomorphism K ( X ) ⊗ Q ∼ = −→ CH ∗ ( X ) ⊗ Q ). For each i , there is a cycle class map CH i ( X ) ⊗ Q → H idR ( X/k )taking values in the algebraic de Rham cohomology of X relative to the base field k , and the algebraicde Rham Chern character map is the composition: K ( X ) ch dR −−−→ M i H idR ( X/k ) . As in the topological setting, ch dR ( V ) may be given an explicit formula by choosing an algebraicconnection ∇ : V → V ⊗ Ω X/k . (When X is affine, every vector bundle admits such a connection,but for vector bundles over non-affine schemes, connections typically do not exist. When X is quasi-projective, this difficulty may be circumvented by using Jouanolou’s device to reduce to the affinecase.)The algebraic de Rham Chern character for smooth k -varieties can be recast in another way usingthe Hochschild-Kostant-Rosenberg isomorphismHP ( X ) ∼ = M i H idR ( X/k ) , where HP denotes periodic cyclic homology . This gives a map ch HP : K ( X ) → HP ( X ) . In fact, ch HP factors through the canonical map HN ( X ) → HP ( X ), where HN denotes negativecyclic homology , giving a map ch HN : K ( X ) ch HN −−−−→ HN ( X ) . The map ch HN may be vastly generalized: one may replace X by any k -linear differential graded(dg) category , i.e., a category enriched over complexes of k -vector spaces. See Section 2.1 below forthe precise definition of a dg category, and see Section 4.1 for the definition of the map ch HN in thesetting of dg categories. CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 3
There is a variant of negative cyclic homology called negative cyclic homology of the second kind (see Section 3.2 below), and there is a canonical mapcan : HN ( A ) → HN II ( A )for any dg category A . By composing ch HN with the map can, we obtain the Chern character mapof the second kind ch IIHN : K ( A ) → HN II ( A )for any dg-category A . The Chern character map of the second kind admits an even further general-ization: we may take A to be a curved dg category. Hochschild-type invariants of curved dg categories(or cdg categories , for short) have been intensively studied in recent work of, for instance, C˘ald˘araru-Tu [CT13], Efimov [Efi12], Platt [Pla12], Polishchuk-Positselski [PP12] and Segal [Seg13]; we refer thereader to Section 2.1 below for background on cdg categories.We will mostly be interested in a certain family of cdg categories called essentially smooth curved k -algebras, which are cdg categories having only one object satisfying an appropriate smoothnesscondition. In detail, an essentially smooth curved k -algebra is a pair ( A, h ), where • A is a commutative Z - or Z / k -algebra that is concentrated in even degrees, • h is an element of degree 2, and • upon forgetting the grading, A is an essentially smooth k -algebra in the classical sense.Fix an essentially smooth curved k -algebra A = ( A, h ). It turns out that the ordinary negativecyclic homology HN ∗ ( A ) is 0 when h = 0 [PP12, Section 2.4], but HN II ∗ ( A ) is typically of interest:there is a variation of the Hochschild-Kostant-Rosenberg isomorphism taking the form ǫ : HN II ( A ) ∼ = −→ H (Ω • A/k [[ u ]] , ud + dh ) , where (Ω • A/k , d ) is the classical de Rham complex, u is a formal parameter of degree 2, and dh is left-multiplication by the element dh ∈ Ω A/k . (With our indexing convention, the de Rham differential d has degree −
1, and hence ud has degree 1.) See Section 3.3 for more details.Our main result gives a Chern-Weil-type formula for the Chern character map of the second kindassociated to A . Before stating it, we recall some more terminology. A perfect right A -module is apair ( P, δ P ), where • P is a finitely generated projective right graded A -module, and • δ P is a degree 1 A -linear endomorphism of P such that δ P = − ρ h , where ρ h denotes rightmultiplication by h .A perfect right A -module may equivalently be viewed as a “graded matrix factorization”; see Remark2.16 for details.A connection on a perfect right A -module ( P, δ P ) is a k -linear map ∇ : P → P ⊗ A Ω A/k of degree − ∇ ( pa ) = ∇ ( p ) a + ( − | p | p ⊗ d ( a ) for all a ∈ A and p ∈ P . That is, a connectionon ( P, δ P ) is just a connection on the projective A -module P which preserves the grading; whenΓ = Z / Z , such a connection is called a superconnection [Qui85]. A connection on ( P, δ P ) extends ina natural way to a k -linear endomorphism of P ⊗ A Ω • A,k , and, in particular, we have an A -linear map ∇ = ∇ ◦ ∇ : P → P ⊗ A Ω A/k . We define the curvature associated to such a connection to be the A -linear map R : P → P ⊗ A Ω • A/k [[ u ]]given by R = u ∇ + [ ∇ , δ P ] , where [ ∇ , δ P ] := ∇ ◦ δ P + ( δ P ⊗ id) ◦ ∇ . This definition is inspired by Quillen’s formula in [Qui85] forthe Chern character of a relative topological K -theory class. MICHAEL K. BROWN AND MARK E. WALKER
Using the canonical isomorphismHom A ( P, P ⊗ A Ω • A/k [[ u ]]) ∼ = End A ( P ) ⊗ A Ω • A/k [[ u ]] , we regard R as an element of the right-hand side, which has a natural ring structure. In particular,powers R i make sense for i ≥ Theorem (see Theorem 5.7 below) . Let A = ( A, h ) be an essentially smooth Z - or Z / -graded curvedalgebra over a field k of characteristic , and let P = ( P, δ P ) be a perfect right A -module. For anychoice of connection ∇ on P with associated curvature R , the Chern-Weil formula (1.1) ǫ ◦ ch IIHN ( P ) = tr(exp( − R )) holds, where ǫ is the HKR isomorphism, tr(exp( − R )) = id − tr( R ) + tr( R )2! − tr( R )3! + · · · ∈ H (Ω • A/k [[ u ]] , ud + dh ) , and tr denotes the trace map tr End A ( P ) ⊗ id : End A ( P ) ⊗ A Ω • A/k [[ u ]] → Ω • A/k [[ u ]] .Remark . The alternating sumid − tr( R ) + tr( R )2! − tr( R )3! + · · · is finite, since Ω iA/k = 0 for i > dim( A ). The image of ǫ ◦ ch IIHN ( P ) is therefore contained in the imageof the canonical map H (Ω • A/k [ u ] , ud + dh ) → H (Ω • A/k [[ u ]] , ud + dh ) . Remark . The formula in the main theorem closely resembles Quillen’s formula in [Qui85] for theChern character of a relative K -theory class. We have not yet been able to make the relationshipbetween the two precise.There is a commutative square(1.4) HN II ( A ) p (cid:15) (cid:15) ǫ / / H (Ω • A/k [[ u ]] , ud + dh ) u =0 (cid:15) (cid:15) HH II ( A ) ǫ / / H (Ω • A/k , dh ) , where HH II denotes Hochschild homology of the second kind (see Section 3.1), p is a canonical map,and the bottom map is the HKR isomorphism for Hochschild homology of the second kind, which wealso denote by ǫ . Set ch IIHH := p ◦ ch IIHN . Using the main theorem and (1.4), we obtain the followingformula for ǫ ◦ ch IIHH : ǫ ◦ ch IIHH ( P ) = tr(exp( − [ ∇ , δ P ])) ∈ H (Ω • A/k , dh ) . Polishchuk-Vaintrob and Segal also obtain formulas for ǫ ◦ ch IIHH in [PV12, Corollary 3.2.4] and [Seg13,3.2], using different methods, and Platt generalizes these results to the global setting in [Pla12]. SeeExample 6.1 and Remark 5.23 for comparisons between our results and those of Polishchuk-Vaintroband Segal.The main theorem recovers the classical Chern-Weil formula for the Chern character of a projectivemodule over a commutative ring:
Example 1.5.
Suppose A = ( A, A is a smooth Z -graded k -algebra concentrated in degree0. Then one has isomorphismsHN ( A ) ∼ = −→ HN II ( A ) ∼ = −→ H (Ω • A/k [ u ] , ud ) = M j H jdR ( A ) u j . CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 5 If P is a finitely generated projective A -module, we may regard P as an object P = ( P, ∈ Perf( A ) =Perf( A op ). Choose a connection ∇ on P . The main theorem implies( ǫ ◦ ch IIHN )( P ) = tr(exp( −∇ u )) = X j ( − j tr( ∇ j ) j ! u j ∈ M j H jdR ( A ) u j . This coincides, up to a sign, with the classical formula for the Chern character of a projective moduleover a commutative ring (see, for instance, [Lod13, Section 8.1]).Concerning the sign difference: let τ be the endomorphism of H (Ω • A/k [[ u ]] , ud + dh ) that sends u to − u . We have τ ( ǫ ( ch IIHN ( P ))) = tr(exp( R ′ )) , where R ′ := u ∇ − [ ∇ , δ ]. So, in the setting of Example 1.5, applying τ recovers the classical formulafor the Chern character of a projective module on the nose. The reason we do not just define the HKRmap to be τ ◦ ǫ , thereby eliminating this sign discrepancy, is that applying τ destroys the k [[ u ]]-linearityof ǫ .In Sections 2 and 3, we recall some basic definitions and results concerning curved dg categories andtheir associated Hochschild-type invariants. All of the results in these sections have appeared, or havebeen hinted at, in the literature. We define the Chern character of the second kind in Section 4, and weprove the main theorem in Section 5. Section 6 contains several examples. In Appendix A, we justifyour choice of sign convention in the Hochschild complex of the second kind; our convention differsfrom the one in [PP12]. Appendix B is devoted to a proof of a key technical result (Theorem 5.19)needed for the proof of the main theorem. Notational conventions: • k denotes a field. • Throughout, we wish to work simultaneously with both Z -graded and Z / k -vectorspaces, algebras, etc., and so we let Γ denote either Z or Z / k -vectorspaces, k -algebras, etc. We adhere to cohomological indexing, so that the degree j part of a Z -graded vector space V is written V j . When subscripts are used, it is understood that V j denotes V − j . • A differential on a Γ-graded k -vector space is a homogeneous endomorphism d of degree 1.Note that we do not assume d = 0 in general. • An element of a Γ-graded k -vector space V shall always be taken to mean a homogeneouselement, unless otherwise stated. • [ − , − ] means graded commutator: for any Γ-graded ring A and (homogeneous) elements x, y ∈ A , we set [ x, y ] := xy − ( − | x || y | yx . We say a graded ring is commutative if [ x, y ] = 0for all (homogeneous) elements x and y , and x = 0 for all elements x of odd degree (thelatter is redundant if 2 is invertible in A ). • If V is a Γ-graded k -vector space, the suspension Σ V of V is given by setting (Σ V ) i = V i +1 .If V is equipped with a differential d , then Σ V is equipped with the differential − d . We write s : V → Σ V for the evident degree − v ∈ V i to itself but regardedas an element of (Σ V ) i − . Acknowledgements.
We thank the referee for catching an error in the original proof of Theorem5.19. 2.
Curved differential graded categories
We recall some general notions concerning curved differential graded categories, closely followingPolishchuk-Positselski [PP12, Section 1].
MICHAEL K. BROWN AND MARK E. WALKER
Basic definitions.Definition 2.1.
Let k be a field, and let Γ be either Z or Z /
2. A k -linear curved differential Γ -gradedcategory A (or just “cdg category”, for short) consists of the following data: • a collection of objects Ob( A ); • for each pair X, Y ∈ Ob( A ) of objects, a Γ-graded k -module Hom A ( X, Y ) equipped with a k -linear differential δ X,Y (i.e., a degree one endomorphism that need not square to 0); • a composition law: that is, for any X, Y, Z ∈ Ob( A ), a morphism of Γ-graded k -modules µ X,Y,Z : Hom A ( Y, Z ) ⊗ k Hom A ( X, Y ) → Hom A ( X, Z ); • for all X ∈ Ob( A ), an identity morphism 1 X ∈ Hom A ( X, X ); • for every X ∈ Ob( A ), a degree 2 element h X ∈ Hom A ( X, X ) called a curvature element .We will write δ for δ X,Y and µ for µ X,Y,Z when the underlying objects are understood. We willsometimes also use the usual − ◦ − notation for composition, rather than µ .The above data is required to satisfy the following conditions: • For all
W, X, Y, Z ∈ Ob( A ), the diagramHom A ( Y, Z ) ⊗ Hom A ( X, Y ) ⊗ Hom A ( W, X ) id ⊗ µ / / µ ⊗ id (cid:15) (cid:15) Hom A ( Y, Z ) ⊗ Hom A ( W, Y ) µ (cid:15) (cid:15) Hom A ( X, Z ) ⊗ Hom A ( W, X ) µ / / Hom A ( W, Z )commutes. • For each X ∈ Ob( A ), the compositionsHom A ( X, Y ) ∼ = −→ Hom A ( X, Y ) ⊗ k id ⊗ X −−−−→ Hom A ( X, Y ) ⊗ Hom A ( X, X ) µ −→ Hom A ( X, Y )andHom A ( X, Y ) ∼ = −→ k ⊗ Hom A ( X, Y ) Y ⊗ id −−−−→ Hom A ( Y, Y ) ⊗ Hom A ( X, Y ) µ −→ Hom A ( X, Y )are equal to the identity. • For all objects X , Y , and Z , the squareHom A ( X, Y ) ⊗ Hom A ( Y, X ) δ ⊗ id + id ⊗ δ / / µ (cid:15) (cid:15) Hom A ( X, Y ) ⊗ Hom A ( Y, X ) µ (cid:15) (cid:15) Hom A ( X, Z ) δ / / Hom A ( X, Z ) . commutes. Explicitly, given elements f ∈ Hom A ( X, Y ) and g ∈ Hom A ( Y, Z ), we have δ ( µ ( g ⊗ f )) = µ ( δ ( g ) ⊗ f ) + ( − | g | µ ( g ⊗ δ ( f )) . • For all objects X and Y and all f ∈ Hom A ( X, Y ), we have δ ( f ) = µ ( h Y ⊗ f ) − µ ( f ⊗ h X ) . • δ ( h X ) = 0 for all X ∈ Ob( A ).A cdg category in which h X = 0 for all X is called a k -linear differential Γ -graded category , or just a dg category for short. Example 2.2.
A basic example of a cdg category is given by precomplexes of k -vector spaces : • an object of Pre( k ) is a Γ-graded k -vector space V equipped with a k -linear endomorphism d V of degree 1; • Hom
Pre( k ) ( W, V ) is the Γ-graded k -vector space whose degree n component is Q j Hom k ( W j , V j + n ),and it is equipped with the differential which sends f to d V f − ( − | f | f d W ; • the curvature element h V ∈ Hom
Pre( k ) ( V, V ) is d V ; and CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 7 • the unit and composition maps are the obvious ones. Definition 2.3.
Let A and A ′ be cdg categories. A cdg functor F : A → A ′ is given by the followingdata: • a function F : Ob( A ) → Ob( A ′ ); • for all X, Y ∈ Ob( A ), a morphism of graded k -vector spaces F X,Y : Hom A ( X, Y ) → Hom A ′ ( F ( X ) , F ( Y )); • for all X ∈ Ob( A ), a degree 1 element a X ∈ Hom A ′ ( F ( X ) , F ( X )).We will write F for F X,Y when the underlying objects are understood. These data must satisfythe following conditions: • F preserves composition: that is, for all X, Y, Z ∈ Ob( A ), µ F ( X ) ,F ( Y ) ,F ( Z ) ◦ ( F Y,Z ⊗ F X,Y ) = F X,Z ◦ µ X,Y,Z ; • F preserves identities: i.e., for each X ∈ Ob( A ) we have F X,X ◦ X = 1 F ( X ) ; • for all X, Y ∈ Ob( A ) and f ∈ Hom A ( X, Y ), F ( δ ( f )) = δ ( F ( f )) + µ ( a Y ⊗ F ( f )) − ( − | f | µ ( F ( f ) ⊗ a X ); • for all X ∈ Ob( A ), F ( h X ) = h F ( X ) + δ ( a X ) + a X . A cdg functor is called strict if a X = 0 for all X . A strict cdg functor between dg categories iscalled a dg functor .A (strict) quasi-cdg functor , or (strict) qdg functor , F : A → A ′ consists of the same data as a(strict) cdg functor, and it satisfies the same conditions as a cdg functor except for the last one: we donot require the equation relating F ( h X ) and h F ( X ) to hold. Composition of qdg functors F : A → A ′ and G : A ′ → A ′′ is defined in the obvious way on objects and morphisms, and, for each X ∈ A , thedistinguished degree 1 element of Hom A ′′ (( G ◦ F )( X ) , ( G ◦ F )( X )) is given by G ( a X ) + b F ( X ) , where { a X } X ∈A and { b X ′ } X ′ ∈A ′ are the families of distinguished elements associated to F and G . Example 2.4.
A cdg category with just one object is a called a curved differential graded algebra ,or cdga . This amounts to a triple (
A, d, h ), where A is a unital Γ-graded k -algebra, d is a degree one k -linear endomorphism of A , and h is a degree 2 element of A such that • d satisfies the Leibniz rule: d ( ab ) = d ( a ) b + ( − | a | ad ( b ); • d ( a ) = ha − ah ; • d ( h ) = 0.A curved algebra is a cdga with trivial differential, i.e. a Γ-graded k -algebra A equipped with aspecified degree 2 element h such that ha = ah for all a ∈ A . We will often write a cdga as a triple( A, d, h ) and a curved algebra as a pair (
A, h ).Unravelling the definitions, a morphism (
A, d, h ) → ( A ′ , d ′ , h ′ ) of cdga’s is given by a pair ( ρ, β ),with ρ : A → A ′ a homomorphism of unital Γ-graded k -algebras and β ∈ A ′ 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 .A strict morphism ( A, d, h ) → ( A ′ , d ′ , h ′ ) is given by a map ρ : A → A ′ of Γ-graded k -algebras suchthat ρ ◦ d = d ′ ◦ ρ and ρ ( h ) = h ′ .As a simple example of a non-strict morphism of cdga’s, take A to be an exterior algebra on a setof degree 1 variables, and consider it as a curved algebra with trivial curvature. Let β ∈ A be a degree1 element. Then (id A , β ) : ( A, → ( A, MICHAEL K. BROWN AND MARK E. WALKER
Example 2.5.
The following example of a cdga explains the presence of the word “curved” in theterminology introduced above. Let X be a smooth manifold and V a smooth complex vector bundleon X . Let Ω • ( X ) denote the complexified de Rham complex of X , and let ∇ be a connection on V ,i.e. a C -linear map Γ( V ) → Γ( V ) ⊗ Ω ( X ) Ω ( X )such that ∇ ( f s ) = f ∇ ( s ) + s ⊗ df for s ∈ Γ( V ) and f ∈ Ω ( X ). ∇ extends uniquely to a C -linearderivation e ∇ of Γ( V ) ⊗ Ω ( X ) Ω • ( X ). Set R := e ∇ | Γ( V ) ; R is called the curvature of ∇ . Note that R may be identified with an element of the C -algebra End Ω ( X ) (Γ( V )) ⊗ Ω ( X ) Ω • ( X ).The connection ∇ on V induces a connection ∇ ′ on End( V ) given by [ ∇ , − ] which yields a derivation f ∇ ′ of End Ω ( X ) (Γ( V )) ⊗ Ω ( X ) Ω • ( X ). The triple(End Ω ( X ) (Γ( V )) ⊗ Ω ( X ) Ω • ( X ) , f ∇ ′ , R )is a curved differential C -algebra [Pos93, Section 4.1]. Notice that the curvature of this cdga coincideswith the curvature of ∇ . Definition 2.6. If A is a cdg category, its opposite , A op , is defined as follows: • Ob( A op ) = Ob( A ); • Hom A op ( X, Y ) := Hom A ( Y, X ), equipped with the differential δ Y,X ; • for X, Y, Z ∈ Ob( A op ) and elements f ∈ Hom A op ( X, Y ) and g ∈ Hom A op ( Y, Z ), µ op X,Y,Z ( g ⊗ f ) = ( − | f || g | µ Z,Y,X ( f ⊗ g ); • for all X ∈ Ob( A op ), the curvature element in Hom A op ( X, X ) is − h X .Given a cdg functor F : A → A ′ , the opposite functor F op : A op → ( A ′ ) op is defined in the evidentway on objects and morphisms, and, for each object X ∈ A , the distinguished degree 1 element ofHom ( A ′ ) op ( F ( X ) , F ( X )) is − a X . Example 2.7. If A = ( A, d, h ) is a cdga, A op is the cdga ( A op , d, − h ), where A op has the multiplicationrule ⋆ given by x ⋆ y = ( − | x || y | yx . Given a morphism ( ρ, β ) : ( A, d, h ) → ( A ′ , d ′ , h ′ ) of cdga’s,( ρ, − β ) : ( A op , d, − h ) → (( A ′ ) op , d ′ , − h ′ )is the induced opposite map.Polishchuk-Positselski introduce in [PP12] the notion of a pseudo-equivalence of cdg categories, andit will be a useful notion in this paper: Definition 2.8.
Let A be a cdg category, and fix an object X ∈ A . • X is called the direct sum of a family of objects { X α } α ∈ I of A if there exist degree 0 morphisms i α : X α → X for each α ∈ I such that(1) δ X α ,X ( i α ) = 0 for all α ;(2) the induced map Hom A ( X, Y ) → Q α Hom A ( X α , Y ) is an isomorphism of Γ-graded k -vector spaces for all Y .In this situation, each X α is referred to as a direct summand of X . • Fix a degree 1 endomorphism τ of X . An object Y ∈ A is called a twist of X with τ if thereexist degree 0 morphisms i : X → Y and j : Y → X such that j ◦ i = id X , i ◦ j = id Y , and j ◦ δ X,Y ( i ) = τ . • Given n ∈ Z , an object Y ∈ A is called an n th shift of X if there exist morphisms i : X → Y and j : Y → X of degrees n and − n , respectively, such that ji = id X , ij = id Y , and δ X,Y ( i ) = 0 = δ Y,X ( j ). • A cdg functor F : A → A ′ is a pseudo-equivalence if F induces isomorphismsHom A ( Y, Z ) ∼ = −→ Hom A ′ ( F ( Y ) , F ( Z ))for all Y, Z ∈ A , and every object of A ′ may be obtained from objects of the form F ( X )for X ∈ Ob( A ) via the operations of finite direct sum, shift, twist, and passage to a directsummand. CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 9
Modules over cdg categories.
Fix a cdg category A . Definition 2.9. A left A -module (resp. left quasi- A -module ) is a strict cdg (resp. qdg) functor F : A →
Pre( k ) . Right A -modules and quasi- A -modules are defined similarly, replacing F : A →
Pre( k ) with F : A op → Pre( k ). Example 2.10.
Fix an object X of A . The representable functor given, on objects, by Y Hom A ( X, Y ) (resp. Y Hom A ( Y, X )) is a left (resp. right) quasi-module. Each is a moduleprovided h X = 0. Definition 2.11.
A (left or right) quasi- A -module is called perfect if the induced ordinary functorbetween the k -linear Γ-graded categories underlying A and Pre( k ) is, up to a shift, a direct summandof a finite direct sum of representable functors.Let us unravel these definitions in the case where A is a cdga: Example 2.12.
Suppose A is a cdga ( A, d, h ). Then we identify a left quasi-module F : A →
Pre( k )with the value of F on the unique object of A . That is, a left quasi-module over A is pair ( M, δ M ),where M is a graded left A -module, and δ M : M → M is a k -linear map of degree 1 satisfying theLeibniz rule: δ M ( am ) = d ( a ) m + ( − | a | aδ M ( m ) for all a ∈ A , m ∈ M .A left quasi-module ( M, δ M ) is a left module if δ M = λ h , where λ h denotes left multiplication bythe curvature element h of A . A left quasi-module ( M, δ M ) is perfect if M is finitely generated andprojective as a graded left A -module.Using Example 2.7, one sees that a right quasi-module/module over ( A, d, h ) is the same thingas a left quasi-module/module over ( A op , d, − h ). In other words, a right quasi-module over ( A, d, h )consists of a graded right A -module M and a k -linear map δ M : M → M of degree one that satisfies δ M ( ma ) = δ M ( m ) a + ( − | m | md ( a ). Such a pair ( M, δ M ) is a right module if δ M = − ρ h , where − ρ h ( m ) = − mh .Notice that there is no natural sense in which ( A, d ) is a left or right A -module, unless h = 0,but ( A, d ) is both a left and right quasi- A -module in the standard ways. Henceforth, we will abusenotation slightly and say that A is a (left and right) quasi- A -module. Remark . Recall that, given a Γ-graded ring R and Γ-graded left R -modules N, N ′ , a degree e map f : N → N ′ of Γ-graded abelian groups is R -linear if f ( rn ) = ( − | r | e rf ( n )for all elements n ∈ N and r ∈ R . Observe that, if A is a curved algebra and ( M, δ M ) is a left A -module, then δ M is A -linear.It is observed in [PP12, Section 1] that left quasi- A -modules form a cdg category qMod( A ), and left A -modules form a dg category Mod( A ). Let qPerf( A ) denote the cdg subcategory of qMod( A ) con-sisting of perfect left quasi-modules, and let Perf( A ) denote the dg subcategory of Mod( A ) consistingof perfect left modules. Example 2.14.
When A is a cdga ( A, d, h ), the cdg category qPerf( A ) is defined as follows: • Objects are pairs (
P, δ P ) that are isomorphic to summands of quasi-modules of the formΣ n A ⊕ · · · ⊕ Σ n r A for some r ≥ n i ∈ Z . Note that, in particular, P is a finitelygenerated graded projective left A -module. • The morphisms (
P, δ P ) → ( P ′ , δ P ′ ) are given by the Γ-graded k -module Hom A ( P, P ′ ) of A -linear maps from P to P ′ , equipped with the differential ∂ = ∂ P,P ′ defined by ∂ ( g ) = δ P ′ ◦ g − ( − | g | g ◦ δ P on homogeneous maps g ; • composition is defined in the obvious way; • the curvature element associated to an object ( P, δ P ) is h P = δ P − λ h , where λ h denotes leftmultiplication by the element h .Notice that an object P ∈ qPerf( A ) has trivial curvature if and only if δ P = λ h , which is preciselythe extra property required of a quasi-module to make it a module. Thus, Perf( A ) is the full cdgsubcategory of qPerf( A ) consisting of those objects whose endomorphism dga’s have trivial curvature.In particular, Perf( A ) is a dg category. Remark . If P = ( P, δ P ) is a perfect right quasi-module over A = ( A, d, h ) (i.e., an objectin qPerf( A op )), then its endomorphism cdga is (End A ( P ) , [ δ P , − ] , δ P + ρ h ), where ρ h denotes rightmultiplication by h . Remark . If A = ( A, h ) is a curved algebra, and A is commutative upon forgetting the grading,a perfect right A -module ( P, δ P ) is precisely the data of a “graded matrix factorization of − h ”. Indetail, set P even = L j P j and P odd = Σ − (cid:16)L j P j +1 (cid:17) , each of which is a graded A -module in theevident way. Then the differential δ P determines a pair of A -linear maps α : P odd → P even and β : P even → Σ P odd such that β ◦ α = − h and Σ ( α ) ◦ β = − h . In particular, when Γ = Z / Z , the category of perfect right A -modules is identical to the differential Z / A, − h ) of matrix factorizations of − h , as defined in [Dyc11, Definition 2.1]. Example 2.17.
Let A be a cdga, let P = ( P, δ P ) be an object of Perf( A ), and define P ♮ := ( P, ∈ qPerf( A ). Then P ♮ and P are twists of one another. Thus, letting {P , P ♮ } denote the full cdgsubcategory of qPerf( A ) consisting of these two objects, we see that the inclusions {P} ֒ → {P , P ♮ }{P ♮ } ֒ → {P , P ♮ } are pseudo-equivalences of cdg categories. This fact will be used in the proof of Proposition 4.11.For any cdg category A , there is a “quasi-Yoneda” embedding(2.18) qY A : A ֒ → qPerf( A op )given, on objects, by X Hom A ( − , X ). If A has trivial curvature, then the image of qY A lands inthe full subcategory Perf( A op ), giving the more classical Yoneda embedding. We will write Y A forthis functor. Example 2.19.
Suppose A is a cdga ( A, d, h ), which we regard as a cdg category with one object ∗ whose endomorphism cdga is A . Under the identification of Example 2.12, the object qY A ( ∗ ) ∈ qPerf( A op ) is given by the pair ( A, d ), where A is regarded as a right module over itself. Moreover,the endomorphism cdga End qPerf( A op ) ( qY A ( ∗ )) has as its underlying algebra the collection End right A ( A )of right A -module endomorphisms of A with multiplication given by composition. There is a strictisomorphism of cdga’s A ∼ = −→ End qPerf( A op ) ( qY A ( ∗ )) = (End right A ( A ) , [ d, − ] , ρ h )that sends a ∈ A to λ a , where λ a ( x ) = ax .The following result is proven in [PP12, Lemma A, page 5319]: Proposition 2.20 (Polishchuk-Positselski) . For any cdg category A , the inclusions Perf( A op ) ֒ → qPerf( A op ) ← ֓ A are pseudo-equivalences. CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 11 Mixed Hochschild complexes and the HKR theorem
In this section, we define the mixed Hochschild complexes of the first and second kinds associatedto a curved differential Γ-graded category, and we discuss some of their properties. All of the resultsin this section are stated, or at least alluded to, in the literature; see, for instance, [Efi12], [PP12], or[Shk14].3.1.
Mixed Hochschild complexes.
We begin by recalling the notion of a mixed complex. Recallthat k denotes a field, and Γ ∈ { Z , Z / } . Let k [ B ] be the commutative Γ-graded k -algebra freelygenerated by a degree − B , so that k [ B ] is a two dimensional k -vector space, spanned by1 and B , with B = 0. A mixed complex over k is a dg- k [ B ]-module, where k [ B ] is considered as adg algebra with trivial differential. Equivalently, a mixed complex is a Γ-graded k -vector space M equipped with endomorphisms b and B of degrees 1 and −
1, respectively, such that b = 0 = B and[ b, B ] = bB + Bb = 0. We will write mixed complexes as triples ( M, b, B ). The notions of morphismsand quasi-isomorphisms of mixed complexes are the standard ones for dg- k [ B ]-modules. In detail, amorphism ( M, b, B ) → ( M ′ , b ′ , B ′ ) of mixed complexes is given by a graded map of k -vector spacesof degree 0 that commutes with b, b ′ and B, B ′ . Such a morphism is a quasi-isomorphism if it is soregarded as a map ( M, b ) → ( M ′ , b ′ ) of k -complexes.We associate to any curved differential Γ-graded category A two mixed complexes, Hoch( A ) andHoch II ( A ) (cf. [Efi12]), called the (ordinary) Hochschild complex and Hochschild complex of the secondkind . We give detailed constructions when A is a cdga ( A, d, h ) (i.e., a cdg category with just oneobject), and then we sketch how to extend the definitions to the general setting.For a cdga A = ( A, d, h ), define the Γ-graded k -vector spaceHoch( A ) = M n ≥ A ⊗ k (Σ A ) ⊗ n , where Σ A is the shift of A , i.e. (Σ A ) i = A i +1 . As a standard bit of shorthand, for elements a , a , . . . , a n ∈ A , we set a [ a | . . . | a n ] := a ⊗ sa ⊗ · · · ⊗ sa n ∈ A ⊗ k (Σ A ) ⊗ n where s : A → Σ A is the canonical degree − e in Hoch( A ) is afinite sum of elements of the form a [ a | . . . | a n ] such that | a | + | a | + · · · + | a n | − n = e .Equip Hoch( A ) with a differential b = b + b + b , where(3.1) b ( a [ a | . . . | a n ]) = ( − | a | a a [ a | · · · | a n ]+ n − X j =1 ( − | a | + ··· + | a j |− j a [ a | · · · | a j a j +1 | · · · | a n ] − ( − ( | a n |− | a | + ··· + | a n − |− ( n − a n a [ a | · · · | a n − ] , (3.2) b ( a [ a | . . . | a n ]) = d A ( a )[ a | · · · | a n ]+ n X j =1 ( − | a | + ··· + | a j − |− j a [ a | · · · | d ( a j ) | · · · | a n ] , and(3.3) b ( a [ a | . . . | a n ]) = n X j =0 ( − | a | + | a | + ··· + | a j |− j a [ a | · · · | a j | h | a j +1 | · · · | a n ] . Define Hoch II ( A ) to be the Γ-graded k -vector spaceHoch II ( A ) = Y n ≥ A ⊗ k (Σ A ) ⊗ n . The relationship between Hoch( A ) and Hoch II ( A ) can be described in terms of the descending filtra-tion F ⊇ F ⊇ · · · on Hoch( A ) defined by(3.4) F j = M n ≥ j A ⊗ k (Σ A ) ⊗ n . The complex Hoch II ( A ) is the completion of Hoch( A ) for the topology defined by this filtration. Sincethe differential b on Hoch( A ) is continuous with respect to this topology, it induces a differential onHoch II ( A ), which we also write as b . Remark . The complex Hoch( A ) (resp. Hoch II ( A )) also appears in [PP12, Section 2.4] under thename Hoch ⊕• ( A , A ) (resp. Hoch ⊓• ( A , A )), but with a differential which differs from ours by a sign.The map a [ a | · · · | a n ] ( − P n − j =0 ( n − j ) | a j | a [ a | · · · | a n ]yields isomorphisms of complexes of k -vector spacesHoch ⊕• ( A , A ) ∼ = −→ Hoch( A ) and Hoch ⊓• ( A , A ) ∼ = −→ Hoch II ( A ) . The reason for the discrepancy is that the underlying graded k -vector space of Hoch ⊕• ( A , A ) is L n ≥ A ⊗ Σ n ( A ⊗ n ), rather than L n ≥ A ⊗ k (Σ A ) ⊗ n (and similarly for Hoch ⊓• ( A , A )). Appendix Acontains an explanation for the signs appearing in our formulas for b , b , and b .More generally, if A is any cdg category whose objects form a set, define Hoch( A ) and Hoch II ( A )to be the graded vector spaces M X ∈A Hom A ( X ) ⊕ M n ≥ M X ,...,X n ∈A Hom A ( X , X ) ⊗ Σ Hom A ( X , X ) ⊗ · · · ⊗ Σ Hom A ( X , X n )and M X ∈A Hom A ( X ) ⊕ Y n ≥ M X ,...,X n ∈A Hom A ( X , X ) ⊗ Σ Hom A ( X , X ) ⊗ · · · ⊗ Σ Hom A ( X , X n ) , equipped with differentials defined using the same formulas, suitably interpreted, as in the cdgacase. When A is essentially small , so that the isomorphism classes of objects in the Γ-graded categoryunderlying A form a set (see [PP12, Section 2.6]), we define Hoch( A ) and Hoch II ( A ) by first replacing A with a full subcategory consisting of a single object from each isomorphism class. Definition 3.6.
The (ordinary) Hochschild homology groups and
Hochschild homology groups of thesecond kind of a cdg category A are given byHH q ( A ) := H − q Hoch( A ) and HH IIq ( A ) := H − q Hoch II ( A ) . Remark . As discussed in [PP12, Section 2.4], ordinary Hochschild homology has limited valuein the presence of non-trivial curvature. For example if A = ( A, d, h ) is a cdga with h = 0, thenHH ∗ ( A ) = 0. Remark . We have chosen to define Hochschild homology of the second kind in terms of the complexHoch II , but, in [PP12], it is defined in a different manner, using Tor of the second kind. By [PP12,Proposition A in Section 2.4], their definition coincides with the one given here.It will be useful to also have “reduced” versions of Hochschild homology and Hochschild homologyof the second kind. If A = ( A, d, h ) is a cdga, let A denote the graded k -vector space A/k · A . SetHoch( A ) := M n ≥ A ⊗ k (Σ A ) ⊗ n , and similarly for Hoch II ( A ). These are quotients of Hoch( A ) and Hoch II ( A ), respectively, and thedifferentials on these non-reduced complexes descend to differentials on Hoch( A ) and Hoch II ( A ),which we also write as b . The definitions of Hoch( A ) and Hoch II ( A ) extend to the case where A is ageneral cdg category by modding out by id X ∈ Hom A ( X, X ) for all X in A . CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 13
Proposition 3.9. If A is a curved differential Γ -graded category, the canonical surjection Hoch II ( A ) ։ Hoch II ( A ) is a quasi-isomorphism.Proof. We give a proof in the case where A is a cdga ( A, d, h ); the general case differs only in itsnotational complexity. For j ∈ Z , let G j ⊆ Hoch II ( A ) denote the k -vector space comprised ofcochains ( α , α , . . . ) ∈ Q n ≥ A ⊗ k (Σ A ) ⊗ n such that each α n is of the form P l n i =1 a ,i [ a ,i | . . . | a n,i ],where | a ,i | + · · · + | a n,i | ≥ j for each i . Notice that b ( G j ) ⊆ G j , and we have a filtration · · · ⊆ G j +1 ⊆ G j ⊆ · · · ⊆ (Hoch II ( A ) , b )of complexes. Let π denote the canonical surjection Hoch II ( A ) ։ Hoch II ( A ), and define G j = π ( G j ),so that G • is a filtration of (Hoch II ( A ) , b ) and π induces a map of filtrations G • → G • .We have colim( · · · ֒ → G j +1 ֒ → G j ֒ → · · · ) = Hoch II ( A ), and similarly for the filtration ofHoch II ( A ). Fix j ∈ Z ; we need only show G j → G j is a quasi-isomorphism for all j . We havethat G j is the inverse limit of the tower · · · ։ G j /G j +2 ։ G j /G j +1 , and G j is the inverse limit of the tower · · · ։ G j /G j +2 ։ G j /G j +1 . The evident map joining these two towers induces the map π : G j → G j , and so it suffices to prove G j /G j + m → G j /G j + m is a quasi-isomorphism for all m ≥
1. For each such m , we have a morphismof short exact sequences0 / / G j + m /G j + m +1 (cid:15) (cid:15) / / G j /G j + m +1 / / (cid:15) (cid:15) G j /G j + m / / (cid:15) (cid:15) / / G j + m /G j + m +1 / / G j /G j + m +1 / / G j /G j + m / / π . Thus, by induction on m , it suffices to show that π : G l /G l +1 → G l /G l +1 is a quasi-isomorphism for all l ∈ Z . This allows us to assume d = 0 = h and that A is concentratedin degree 0 (and so, in particular, Hoch II ( A ) = Hoch( A )). But in this case, the result is well-known;see, for instance, [Lod13, Proposition 1.6.5]. (cid:3) Remark . The analogous statement for the ordinary Hochschild complex cannot be proven in theabove manner, because the analogue of G j is not the inverse limit of the analogous tower.We next endow each of Hoch( A ), Hoch II ( A ), Hoch( A ) and Hoch II ( A ) with the structure of mixedcomplexes by introducing the Connes B operator. The following construction generalizes the standardone for ordinary algebras; see, for instance, [Lod13, Section 2.1.7].As before, we start with the case where A = ( A, d, h ) is a cdga. For each n ≥
0, define(3.11) τ n +1 : A ⊗ k (Σ A ) ⊗ n → A ⊗ k (Σ A ) ⊗ n by τ n +1 ( a [ a | · · · | a n ]) = ( − ( | a |− | a | + ··· + | a n |− n ) a [ a | · · · | a n | a ] , and define s : Hoch( A ) → Hoch( A ) by s ( a [ a | · · · | a n ]) = 1[ a | a | · · · | a n ] . The
Connes B operator is the degree − B of Hoch( A ) whose restriction to A ⊗ (Σ A ) ⊗ n is given by B | A ⊗ (Σ A ) ⊗ n = (1 − τ − n +2 ) ◦ s ◦ n X l =0 τ ln +1 . The Connes B operator induces an endomorphism on the quotient Hoch( A ) of Hoch( A ), and wewrite the induced map also as B . It is given by the simpler formula B | A ⊗ (Σ A ) ⊗ n = s ◦ n X l =0 τ ln +1 , since τ − n +2 ◦ s ◦ P nl =0 τ ln +1 is the zero map on Hoch( A ). In detail:(3.12) B ( a [ a | · · · | a n ]) = n X l =0 ( − ( | a l | + ··· + | a n |− ( n − l +1))( | a | + ··· + | a l − |− l ) a l | · · · | a n | a | · · · | a l − ] . For example, if each a i has even degree, then B ( a [ a | · · · | a n ]) = n X l =0 ( − nl a l | · · · | a n | a | · · · | a l − ] . On Hoch( A ), we can equivalently define B as follows. For each m ≥
0, define σ m : (Σ A ) ⊗ m → (Σ A ) ⊗ m by σ m ([ a | · · · | a m ]) = ( − | sa | ( | sa | + ··· + | sa m | ) [ a | · · · | a m | a ]= ( − ( | a |− | a | + ··· + | a m |− ( m − [ a | · · · | a m | a ] . Then B | A ⊗ (Σ A ) ⊗ n = n X l =0 (id A ⊗ σ ln +1 ) ◦ s . The Connes B operator is continuous for the topology on Hoch( A ) given by the filtration F ⊇ F ⊇ · · · defined in (3.4), and we also write B for the induced map on the completion Hoch II ( A ),and on Hoch II ( A ) as well.The definition of the operator B extends to cdg categories in an evident manner. We record thefollowing (cf. [Efi12, Section 3.1]): Proposition 3.13. If A is a curved differential Γ -graded category, we have bB + Bb = 0 and B = 0 ,so that each of (Hoch( A ) , b, B ) , (Hoch( A ) , b, B ) , (Hoch II ( A ) , b, B ) and (Hoch II ( A ) , b, B ) is a Γ -gradedmixed complex over k . Definition 3.14.
For a cdg category A , define mixed complexesMC( A ) := (Hoch( A ) , b, B )MC II ( A ) := (Hoch II ( A ) , b, B ) , MC( A ) := (Hoch( A ) , b, B ) , andMC II ( A ) := (Hoch II ( A ) , b, B ) . The following is an immediate consequence of Proposition 3.9:
Proposition 3.15. If A is a curved differential Γ -graded category, the canonical morphism α : MC II ( A ) → MC II ( A ) of mixed complexes is a quasi-isomorphism. CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 15
We now discuss functoriality. It is established in [PP12, Section 2.4] that, for each functor (resp.strict functor) F : A → A ′ of cdg categories, there exists a morphism of complexes F ∗ : Hoch II ( A ) → Hoch II ( A ′ ) (resp., F ∗ : Hoch( A ) → Hoch( A ′ )). Thus, Hoch II ( − ) (resp., Hoch( − )) is a functorfrom the category of curved differential Γ-graded categories, with morphisms given by cdg functors(respectively, strict cdg functors), to the category of complexes of k -vector spaces.We recall the formula for F ∗ in the case where A and A ′ are cdga’s; the general case is similar.Recall that a morphism of cdga’s from ( A, d, h ) to ( A ′ , d ′ , h ′ ) is a pair ( ρ, β ), where ρ : A → A ′ is amorphism of graded algebras and β ∈ ( A ′ ) , satisying ρ ◦ d − d ′ ◦ ρ = [ β, ρ ] and ρ ( h ) = h ′ + d ′ ( β ) + β . The induced morphism ( ρ, β ) ∗ : Hoch II ( A, d, h ) → Hoch II ( A ′ , d ′ , h ′ )sends a class of the form a [ a | . . . | a n ] to(3.16) 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 ] , where the sum ranges over all ( n + 1)-tuples of non-negative integers. The map ( ρ, β ) ∗ extends toinfinite sums in the evident way. Henceforth, for brevity, we will write ( ρ, β ) ∗ ( a [ a | . . . | a n ]) in thefollowing way: X ( − i + ··· + i n ρ ( a )[ β i | ρ ( a ) | β i | ρ ( a ) | · · · | ρ ( a n ) | β i n ] . If β = 0 (i.e., when the morphism is strict), the map ( ρ, ∗ restricts to a map Hoch( A, d, h ) → Hoch( A ′ , d ′ , h ′ ), but, since the sum occurring above is an infinite one when β = 0, Hoch( − ) is notfunctorial for arbitrary functors of cdg categories. Remark . The sign appearing in the formula for ( ρ, β ) ∗ corresponds to the sign in line (15) of[PP12, Section 2.4] via the isomorphism discussed in Remark 3.5. Proposition 3.18. If F : A → A ′ is a cdg functor (resp. strict cdg functor), then there is a morphismof mixed complexes F ∗ : MC II ( A ) → MC II ( A ′ ) ( respectively, F ∗ : MC( A ) → MC( A ′ )) determined by the formula (3.16) .Proof. By results of Polishchuk-Positselski quoted above, we need only show that the maps F ∗ :MC II ( A ) → MC II ( A ′ ) and F ∗ : MC( A ) → MC( A ′ ) preserve Connes B operators. It suffices to provethe statement involving MC II ( − ); the statement involving MC( − ) is an immediate consequence.Moreover, we will assume A and A ′ are cdga’s; the general case is similar.Given a morphism ( ρ, β ) : ( A, d, h ) → ( A ′ , d ′ , h ′ )of cdga’s, let us verify that ( ρ, β ) ∗ : Hoch II ( A, d, h ) → Hoch II ( A ′ , d ′ , h ′ )commutes with the Connes B operators. Throughout this proof, we shall use the following notation:given a [ a | · · · | a n ], • write a ′ i for ρ ( a i ), and • ǫ l = ( − ( | sa l | + ··· + | sa n | )( | sa | + ··· + | sa l − | ) = ( − ( | a l | + ··· + | a n |− ( n − l +1))( | a | + ··· + | a l − |− l ) . We have ( B ◦ ( ρ, β ) ∗ )( a [ a | · · · | a n ]) = ∞ X N =0 X | ~i | = N ( − N B ( a ′ [ β i | a ′ | β i | · · · | a ′ n | β i n ]) , where the inner sum ranges over all ~i = ( i , . . . , i n ) such that | ~i | := i + · · · + i n = N . Evaluating B ,we obtain( B ◦ ( ρ, β ) ∗ )( a [ a | · · · | a n ]) = ∞ X N =0 X | ~i | = N n + N X l =0 ( − N σ ln + N +1 ([ a ′ | β i | a ′ | β i | · · · | a ′ n | β i n ]) , where σ m : (Σ A ) ⊗ m → (Σ A ) ⊗ m is as defined above. On the other hand,(( ρ, β ) ∗ ◦ B )( a [ a | · · · | a n ]) = ( ρ, β ) ∗ ( n X l =0 ǫ l [ a l | · · · | a n | a | · · · | a l − ])= ∞ X N =0 X | ~j | = N n X l =0 ( − N ǫ l [ β j | a ′ l | β j | · · · | a ′ n | β j n − l +1 | a ′ | β j n − l +2 | · · · | a ′ l − | β j n +1 ] , where the middle sum ranges over all ~j = ( j , . . . , j n +1 ) such that | ~j | := j + · · · + j n +1 = N . Fix N ≥
0. It will suffice to prove that(3.19) X | ~i | = N n + N X l =0 · σ ln + N +1 ([ a ′ | β i | a ′ | β i | · · · | a ′ n | β i n ])is equal to(3.20) X | ~j | = N n X l =0 ǫ l [ β j | a ′ l | β j | · · · | a ′ n | β j n − l +1 | a ′ | β j n − l +2 | · · · | a ′ l − | β j n +1 ] . Since there are (cid:0) N + m − m − (cid:1) ways of adding m nonnegative integers up to N , there are ( N + n +1) (cid:0) N + nn (cid:1) terms in (3.19) and ( n + 1) (cid:0) N + n +1 n +1 (cid:1) terms in (3.20). Using that ( N + n + 1) (cid:0) N + nn (cid:1) = ( n + 1) (cid:0) N + n +1 n +1 (cid:1) ,we need only associate to each summand of (3.19) a distinct summand of (3.20) to which it is equal,and it is easy to check that the summand of (3.20) corresponding to the pair( ~j = ( j , . . . , j n +1 ) , l )is equal to the summand of (3.19) corresponding to the pair( ~i = ( j n − l +2 , . . . , j n , j + j n +1 , j , . . . , j n − l +1 ) , j n − l +2 + · · · + j n +1 + l ) . (cid:3) Remark . Let F : A → A ′ be a cdg functor. The induced map F ∗ : Hoch II ( A ) → Hoch II ( A ′ )does not necessarily commute with B operators, and it therefore does not induce a morphism ofmixed complexes MC II ( A ) → MC II ( A ′ ). For example, take A to be the curved algebra (with trivialcurvature) ( A, A is an exterior algebra on two degree 1 variables e , e , and consider themorphism (id A , e ) : ( A, → ( A, e as an element of Hoch II ( A ). Then((id A , e ) ◦ B )( e ) = (id A , e )(1[ e ] − e [1])(3.22) = X N ≥ X i + i = N ( − N (1[ e i | e | e i ] − e [ e i | | e i ]) . On the other hand, ( B ◦ (id A , e ))( e ) = X N ≥ ( − N B ( e [ e N ]) CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 17 (3.23) = X N ≥ ( − N ((1 − τ − N +2 ) ◦ s ◦ N X l =0 τ lN +1 )( e [ e N ]) . The summand of (3.22) corresponding to N = 1 is − e | e ] + e [1 | e ] − e | e ] + e [ e | , and the summand of (3.23) corresponding to N = 1 is(( τ − − ◦ s ◦ X l =0 τ l )( e [ e ]) = ( τ − − e | e ] + 1[ e | e ])= e [1 | e ] + e [1 | e ] − e | e ] − e | e ] . Since these summands differ, ((id A , e ) ◦ B )( e ) = ( B ◦ (id A , e ))( e ). Notice that these summandscoincide upon passing to Hoch II ( A, II ( − ) is that it sends pseudo-equivalences to quasi-isomorphisms.The following statement is proven in [PP12, p. 5326]: Theorem 3.24 (Polishchuk-Positselski) . If F : A → A ′ is a pseudo-equivalence of cdg categories,then F ∗ : MC II ( A ) → MC II ( A ′ ) is a quasi-isomorphism. It is well-known that Hochschild homology of dg categories is Morita invariant. Combining Proposi-tion 2.20 and Theorem 3.24, we obtain a version of Morita invariance for mixed Hochschild complexesof cdg categories which we dub “quasi-Morita invariance”:
Proposition 3.25.
Let A be a cdg category. The canonical inclusion Perf( A ) op ֒ → qPerf( A ) op and the quasi-Yoneda embedding qY A : A ֒ → qPerf( A ) op defined in (2.18) both induce quasi-isomorphisms of mixed Hochschild complexes of the second kind: MC II (Perf( A ) op ) ≃ −→ MC II (qPerf( A ) op ) ≃ ←− MC II ( A ) . Negative cyclic homology of cdg categories.
Let (
M, b, B ) be a mixed complex. The neg-ative cyclic complex associated to (
M, b, B ) is defined to be ( M [[ u ]] , b + uB ), where u is a degree 2indeterminant, and M [[ u ]] is the direct product totalization of the (Γ × Z )-graded k -vector space M, , M u, , M u , . . . So, an element of M [[ u ]] of degree e is a formal power series P i m i u i such that m i ∈ M has degree e − i ∈ Γ for each i . Since b = B = [ b, B ] = 0, | b | = 1 and | B | = −
1, ( M [[ u ]] , b + uB ) is indeed aΓ-graded complex of k -vector spaces. Example 3.26.
The negative cyclic complex of the mixed complex ( k, ,
0) has trivial differentialand is k [[ u ]]. An element of k [[ u ]] of degree e is a formal power series P i α i u i such that α i ∈ k hasdegree e − i ∈ Γ, for each i , and multiplication is given by the usual rule for multiplying power series.If Γ = Z , then k [[ u ]] is just the usual polynomial ring k [ u ] with | u | = 2 (and so the notation k [[ u ]] ismisleading in this case). But if Γ = Z /
2, then k [[ u ]] is the ring of formal power series, concentratedin degree 0.The graded k -module k [[ u ]] is a commutative Γ-graded k -algebra, with polynomial/power seriesmultiplication. For any mixed complex ( M, b, B ), the complex ( M [[ u ]] , b + uB ) is a dg- k [[ u ]]-modulein an evident way.We define k (( u )) to be the Γ-graded ring obtained from k [[ u ]] by inverting the degree two element u . So, if Γ = Z , then k (( u )) is the ring of Laurent polynomials k [ u, u − ] with | u | = 2, and if Γ = Z / then k (( u )) is the ring of Laurent power series concentrated in degree 0. The periodic cyclic complex of ( M, b, B ) is the Γ-graded dg- k (( u ))-module( M (( u )) , b + uB ) = ( M [[ u ]] , b + uB ) ⊗ k [[ u ]] k (( u ))given by inverting u . Definition 3.27.
Let A be a cdg category. The (ordinary) negative cyclic complex and the negativecyclic complex of the second kind associated to A are the negative cyclic complexes associated to themixed complexes MC( A ) and MC II ( A ), respectively. We write them asHN( A ) and HN II ( A ) . Similarly, the (ordinary) periodic cyclic complex and the periodic cyclic complex of the second kind associated to A are the periodic cyclic complexes associated to MC( A ) and MC II ( A ), and they arewritten as HP( A ) and HP II ( A ) . Likewise, HN( A ), HN II ( A ), HP( A ) and HP II ( A ) are obtained from MC( A ) and MC II ( A ). ByProposition 3.15, there are natural quasi-isomorphisms HN( A ) ∼ −→ HN( A ), etc. The negative andperiodic cyclic homology groups, of both kinds, are defined as the homology of the complexes HN( A ),HN II ( A ), HP( A ) and HP II ( A ): HN q ( A ) := H − q HN( A ) , HN IIq ( A ) := H − q HN II ( A ) , HP q ( A ) := H − q HP( A ) , andHP IIq ( A ) := H − q HP II ( A ) . (Since H − q (HN( A )) is canonically isomorphic to H − q (HN( A )), we do not introduce notation for thelatter and its variants.)3.3. The HKR theorem.
Recall that if A is a classical k -algebra (i.e., an algebra concentrated indegree 0), we say A is smooth (over k ) if it is isomophic to a commutative k -algebra of the form B = k [ x , . . . , x n ] / ( f , . . . , f m ) such that the Jacobian matrix ( ∂f i /∂x j ) has maximal rank at everypoint in Spec( B ), and we say A is essentially smooth (over k ) if it is isomorphic to S − B for somesmooth k -algebra B and some multiplicatively closed subset S . Definition 3.28.
A Γ-graded curved algebra A = ( A, h ) (with trivial differential) is essentially smoothover k if • A is commutative and concentrated in even degrees, and • upon forgetting the grading, A is an essentially smooth k -algebra in the classical sense. Example 3.29.
Suppose A is an essentially smooth k -algebra in the classical sense, and let h ∈ A .Then ( A, h ) is a Z / k . Example 3.30.
Suppose Q is an essentially smooth k -algebra in the classical sense, let f , . . . , f c ∈ Q ,and let T , . . . , T c be indeterminants of degree 2. Then ( Q [ T , . . . , T c ] , P i f i T i ) is an essentially smoothcurved Z -graded algebra over k .Let A = ( A, h ) be a Γ-graded, essentially smooth curved k -algebra, and let Ω • A/k denote the gradedring given by the exterior powers of the module of K¨ahler differentials of A over k , Γ-graded so that a da ∧ · · · ∧ da j has degree P | a i | − j . We consider the mixed complex(Ω • A/k , dh, d ) , where dh denotes the map given by left multiplication by the degree 1 element dh , and d is given bythe de Rham differential (which, by our indexing convention, has degree − CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 19
Theorem 3.31 (“The HKR Theorem”) . Assume char( k ) = 0 . If A = ( A, h ) is a Γ -graded, essentiallysmooth curved k -algebra, then there is a quasi-isomorphism of mixed complexes ǫ : MC II ( A ) ≃ −→ (Ω • A/k , dh, d ) which sends elements of the form a [ a | · · · | a n ] to n ! a da · · · da n . In particular, we have an isomor-phism HN IIq ( A ) ∼ = −→ H − q (Ω • A/k [[ u ]] , ud + dh ) of graded k [[ u ]] -modules for all q ∈ Z .Remark . Efimov’s model for the Hochschild complex of the second kind associated to a curved al-gebra A = ( A, W ), written as Hoch Q ( A, W ) in [Efi12, Proposition 3.14], is identical to our Hoch II ( A ),except the differential is negated. Further, the Connes B operators on both complexes are the same.This is why the differential in the target of the HKR map in loc. cit. is the negative of ours, but thestatements are otherwise the same.4. Chern character maps for curved differential graded categories
The
Grothendieck group of a triangulated category T is the abelian group generated by isomorphismclasses [ X ] of objects of T modulo the relations [ X ] = [ X ′ ] + [ X ′′ ] whenever there is a distinguishedtriangle X ′ → X → X ′′ → Σ X . We write this group as K ∆0 ( T ). If C is a dg category, the associated homotopy category , written [ C ], is the ordinary category having the same objects as C and with homsets given by Hom [ C ] ( X, Y ) := H Hom C ( X, Y ). When C is a pretriangulated dg category, [ C ] inheritsa canonical triangulated structure; this occurs, for instance, when C = Perf( A ) for some cdg category A . See [Kel06, Section 4.5] for a discussion of pretriangulated dg categories. Definition 4.1.
Let A be a curved differential Γ-graded category. The Grothendieck group of A ,written K ( A ), is the Grothendieck group of the (triangulated) homotopy category [Perf( A op )] of thedg category of perfect right A -modules: K ( A ) := K ∆0 ([Perf( A op )]) . So, K ( A ) is generated by isomorphism classes of perfect right A -modules modulo relations comingfrom homotopy equivalences and distinguished triangles.In this section, we define and develop the (ordinary) Chern character map and the Chern charactermap of the second kind for cdg categories. The former is a homomorphism of the form ch HN : K ( A ) → HN ( A ) , where A is any dg category (i.e., a cdg category with trivial curvatures). The latter is a homomorphismof the form ch IIHN : K ( A ) → HN II ( A ) , and it is defined for an arbitrary cdg category. For a dg category A , the two are related by theequation ch IIHN = can ◦ ch HN , where can is the canonical map from HN to HN II . Remark . If A has non-trivial curvature, then the map ch HN is undefined, and no such factorizationexists. This is as expected: if, for example, A = ( A, d, h ) is a cdga with h = 0, then HN ( A ) = 0, butthe map ch IIHN is often non-trivial.4.1.
The Chern character map for dg categories.
We first review the construction of the ordinaryChern character map ch HN : K ( A ) → HN ( A )for a dg category A , as defined, for instance, in [Kel98]. Merely knowing that such a map exists andis natural is enough to determine it, as we now explain.There are isomorphisms Z ∼ = −→ K ( k ) and k [[ u ]] ∼ = −→ HN ( k )given by the ring map sending 1 to [ k ] and the k [[ u ]] -linear map sending 1 to the class γ representedby the constant power series 1 ∈ Hoch( k )[[ u ]], respectively. Recall that if Γ = Z , then k [[ u ]] = k ,and if Γ = Z / k [[ u ]] = k [[ u ]], the ring of formal power series. Under these isomorphisms, theChern character(4.3) ch HN : K ( k ) → HN ( k )is the unique ring map Z → k [[ u ]] . Remark . The element 1 is a cycle in the complex (Hoch II ( k )[[ u ]] , b + uB ), but not in (Hoch II ( k )[[ u ]] , b + uB ) (unless char( k ) = 2). One easily checks that ∞ X i =0 c i | · · · | | {z } i copies ] u i ∈ (Hoch II ( k )[[ u ]] , b + uB ) , where c i := ( − i i Q i − j =0 (2 j + 1), is a cycle; the projection(Hoch II ( k )[[ u ]] , b + uB ) ։ (Hoch II ( k )[[ u ]] , b + uB )sends this cycle to 1.Both K ( − ) and HN ( − ) are Morita invariant ; that is, the Yoneda map Y A : A →
Perf( A op ) , X Hom A ( − , X ) . induces isomorphisms upon applying K ( − ) and HN ( − ). Let X be an object of Perf( A op ). Thereis a canonical map α X : k → End( X ) := End Perf( A op ) ( X ) sending 1 to id X and an inclusion inc :End Perf( A op ) ( X ) ֒ → Perf( A op ). The naturality of ch yields the commutative diagram(4.5) K ( k ) ( α X ) ∗ / / ch HN (cid:15) (cid:15) K (End( X )) inc ∗ / / ch HN (cid:15) (cid:15) K (Perf( A op )) ch HN (cid:15) (cid:15) K ( A ) ∼ =( Y A ) ∗ o o ch HN (cid:15) (cid:15) HN ( k ) ( α X ) ∗ / / HN (End( X )) inc ∗ / / HN (Perf( A op ) HN ( A ) . ∼ =( Y A ) ∗ o o From this diagram we deduce:
Proposition 4.6. If A is a differential Γ -graded category, and X ∈ Perf( A op ) is a perfect right A -module, then ch HN ( X ) is the image of γ ∈ HN ( k ) under ( Y A ) − ∗ ◦ inc ∗ ◦ ( α X ) ∗ . The Chern character map of the second kind.
We now wish to define a Chern charactermap of the second kind ch IIHN : K ( A ) → HN II ( A )for a cdg category A . We cannot proceed exactly as in the previous section, since there is no Yonedaembedding A ֒ → Perf( A op ) (Example 2.10). Instead, we use the quasi-Yoneda embedding qY A : A ֒ → qPerf( A op ) , X Hom A ( − , X ) , and the quasi-Morita invariance of mixed Hochschild complexes of the second kind (Proposition 3.25).Recall that K ( A ) is defined to be K ∆0 ([Perf( A op )]). The Yoneda embedding Y = Y Perf( A op ) : Perf( A op ) ֒ → Perf(Perf( A op ) op )induces a triangulated functor on homotopy categories[ Y ] : [Perf( A op )] −→ [Perf(Perf( A op ) op )];the target is the idempotent completion of the source [Kel06, Section 4.6]. Note that [Perf( A op )] neednot be idempotent complete; for instance, take A to be the Z / C [ x, y ] ( x,y ) , − y + CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 21 x ( x + 1)), where C [ x, y ] ( x,y ) is concentrated in degree 0. Then Perf( A op ) coincides with the matrixfactorization category MF( C [ x, y ] ( x,y ) , y − x ( x + 1)), and, as shown in [Bro15, Section 2.5], thehomotopy category of this matrix factorization category is not idempotent complete.The triangulated functor [ Y ] induces a canonical map K ∆0 ([ Y ]) : K ( A ) → K (Perf( A op )) . Definition 4.7.
For a cdg category A , the Chern character map of the second kind is the homomor-phism ch IIHN : K ( A ) → HN II ( A )given by the composition K ( A ) K ∆0 ([ Y ]) −−−−−→ K (Perf( A op )) ch HN −−−−→ HN (Perf( A op )) can −−→ HN II (Perf( A op )) → HN II (qPerf( A op )) ( qY A ) − ∗ −−−−−→ HN II ( A ) , where the penultimate map is induced by the inclusion Perf( A op ) ⊆ qPerf( A op ), and the last map isthe inverse of the isomorphism induced by the quasi-Yoneda embedding qY A (see Proposition 3.25). Remark . If A is a dg category, then, since the quasi-Yoneda embedding lands in Perf( A ), adiagram chase shows that ch IIHN = can ◦ ch HN , where can : HN ( A ) → HN II ( A ) is the canonical map.4.3. The Chern character map of the second kind for cdg modules over a cdga.
We nowspecialize our general construction to the case of a cdga A = ( A, d, h ). In the previous section, weestablished a Chern character map of the second kind ch IIHN : K ( A ) → HN II ( A ) . In this section, we extend this construction slightly.Given a perfect right quasi-module P = ( P, δ P ) ∈ qPerf( A op ), recall that End( P ) := End qPerf( A op ) ( P )is the cdga (End A ( P ) , [ δ P , − ] , δ P + ρ h ) where ρ h is right multiplication by h ; see Remark 2.15. Definition 4.9.
For a perfect right quasi-module P = ( P, δ P ) ∈ qPerf( A op ), define the map ch II P : HN II (End( P )) → HN II ( A )to be the composition of the maps induced by the inclusion of End( P ) into qPerf( A op ) and the inverseof the quasi-Yoneda isomorphism:HN II (End( P )) → HN II (qPerf( A op )) ∼ = −→ HN II ( A ) . If P ∈
Perf( A op ), so that End( P ) has trivial curvature, then there is a distinguished class γ P ∈ HN II (End( P ))represented by the constant power series id P ∈ Hoch II (End( P ))[[ u ]]. The reason we use the notation γ P rather than id P for this distinguished class is that id P is not a cycle in (Hoch II (End( P ))[[ u ]] , b + uB )(see Remark 4.4).The following is immediate from the definitions: Proposition 4.10.
For any cdga A and any P ∈
Perf( A op ) , we have ch IIHN ([ P ]) = ch II P ( γ P ) . For
P ∈ qPerf( A op ), recall from Example 2.17 the notation P ♮ := ( P, ∈ qPerf( A op ). We havean isomorphism of cdga’s (id , δ P ) : End( P ) → End( P ♮ )with inverse given by (id , − δ P ) : End( P ♮ ) → End( P ) . The following proposition, which relates the Chern character maps for P and P ♮ , will play a crucialrole later on: Proposition 4.11.
For a cdga A and P ∈ qPerf( A op ) , we have ch II P ♮ ◦ (id , δ P ) ∗ = ch II P . In particular, if
P ∈
Perf( A op ) , we have ch IIHN ( P ) = ch II P ♮ X j ( − j [ δ P | · · · | δ P | {z } j copies ] ∈ H (Hoch II ( A )[[ u ]] , b + uB ) = HN II ( A ) . Proof.
Let {P , P ♮ } denote the full subcategory of qPerf( A op ) consisting of just the two indicatedobjects, and let ι : {P ♮ } → {P , P ♮ } and ι ′ : {P} → {P , P ♮ } denote the inclusion functors. The functor ι admits a left inverse; that is, there is a cdg functor G : {P , P ♮ } → {P ♮ } such that G ◦ ι = id. The functor G is defined in the only way possible on objects; on morphisms,the map End( P ) → End( P ♮ ) is (id , δ P ), the map End( P ♮ ) → End( P ♮ ) is (id , P , P ♮ ) → End( P ♮ ) and Hom( P ♮ , P ) → End( P ♮ ) are the identity. Observe that G ◦ ι ′ = (id , δ P ).Consider the diagram of graded k [[ u ]]-modulesHN II ∗ ( {P} ) (id ,δ P ) ∗ / / ι ′∗ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ ! ! ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ HN II ∗ ( {P ♮ } ) } } ④④④④④④④④④④④④④④④④④④④④④ ι ∗ v v ♠♠♠♠♠♠♠♠♠♠♠♠♠ HN II ∗ ( {P , P ♮ } ) (cid:15) (cid:15) HN II ∗ (qPerf( A op ))in which each map, except for the horizontal one, is induced by the evident inclusion of cdg categories.To prove ch P ♮ ◦ (id , δ P ) ∗ = ch P , it suffices to show that the exterior triangle commutes. Clearly thebottom two interior triangles commute, so we need only show ι ′∗ = ι ∗ ◦ (id , δ P ) ∗ . By Example 2.17 andProposition 3.18, ι ∗ and ι ′∗ are isomorphisms. Thus, we have G ∗ = (id , δ P ) ∗ ◦ ( ι ′∗ ) − , and composingboth sides with ι ∗ on the left yields the result.Finally, by (3.16), we have (id , δ P ) ∗ ( γ P ) = X j ( − j [ δ P | · · · | δ P | {z } j copies ]) . (cid:3) The main theorem
Assume char( k ) = 0, and let A = ( A, h ) be an essentially smooth curved k -algebra (Definition 3.28).Upon composing the Chern character map ch IIHN : K ( A ) → HN II ( A )with the HKR isomorphism ǫ : HN II ( A ) ∼ = −→ H (Ω • A/k [[ u ]] , ud + dh ) , we get a map ch IIHKR : K ( A ) → H (Ω • A/k [[ u ]] , ud + dh ) . The main theorem of this paper provides a Chern-Weil-type formula for the class ch IIHKR ( P ) associatedto a perfect right module ( P, δ P ) ∈ Perf( A op ). CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 23
Connections, curvature, and the trace map.
Before giving the precise statement of themain theorem, we establish some terminology.
Definition 5.1. If P = ( P, δ P ) ∈ qPerf( A ), a connection on P is a degree − ∇ : P → P ⊗ A Ω A/k of graded k -vector spaces such that ∇ ( pa ) = ∇ ( p ) a + ( − | p | p ⊗ d ( a ) for all a ∈ A and p ∈ P , where d denotes the de Rham differential on Ω • A/k . That is, a connection on P is just a connection on P which respects the grading. Remark . Our grading convention for Ω • A/k is the same as the one used in Section 3.3, so a da ∧· · · ∧ da j has degree P | a j | − j .A connection ∇ on ( P, δ P ) ∈ qPerf( A ) extends uniquely to a degree − e ∇ : P ⊗ A Ω • A/k → P ⊗ A Ω • A/k of graded k -vector spaces such that e ∇ ( p ⊗ ω ) = e ∇ ( p ) ω + ( − | p | p ⊗ dω . Let ∇ : P → P ⊗ A Ω A/k denote the restriction of ( e ∇ ) to P . Both ∇ and the map[ ∇ , δ P ] : P → P ⊗ A Ω A/k are A -linear maps, of degrees − A is commutative and P is projective,there is a canonical isomorphismHom A ( P, P ⊗ A Ω • A/k ) ∼ = −→ End A ( P ) ⊗ A Ω • A/k , and thus we may identify ∇ and [ ∇ , δ P ] with elements of End A ( P ) ⊗ A Ω • A/k . Definition 5.3.
The curvature of a connection ∇ on ( P, δ P ) ∈ qPerf( A op ) is the class R := ∇ u + [ ∇ , δ P ] ∈ End A ( P ) ⊗ A Ω • A/k [[ u ]] . Remark . Recall that, classically, the curvature of a connection ∇ on an ordinary algebraic vectorbundle is given by just ∇ . As discussed in the introduction, the presence of the extra term [ ∇ , δ P ]is motivated by Quillen’s formula for the Chern character of a relative topological K -theory class in[Qui85].For a projective A -module P , we have a canonical isomorphism P ⊗ A P ∗ ∼ = −→ End A ( P )given by p ⊗ γ ( x pγ ( x )). We define the trace maptr : End A ( P ) → A to be the map corresponding, via this isomorphism, toev : P ⊗ A P ∗ → A, where ev( p ⊗ γ ) = ( − | p || γ | γ ( p ) . We extend tr to a maptr : End A ( P ) ⊗ A Ω • A/k [[ u ]] → Ω • A/k [[ u ]]by extension of scalars: tr( α ⊗ ω ) = tr( α ) ω . Remark . If P is a graded free A -module with basis e , . . . , e r , a homogeneous element of End A ( P )of degree m may be identified with an r × r matrix ( a i,j ) of homogeneous elements with | a i,j | = | e i | − | e j | + m . In this case, tr is the Ω • A/k [[ u ]]-linear map which sends ( a i,j ) ⊗ P i ( − | e i | a i,i ∈ Ω • A/k [[ u ]]. When P is not necessarily free, tr may be described locally in the above manner. Usingthis observation, we conclude that (1) tr( α ⊗ ω ) = 0 if α ∈ End A ( P ) has odd degree (since A has no non-zero odd degree elements),and(2) tr ◦ [ − , − ] = 0, where [ − , − ] denotes the graded commutator in the algebra End A ( P ) ⊗ A Ω • A/k [[ u ]].The following lemma is adapted directly from classical Chern-Weil theory; see, for instance, [Lod13,Lemma 8.1.5]. Lemma 5.6.
Let A = ( A, h ) be an essentially smooth curved k -algebra, let P = ( P, be an object in qPerf( A ) , and let ∇ be a connection on P . We have tr ◦ [ ∇ , − ] = d ◦ tr , where tr : End A ( P ) ⊗ A Ω • A/k → Ω • A/k is the trace map, and d is the de Rham differential. In fact, the curvature plays no role here, so this lemma is almost identical to the classical version.But, since there is a Γ-grading to keep track of, our statement is slightly more general, and so weprovide a proof.
Proof.
Localizing at a homogeneous prime ideal of A , we may assume P is graded free [BH98, Propo-sition 1.15(d)]. Let n denote the rank of P . Identifying End A ( P ) ⊗ A Ω • A/k with Mat n × n (Ω • A/k ), theconnection ∇ is given by ∇ ( v ) = dv + θv for some matrix of one-forms θ ∈ Mat n × n (Ω A/k ). Let X ∈ Mat n × n (Ω • A/k ). Noting that, for anycolumn vector v ∈ (Ω mA/k ) ⊕ n , one has d ( Xv ) = ( dX )( v ) + ( − m X ( dv ), it follows that[ ∇ , X ] = dX + [ θ, X ] . Thus, since tr([ θ, X ]) = 0 (Remark 5.5), we get tr([ ∇ , X ]) = tr( dX ) = d tr( X ) . (cid:3) Statement of the main theorem.
The following theorem is the main result of this paper:
Theorem 5.7.
Let k be a field of characteristic . Assume A = ( A, h ) is a Γ -graded, essentiallysmooth curved k -algebra, and let P = ( P, δ P ) ∈ Perf( A op ) be a perfect right A -module. For anyconnection ∇ on P , we have ch IIHKR ( P ) = tr(exp( − R )) ∈ H (Ω • A/k [[ u ]] , ud + dh ) , where R = ∇ u + [ ∇ , δ P ] , and exp( − R ) = id − R + R − R
3! + · · · ∈
End A ( P ) ⊗ A Ω • A/k [[ u ]] . Remark . As a reality check, let us apply Lemma 5.6 to show that the element tr(exp( − R )) ∈ (Ω • A/k [[ u ]] , ud + dh ) in the statement of Theorem 5.7 is indeed a cycle. We must show ud (tr(exp( − R ))) = − dh ∧ tr(exp( − R )) . Applying Lemma 5.6, we have ud (tr(exp( − R ))) = u tr ◦ [ ∇ , exp( − R )] . Since, δ P is A -linear (Remark 2.13), line (2) in Remark 5.5 implies u tr ◦ [ ∇ , exp( − R )] = tr ◦ [ u ∇ + δ P , exp( − R )] . Set D := u ∇ + δ P . It’s easy to check that [ D, R ] = dh (here, dh denotes left multiplication by theform dh ), and an easy induction argument shows[ D, ( − R ) i ] = − i ( − R ) i − [ D, R ] = − i ( − R ) i − dh. Thus, tr ◦ [ D, exp( − R )] = − dh ∧ tr(exp( − R )) . In a similar way, one can prove directly that the class represented by tr(exp( − R )) in H (Ω • A/k [[ u ]] , ud + dh ) is independent of the choice of connection ∇ (of course, this is also a consequence of Theorem5.7). CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 25
Remark . For any integer m ≥ R m = m X p =0 X j + ··· + j p = m − p ∇ j δ ′ P ∇ j δ ′ P · · · δ ′ P ∇ j p u m − p , where δ ′ P := [ ∇ , δ ], and the inner sum ranges over all p + 1 tuples of non-negative integers that sumto m − p . It follows that(5.10) tr(exp( − R )) = ∞ X n,J ≥ ( − J + n X j + ··· + j n = J tr (cid:0) ∇ j δ ′ P ∇ j δ ′ P · · · δ ′ P ∇ j n (cid:1) ( J + n )! u J . Remark . Every (
P, δ P ) ∈ qPerf( A ) may be equipped with a connection, and so Theorem 5.7does, in fact, give a formula for ch IIHKR in general. Indeed, assume first that F is a free A -moduleof finite rank. Equip F with a basis { e , . . . , e r } , and define ∇ F : F → F ⊗ A Ω A/k by P a i e i P ( − | e i | e i ⊗ da i . When P is an arbitrary finitely generated projective A -module, we may choose afinite rank free A -module F and maps i : P → F , π : F → P such that π ◦ i = id P ; then ∇ P := (id ⊗ π ) ◦ ∇ F ◦ i is a connection on P . A connection constructed inthis manner is sometimes called a Levi-Civita connection (cf. [Lod13, 8.1.1]).
Corollary 5.12.
Given P = ( P, δ P ) ∈ Perf( A op ) , if P admits a flat connection ∇ (that is, a connec-tion ∇ such that ∇ = 0 ), then ch IIHKR ( P ) = tr(exp( − δ ′ P )) = X j ≥ tr (cid:0) [ ∇ , δ P ] j (cid:1) (2 j )! ∈ H (Ω • A/k [[ u ]] , ud + dh ) . Proof.
This follows from Theorem 5.7, using that tr( α ⊗ ω ) = 0 for α ∈ End A ( P ) and ω ∈ Ω • A/k whenever α has odd degree (see Remark 5.5). (cid:3) Key technical result.
Let D denote a full cdg subcategory of qPerf( A op ) consisting of objectswith trivial differential. Let P be an object in D and ∇ a connection on P . The key to provingTheorem 5.7 is the construction of a mapHN II ( D ) → (Ω • A/k [[ u ]] , ud + dh )of dg- k [[ u ]]-modules which sends id P ∈ HN II (End( P )) ⊆ HN II ( D ) to tr(exp( − R )). The existence ofsuch a map is the content of Theorem 5.19. Definition 5.13.
Let D be a full subcategory of qPerf( A op ) consisting of objects with trivial differ-entials. Let ∇ denote a family of connections ∇ P : P → P ⊗ A Ω A/k indexed by the objects ( P, ∈ D . Definetr ∇ : Hoch II ( D )[[ u ]] → Ω • A/k [[ u ]]to be the k [[ u ]]-linear map given as follows: for n ≥
0, objects P , . . . , P n of D , and morphisms P α ←− P α ←− P α ←− · · · α n − ←−−− P n α n ←−− P , define(5.14) tr ∇ ( α [ α | · · · | α n ]) = ∞ X J =0 X j + ··· + j n = J ( − J tr (cid:16) α ∇ j α ′ ∇ j α ′ · · · ∇ j n − n α ′ n ∇ j n (cid:17) ( J + n )! u J , where ∇ i := ∇ P i (with ∇ n +1 = ∇ ), α ′ i := ∇ i ◦ α i − ( − | α i | α i ◦ ∇ i +1 , and the inner sum ranges over( n + 1)-tuples of non-negative integers that sum to J . Remark . Note that the coefficient of u J in formula (5.14) is 0 for J ≫
0, since Ω iA/k = 0 for i > dim( A ) (cf. Remark 1.2). Example 5.16. If D consists of just one object ( P, ∇ consists of a choice of connection for P , and we havetr ∇ ( α [ α | · · · | α n ]) = ∞ X J =0 X j + ··· + j n = J ( − J tr (cid:0) α ∇ j α ′ ∇ j α ′ · · · ∇ j n − α ′ n ∇ j n (cid:1) ( J + n )! u J , where α i ∈ End A ( P ) for all i , and α ′ i is the derivative of α i with respect to ∇ . Example 5.17.
If each ∇ i is a flat connection, thentr ∇ ( α [ α | · · · | α n ]) = tr ( α α ′ · · · α ′ n ) n ! . Example 5.18.
Suppose D consists of just the object P = ( A, ∈ qPerf( A op ). Let θ : A ∼ = −→ End A ( A ) denote the isomorphism of k -algebras given by θ ( a )( x ) = ax , and let ∇ be the de Rhamdifferential A d −→ Ω A/k . For any a ∈ A , we have[ ∇ , θ ( a )]( x ) = d ( ax ) − ( − | a | adx = da · x + ( − | a | adx − ( − | a | adx = da · x for all x ∈ A , and thus [ ∇ , θ ( a )] coincides with left multiplication by da in End A ( A ) ⊗ Ω • A/k . Since ∇ is flat, Example 5.17 impliestr ∇ : Hoch II ( A )[[ u ]] θ ∗ −→ Hoch II (End A ( A ))[[ u ]] tr ∇ −−→ Ω • A/k [[ u ]]is given by tr ∇ ( a [ a | · · · | a n ]) = a da · · · da n n ! . Thus, tr ∇ coincides, in this case, with the Hochschild-Kostant-Rosenberg isomorphism ǫ (see Theo-rem 3.31).The key result concerning the map tr ∇ is the following: Theorem 5.19.
Let k be a field of characteristic , let A = ( A, h ) be an essentially smooth curved k -algebra, and let D be a full subcategory of qPerf( A op ) consisting of objects with trivial differentials.For any choices of connections ∇ on the objects of D , the map tr ∇ induces a morphism HN II ( D ) → (Ω • A/k [[ u ]] , ud + dh ) of dg- k [[ u ]] -modules; that is, tr ∇ ◦ ( b + b + uB ) = ( ud + dh ) ◦ tr ∇ . Remark . It is clear from the definition of tr ∇ that it is natural with respect to inclusions of fullsubcategories (provided the same connections are used on the smaller category). Remark . The proof shows thattr ∇ ◦ ( b + uB ) = ud ◦ tr ∇ and tr ∇ ◦ b = dh ◦ tr ∇ both hold.We relegate the highly technical proof of Theorem 5.19 to the appendix; see Corollaries B.5 andB.7. CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 27
Proof of the main theorem.
Recall the map ch II P : HN II (End( P )) → HN II ( A )given in Definition 4.9. When A is an essentially smooth curved k -algebra, we compose ch II P with theHKR isomorphism to obtain the map ch II P ,HKR : HN II (End( P )) → (Ω • A/k [[ u ]] , ud + dh ) . Corollary 5.22.
Let A be an essentially smooth curved k -algebra, P = ( P, ∈ qPerf( A op ) a perfectright A -module with trivial differential, and ∇ a connection on P . Then the map HN II (End( P )) → H (Ω • A/k [[ u ]] , ud + dh ) induced by tr ∇ is equal to ch II P ,HKR .Proof. Let D be the full subcategory of qPerf( A op ) consisting of just the two objects ( P,
0) and( A, ∇ P = ∇ (the connection in the statement) and ∇ A = d , the de Rham differential. ByTheorem 5.19, the map tr ∇ : HN II ( D ) → H (Ω • A/k [[ u ]] , ud + dh )defined by (5.14) is a map of dg- k [[ u ]]-modules. Moreover, using the naturality of tr ∇ (Remark 5.20)and Example 5.18, we see that the diagramHN II (End( P )) tr ∇ P ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ ( ι ) ∗ / / HN II ( D ) tr ∇ (cid:15) (cid:15) HN II ( A ) ( ι ) ∗ o o ǫ v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠ H (Ω • A/k [[ u ]] , ud + dh )commutes, where ǫ is the HKR map, and ι , ι are the evident inclusion functors. These inclusionfunctors are pseudo-equivalences; thus, by Proposition 3.18, the horizontal maps in the diagram areisomorphisms. Since the HKR map ǫ is an isomorphism, so are tr ∇ and tr ∇ P . Now, consider thecomposition T : HN II ( D ) → HN II (qPerf( A op )) ∼ = ←− HN II ( A ) ǫ −→ H (Ω • A/k [[ u ]] , ud + dh ) , where the first map is induced by inclusion. The map T also satisfies T ◦ ( ι ) ∗ = ǫ , and hence we musthave T = tr ∇ . It follows that tr ∇ P = tr ∇ ◦ ( ι ) ∗ = T ◦ ( ι ) ∗ , and it is clear from the definitions that T ◦ ( ι ) ∗ = ch II P ,HKR . (cid:3) We now prove the main theorem:
Proof of Theorem 5.7.
By Proposition 4.11, we have ch IIHKR ( P ) = ch II P ♮ ,HKR ( X n ( − n [ δ P | · · · | δ P | {z } n copies ]) . Corollary 5.22 applied to P ♮ thus yields ch IIHKR ( P ) = X n X j ,...,j n ( − j + ··· + j n + n tr( ∇ j δ ′ P ∇ j δ ′ P · · · δ ′ P ∇ j n )( j + · · · + j n + n )! u j + ··· + j n , which, by (5.10), is equal to tr(exp( − R )). (cid:3) Remark . In the setting of Theorem 5.7, it seems likely that one can obtain an explicit formulafor ch IIHKR using instead the results of Segal in [Seg13], in the following way. In [Seg13, Proposition2.13], Segal constructs an explicit quasi-isomorphism
T r : Hoch II ( D ) ≃ −→ Hoch II ( A ) , where D is the full cdg subcategory of qPerf( A op ) spanned by objects with trivial differential. The map T r is an adaptation of the “generalized trace map” (cf. [Lod13, Section 1.2]) to the setting of curvedmodules. Unfortunately,
T r does not induce a map on negative cyclic homology of the second kind,because it does not commute with the B operator. For example, take A = C [ x , x , x ] / ( x + x + x − h = 0, and P the image of the idempotent12 (cid:18) − x − x − ix − x + ix x (cid:19) : A ⊕ → A ⊕ . Then ( B ◦ T r )(id P ) = ( T r ◦ B )(id P ).We believe that this problem may be rectified by working in the more general setting of non-unitalcdga’s, and slightly modifying Segal’s map so that it is defined using the version of the cyclic barcomplex for nonunital cdga’s (cf. [Shk14, Section 3.2]). Granting this, it follows that the modifiedversion of the map T r induces a map HN II ( D ) → HN II ( A ). Moreover, the proof of Corollary 5.22would then hold when tr ∇ is replaced by ǫ ◦ T r (except there is no need to choose any connections),and the above proof of Theorem 5.7 would therefore yield another explicit formula for ch IIHKR .In more detail, given a perfect left A -module ( P, δ P ), one may realize it as a summand of ( F, δ F )with F a free A -module. Let e be the idempotent endomorphism of F with image P . Choose a basisof F and, for any endomorphism γ of F , write γ ′ for its derivative with respect to the associatedLevi-Cevita connection (that is, if we represent γ as a matrix, then γ ′ = dγ , the result of applying thede Rham differential to the entries of this matrix). Then, assuming the extension of Segal’s map tothe non-unital setting works out as we expect, the formula for the Chern character of ( P, δ P ) arisingfrom Segal’s map would be(5.24) X j j ! tr ( e ( δ ′ F ) j )+ X i ≥ X j ,...,j i ( − i (2 i − i − i + J )! tr (cid:0) (2 e − δ ′ F ) j e ′ ( δ ′ F ) i e ′ · · · e ′ ( δ ′ F ) j i (cid:1) , where j , . . . , j i range over all non-negative integers, and J := j + · · · + j i .Assuming the details check out, the complicated formula (5.24) and the formula of Theorem 5.7must agree as homology classes, since they both coincide with ch IIHKR ( P, δ P ). It would be pleasing togive a direct proof of this fact, but we have been unable to do so.6. Examples
Throughout this section, assume char( k ) = 0. Example 6.1.
Suppose Γ = Z / A = ( A, − f ), where A is the localization of a smooth k -algebraat a maximal ideal m , and f ∈ m ⊆ A is a non-zero-divisor. Recall that Perf( A op ) is identical tothe differential Z / A, f ) of matrix factorizations of f . The HKR isomorphismyields HN II ( A ) ∼ = −→ H (Ω • A/k [[ u ]] , ud − df ) . Let P = ( P, δ P ) ∈ Perf( A op ), and write P = P ⊕ P , where P (resp. P ) is the even (resp. odd)degree component of P . Since f is a non-zero-divisor, r := rank( P ) = rank( P ). Upon choosingbases of P and P , we may identify δ P with a matrix of the form (cid:18) αβ (cid:19) , where α : P → P and β : P → P are ( r × r ) matrices over A . Using this choice of basis, we also construct a Levi-Civitaconnection ∇ on P (Remark 5.11). Since this connection is flat, we have R = [ ∇ , δ P ] = (cid:18) dαdβ (cid:19) ∈ End A ( P ) ⊗ A Ω • A/k [[ u ]] , CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 29 where dα, dβ denote the matrices obtained by applying the de Rham differential entry-wise to α, β .Thus, for any j ≥
0, we have R j = [ ∇ , δ P ] j = (cid:18) ( dαdβ ) j
00 ( dβdα ) j (cid:19) , and so tr( R j ) = tr(( dαdβ ) j ) − tr(( dβdα ) j ). By Remark 5.5,tr([ ∇ , δ P ] j ) = ( j = 02 tr(( dαdβ ) j ) if j > ch IIHKR ( P ) = X j ≥ dαdβ ) j )(2 j )! ∈ H (Ω • A/k [[ u ]] , ud − df ) . Recall that the canonical map HN II ( A ) → HH II ( A ) is given by setting u = 0, and, in this case,this map may be identified, via the HKR isomorphism, with H (Ω • A/k [[ u ]] , ud − df ) u −−−→ H (Ω • A/k , − df ) . Let ch IIHH ( P ) denote the image of ch IIHN ( P ) in HH II ( A ). The class ch IIHH ( P ) corresponds, via theHKR isomorphism, to the class(6.2) X j ≥ dαdβ ) j )(2 j )! ∈ H (Ω • A/k , − df ) . As discussed in the introduction, Segal obtains formula (6.2) in [Seg13, 3.1].When
A/f has at most an isolated singularity (meaning that ( A p /f ) is smooth for all primes p = m ), then, letting n = dim( A ), we have H (Ω • A/k , − df ) ∼ = Ω nA/k df ∧ Ω n − A/k , n even0 , n odd . Moreover, when n is even, ch IIHN ( P ) corresponds to(6.3) 2 tr (( dαdβ ) n ) n ! ∈ Ω nA/k df ∧ Ω n − A/k . In this case, the canonical map HH (MF( A, f )) → HH II (MF( A, f )) is an isomorphism [PP12, Section4.8]. Thus, (6.3) coincides with the formula, due to Polishchuk-Vaintrob, of the Chern character of amatrix factorization of an isolated singularity f ∈ k [ x , . . . , x n ] taking values in HH (MF( A, f )); see[PV12, Corollary 3.2.4] for the precise statement of their result.
Example 6.4.
Let A be an essentially smooth k -algebra, let f , . . . , f c ∈ A be a regular sequence,and set R := A/ ( f , . . . , f c ). Take Γ = Z . Let D bdg ( R ) denote the differential Z -graded categorywhose objects are bounded below complexes of finitely generated projective R -modules whose totalhomology is finitely generated, so that D bdg ( R ) is a dg enhancement of the ordinary derived category D b ( R ). In this example, we give a formula for the Chern character map K ( D bdg ( R )) → HN ( D bdg ( R )) . Let G ( R ) denote the Grothendieck group of the exact category mod( R ) of finitely generated R -modules. Then there is an isomorphism(6.5) G ( R ) ∼ = −→ K ( D bdg ( R )) = K ∆0 ( D b ( R ))given by sending the class of a module M to the class of M regarded as a complex concentrated indegree 0. The inverse sends the class of a complex C to P i ( − i [ H i ( C )]. Note that the Grothendieckgroup of D bdg ( R ) coincides, by definition, with K ∆0 ([ D bdg ( R )]) = K ∆0 ( D b ( R )) [Sch11, Section 3.2.32]. Let B denote the curved differential Z -graded algebra ( A [ T , . . . , T c ] , − ( f T + · · · + f c T c )), where T , . . . , T c are degree two indeterminants. A theorem of Burke-Stevenson [BS13, Theorem 7.5] givesa quasi-equivalence of dg categories(6.6) D dg ( R ) ≃ −→ Perf( B )and therefore a commutative diagram(6.7) K ( D bdg ( R )) / / ∼ = (cid:15) (cid:15) HN ( D bdg ( R )) / / ∼ = (cid:15) (cid:15) HN II ( D bdg ( R )) (cid:15) (cid:15) K (Perf( B )) / / HN (Perf( B )) ∼ = / / HN II (Perf( B )) . The two left vertical arrows are isomorphisms because K ( − ) and HN ( − ) send quasi-equivalences toisomorphisms. The bottom-right map is an isomorphism by [PP12, Corollary A in Section 4.7].Combining 6.5, Diagram (6.7), Proposition 3.25, and the HKR theorem, one sees that the Cherncharacter map K ( D bdg ( R )) → HN ( D bdg ( R ))is given, up to the canonical isomorphisms described above, by a map(6.8) ch : G ( R ) → H (Ω • A [ T ,...,T c ] /k [ u ] , ud − d ( f T + · · · + df c T c )) . In detail, ch is given by the composition G ( R ) ( . ) −−−→ K ( D bdg ( R )) ( . ) −−−→ K (Perf( B )) ch IIHKR −−−−−→ H (Ω • A [ T ,...,T c ] /k [ u ] , ud − d ( f T + · · · + df c T c )) . We proceed to describe this map explicitly.Let M be a finitely generated R -module. We first describe the image of [ M ] under the map ψ : G ( R ) ∼ = −→ K (Perf( B ))induced by (6.5) and (6.6).For any J = ( j , . . . , j c ) ∈ Z c ≥ , set | J | = P ci =1 j i . For each 1 ≤ i ≤ c , let e i denote the element of Z c ≥ with 1 in the i th component and 0 elsewhere. Given a finite A -projective resolution P of M , weequip P with a system of higher homotopies ; i.e., a family of endomorphisms σ J : P → Σ − | J | +1 P indexed by J ∈ Z c ≥ such that • σ is the differential on P , • for each 1 ≤ i ≤ c , the map σ σ e i + σ e i σ is given by multiplication by f i , and • for each J with | J | ≥ P J + J = J σ J σ J = 0.(See [EP16, Proposition 3.4.2] for a proof that such a system of higher homotopies exists.) Then ψ ([ M ]) = [( P [ T , . . . , T c ] , δ )] , where δ := P J ∈ Z c ≥ σ J T J . (For J = ( i , . . . , i c ), T J denotes T i · · · T i c c .) Thus,(6.9) ch ([ M ]) = ( ch IIHKR ◦ ψ )([ M ]) = ch IIHKR ( P [ T , . . . , T c ] , δ ) , and one may compute ch IIHKR ( P [ T , . . . , T c ] , δ ) using Theorem 5.7.We work out this computation in detail in the case where A is local and c = 1. Let M be afinitely generated R -module. The group G ( R ) is generated by classes represented by maximal CohenMacaulay (MCM) R -modules, i.e. R -modules whose projective dimension over A is 1, so we mayassume M is MCM. Let P α −→ P be the minimal A -free resolution of M ; α was denoted by σ in the CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 31 more general setup above. Then there is a unique map β : P → P such that α ◦ β = f = β ◦ α ; thisis the map written as σ e above. Therefore, by (6.9), ch ([ M ]) = ch IIHKR ( P [ T ] ⊕ Σ P [ T ] , δ ) , where δ := (cid:18) αβT (cid:19) . We compute ch IIHKR ( P [ T ] ⊕ Σ P [ T ] , δ ) using Theorem 5.7. It is convenient to use the canonicalisomorphism Ω • A [ T ] /k ∼ = Ω • A/k [ T ] ⊕ Ω • A/k [ T ] dT. Under this identification, the differential ud − d ( f T ) = ud − df T − f dT on Ω • A [ T ] /k becomes (cid:18) ud A − ( df ) T ∂∂T dT − f dT ud A − ( df ) T (cid:19) , where d A is the de Rham differential on Ω • A/k .Choose bases of P and P , so that we may represent α, β as matrices. Let ∇ denote the Levi-Civitaconnection on P [ T ] ⊕ Σ P [ T ] corresponding to this choice of basis. Then the associated curvature,regarded as a matrix with entries in Ω • A/k [ T ] ⊕ Ω • A/k [ T ] dT , is R = [ ∇ , δ ] = (cid:18) dαT dβ (cid:19) + (cid:18) β (cid:19) dT. For brevity, we set R = (cid:18) dαT dβ (cid:19) and R = (cid:18) β (cid:19) , so that R = R + R dT . For any l , we have R l = R l + l − X i =0 R i R dT R l − i − . By properties of the trace map, tr( R l ) = tr( R l ) + l · tr( R l − R ) dT, which gives tr( e − R ) = tr( e − R ) − tr( e − R R ) dT. Therefore, ch ([ M ]) = X j ≥ dα · dβ ) j )(2 j )! T j + X j ≥ tr (cid:0) ( dα · dβ ) j · dα · β (cid:1) (2 j + 1)! T j dT, regarded as a cochain in the complex (cid:18) Ω • A/k [ T, u ] ⊕ Ω • A/k [ T, u ] dT, (cid:18) ud A − T df ∂∂T dT − f dT ud A − T df (cid:19)(cid:19) . Observe that the computation in the previous example is recovered from this one by composing withthe map to the Z / • A/k [[ u ]] , ud A + df ) given by setting T = 1 and dT = 0. Example 6.10.
As a final example, we compute the Chern character of a perfect curved module P which does not admit a flat connection. Take Γ = Z / k = C , and A = ( A, − h ), where A isthe Γ-graded smooth C -algebra C [ x , . . . , x ] / ( x + · · · + x −
1) concentrated in even degree, and h = (1 − x )( x x + x x ). Recall that Perf( A op ) coincides with the dg category MF( A, h ) of matrixfactorizations of h . Let P denote the image of the idempotent e := 12 − x − x − ix x + ix − x + ix x x + ix x − ix x x + ix x − ix x − ix − x of A ⊕ . A direct calculation shows that the top Chern class of the ordinary projective A -module P ,thought of as an element of H dR ( A/k ), is32 X i =1 ( − i +1 x i dx · · · c dx i · · · dx . Stokes’ Theorem implies that the integral of this Chern class, thought of as an element of H dR ( S ), is vol( B ) = 0, where B denotes the unit ball in R . In particular, P does not admit a flat connection.Interpret the idempotent e as a map π : A ⊕ → P , and let ι : P ֒ → A ⊕ denote the inclusion.Similarly, let P ′ denote the image of 1 − e , interpret 1 − e as a map π ′ : A ⊕ → P ′ , and let ι ′ : P ′ ֒ → A ⊕ denote the inclusion. We now construct a matrix factorization of h with underlying projective A -module P ⊕ Σ P ′ .Set v := − x , and consider the matrices S := v ( x + ix ) v ( − x − ix ) v ( − x + ix ) v ( x + ix ) 1 0 v ( x − ix ) v ( x + ix ) 0 10 1 v ( − x + ix ) v ( − x + ix ) U := 12 − x − x − ix x + ix x − ix x − ix − x x − ix − x − x − ix − x + ix − x − x − ix . Columns 1 and 2 (resp. 3 and 4) of S give a basis for P ⊗ A [ v ] (resp. P ′ ⊗ A [ v ]) over A [ v ], and U = S − .Set T := (1 − x ) S . Then T and U are matrices over A , and of course we have T U = (1 − x ) id A ⊕ = U T.
Let C denote the matrix , and set α := x − x x x x − x x x , β := x x − x x x x − x x . Letting δ := (cid:18) πT βCU ι ′ π ′ T CαU ι (cid:19) , one sees that P := ( P ⊕ P ′ [1] , δ ) is a matrix factorization of h = (1 − x ) ( x x + x x ).We observe that there is nothing special about x x + x x in this construction, and the onlything special about the matrices α and β are that the top left and bottom right corners come from a2 × x x + x x . So we could have taken any h ′ ∈ A and any 2 × α ′ , β ′ ) of h ′ and obtained, in the same way, a matrix factorization of (1 − x ) h ′ withcomponents given by P and P ′ . CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 33
Now, let’s use Theorem 5.7 to compute ch IIHKR ( P ). Equip A ⊕ with a Levi-Civita connection D corresponding to the standard basis, and let ∇ := (cid:18) eDι e ′ Dι ′ (cid:19) be the induced connection on P ⊕ P ′ [1]. Theorem 5.7 tells us ch IIHKR ( P ) = tr(exp( − R )) = ∞ X n,J ≥ ( − J + n X j + ··· + j n = J tr( ∇ j δ ′ ∇ j δ ′ · · · δ ′ ∇ j n )( J + n )! u J , where δ ′ := [ ∇ , δ ]. Collecting the terms which are nonzero in Ω • A/k [[ u ]], we gettr(exp( − R )) = tr(( δ ′ ) )24 − tr( δ ′ ∇ δ ′ )2 u + tr( ∇ )2 u . The first term is the image of ch
IIHH ( P ) under the HKR map, and the last term is the difference ofthe top Chern classes of the components P and P ′ . Appendix A. Hochschild homology and cohomology via twisting cochains
Throughout, k is an arbitrary field, Γ = Z or Z /
2, and “graded” means Γ-graded. Let A = ( A, d A , h )be a cdga over k . In this section, we use the notion of a “twisting cochain” to construct a dga(Hom k ( T, A ) , d τ ) whose cohomology is the Hochschild cohomology of A (cf. [PP12, Section 2.4]), andwe exhibit the complex Hoch( A ) defined in Section 3.1 as a left dg-module over (Hom k ( T, A ) , d τ ).Our approach uses ideas of Negron in [Neg17].Define a graded k -vector space T = T ( A ) = M n ≥ (Σ A ) ⊗ n , where Σ A is the shift of the graded vector space A , so (Σ A ) i = A i +1 . For a , . . . , a n ∈ A , let[ a | · · · | a n ] := sa ⊗ · · · ⊗ sa n , where s : A → Σ A is the canonical map of degree −
1. Thus, atypical element of T is a k -linear combination of expressions of the form [ a | · · · | a n ]. Observe that[ a | · · · | a n ] ∈ T has degree | a | + · · · + | a n | − n . As a matter of convention, for n = 0, the symbol []denotes the element 1 in k = (Σ A ) ⊗ . T is a counital coalgebra over k under the “splitting of tensors” coproduct map∆ : T → T ⊗ k T defined by∆([ a | . . . | a n ]) = n X i =0 [ a | · · · | a i ] ⊗ [ a i +1 | · · · | a n ]= [] ⊗ [ a | . . . | a n ] + ( n − X i =1 [ a | · · · | a i ] ⊗ [ a i +1 | · · · | a n ]) + [ a | . . . | a n ] ⊗ [] . Define a coderivation on T to be a degree one map δ : T → T such that(1 ⊗ δ + δ ⊗ ◦ ∆ = ∆ ◦ δ. Let CoDer(T) denote the set of (degree one) coderivations on T . Then there is a bijectionCoDer(T) ∼ = −→ Hom k ( T, Σ A ) given by δ π ◦ δ , where π : T ։ Σ A is the canonical projection. The inverse is given by sending g ∈ Hom k ( T, Σ A ) to the coderivation[ a | · · · | a n ] X ≤ j ≤ n − ss ≥ ( − ( | a | + ··· + | a j |− j ) [ a | · · · | a j | g s ([ a j +1 | · · · | a j + s ]) | · · · | a n ] , where g s is the restriction of g to (Σ A ) ⊗ s ; see [Mar12, page 41].We define coderivations δ (2) T , δ (1) T , and δ (0) T on T using the multiplication map µ , differential d A ,and curvature h of the cdga A . In detail, for i = 0 , ,
2, let δ ( i ) T be the unique coderivation such that π ◦ δ ( i ) T vanishes on (Σ A ) ⊗ n for all n = i and such that the restriction of π ◦ δ ( i ) T to (Σ A ) ⊗ i is • the map [ a | b ] ( − | a |− [ ab ] for i = 2; • the map [ a ] [ − da ] for i = 1; and • the map (Σ A ) ⊗ → Σ A sending [] ∈ (Σ A ) ⊗ to [ h ] for i = 0.The sign for δ (2) T is explained by the fact that it is the map s ◦ µ ◦ ( s − ⊗ s − ). Applying this to[ a | b ] = sa ⊗ sb introduces a sign of ( − | s − || sa | = ( − | a |− , because s − and sa are interchanged. Thesign in the formula for δ (1) T is justified since [ a ]
7→ − [ da ] is the standard differential on the suspensionΣ A of A .Explicitly, for any n and a , . . . , a n ∈ A , we have: δ (2) T ([ a | · · · | a n ]) = n − X j =1 ( − | a | + ··· + | a j |− j [ a | · · · | a j a j +1 | · · · | a n ] ,δ (1) T ([ a | · · · | a n ]) = n X j =1 ( − | a | + ··· + | a j − |− j [ a | · · · | d ( a j ) | · · · | a n ] , and δ (0) T ([ a | · · · | a n ]) = n X j =0 ( − | a | + ··· + | a j |− j [ a | · · · | a j | h | a j +1 | · · · | a n ] . We endow T with the coderivation δ T = δ (2) T + δ (1) T + δ (0) T . Lemma A.1. ( T, ∆ , δ T ) is a dg-coalgebra; that is, δ T is a coderivation, and δ T = 0 .Proof. δ T is a coderivation since each of δ (2) T , δ (1) T and δ (0) T are coderivations. The condition δ T ◦ δ T = 0is equivalent to the defining relations for a curved dga. In detail, we have δ T ◦ δ T = δ (2) T ◦ δ (2) T + [ δ (2) T , δ (1) T ] + [ δ (2) T , δ (0) T ] + δ (1) T ◦ δ (1) T + [ δ (1) T , δ (0) T ] + δ (0) T ◦ δ (0) T , and each of the six terms on the right is a coderivation of T . In general, a coderivation δ ′ of T vanishesif and only if the k -linear map π ◦ δ ′ = 0. We have(1) π ◦ δ (2) T ◦ δ (2) T = 0, since multiplication is associative;(2) π ◦ [ δ (2) T , ◦ δ (1) T ] = 0, since d satisfies the Leibniz rule;(3) π ◦ (cid:16) [ δ (2) T , δ (0) T ] + δ (1) T ◦ δ (1) T (cid:17) = 0, since d = [ h, − ];(4) π ◦ [ δ (1) T , δ (0) T ] = 0, since d ( h ) = 0; and(5) π ◦ δ (0) T ◦ δ (0) T = 0, since δ (0) T maps (Σ A ) ⊗ n to (Σ A ) ⊗ n +1 for all n . (cid:3) We now consider the graded k -vector space Hom k ( T, A ). There is a canonical isomorphismHom k ( T, A ) ∼ = Y n ≥ Hom k ((Σ A ) ⊗ n , A ) , so we will write a typical element of Hom k ( T, A ) as an infinite sum f = P n f n for maps f n : (Σ A ) ⊗ n → A , n ≥
0. The maps δ T and d A endow Hom k ( T, A ) with a differential given by d Hom ( f ) = d A ◦ f − ( − | f | f ◦ δ T , making it into a chain complex.Using that T is a coalgebra and A is an algebra, Hom k ( T, A ) admits a “convolution” product ⋆ defined as f ⋆ g = µ ◦ ( f ⊗ g ) ◦ ∆ . CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 35
Explicitly, given f m : (Σ A ) ⊗ m → A and g n : (Σ A ) ⊗ n → A , the map f m ⋆ g n : (Σ A ) ⊗ ( m + n ) → A isgiven by( f m ⋆ g n )([ a | · · · | a m + n ]) = ( − | g | ( | a | + ··· + | a m |− m ) f ([ a | · · · | a m ]) g ([ a m +1 | · · · | a m + n ]) . For f = P m f m and g = P n g n we have f ⋆ g = X k X m + n = k f m ⋆ g n . The following is an immediate consequence of the facts that ∆ is co-associative, δ T is a coderivationfor ∆, µ is associative, and d A is a derivation for A : Lemma A.2.
The endomorphism d Hom on Hom k ( T, A ) is a derivation relative to the convolutionproduct. In general, (Hom k ( T, A ) , d Hom ) is not a dga, because d Hom does not square to 0. Instead, d Hom ◦ d Hom = [ h ◦ ǫ, − ] , where ǫ : T → k is the counit for T and we interpret h as a map k → A . Since d Hom ( h ◦ ǫ ) = 0, wehave: Lemma A.3. (Hom k ( T, A ) , ⋆, d Hom , h ◦ ǫ ) is a cdga over k . Let τ ∈ Hom k ( T, A ) be the degree 1 element that is zero on (Σ A ) ⊗ n for all n = 1 and is the map s − on Σ A . Then τ is a “twisting cochain” for the cdga (Hom k ( T, A ) , ⋆, d Hom , h ◦ ǫ ); that is, it satisfiesthe Maurer-Cartan equation: τ ⋆ τ − d Hom ( τ ) + h ◦ ǫ = 0 . As with any twisting cochain, we may use τ to deform the differential by setting d τ = d Hom − [ τ, − ] , where [ − , − ] denotes the commutator with respect to the convolution product. Explicitly, d τ ( f ) = d A ◦ f − ( − | f | f ◦ δ T − τ ⋆ f + ( − | f | f ⋆ τ. Lemma A.4. (Hom k ( T, A ) , ⋆, d τ ) is a dga over k . That is, d τ ◦ d τ = 0 , and d τ satisfies the Leibnizrule relative to the convolution product ⋆ Proof.
For any degree one element β , the endomorphism [ β, − ] satisfies the Leibniz rule; hence, sodoes d τ . Using that τ is a twisting cochain, we get d τ = d − d Hom ◦ [ τ, − ] − [ τ, − ] ◦ d Hom + [ τ ⋆ τ, − ]= [ h ◦ ǫ, − ] − [ d Hom ( τ ) , − ] + [ τ ⋆ τ, − ]= [ h ◦ ǫ − d Hom ( τ ) + τ ⋆ τ, − ]= 0 . (cid:3) Let us write Hom τk ( T, A ) for the dga (Hom k ( T, A ) , ⋆, d τ ). Explicitly, for a , . . . , a n +1 ∈ A and f = P n f n ∈ Hom k ( T, A ), we have d τ ( f )([ a | · · · | a n +1 ]) = − ( − | f | ( | a |− a f n ([ a | · · · | a n +1 ]) − n X j =1 ( − | f | + | a | + ··· + | a j |− j f n ([ a | · · · | a j a j +1 | · · · | a n +1 ])+ ( − | f | + | a | + ··· + | a n |− n f n ([ a | · · · | a n ]) a n +1 − n +1 X j =1 ( − | f | + | a | + ··· + | a j − |− j f n +1 ([ a | · · · | d A ( a j ) | · · · | a n +1 ]) − n +1 X j =0 ( − | f | + | a | + ··· + | a j |− j f n +2 ([ a | · · · | a j | h | a j +1 | · · · | a n +1 ])+ d A ( f n +1 ([ a | · · · | a n +1 ])) . Thus, Hom τk ( T, A ) is the standard Hochschild cohomology complex of the cdga A (cf. [PP12, Section2.4]).We can also build the Hochschild homology complex of A from T , δ T and τ . We define both a leftand a right action of the algebra Hom k ( T, A ) on the graded k -vector space A ⊗ k T as follows. As amatter of shorthand, given an element t ∈ T , we write t ′ i , t ′′ i for the various elements of T appearingin ∆( t ) = P i t ′ i ⊗ t ′′ i . In detail, if t = [ a | · · · | a n ], then t ′ i = [ a | · · · | a i ] and t ′′ i = [ a i +1 | · · · | a n ], for i = 0 , . . . , n . Given f ∈ Hom k ( T, A ), a ∈ A , and t ∈ T , we define(A.5) f · ( a ⊗ t ) = X i ( − | a || t | + | t ′ i | ( | t ′′ i | + | a | ) f ( t ′′ i ) a ⊗ t ′ i and(A.6) ( a ⊗ t ) · f = X i ( − | f || t | af ( t ′ i ) ⊗ t ′′ i . Equivalently, left multiplication by f is the function(A.7) σ ◦ (id T ⊗ µ ) ◦ (id T ⊗ f ⊗ id A ) ◦ (∆ ⊗ id A ) ◦ σ, where σ : A ⊗ k T → T ⊗ k A is the switching map given by σ ( a ⊗ t ) = ( − | a || t | t ⊗ a . Recall that giving A ⊗ k T the structure of a right module over Hom k ( T, A ) is equivalent to giving it the structure of aleft module over Hom k ( T, A ) op , and that the two are related by( a ⊗ t ) · f = ( − | f | ( | a | + | t | ) f op · ( a ⊗ t ) , where, by f op , we mean the element f ∈ Hom k ( T, A ) regarded as an element of Hom k ( T, A ) op . Withthis in mind, we can see that right multiplication by f ∈ Hom k ( T, A ) on A ⊗ k T corresponds to theleft action of f op given by the function(A.8) ( µ ⊗ id T ) ◦ (id A ⊗ f op ⊗ id T ) ◦ (id A ⊗ ∆) . We also equip A ⊗ k T with the differential d ⊗ = d A ⊗ id + id ⊗ δ T . Lemma A.9.
The actions described in (A.5) and (A.6) give ( A ⊗ k T, d ⊗ ) the structure of a bimoduleover the cdga (Hom k ( T, A ) , ⋆, d Hom , h ◦ ǫ ) . By definition, this means A ⊗ k T is a Hom k ( T, A ) − Hom k ( T, A ) bimodule in the usual sense, d ⊗ satisfies the Leibniz rule for both the left and rightactions, and d ⊗ ◦ d ⊗ = [ h ◦ ǫ, − ] where [ h ◦ ǫ, − ] is the endomorphism of A ⊗ k T determined by the two actions given by [ h ◦ ǫ, ω ] = ( h ◦ ǫ ) · ω − ω · ( h ◦ ǫ ) . CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 37
Proof.
The actions defined in (A.5) and (A.6) are obviously additive. It is a straightforward (buttedious) exercise to check the that they are associative, and also that the relation( f · ( a ⊗ t )) · g = f · (( a ⊗ t ) · g )holds. It is clear that d ⊗ ◦ d ⊗ = [ h ◦ ǫ, − ], and the left and right Leibniz rules follow in a mannersimilar to the proof of Lemma A.2 outlined above; that is, one applies the abstract formulas (A.7)and (A.8) for the left and right actions, recalling that d A is a derivation and δ T is a coderivation. (cid:3) We now deform the differential on ( A ⊗ k T, d ⊗ ) to make it a dg-module over the dga Hom τk ( T, A ). Lemma A.10.
The endomorphism d ⊗ − [ τ, − ] of A ⊗ k T makes it a left dg-module over the dga Hom τk ( T, A ) .Proof. Proceed exactly as in the proof of Lemma A.4, applying Lemma A.9. (cid:3)
Let us write A ⊗ τk T for the complex ( A ⊗ k T, d ⊗ − [ τ, − ]). Explicitly, the differential d ⊗ − [ τ, − ]sends an element α ∈ A ⊗ k T to( d A ⊗ id T )( α ) + (id A ⊗ δ (2) T )( α ) + (id A ⊗ δ (1) T )( α ) + (id A ⊗ δ (0) T )( α ) − τ · α + ( − | α | α · τ. For α = a [ a | · · · | a n ] we have( d A ⊗ id T )( α ) = d ( a )[ a | · · · | a n ] , (id A ⊗ δ (2) T )( α ) = n − X j =1 ( − | a | + | a | + ···| a j |− j a [ a | · · · | a j a j +1 | · · · | a n ] , (id A ⊗ δ (1) T )( α ) = n X j =1 ( − | a | + | a | + ··· + | a j − |− j a [ a | · · · | d ( a j ) | · · · | a n ] , (id A ⊗ δ (0) T )( α ) = n X j =0 ( − | a | + | a | + ··· + | a j |− j a [ a | · · · | a j | h | a j +1 | · · · | a n ] . In the formulas for the left and right actions of τ ∈ Hom k ( T, A ) on A ⊗ k T , the only terms that donot vanish are those when i = n − i = 1, respectively, and so we have − τ · a [ a | · · · | a n ] = − ( − ( | a | + ··· + | a n − |− n − | a n |− a n a [ a | · · · | a n − ] , and ( − | α | a [ a | · · · | a n ] · τ = ( − | a | + | a | + ··· + | a n |− n + | a | + ··· + | a n |− n a a [ a | · · · | a n ]= ( − | a | a a [ a | · · · | a n ] . Putting these equations together gives that d ⊗ − [ τ, − ] coincides with the formula for the Hochschilddifferential b = b + b + b for A given in Section 3.1. That is, we have an identity A ⊗ τk T = (Hoch( A ) , b )of chain complexes.In summary, the Hochschild homology of A is (Hoch( A ) , b ) = A ⊗ τk T , and it is a left dg-moduleover Hochschild cohomology of A , which is the dga Hom τ ( T, A ). Appendix B. Proof of Theorem 5.19
Assume k is a field of characteristic 0 and C is a curved differential Γ-graded category over k , whereΓ = Z or Γ = Z /
2. Define the graded k -vector space C / [ C , C ] = M X End( X ) / ∼ , where ∼ is the equivalence relation given by modding out by commutators; that is, it is generated by αβ ∼ ( − | α || β | βα for all pairs of maps of the form α : X → Y and β : Y → X . The graded k -vector space C / [ C , C ] is not an algebra, but it does enjoy the following “(graded) cyclic invariance” property:given composable morphisms X m α m −−→ X m − α m −−→ · · · α −→ X α −→ X m we have α ◦ · · · ◦ α m = ( − | α m | ( | α | + ··· + | α m − | ) α m ◦ α ◦ · · · ◦ α m − (mod [ C , C ]) . The differentials in C induce a map d on C / [ C , C ] that squares to 0, making C / [ C , C ] into a complexof k -vector spaces. Let v be a formal parameter of degree −
2, and let C / [ C , C ][[ v ]] be the complexof k [[ v ]]-modules obtained from C / [ C , C ] by extension of scalars along k → k [[ v ]]. Define a k [ v ]-linearmap of degree 0 φ : Hoch( C )[ v ] → C / [ C , C ][[ v ]]as follows: for any integer n ≥
0, objects X , . . . , X n of C and morphisms a i : X i +1 → X i , where X n +1 := X , we set φ ( a [ a | · · · | a n ]) = X ~j a h j d ( a ) · · · h j n − n d ( a n ) h j n ( J + n )! v J + n (mod [ C , C ]) , where h , . . . , h n are the curvatures of X , . . . , X n , ~j = ( j , . . . , j n ) ranges over all ( n + 1)-tuples ofnon-negative integers, and J = j + · · · + j n . To ease notation in this section, we omit the bars overthe a i when we write elements of the reduced Hochschild homology complex.This map extends in the evident way to a mapHoch II ( C )[[ v ]] → C / [ C , C ][[ v ]]which we also call φ . In more detail, the first map is continuous for the topology on the sourcedetermined by the (reduced version of the) filtration F j defined in (3.4) and the usual topology forpower series on the target, and hence it induces a map on completions. Proposition B.1.
With the notation above, we have vd ◦ φ = − vφ ◦ b + φ ◦ B. Remark
B.2 . The Proposition implies that φ ◦ B is divisible by v ; this is clear from the formula for φ , since the only way to get a term not involving v occurs when n = 0. Proof of Proposition.
We have d ( φ ( a [ a | · · · | a n ])) = β ( a [ a | · · · | a n ]) + γ ( a [ a | · · · | a n ]) , where β ( a [ a | · · · | a n ]) = X j ,...,j n d ( a ) h j d ( a ) · · · d ( a n ) h j n ( J + n )! v J + n , and γ ( a [ a | · · · | a n ]) = X j ,...,j n n X i =1 σ i a h j d ( a ) · · · d ( a i ) · · · d ( a n ) h j n ( J + n )! v J + n . Here, σ i ∈ {± } is given by the formula σ i = σ i ( a , . . . , a n ) = ( − | a | + | sa | + ···| sa i − | = ( − | a | + | a | + ···| a i − | + i − . Note that σ = ( − | a | and σ i +1 = − ( − | a i | · σ i for i = 1 , . . . n − vγ + vβ = − vφ ◦ b + φ ◦ B. In fact, we show that γ = − φ ◦ b and vβ = φ ◦ B. CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 39
We first show γ = − φ ◦ b . Since d ( a i ) = h i a i − a i h i +1 we get γ ( a [ a | · · · | a n ]) = n X i =1 σ i γ i ( a [ a | · · · | a n ]) , where γ i ( a [ a | · · · | a n ]) = X ~j a h j d ( a ) · · · h j i − +1 i a i h j i i +1 · · · d ( a n ) h j n ( J + n )! v J + n − X ~j a h j d ( a ) · · · h j i − i a i h j i +1 i +1 · · · d ( a n ) h j n ( J + n )! v J + n = X ~j,j i =0 a h j d ( a ) · · · h j i − +1 i a i d ( a i +1 ) · · · d ( a n ) h j n ( J + n )! v J + n − X ~j,j i − =0 a h j d ( a ) · · · d ( a i − ) a i h j i +1 i +1 · · · d ( a n ) h j n ( J + n )! v J + n . Upon reindexing the first summation by n -tuples ~l = ( l , . . . , l n − ) with l = j , . . . , l i − = j i − + 1 , l i = j i +1 , . . . , l n − = j n , and similarly for the second summation, we obtain γ i ( a [ a | · · · | a n ]) = X ~l,l i − > a h l d ( a ) · · · h l i − i a i d ( a i +1 ) h l i i +2 · · · d ( a n ) h l n − ( L − n )! v L − n − X ~l,l i − > a h l d ( a ) · · · d ( a i − ) a i h l i − i · · · d ( a n ) h l n − ( L − n )! v L − n , where L = l + · · · + l n − = J + 1. The terms added by omitting the restriction l i − > γ i ( a [ a | · · · a n ]) = γ ′ i ( a [ a | · · · | a n ]) − γ ′′ i ( a [ a | · · · | a n ]) , where we define(B.3) γ ′ i ( a [ a | · · · | a n ]) := X ~l a h l d ( a ) · · · d ( a i − ) h l i − i a i d ( a i +1 ) h l i i +2 · · · d ( a n ) h l n − ( L − n )! v L − n γ ′′ i ( a [ a | · · · | a n ]) := X ~l a h l d ( a ) · · · d ( a i − ) a i h l i − i d ( a i +1 ) · · · d ( a n ) h l n − ( L − n )! v L − n . Note that γ ′ n = X l ,...,l n − a h l d ( a ) · · · d ( a n − ) h l n − n a n ( L − n )! v L − n and γ ′′ = X l ,...,l n − a a h l d ( a ) · · · d ( a n ) h l n − ( L − n )! v L − n . Thanks to the cyclic invariance of C / [ C , C ], the former can be rewritten as γ ′ n = X l ,...,l n − ( − | a n | ( | a | + | a | + ··· + | a n − | + n − a n a h l d ( a ) · · · d ( a n − ) h l n − n ( L − n )! v L − n . In summary, we have γ = n X i =1 σ i γ i = X i σ i ( γ ′ i − γ ′′ i ) , with γ ′ i , γ ′′ i defined in (B.3).On the other hand, rewriting (3.1) using σ i gives(B.4) φ ( b ( a [ a | . . . | a n ])) = σ φ ( a a [ a | · · · | a n ])+ n − X j =1 σ j +1 φ ( a [ a | · · · | a j a j +1 | · · · | a n ]) − ( − ( | a n |− | a | + ··· + | a n − |− ( n − φ ( a n a [ a | · · · | a n − ]) . For 1 ≤ i ≤ n −
1, using d ( a i a i +1 ) = d ( a i ) a i +1 + ( − | a i | a i d ( a i +1 ) gives φ ( a [ a | · · · | a i a i +1 | · · · | a n ])= X ~l a h l d ( a ) · · · h l i − i d ( a i a i +1 ) h l i i +2 · · · d ( a n ) h l n − ( L − n )! v L − n = X ~l a h l d ( a ) · · · h l i − i d ( a i ) a i +1 h l i i +2 · · · d ( a n ) h l n − ( L − n )! v L − n + ( − | a i | X ~l a h l d ( a ) · · · h l i − i a i d ( a i +1 ) h l i i +2 · · · d ( a n ) h l n − ( L − n )! v L − n = γ ′′ i +1 + ( − | a i | γ ′ i We also have σ φ ( a a [ a | · · · | a n ]) = σ X ~l a a h l d ( a ) · · · d ( a n ) h l n − ( L + n − v L + n − = σ γ ′′ , and − ( − ( | a n |− | a | + ··· + | a n − |− ( n − φ ( a n a [ a | · · · | a n − ])= − ( − | a n | ( | a | + ··· + | a n − |− ( n − σ n X ~l a n a h j d ( a ) · · · d ( a n − ) h j n − n ( L + n − v L + n − = − σ n γ ′ n . Substituting these equations into (B.4) and using σ i +1 = − ( − | a i | · σ i yields φ ( b ( a [ a | . . . | a n ])) = σ γ ′′ + n − X i =1 ( σ i +1 γ ′′ i +1 − σ i γ ′ i ) − σ n γ ′ n = − n X i =1 σ i ( γ ′ i − γ ′′ i )= − γ ( a [ a | · · · | a n ]) . This proves γ = − φ ◦ b .We now prove vβ = φ ◦ B . We have φ ( B ( a [ a | · · · | a n ])= n X i =0 τ i X ~j h j i d ( a i ) h j i +1 d ( a i +1 ) · · · h j n − i n d ( a n ) h j n − i +1 d ( a ) h j n − i +2 · · · h j n i − d ( a i − ) h j n +1 i )( J + n + 1)! v J + n +1 , CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 41 where the signs τ i are given by τ i = ( − ( | sa i | + ··· + | sa n | )( | sa | + ··· + | sa i − | ) = ( − ( | a i | + ··· + | a n |− n + i − | a | + ··· + | a i − |− i ) . The cyclic invariance of C / [ C , C ] gives φ ( B ( a [ a | · · · | a n ]) = n X i =0 τ i X ~m h m i d ( a i ) h m i +1 · · · d ( a n ) h m n − i +1 d ( a ) h m n − i +2 · · · d ( a i − ) h m n +1 i ( M + n + 1)! v M + n +1 = n X i =0 X ~m d ( a ) h m n − i +2 d ( a ) · · · d ( a i − ) h m n +1 + m i d ( a i ) · · · h m n − i n d ( a n ) h m n − i +1 ( M + n + 1)! v M + n +1 , where ~m = ( m , . . . , m n +1 ) ranges over all ( n + 2)-tuples of non-negative integers, and M := m + · · · + m n +1 . In the inner summation indexed by i , make the substitutions j = m n − i +2 , j = m n − i +3 , . . . , j i − = m n , j i − = m n +1 + m , j i = m , . . . , j n = m n − i +1 to get φ ( B ( a [ a | · · · | a n ]) = n X i =0 X ~j ( j i − + 1) d ( a ) h j d ( a ) · · · h j i − i d ( a i ) · · · d ( a n ) h j n ( J + n + 1)! v J + n +1 = X ~j ( j i − + 1) n X i =0 d ( a ) h j d ( a ) · · · h j i − i d ( a i ) · · · d ( a n ) h j n ( J + n + 1)! v J + n +1 = X ~j ( J + n + 1) d ( a ) h j d ( a ) · · · h j i − i d ( a i ) · · · d ( a n ) h j n ( J + n + 1)! v J + n +1 = X ~j d ( a ) h j d ( a ) · · · h j i − i d ( a i ) · · · d ( a n ) h j n ( J + n )! v J + n +1 = vβ ( a [ a | · · · | a n ]) (cid:3) Corollary B.5.
With the notation of Theorem 5.19, we have tr ∇ ◦ ( b + uB ) = ud ◦ tr ∇ . Proof.
Recall that A = ( A,
0) is an essentially smooth curved k -algebra, and D is a full cdg subcategoryof qPerf( A ) consisting of objects with trivial differentials. From D we form a new cdg category, C ,that has the same objects as D but with morphism spaces given byHom C ( P, P ′ ) := Hom A ( P, P ′ ) ⊗ A Ω • A/k [[ u ]] , where u has degree 2. We equip these with the k [[ u ]]-linear differential δ given by u [ ∇ , − ]; that is, δ ( α ) = u ∇ P ′ ◦ α − ( − | α | uα ◦ ∇ P . For each P , set the curvature of P in C to be u ∇ P . These data make C into a k [[ u ]]-linear curveddifferential Γ-graded category.Let D ♮ and C ♮ denote the k -linear categories underlying D and C , obtained by forgetting differentialsand curvatures. There is an evident functor F : D ♮ → C ♮ which induces a map F ∗ : Hoch II ( D ♮ ) → Hoch II ( C ♮ )that commutes with b and B . Moreover, the collection of trace mapsEnd A ( P ) ⊗ A Ω • A/k [[ u ]] → Ω • A/k [[ u ]]induce a map π : C / [ C , C ][[ v ]] → Ω • A/k [[ u, v ]] . Recall that, by Lemma 5.6, π ◦ δ = ud ◦ π, where d is the de Rham differential.Thus, we get an induced map π ◦ φ ◦ F ∗ : Hoch II ( D ♮ ) → Ω • A/k [[ u, v ]]given by a [ a | · · · | a n ] X ~j π ( a ( u ∇ ) j ua ′ ( u ∇ ) j · · · ua ′ n ( u ∇ ) j n )( J + n )! v J + n = X ~j π ( a ∇ j a ′ ∇ j · · · a ′ n ∇ j n )( J + n )! u J + n v J + n , where the a ′ i are as defined in Definition 5.13. Notice that the coefficient of u N v M in ( π ◦ φ ◦ F ∗ )( a [ a | · · · | a n ]) is 0 when N or M is greater than dim( A ), and so π ◦ φ ◦ F ∗ takes values inΩ • A/k [ u, v ]. Composing π ◦ φ ◦ F ∗ with the composition l : Ω • A/k [ u, v ] → Ω • A/k [ u, u − ] ֒ → Ω • A/k [[ u, u − ]] , where the first map sends v to − u − and the second is the inclusion, we obtain a map e tr ∇ : Hoch II ( D ) → Ω • A/k [[ u, u − ]]given by the formula e tr ∇ ( a [ a | · · · | a n ]) = X ~j ( − J + n π ( a ∇ j a ′ ∇ j · · · a ′ n ∇ j n )( J + n )! u J . In other words, e tr ∇ differs from the map tr ∇ defined in (5.14) by a sign of ( − n . By the Propositionand the calculations above, we have ud e tr ∇ = udlπφF ∗ = ludπφF ∗ = lπδφF ∗ = ulπvδφF ∗ = ulπ ( − vφb + φB ) F ∗ = ulπ ( − vφF ∗ b + φF ∗ B )= − e tr ∇ ( b + uB ) . Since b and uB each increase the index n by one, it follows thattr ∇ ◦ ( b + uB ) = ud ◦ tr ∇ . (cid:3) For a cdg category C , a central element of degree z of degree 2 elements z X ∈ End C ( X )ranging over all X ∈ C such that, for all morphisms α : X → Y , we have αz X = z Y α and αd ( z X ) =( − | α | d ( z Y ) α . Given such a collection z , we define a degree 1 endomorphism b z of Hoch II ( C ) by theformula b z ( a [ a | . . . | a n ]) = n X j =0 ( − | a | + | a | + ··· + | a j |− j a [ a | · · · | a j | z j +1 | a j +1 | · · · | a n ] , where if α j +1 : X j +2 → X j +1 and α j : X j +1 → X j , then z j +1 = z X j +1 . CHERN-WEIL FORMULA FOR THE CHERN CHARACTER OF A PERFECT CURVED MODULE 43
Write d ( z ) for the collection of degree 3 elements d ( z X ) ∈ End C ( X ), X ∈ C . Left multiplication by d ( z ) determines a degree 3 endomorphism of L X End C ( X ). Given maps α : X → Y and β : Y → X ,we have d ( z X ) · [ α, β ] = d ( z Y ) αβ − ( − | α || β | d ( z X ) βα = d ( z Y ) αβ − ( − | α || β | + | β | βd ( z Y ) α = d ( z Y ) αβ − ( − ( | d ( z Y ) | + | α | ) | β | βd ( z Y ) α = [ d ( z Y ) α, β ] . In particular, left multiplication by d ( z ) descends to a degree 3 endomorphism of C / [ C , C ] which weshall write as λ d ( z ) . Proposition B.6.
For any central curvature element z of C , we have φ ◦ b z = vλ d ( z ) ◦ φ Proof.
For a i : X i +1 → X i , i = 0 , . . . n , with X n +1 = X , set z i = z X i and h i = h X i . Then φ ( b z ( a [ a | · · · | a n ]))= φ ( n X i =0 ( − | a | + ··· + | a i |− i a [ a | · · · | a i | z i +1 | a i +1 | · · · | a n ]))= X i ( − | a | + ··· + | a i |− i X ~m a h m d ( a ) · · · d ( a i ) h m i i +1 d ( z i +1 ) h m i +1 i +1 d ( a i +1 ) · · · d ( a n ) h m n +1 ( M + n + 1)! v M + n +1 = d ( z ) X i X ~m a h m · · · d ( a i ) h m i + m i +1 i +1 d ( a i +1 ) · · · d ( a n ) h m n +1 ( M + n + 1)! v M + n +1 where ~m = ( m , . . . , m n +1 ) ranges over all ( n + 1)-tuples, and M := m + · · · + m n +1 . By substituting j = m , . . . , j i − = m i − , j i = m i + m i +1 , j i +1 = m i +2 , . . . , j n = m n +1 , we obtain φ ( b z ( a [ a | · · · | a n ]))= d ( z ) X ~j X i ( j i + 1) a h j · · · d ( a i ) h j i d ( a i +1 ) · · · d ( a n ) h j n ( J + n + 1)! v J + n +1 = d ( z ) X ~j ( J + n + 1) a h j · · · d ( a i ) h j i d ( a i +1 ) · · · d ( a n ) h j n ( J + n + 1)! v J + n +1 = vd ( z ) X ~j a h j · · · d ( a i ) h j i d ( a i +1 ) · · · d ( a n ) h j n ( J + n )! v J + n = vλ d ( z ) φ ( a [ a | · · · | a n ]) . (cid:3) Corollary B.7.
With the notation of Theorem 5.19, we have tr ∇ ◦ b = λ dh ◦ tr ∇ . Proof.
We adopt the notation in the proof of Corollary B.5. The curvature element h ∈ A determinesa central element of C by setting z P ∈ End A ( P ) ⊆ End A ( P ) ⊗ A Ω • A/k to be multiplication by h on P .Note that d ( z P ) is left multiplication by dh ∈ Ω A/k . Using the Proposition, we have an equality ofmaps π ◦ φ ◦ b z ◦ F ∗ = vπ ◦ λ dh ◦ φ ◦ F ∗ from Hoch II ( D ) to Ω • A/k [[ u, v ]]. We have b z ◦ F ∗ = F ∗ ◦ b and π ◦ λ dh = λ dh ◦ π , and so, upon setting v = − u − , we arrive at e tr ∇ ◦ b = − u − λ dh e tr ∇ . As before, since b shifts the index n by 1, this yields u tr ∇ ◦ b = λ dh ◦ tr ∇ . (cid:3) Observe that Corollaries B.5 and B.7 together establish Theorem 5.19.
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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
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