Hochschild and cyclic (co)homology of superadditive categories
aa r X i v : . [ m a t h . C T ] D ec Hochschild and cyclic (co)homology of superadditive categories
Deke Zhao
School of Applied Mathematics,Beijing Normal University at Zhuhai,No. 18 Jinfeng Road,Zhuhai City 519087, Chinae-mail:[email protected]
Abstract
We define the Hochschild and cyclic (co)homology groups for superadditive categories andshow that these (co)homology groups are graded Morita invariants. We also show that theHochschild and cyclic homology are compatible with the tensor product of superadditive cate-gories.
Keywords: K -categories; Superadditive categories; Hochschild (co)homology; Cyclic (co)homology; Graded Moritaequivalence; K¨unneth formula Mathematics Subject Classifications (2010)
Let K be a field. Recall that a K -linear additive category (or simply K -category) is a categorytogether with a K -vector space structure on each of its homomorphism sets such that compositionis bilinear, which is defined and investigated by Mitchell in [14]. Since their appearance, theyhave been used as a very important tool not only in algebra, but in many other fields, includ-ing algebraic topology, logic, computer science, etc. The Hochschild homology and cohomologytheories HH ∗ ( C , M ) and HH ∗ ( C , M ) of a K -category C with coefficients in a bimodule M over C were introduced by Mitchell in [14], and are closely related with theories studied by Keller [7]and McCarthy [12]. It is well-known that in case of a K -category with finite number of objects,the Hochschild (co)homology coincides with the usual Hochschild (co)homology of the K -algebraassociated to the K -category (ref. [3, Proposition 2.7]).A ( K -linear) superadditive category is a K -category A with each morphism set is a Z -graded K -vector space (or simply superspace) such that the composition of morphisms is compatible withthis Z -grading. The purpose of this paper is to define the Hochschild and cyclic (co)homology ofsuperadditive categories and investigate their behaviors with respect to the graded Morita equiva-lence and tensor product of superadditive categories.The Hochschild and cyclic (co)homology of superadditive categories enjoy a number of desirableproperties, the most basic being the agreement properties, i.e. the facts that if the superadditivecategory is trivially grading they coincides with the Hochschild and cyclic (co)homology theories of K -categories and that when applied to the superadditive category with a finite number of objectsthey specialize to the Hochschild and cyclic (co)homology of the corresponding superalgebra of thesuperadditive category (Proposition 3.13). In particular, when the superadditive category is thesuperadditive category of finitely generated projective modules over a finite dimensional superalge-bra they are exactly the Hochschild and cyclic (co)homology of this superalgebra. Furthermore, weshow that the Hochschild and cyclic (co)homology of superadditive categories are graded Morita1quivalent invariants (Theorem 4.9) and prove the Eilenberg-Zilber Theorem for Hochschild homol-ogy and the K¨unneth exact sequence for cyclic homology of superadditive categories (Theorems 5.4and 5.11).The paper is organized as follows. We begin in Section 2 with preliminaries on superadditivecategories and fix our notations. In Section 3 we define the Hochschild and cyclic (co)homologyof superadditive categories and show that they are coincide with the usual Hochschild and cyclic(co)homology of superalgebras. In Section 4 we give a brief description of graded Morita equivalenceof superadditive categories and prove the graded Morita equivalent invariance of the Hochschild andcyclic (co)homology. In Section 5, we define the shuffle and cyclic shuffle product for superadditivecategories and prove the Eilenberg-Zilber Theorem for Hochschild homology and the K¨unneth exactsequence for cyclic homology. Acknowledgements.
The author would like to thank the Institute of Mathematics of the Mathematics Institute ofthe Chinese Academy of Sciences in Beijing and the Chern Institute of Mathematics in NankaiUniversity for their hospitality and support while part of this work was carried out. The authorwas supported by the National Natural Science Foundation of China (Grant No. 11101037).
In this section, we recall some facts on superadditive category and fix the notations.
Throughout this paper we denote by K a filed and by Z = { , } . All unadorned tensorproducts will be over the field K , i.e., ⊗ = ⊗ K . The composition αβ of two morphisms is to beread as first α and then β . However, this will not present us from sometimes writing α ( x ) in placeof xα when α happens to be a function. When this is done, one must take into the account theswitch ( αβ )( x ) = β ( α ( x )).In certain case when there can be no confusion, we shall write X ∈ A to denote that X is anobject of A . Occasionally, we shall identify an object X with its identity morphism 1 X . For twoobjects X, Y of a category A , we shall write A ( X, Y ) for the set of morphisms form X to Y andwrite X α Y for α ∈ A ( X, Y ) to indicate the subscript. By a category we always mean a K -category,that is every Hom-set in it is a K -linear space and the composition maps are bilinear. Recall that a superspace is a Z -graded K -vector space, namely a K -vector space V with adecomposition into two subspaces V = V ⊕ V . A nonzero element v of V i will be called homogeneous and we denote its degree by | v | = i ∈ Z . We will view K as a superspace concentrated in degree0. Given superspaces V and W , we view the direct sum V ⊕ W and the tensor product V ⊗ W assuperspaces with ( V ⊕ W ) i = V i ⊕ W i , and ( V ⊗ W ) i = V ⊗ W i ⊕ V ⊗ W − i for i ∈ Z . With thisgrading, V ⊗ K W is called the graded tensor product of V and W . Also, we make the vector spaceHom K ( V, W ) of all K -linear maps from V to W into a superspace by setting that Hom K ( V, W ) i consists of all the K -linear maps f : V → W with f ( V j ) ⊆ W i + j for i, j ∈ Z . Elements ofHom K ( V, W ) (resp. Hom K ( V, W ) ) will be referred to as even (resp. odd) linear maps . We denoteby Gr K the category whose objects are superspaces over K and morphisms are even linear maps.Now the Koszul sign rule, v ⊗ w ( − | v || w | w ⊗ v (Note this and other such expressions only makesense for homogeneous elements.), implies that V ⊗ W ∼ = W ⊗ V in Gr K . The parity charge functor Π from Gr K to itself is an auto-equivalence defined as follows:(Π V ) i = V i +1 for V = V ⊕ V ∈ Gr K and i ∈ Z ; Π( α ) = α for α ∈ Gr K. efinition 2.3 ([11, Chapter 3.2.7]) . A superadditive category A is a K -category with each mor-phism set A ( X, Y ) is provided a Z -grading compatible with composition of morphisms, that is,for each pair X, Y ∈ A we have A ( X, Y ) = A ( X, Y ) ⊕ A ( X, Y ) such that if α ∈ A ( X, Y ) i and β ∈ A ( Y, Z ) j , then αβ ∈ A ( X, Z ) i + j for i, j ∈ Z and Z ∈ A . Asusual, A op denotes the opposite category of A , which is the superadditive category has the sameobjects and morphisms as A but the composition of morphisms is given by φ o ψ o := ( − | φ || ψ | ( ψφ ) o . Example 2.4.
Let A be a finite dimensional superalgebra over K and denote by Gr A the categoryof all (left) Z -graded A -modules with even homomorphismsHom Gr A ( X, Y ) = { f ∈ Hom A ( X, Y ) | f ( X i ) ⊆ Y i for all i ∈ Z } . Denote by gr A the full subcategory of Gr A consisting of all finitely generated G -graded A -modules.Then Gr A and gr A are abelian categories. Let Gr • A be the category with the same objects as Gr A but with extended Z -graded Hom-setsHom Gr • A ( X, Y ) := Hom Gr • A ( X, Y ) ⊕ Hom Gr • A ( X, Y ) = Hom Gr A ( X, Y ) ⊕ Hom Gr A ( X, Π Y ) , where Π is the parity change functor from Gr A to itself (ref. [11, Chapter 3]). Then Gr • A is asuperadditive category. Similarly, we can obtain the superadditive category gr • A . Let A and B be superadditive categories. A (covariant Z -) graded functor F : A → B is an additive functor between the (ungraded) categories A and B such that F induces an evenlinear map between A ( − , − ) and B ( F ( − ) , F ( − )); that is, for all X, Y ∈ A , F XY : A ( X, Y ) = M i ∈ Z A ( X, Y ) i −→ B ( F ( X ) , F ( Y )) = M i ∈ Z B ( F ( X ) , F ( Y )) i is an even linear map, or equivalently, there is a family { M ( X ) } X ∈ A of superspaces together witheven linear map A ( X, Y ) ⊗ M ( Y ) → M ( X )such that 1 Z · m = m , φ · ( ψ · m ) = ( φψ ) m for all m ∈ M ( Z ), φ ∈ A ( X, Y ), ψ ∈ A ( Y, Z ). A ( X, − ) : A → Gr K is a covariant Z -graded functor for each X ∈ A . Remark 2.6.
Let S be the suspension form A to itself, which is defined by S ( X ) = X and S ( α i ) = α i +1 for X ∈ A , α ∈ A ( − , − ), and i ∈ Z . Then a functor F : A → B is a gradedfunctor provided F commutes with the suspension.Let F , G : A → B be graded functors. Then a natural transformation η : F → G is said to even if it commutes with the suspension S . We denote by Gr( A , B ) the category of all gradedfunctors from A to B and morphisms being the graded natural transformations.By a graded (left) A -module we means a graded functor M : A → Gr K . We denote by Gr A the category of all graded A -modules and morphisms being the graded natural transformations.By definition, a ( Z )- graded A -bimodule is a ( Z -)graded bifunctor M : A × A op → Gr K . Inother words M is given by a set of superspaces {M ( X, Y ) } X,Y ∈ A and left and right actions M ( X, Y ) ⊗ A ( Y, Z ) −→ M ( X, Z ) and A ( Z, X ) ⊗ M ( X, Y ) −→ M ( Z, Y )satisfying the usual associativity conditions for all
X, Y, Z ∈ A . For instance the standard gradedbimodule over A is itself. 3 .7. The (naive) tensor product category A ⊗ B of superadditive categories A and B is thecategory whose objects are X ⊗ Y for X ∈ A and Y ∈ B and whose morphisms from X ⊗ Y to X ′ ⊗ Y ′ is the graded tensor product of superspaces A ( X, X ′ ) ⊗ B ( Y, Y ′ ). The composition isgiven by the rule ( φ ⊗ φ )( ψ ⊗ ψ ) = ( − | φ | ψ | φ ψ ⊗ φ ψ . Clearly A ⊗ B is again a superadditive category and we have the following canonical isomorphismsof superadditive categories( A ⊗ B ) ⊗ C ≃ A ⊗ ( B ⊗ C ) , A ⊗ B ≃ B ⊗ A . The category A e := A ⊗ A op will be called the enveloping category of A . Then graded A -bimodules are equivalent to graded A e -modules, which are given by φ m ψ = ( − | m | ψ ( φ ⊗ ψ ) m = ( − | φ | ( | m | + | ψ | ) m ( ψ ⊗ φ ) . Recall that the parity charge Π on Gr K defined in Section 2.2. Given a graded A -module F we define a shift operation π on F by letting π ( F )( X ) := Π F ( X ) for X ∈ A and π ( F )( α ) := Π F ( α ) for morphism α ∈ A . Now we define the category Gr • A to be the category with the same objects as Gr A but with theextended ( Z -graded) Hom-setsHom Gr • A ( F , G ) := Hom Gr A ( F , G ) M Hom Gr A ( F , π ( G )) for F , G ∈ Gr A , alternatively, Gr • A is the category of all graded functors F : A → Gr K and morphism being thenatural transformation η : F → G with η X : F ( X ) → G ( X ) is a linear map for each X in A . Inthis way Gr • A becomes a superadditive category. Remark 2.8. (i) If A is a superadditive category with finite number of objects, then the corre-sponding superalgebra of A is the unital superalgebra Λ A := L X,Y ∈ A A ( X, Y ) with a well-definedmatrix product given by the composition of A and the identity is the diagonal matrix with 1 X in the diagonal. Furthermore, the usual Z -graded Λ A -(bi)modules and Z -graded A -(bi)modulescoincide.(ii) Let A be a superalgebra equipped with a finite complete set E of orthogonal idempotents(non necessarily primitive). Then there is a finite superadditive category C ( A, E ) associated to(
A, E ) with objects E and morphisms form x to y is xAy . Composition is given by the product in A . Furthermore, the Z -graded C ( A, E )-(bi)modules and Z -graded A -(bi)modules coincide. In this section we define Hochschild and cyclic (co)homology groups for superadditive categoriesby applying the ( Z -graded) Hochschild-Mitchell complex. Throughout this section, A is a super-additive category unless otherwise stated. Definition 3.1.
Let X ( n ) := ( X n , . . . , X , X ) be an n + 1-tuple of objects of A . The K -nerve associated to X ( n ) is the superspace A ( X n , X n − ) ⊗ · · · ⊗ A ( X , X ) . The K -nerve N n of degree n is the direct sum of all the K -nerves associated to n + 1-tuples ofobjects N n ( A ) = M n +1-tuples A ( X n , X n − ) ⊗ · · · ⊗ A ( X , X ) .
4n element a ∈ N n ( A ) is a chain of degree n and denote it by a = ( a , a , · · · , a n ). Then N n ( A )is a graded A -bimodule defined by N n ( A )( X, Y ) := M n +1-tuples A ( X, X n ) ⊗ A ( X n , X n − ) ⊗ · · · ⊗ A ( X , X ) ⊗ A ( X , Y ) . The ( Z -graded) Hochschild-Mitchell complex or standard complex N ( A ) of A is · · · ∂ n +1 −→ N n ( A ) ∂ n −→ · · · ∂ −→ N ( A ) ∂ −→ N ( A ) ∂ −→ A −→ . where ∂ n = P ni =0 ( − i d in with d in : N n ( A ) → N n − ( A ) defined as follows: d in ( α ⊗ α ⊗ · · · ⊗ α n ) = a a ⊗ · · · ⊗ a n if i = 0 a ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n if 1 ≤ i < n ( − | a n || a a ··· a n − | a n a ⊗ a ⊗ · · · ⊗ a n − if i = n. (3.2)Note that each d in is well-defined and ∂ n +1 ∂ n = 0. This complex is a projective resolution of thestandard graded A -bimodule A . Definition 3.3.
Let M be a graded A -bimodule. The Hochschild homology groups HH ∗ ( A , M )of A with coefficients in M are defined to the homology groups of the chain complex · · · ∂ n +1 −→ C n ( A , M ) ∂ n −→ · · · ∂ −→ C ( A , M ) ∂ −→ C ( A , M ) −→ , where ∂ is defined as (3.2) and C n ( A , M ) = M ⊗ A e N n ( A ) = M n +1-tuples M ( X , X n ) ⊗ A ( X n , X n − ) ⊗ · · · ⊗ A ( X , X ) . Remark 3.4. If A is trivially graded, then the Hochschild homology groups HH ∗ ( A , A ) coincidewith the ones defined in the ungraded situation (ref. [14, § ( A , A ) = A / [ A , A ] gr , where [ A , A ] gr is the subsuperspace spanned by all graded commutators[ α, β ] gr = αβ − ( − | α || β | βα for morphisms α, β of A . As in the ungraded case, the Hochschild homology of a superadditive category can be defined as aderived functor. Indeed, HH ∗ ( A , A ) = Tor A e ∗ ( A , A ). Definition 3.5.
Let M be a graded A -bimodule. The Hochschild cohomology groups HH ∗ ( A , M )of A with coefficients in M are defined to the cohomology groups of the cochain complex0 −→ C ( A , M ) ∂ −→ C ( A , M ) ∂ −→ · · · ∂ n − −→ C n ( A , M ) ∂ n −→ . . . , where C n ( A , M ) = Hom( N n , M ) = Y n +1-tuples Hom K ( A ( X n , X n − ) ⊗ · · · ⊗ A ( X , X ) , M ( X n , X ))and ∂ n = P ni =0 ( − i d ni with d in : C n ( A , M ) → C n +1 ( A , M ) defined as following:( d in φ )( a , · · · , a n +1 ) = ( − | a || φ | a φ ( a , · · · , a n +1 ) if i = 0 φ ( a , · · · , a i a i +1 , · · · , a n +1 ) if 1 ≤ i < nφ ( a , · · · , a n ) a n +1 if i = n. Note that each linear map of the family d in ( φ ) is well-defined for φ ∈ C n ( A , M ) and ∂ n ∂ n +1 = 0.5ecall that the graded center Z gr ( A ) of A is the ring of endomorphisms of the identity functorto itself. More concretely Z gr ( A ) is a graded ring whose homogeneous elements consist of tuplesof homogeneous elements ( φ X ) X with X ∈ A and φ X ∈ A ( X, X ) such that for any homogeneous ψ ∈ A ( X, Y ) one has φ X ψ = ( − | ψ || φ X | ψφ Y . For example, if A is a finite dimensional superalgebrathen Z gr (gr • A ) is canonically isomorphic to the graded center of the superalgebra A itself, that is, Z gr (gr • A ):= Span K { a ∈ A | ab = ( − ) | a || b | ba for all b ∈ A } . It is clear thatHH ( A , M ) = { ( m X ) X | m X ∈ M ( X, X ) and φ m X = ( − | m X || φ | m Y φ for all φ ∈ A ( Y, X ) , Y ∈ A } . Hence HH ( A , A ) = Z gr ( A ). Note that the standard bimodule A has a resolution by tensorpowers of A which are projective graded bimodules. Applying the functor Hom( − , M ) providesprecisely the cochain complex above. Consequently Hochschild cohomology is an instance of anExt functor, namely HH ∗ ( A , A ) = Ext ∗ A e ( A , A ). Now let t n be the generator of Z / ( n + 1) Z and define its action on N n +1 ( A ) by t n ( a , a , · · · , a n ) = ( − n + | a n | ( | a | + ··· + | a n − | ) ( a n , a , · · · , a n − ) . Then for all n ≥
1, the operators t n , N n := 1 + t n + · · · + t nn , ∂ n , and ¯ ∂ n := P n − i =0 d in satisfy identities t n +1 n = id, (1 − t n − ) ¯ ∂ n = ∂ n (1 − t n ), and ¯ ∂ n N n = N n − ∂ n . Now we get the following cyclic bicomplex CC ∗∗ ( A )... ∂ (cid:15) (cid:15) ... − ¯ ∂ (cid:15) (cid:15) ... ∂ (cid:15) (cid:15) ... − ¯ ∂ (cid:15) (cid:15) N ( A ) ∂ (cid:15) (cid:15) N ( A ) − ¯ ∂ (cid:15) (cid:15) − t o o N ( A ) ∂ (cid:15) (cid:15) N o o N ( A ) − ¯ ∂ (cid:15) (cid:15) − t o o · · · N o o N ( A ) ∂ (cid:15) (cid:15) N ( A ) − ¯ ∂ (cid:15) (cid:15) − t o o N ( A ) ∂ (cid:15) (cid:15) N o o N ( A ) − ¯ ∂ (cid:15) (cid:15) − t o o · · · N o o A A − t o o A N o o A − t o o · · · N o o By convention, the standard graded bimodule A being in the left-hand corner is of bidegree (0 , CC pq ( A ) = N q ( A ). Note that columns of CC ∗∗ ( A ) are copies of the Hochschild-Mithcellcomplex N ( A ) of A . The cyclic complex C ( A ) is by definition the total complex of this doublecomplex, that is, the space of cyclic n -chains C ( A ) n = L k ≥ N n − k ( A ) with cyclic boundary ∂ + B where B = (1 − t ) s P ni =0 t i and s is the extra degeneracy defined by s ( α ⊗ α ⊗ · · · ⊗ α n ) = 1 ⊗ α ⊗ α ⊗ · · · ⊗ α n . Definition 3.9.
The cyclic homology groups HC ∗ ( A ) of the superadditive category A is definedto be the homology groups of the cyclic complex C ( A ), i.e., HC n ( A ) := H n (Tot CC ( A )).Now the arguments of [9, §
1] can be adopted to the case of superadditive categories. Moreprecisely there is a so-called
Connes periodicity operator S : C n ( A ) → C n − ( A ) , ( α n , α n − , α n − , · · · ) ( α n − , α n − , · · · ) where α i ∈ N i ( A ) . Let I be the inclusion of the Hochschild-Mitchell complex as the first column of the cyclic complex.Then the Connes periodicity operator induces the following short exact sequence of complexes0 / / N ( A ) I / / C ( A ) S / / C ( A )[2] / / Gysin-Connes exact sequence · · · / / HH n ( A ) I / / HC n ( A ) S / / HC n − ( A ) B / / HH n − ( A ) I / / · · · . (3.11)We construct now the cyclic cohomology of a superadditive category. Dualizing the cyclic bicomplex CC ∗∗ ( A ) introduced in Section 3.8, we obtain a bicomplexof cochains CC ∗∗ ( A ) such that CC pq ( A ) = C q ( A ). It has vertical differential maps ∂ ∗ or ¯ ∂ ∗ : CC pq ( A ) → CC pq +1 ( A ) and horizontal differential maps (1 − t ) ∗ or N ∗ : CC pq ( A ) → CC p +1 q ( A ).By definition the cyclic cohomology groups of A is the cohomology groups of the cochain complexTot CC ∗∗ ( A ), that is, HC n ( A ) := H n (Tot CC ∗∗ ( A )).Note that if A has just one object, then the Hochschild and cyclic (co)homology theories of A are the usual Hochschild and cyclic (co)homology theories of superalgebras (ref. [6, 17]). Moregenerally, we have the following facts. Proposition 3.13.
Let A be a superadditive category with finite number of objects and let Λ A bethe corresponding superalgebra. Then1. HH ∗ ( A , M ) = HH ∗ (Λ A , M ) and HH ∗ ( A , M ) = HH ∗ (Λ A , M ) where HH ∗ (Λ A , M ) and HH ∗ (Λ A , M ) )are respectively the Hochschild homology and cohomology of Λ A with coefficients in graded Λ A -bimodule M .2. HC ∗ ( A ) = HC ∗ (Λ A ) and HC ∗ ( A ) = HC ∗ (Λ A ) where HC ∗ (Λ A ) and HC ∗ (Λ A ) are respec-tively the cyclic homology and cohomology of Λ A .Proof. Let Λ := Λ A and consider the semisimple sub-superalgebra E = Q K X (concentrated indegree 0) of Λ. Note that any Z -graded E -bimodule is graded projective since the envelopingsuperalgebra of E is still semisimple. Consequently there is a Z -graded projective resolution of Λas a graded Λ-bimodule given by . . . −→ Λ ⊗ E Λ ⊗ E · · · ⊗ E Λ −→ . . . −→ Λ ⊗ E Λ −→ Λ −→ . Applying the functor M ⊗ Λ e − to this resolution and considering the canonical superspace isomor-phism M ⊗ Λ e (Λ ⊗ E − ⊗ E Λ) = M ⊗ E − , we obtain a chain complex computing HH ∗ (Λ , M ) which coincides with the complex we have definedfor the Hochschild homology of superalgebras (see [17, § Λ e ( − , M ) to this resolution and considering the canonical superspaceisomorphism Hom Λ e (Λ ⊗ E − ⊗ E Λ , M ) = Hom E e ( − , M ) , we obtain a cochain complex computing HH ∗ (Λ , M ) which coincides with the complex we havedefined for the Hochschild cohomology of superalgebras (see [17, § In this section we give a brief description of the graded Morita theory for superadditive categoriesand prove the graded Morita invariance of the Hochschild and cyclic (co)homologies.Let A be a superadditive category. Denote by Mod A the category of all (ungraded) A -modules,that is, the category consists of all additive covariant functors from A to Mod K and morphismsbeing the natural transformations. Denote by F the forgetful functor from Gr A to Mod A . Now7et B be another superadditive category. A linear additive functor F : Gr A −→ Gr B is an evenfunctor provided it commutes with the charge parity π . An even functor F : Gr A −→ Gr B issaid to be an even equivalence provided there is an even functor G : Gr B −→ Gr A such that F G = Id Gr A and GF = Id Gr B . In this case we say that A and B are even equivalent . Definition 4.1.
Two superadditive categories A and B are said to be graded Morita equivalent if Gr A is even equivalent to Gr B .Let A be a superadditive category. An idempotent in A is an even morphism ǫ ∈ A ( X, X )such that ǫ = ǫ . An idempotent ǫ ∈ A ( X, X ) is split if there is an object Y ∈ A and homogeneousmorphisms α ∈ A ( X, Y ), β ∈ A ( Y, X ) such that αβ = ǫ and βα = 1 Y . We say that A is idempotent complete if every idempotent is split. Remark 4.2.
Every superadditive category A be fully faithfully embedded into an idempotentcomplete superadditive category c A .The following construction of c A is essential due to Freyd (cf. [13]): The objects of c A are thepairs ( X, ǫ ) consisting of an object X of A and an idempotent ǫ ∈ A ( X, X ) while the sets ofmorphisms are c A (( X, ǫ ) , ( Y, ε )) = ǫ A ( X, Y ) ε with composition induced by the composition in A . It is easy to see that c A is a superadditivecategory with biproduct ( X, ǫ ) ⊕ ( Y, ε ) = ( X ⊕ Y, ǫ ⊕ ε ) and obviously the functor hat A : A → c A given by hat A ( X ) = ( X, X ) and hat A ( φ ) = φ is is a fully faithful graded additive functor.Furthermore c A is idempotent complete, which is called the idempotent complete of A . Note thatif A is idempotent complete then hat A : A → c A is an equivalence of superadditive categories.The following facts can be proved by the similar arguments as those of [2, §
6] for K -categories. Proposition 4.3.
Let A and B be superadditive categories. If B is idempotent complete then hat A induces an equivalence hat : Gr( c A , B ) ∼ −→ Gr( A , B ) . In particular, hat induces an equivalence ofcategories hat : Gr c A e ∼ −→ Gr A e . Given any superadditive category A , we can form the category Mat A whose objects arefinite sequence of objects in A and where a morphism from ( X , · · · , X m ) to ( Y , · · · , Y n ) is an m × n matrix [ α ij ] with α ij ∈ A ( X i , Y j ), composition is defined by ordinary matrix multiplication.The category Mat( A ) is called an additive completion of A . Then Mat A is a superadditivecategory with finite product and contains A as a full superadditive subcategory by identifying X ∈ A with the matrix whose only nonzero entry is 1 X . Moreover, the functor mat : A → Mat A defined by X ( X ) is a graded functor, and if B is small superadditive category then mat inducesan equivalence mat : Gr(Mat A , B ) ∼ −→ Gr( A , B ) . In particular, mat induces an equivalence of categories mat : Gr(Mat A ) e ∼ −→ Gr A e .By Proposition 4.3 and Section 4.4, two superadditive categories are graded Morita equivalentif and only if their completions (i.e., the additive completion and the idempotent completions) aregraded Morita equivalent. Note that these completions are superadditive categories with finiteproducts, so they are graded Morita equivalent if and only if they are equivalent according toSection 4.4.Using the similar arguments as that of [4], we can prove a super-version of [4, Theorem 4.7].The details will appear elsewhere. Theorem 4.5.
Any graded Morita equivalence between superadditive categories is a compositionof equivalences, additivization and idempotent completions of superadditive categories. A and B , a graded left B -module N and a graded functor F : A → B , we define the graded A -module F ( N ) given by F ( N )( Z ) = N ( F ( Z )) for all Z ∈ A ,where the action is the following φ · n := F ( φ ) · n for n ∈ F ( N )( Z ) and φ ∈ A ( X, Z ). The similarconstruction is made for graded right modules and graded bimodules.
Proposition 4.6.
Let A and B be superadditive categories and let N be a graded B -bimodule. If F : A → B is an even equivalence, then there are isomorphisms of superspaces ( i ) HH ∗ ( A , F ( N )) ∼ = HH ∗ ( B , N ) HH ∗ ( A , F ( N )) ∼ = HH ∗ ( B , N ) . ( ii ) HC ∗ ( A ) ∼ = HC ∗ ( B ) HC ∗ ( A ) ∼ = HC ∗ ( B ) . Proof.
The functor F ∗ : Gr B e → Gr A e induced by F is an even equivalence since F is itself aneven equivalence. Hence the induced functor HH ∗ ( A , − ) is a coeffaceable universal δ -functor. Thusthe following collection of functorsHH n ( A , F ) : Gr B e → Gr K, N 7→ F ( N ) HH n ( A , F ( N ))is a universal δ -functor and the collection of functorsHH n ( B , − ) : Gr B e → Gr K, N 7→ HH n ( B , N )is a universal δ -functor too.In order to prove that two universal δ -functors are isomorphic, it is enough to prove thatHH ( B , N ) ∼ = HH ( A , F ( N )) as superspaces (see [16, Section 2.1]). If Y is in the image of F ,then we define the natural morphism ı : HH ( B , N ) → HH ( A , F ( N )) , N ( Y, Y ) Y ∈ B
7→ N ( F ( X ) , F ( X )) X ∈ A . Since N ( Y, Y ) Y ∈ B ∈ HH ( B , N ), Section 3.6 implies that n Y φ = ( − | φ || n Y | φn Y ′ for all φ ∈ B ( Y, Y ′ ). It follows that n F ( X ) ψ = ψn F ( X ′ ) for ψ ∈ A ( X ′ , X ) and ı is an isomorphism.If Y is not in the image of F , since it is an even equivalence, there exists X ∈ A suchthat there is a homogeneous isomorphism h : Y ∼ = F ( X ) with h ∈ B ( F ( X ) , Y ). Thus we get n Y = hn F ( X ) h − = hı ( n X ) h − . Thus ı is an isomorphism by the above arguments.The homological case is analogous to cohomological one. If Y is in the image of F , it followsby applying Remark 3.4 and the following natural well-defined isomorphism : HH ( B , N ) → HH ( A , F ( N )) , X Y ∈ B n Y X X ∈ B n F ( X ) . If Y is not in the image of F , since F is an equivalence, there exists X ∈ A such that h : Y ∼ = F ( X )with homogeneous isomorphism h ∈ B ( F ( X ) , Y ). Using Remark 3.4, we get n Y = hn F ( X ) h − = h ( n X ) h − . As a consequence, the map is an isomorphism.The assertion (ii) follows by applying the Gysin-Connes exact sequence (3.11) and (i). Proposition 4.7.
Let A be a superadditive category and M a graded c A e -module. Then HH ∗ ( A , hat M ) ∼ = HH ∗ ( c A , M ) and HC ∗ ( A ) ∼ = HC ∗ ( c A ) , HH ∗ ( A , hat M ) ∼ = HH ∗ ( c A , M ) and HC ∗ ( A ) ∼ = HC ∗ ( c A ) . Proof.
Let us define the following collections of functors:hat : Gr c A e → Gr K, M 7→ hat
M 7→ HH ∗ ( A , hat M )hat : Gr c A e → Gr K, M 7→ HH ∗ ( c A , M ) . δ -functors (coeffaceable in the homological case and effaceablein the cohomological one).Now we define the following natural morphisms η : HH ( c A , M ) → HH ( A , hat M ) , ( m ( X,ǫ ) ) ( X,ǫ ) ∈ b A ( m X ) X ∈ A ; θ : HH ( c A , M ) → HH ( A , hat M ) , X ( X,ǫ ) ∈ b A m ( X,ǫ ) X X ∈ A m X . It is immediate to see that they are both well-defined and surjective. Moreover, by applyingProposition 4.3, η and θ are isomorphisms. Proposition 4.8.
Let A be a superadditive category and M a graded c A e -module. Then HH ∗ ( A , hat M ) ∼ = HH ∗ (mat A , M ) and HC ∗ ( A ) ∼ = HC ∗ (mat A ) , HH ∗ ( A , mat M ) ∼ = HH ∗ (mat A , M ) and HC ∗ ( A ) ∼ = HC ∗ (mat A ) . Proof.
Define the following collections of functors:mat : GrMat( A ) e → Gr K, N 7→ mat( M ) HH ∗ ( A , mat M )mat : GrMat A e → Gr K, N 7→ HH ∗ (Mat A , M ) . Notice that both are universal graded δ -functors (coeffaceable in the homological case and effaceablein the cohomological one).Now we define the following natural morphisms µ : HH (Mat A , M ) → HH ( A , mat M ) , ( m X , ··· ,X n ) ( X , ··· ,X n ) ∈ Mat A ( m X ) X ∈ A ; ν : HH (Mat A , M ) → HH ( A , mat M ) , X ( X , ··· ,X n ) ∈ Mat A ( m X , ··· ,X n ) X X ∈ A m X . It is immediate to see that they are both well-defined and surjective. Moreover, by applyingSection 4.4, µ and ν are isomorphisms.The following result extends [5, Theorem 2.12] and [17, Theorem 3.6]. Theorem 4.9.
The Hochschild and cyclic (co)homology of superadditive categories are gradedMorita equivalent invariants.Proof.
Thanks to Theorem 4.5, it is enough to prove that the Hochschild and cyclic (co)homologyare invariant under equivalences, idempotent and additive completion. By applying Proposi-tions 4.6, 4.7 and 4.8, we complete the proof.
In this section we first introduce the shuffles of superadditive categories and show that Hochschildhomology commute with tensor product. Then we introduce the cyclic shuffles of superadditivecategories and determine the K¨unneth exact sequence of cyclic homology. Throughout this section A and B are superadditive categories. Let S n be the symmetric group on { , · · · , n } . A ( p, q )- shuffle is a permutation σ ∈ S p + q such that σ (1) < σ (2) < · · · < σ ( p ) and σ ( p + 1) < σ ( p + 2) < · · · < σ ( p + q ) . A we let S act on the left on N n +1 = N n +1 ( A ) by σ ( a , a , · · · , a n ) = ( a , a σ − (1) , · · · , a σ − ( n ) ) . Thus if σ is a ( p, q )-shuffle the elements { a , a , · · · , a p } appear in the same order in the sequence σ · ( a , a , · · · , a n ) as do the element { a p +1 , a p +2 , · · · , a p + q } .The shuffle product sh pq : N p ( A ) ⊗ N q ( B ) → N p + q ( A ⊗ B )is defined by the following formula:sh pq (( a , a , · · · , a p ) ⊗ ( b , b , · · · , b q )) = X σ sgn( σ ) σ ( a ⊗ b , a ⊗ , · · · , a p ⊗
1; 1 ⊗ b , · · · , ⊗ b q ) , where the sum is extended over all ( p, q )-shuffles. Note that the same formula defines more generallya shuffle product from N p ( A , M ) ⊗ N q ( B , N ) to N p + q ( A ⊗ B , M ⊗ N ). Proposition 5.2.
For a ∈ N p ( A ) and b ∈ N q ( B ) , we have ∂ p + q ◦ sh pq ( a ⊗ b ) = sh p − q ( ∂ p ( a ) ⊗ b ) + ( − | q | sh pq − ( a ⊗ ∂ q ( b )) . Proof.
Let a = ( a , a , · · · , a p ) and b = ( b , b , · · · , b q ) and writesh pq ( a ⊗ b ) = X ± ( c = a ⊗ b , c , · · · , c p + q ) , where c i is either in I = { a ⊗ , · · · , a p ⊗ } or in I = { ⊗ b , · · · , ⊗ b q } . Fix i (0 ≤ i ≤ n = p + q )and consider the element d in ( c , c , · · · , c n ) appearing in the expansion of ∂ n (sh pq ( a, b )). If c i and c i +1 are in I (resp. I ), or if i = 0 and c is in I (resp. I ), then d in ( c , c , · · · , c n ) appears alsoin the expansion of sh p − q ( ∂ p + q ( a ) , b ) (resp. sh pq − ( a, ∂ q ( b )) and conversely. If c i and c i +1 belongto two different sets, then ( c , c , · · · , c i − , c i +1 , c i , c i +2 , · · · , c p + q ) is also a shuffle and appears inthe expansion of sh pq ( a ⊗ b ). As its sin is the opposite of the sign (in front) of ( c , c , · · · , c p + q ),these two elements cancel after applying d in (because c i c i +1 = c i +1 c i ). Thus we have proved theproposition.Recall that the tensor product of complexes ( C • , ∂ ) and ( e C • , ∂ ) is the complex ( C ⊗ e C ) • with( C ⊗ e C ) n = ⊕ p + q = n C p ⊗ e C q and the differential map is defined by the formula d ( x ⊗ y ) = ( ∂ ⊗ ⊗ ∂ )( x ⊗ y ) = ∂ ( x ) ⊗ y + ( − | x | x ⊗ ∂ ( y ) . Let sh : ( N ∗ ( A ) ⊗ N ∗ ( B )) n = M p + q = n N p ( A ) ⊗ N q ( B ) → N n ( A ⊗ B )be the sum of the shuffle product maps sh pq for p + q = n .The following fact follows readily form Proposition 5.2. Corollary 5.3.
The map sh : N ∗ ( A ) ⊗ N ∗ ( B ) → N ∗ ( A ⊗ B ) is a map of complexes of degree ofzero, that is, [ ∂, sh] := ∂ ◦ sh − sh ◦ ( ∂ ⊗ ⊗ ∂ ) = 0 . The following is a categorical version of the Eilenberg-Zilber Theorem for Hochschild homologyof algebras.
Theorem 5.4.
Assume that K is a field. Then the shuffle map sh induces an isomorphism sh ∗ : HH ∗ ( A ) ⊗ HH ∗ ( B ) ∼ −→ HH ∗ ( A ⊗ B ) . roof. The theorem can be proved by applying the same arguments as that of the Eilenberg-ZilberTheorem for Hochschild homology of algebras (see [10, Chapter 8, Theorem 8.1].
By definition a ( p, q )- cyclic shuffle is a permutation σ ∈ S p + q defined as follows: perform acyclic permutation of any order on the set of { , · · · , p } and a cyclic permutation of any order onthe set { p + 1 , · · · , p + q } . Then shuffle the two results to obtain the permutation. This is a cyclicshuffle if 1 appears before p + 1 in the sequence ( σ (1) , · · · , σ ( p + q )).The cyclic shuffle is a map ⊥ : N p ( A ) ⊗ N q ( B ) → N p + q ( A ⊗ B ) given by( a , a , · · · , a p ) ⊥ ( b , b , · · · , b q )) = X σ sgn( σ ) σ ( a ⊗ b , a ⊗ , · · · , a p ⊗
1; 1 ⊗ b , · · · , ⊗ b q ) , where the sum is extended over all ( p, q )-cyclic shuffles. Note that there is a similar operation ⊥ : N p ( A ) ⊗ N q ( B ) → N n ( A ) ⊗ N n ( B ) for n = p + q .The cyclic shuffle map is a map of degree 2csh pq : N p ( A ) ⊗ N q ( B ) → N p + q +2 ( A ) ⊗ N p + q +2 ( B )defined by the following formula:csh pq ( a, b ) := csh pq ( a ⊗ b ) = s ( a ) ⊥ s ( b ) , where s : N n ( A ) → N n +1 ( A ) is the extra degeneracy of N ( A ) (cf. Section 3.8). Note that sB = 0in the normalized setting, it follows that for any a and b ,csh( B ( a ) , b ) = sB ( a ) ⊥ b and csh( a, B ( b )) = 0 . (5.6) Proposition 5.7.
For a ∈ N p ( A ) and b ∈ N q ( B ) , the following equality holds in N p + q +1 ( A ⊗ B ) : B sh pq ( a, b ) − (cid:16) sh( B ( a ) , b )+( − | a | sh( a, B ( b )) (cid:17) + ∂ csh( a, b ) − csh( ∂ ( a ) , b ) − ( − | a | csh( a, ∂ ( b )) = 0 . Proof.
Let a = ( a , a , · · · , a p ) and b = ( b , b , · · · , b q ). Note that the image of a ⊗ b under anyof these compositions of these maps is the sum of elements of two differential types: either it is apermutation of ( a ⊗ , · · · , a p ⊗ , ⊗ b , · · · , ⊗ b q ), or it is an element of the form (1 ⊗ , · · · )where one the of the other entries is of the form a − i ⊗ a j . We only need to show that the sumof the elements of the first type is 0 (similarly we can show that the sum of the elements of thesecond type is 0).Elements of the first type arise only form ∂ csh pq , sh p +1 q ( B ⊗
1) and sh pq +1 (1 ⊗ B ). Let a =( a , · · · , a n ) and τ the cyclic permutation such that ∂ (1 , a ) = a + sgn( τ ) τ ( a ) modulo the elementsof the form (1 , · · · ). So the permutations coming from ∂ csh pq are of the form (cyclic shuffle) or τ ◦ (cyclic shuffle). The cyclic shuffle which have 1 in the first position cancel with the permutationscoming from sh p +1 q ( B ⊗ σ , consider τ σ . If p + 1 is in the first position, then τ σ cancels with a permutation coming from sh P +1 q ( B ⊗ σ ′ (and itcancels with it). So we are led to examine τ σ ′ for which we play the same game. By the end allthe elements have disappeared. The cyclic shuffle product csh : ( N ( A ) ⊗ N ( B )) n = M p + q = n N p ( A ) ⊗ N q ( B ) −→ csh( N ( A ) , N ( B )) n +2 is the sum of all the ( p, q )-shuffle maps csh pq for p + q = n .12 emma 5.9. The maps ∂ , B , sh , and csh satisfy the following formulas in the normalized setting ( i ) [ ∂, sh] = 0 , ( ii ) [ B, sh] + [ ∂, csh] = 0 , ( iii ) [ B, csh] = 0 . Proof. ( i ) is exactly Corollary 5.3. ( ii ) is an immediate consequence of Proposition 5.7. ( iii ) followsdirectly by applying Equation 5.6. Theorem 5.10.
Let A and B be superadditive categories. Then the shuffle product and the cyclicshuffle product induce a canonical isomorphism Sh : HC ∗ ( N ∗ ( A ) ⊗ N ∗ ( B )) ∼ −→ HC(sh( N ∗ ( A ⊗ B ))) . Furthermore Sh commutes with the morphisms B , I and S of Connes’s exact sequence.Proof. Lemma 5.9(i) implies that the shuffle map sh : N ∗ ( A ) ⊗ N ∗ ( B ) → N ∗ ( A ⊗ B ) is a mapof complexes. However it is not a map of mixed complexes since B does not commute with sh (seeLemma 5.9(ii)). Note that there is following commutative diagram0 / / N ∗ ( A ) ⊗ N ∗ ( B ) sh (cid:15) (cid:15) / / Tot N ∗ ( A ) ⊗ N ∗ ( B ) Sh (cid:15) (cid:15) / / Tot N ∗ ( A ) ⊗ N ∗ ( B )[2] Sh[2] (cid:15) (cid:15) / / N ∗ ( A ⊗ B ) / / Tot N ∗ ( A ⊗ B ) / / Tot N ∗ ( A ⊗ B )[2] . Recall that Tot( N ∗ ( A ) ⊗ N ∗ ( B )) n = ∞ M k =0 ( N ∗ ( A ) ⊗ N ∗ ( B )) n − k Tot( N ∗ ( A ⊗ B )) n = ∞ M k =0 ( N ∗ ( A ) ⊗ N ∗ ( B )) n − k Now let Sh = sh cshsh cshsh csh. . . . . . . By Lemma 5.9, Sh is a morphism of complexes. The commutativity of the right-hand squarecomes form the form of Sh and the commutativity of the left-hand square is immediate. ApplyingTheorem 5.4, we complete the proof of the theorem.We now state our main theorem comparing HC n ( A ⊗ B ) with HC n ( A ) and HC n ( B ). Theorem 5.11.
Let A and B be superadditive categories over a filed K . Then there exists anatural long exact sequence · · · / / HC n ( A ⊗ B ) (cid:15) (cid:15) M p + q = n HC p ( A ) ⊗ HC q ( B ) S ⊗ id − id ⊗ S / / M p + q = n − HC p ( A ) ⊗ HC q ( B ) (cid:15) (cid:15) HC n − ( A ⊗ B ) / / · · · Proof.
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