aa r X i v : . [ m a t h . K T ] A p r Categorified Chern character and cyclic cohomology.
Ilya Shapiro
Abstract
We examine Hopf cyclic cohomology in the same context as the analysis [1, 2, 3] ofthe geometry of loop spaces LX in derived algebraic geometry and the resulting closerelationship between S -equivariant quasi-coherent sheaves on LX and D X -modules.Furthermore, the Hopf setting serves as a toy case for the categorification of Cherncharacter theory as discussed in [20]. More precisely, this examination naturally leadsto a definition of mixed anti-Yetter-Drinfeld contramodules which reduces to that ofthe usual mixed complexes for the trivial Hopf algebra and generalizes the notion ofstable anti-Yetter-Drinfeld contramodules that have thus far served as the coefficientsfor Hopf-cyclic theories [4]. The cohomology is then obtained as a Hom in this dg-category between a Chern character object associated to an algebra and an arbitrarycoefficient mixed anti-Yetter-Drinfeld contramodule.
Mathematics Subject Classification:
This paper is motivated by the connection, observed in [10], between the Hopf-cyclic theorycoefficients and the objects appearing in a twice categorified 1dTFT (one dimensionaltopological field theory) [7]. We begin to explore this link and its implications with thegoal of better understanding and generalizing coefficients, as well as reinterpreting theHopf cyclic cohomology itself. The main concepts involved are as follows. For a monoidalcategory C and its bimodule category M we have the Hochschild homology category HH ( C , M ) and the cyclic homology category HC ( C ). These are categorifications of theHochschild homology of an algebra with coefficients in a bimodule, and the cyclic homologyof an algebra respectively. Furthermore, as the title suggests, the notion of a categorifiedversion of the Chern character is essential for our purposes and is also of independentinterest.We do not actually use the language of differential graded categories, instead everythingtakes place inside the 2-category of categories with the additional requirement that functorspossess right adjoints. Often we consider the dual picture where the adjoints are themselvesthe 1-morphisms and so we have a 2-category of categories with functors possessing leftadjoints. The disadvantage of foregoing dg-categories is that the naive, i.e., resultingfrom the 2-category setting above, HC ( C ) is not sufficient for our purposes; though itrecovers the classical coefficients. The advantage is that the exposition is considerably1ess sophisticated. We get around the shortcomings of the naive approach in an ad hocmanner that is suggested by the dg setting.Briefly, for a Hopf algebra H , motivated by some results in derived algebraic geometry[1, 2, 3], we propose a generalization of stable anti-Yetter-Drinfeld contramodules (with mixed replacing stable) as an analogue of QC ( LX ) S , the S -equivariant quasi-coherentsheaves on the derived loop space of X . This category serves both as the target for thecategorified Chern characters of H -module algebras and also as the source of coefficientsfor the cohomology theory. The Hopf-cyclic cohomology is then recovered as an Ext in thiscategory as was previously done by Connes and Kassel for cyclic cohomology, using cyclicobjects [5], and mixed complexes [12] respectively. This places Hopf-cyclic cohomologyinto the same framework as de Rham cohomology. Roughly speaking, QC ( LX ) S (relatedto D X -modules) is the cyclic homology of the monoidal category QC ( X ), while mixed anti-Yetter-Drinfeld contramodules form the cyclic homology (in the proper sense) of H M , thecategory of modules over the Hopf algebra H . Conventions : All algebras A in monoidal categories C are assumed to be unitalassociative, unless stated otherwise; we say that A ∈ Alg ( C ). We let A C -mod stand for left A -modules in C which is a right C -module category. Similarly, B md C A denotes the dualmonoidal category of A -bimodules in C . Our H is a Hopf algebra over some ground field k which is fixed throughout the paper, thus Vec denotes the category of k -vector spaces. Outline : Section 2 is provided as a guide and contains no proofs. It uses the TFTformalism as motivation. Section 3 discusses, in general terms, the construction of amonad on a bimodule category such that modules over the monad are precisely its naiveHochschild homology category. This construction is not always possible, but proceeds asdescribed provided that, what we call, property A holds. Section 3.3 describes what hap-pens in the cyclic case. Section 4 expands on the discussion began in Section 2 concerningthe adjoint pair of functors relating the Hochschild homology categories of C and D andinduced by an admissible ( C , D )-bimodule M . This helps motivate the Definition 6.3 ofthe (categorified) Chern character, and will also be used in Section 8 in the Hopf case.Section 5 applies the notions of Section 3 to the case of H M , the category of modulesover a Hopf algebra. In particular it is shown that property A holds for H M . Furthermore, HH ( H M ), the naive Hochschild homology of H M coincides with anti-Yetter-Drinfeldcontramodules as seen from Proposition 5.11 and HC ( H M ), the naive cyclic homology of H M coincides with the stable anti-Yetter-Drinfeld contramodules as seen from Theorem5.13.Section 6 begins by addressing the deficiency of HC ( C ) in the Definition 6.1; the HH ( C ) S so defined is certainly the correct cyclic homology of C in the dg-sense. Weproceed by defining the Chern character ch ( A ) in an ad hoc manner, motivated by Sections2 and 4. We conjecture that the definition would be a consequence of those sections, werethey to be made rigorous. In Section 6.1 we prove Theorem 6.9 which relates the usualdefinition [4] of Hopf cyclic cohomology of an algebra A with coefficients in a stable anti-Yetter-Drinfeld contramodule M to an Ext between ch ( A ) and M in the category ofmixed anti-Yetter-Drinfeld contramodules. The stable M can of course be replaced by amixed one, thereby generalizing the coefficients. Alternatively, in Section 6.2 we pursue2sygan’s approach to cyclic homology via a double complex. More precisely, we describea tricomplex that computes Hopf-cyclic cohomology of A with coefficients in a mixed anti-Yetter-Drinfeld contramodule. The advantage is that it does not require A to be unital.We make no claims about the compatibility of this approach with the Ext one.Section 7 is a summary of what would happen to Section 5 were it repeated for M H ,the monoidal category of H -comodules. It is an outline without proofs, though these caneasily be filled in, unlike in the case of the more conjectural sections. The main point isthat stable anti-Yetter-Drinfeld modules arise as HC ( M H ).Section 8 contains a discussion of applications of Sections 2 and 4 to the Hopf setting. Inparticular (8.2), obtained from the relationship between Hochschild homologies of H M and M H via the admissible bimodule category Vec, is exactly that which forms the backboneof [19].Finally, Section 9 is meant to provide a bridge between this paper and the literaturethat motivated it. More precisely, in Section 9.1 we attempt to explain, by sketching thecommutative case, why the mixed anti-Yetter-Drinfeld contramodules in the Hopf settingare exactly analogous to the D -modules in the geometric setting. In Section 9.2 we linkthe monadic approach with the previous attempts [10, 15, 14, 18, 19] to understand Hopfcyclic coefficients as centers of certain bimodule categories. Acknowledgments : The author wishes to thank Masoud Khalkhali and Ivan Kobyzevfor stimulating discussions. This research was supported in part by the NSERC DiscoveryGrant number 406709.
This section is motivational in nature and is concerned with exploring the TFT point ofview that will be helpful in the analysis of Hopf cyclic cohomology.Considering twice categorified local 1-dimensional topological field theories, see [7] for agood overview, we obtain a way to keep track of all the conjectural relationships betweenthe notions considered in the present paper. Namely, such a TFT assigns a monoidalcategory C to a point and consequently HH ( C ) to S , and the non-trivial self-homotopyof the identity on S is assigned an automorphism of the identity functor on HH ( C ), i.e.,the S -action. A map between TFTs C and D is then given by a suitable ( C , D )-bimodulecategory M .More precisely, as suggested by [18], we consider only admissible M , i.e., Definition 2.1.
We say that a ( C , D ) -bimodule category M is admissible if M ≃ B D -mod,for some algebra B ∈ D , as a right D -module category, and the left C -action is given by amonoidal functor C → B md D B that has a right adjoint. Furthermore, we expect that (see Section 4 for constructions) such an M induces anadjoint pair of functors ( HH ( M ) ∗ , HH ( M ) ∗ ): HH ( M ) ∗ : HH ( C ) ⇆ HH ( D ) : HH ( M ) ∗ (2.1)3hat are compatible with the S -actions. Given an A ∈ Alg ( C ), we observe that A C -modis an example of an admissible (Vec , C )-bimodule, thus: Definition 2.2.
Let the Chern character of A be ch ( A ) = HH ( A C -mod ) ∗ k ∈ HH ( C ) S where k ∈ Vec S is the trivial mixed complex. Moreover, given an admissible ( D , E )-bimodule N , it is possible ([18] and Remark 2.4)to define an admissible ( C , E )-bimodule suggestively denoted M ⊠ D N , and we should havean equivalence of S -equivariant functors: HH ( M ⊠ D N ) ∗ ≃ HH ( N ) ∗ HH ( M ) ∗ and thus HH ( M ⊠ D N ) ∗ ≃ HH ( M ) ∗ HH ( N ) ∗ . Definition 2.3.
Given an A ∈ Alg ( C ) , let M ∗ A ∈ Alg ( D ) be the image of A under C act → B md D B U → D , where U forgets the B -bimodule structure. We have A C -mod ⊠ C M = ( M ∗ A ) D -mod and HH ( M ) ∗ ch ( A ) ≃ ch ( M ∗ A ) (2.2)since the former is HH ( M ) ∗ HH ( A C -mod) ∗ k ≃ HH ( A C -mod ⊠ C M ) ∗ k = HH (( M ∗ A ) D -mod) ∗ k which is the latter. Remark 2.4.
Note that there is of course a more conceptual definition of M ⊠ D N asthe Hochschild homology category, i.e., HH ( D , M ⊠ N ) . By [1] the concrete descriptionabove and the conceptual one agree. More precisely, we have A C -mod ⊠ C M = A M -mod,where the former has the Hochschild homology definition. The description of the Chern character in Definition 2.2 is a categorification of the usualone. On the other hand it “extends down”. More precisely, we can continue the TFTpatterns from above to the consideration of what happens to a suitable F ∈ F un C ( M , N ) D where M and N are admissible ( C , D )-bimodules. It should yield an associated naturaltransformation HH ( F ) ∗ : HH ( N ) ∗ → HH ( M ) ∗ . In the case of a suitable M ∈ B md C ( A, B ) for
A, B ∈ Alg ( C ) and the resulting M ⊗ B − : B C -mod → A C -mod we would get HH ( M ⊗ B − ) ∗ : HH ( A C -mod) ∗ → HH ( B C -mod) ∗ M , i.e., ch ( M ) ∈ RHom HH ( C ) S ( ch ( A ) , ch ( B ))where ch ( M ) = HH ( M ⊗ B − ) ∗ k . Note that if N is a suitable right B -module in C then ch ( N ) ∈ RHom HH ( C ) S ( ch (1) , ch ( B )) = Ext mixed ( k, u ∗ ch ( B )), with u ∗ the right adjointof u ∗ : Vec → C (unit inclusion). We remark that despite appearing to, at first glance,be concentrated in degree 0, the Chern character ch ( N ) is spread out as expected (seeRemark 6.11). We will not address the above in this paper as it is not relevant for ourpresent purposes. M ∗ E exist? Above we saw that given A ∈ Alg ( C ) we have a corresponding M ∗ A ∈ Alg ( D ) for M an admissible ( C , D )-bimodule category. We can view − ⊠ C M as a 2-functor from right C -modules, i.e., Mod C , to Mod D that preserves admissibility. As such it has a rightadjoint F un ( M , − ) D . Observe that for E ∈ Alg ( D ) we have F un ( M , E D -mod) D ≃ B md D ( E, B ) with the right C -action via C → B md D B . Unfortunately defining M ∗ E tosatisfy ( M ∗ E ) C -mod ≃ F un ( M , E D -mod) D need not work as such an algebra need notexist in C .We can see the problem more clearly in the special case of M = D , i.e., we havea monoidal f ∗ : C → D with a right adjoint f ∗ . In this case F un ( M , − ) D is simplythe pullback of right D -modules to right C -modules via f ∗ . Now for A ∈ Alg ( C ) thecorresponding algebra in D is f ∗ ( A ) and furthermore, we observe that for E ∈ Alg ( D )the object f ∗ ( E ) also has an algebra structure via f ∗ ( E ) ⊗ f ∗ ( E ) → f ∗ ( E ⊗ E ) → f ∗ ( E ).What we would like is to have an equivalence B md D ( E, f ∗ ( A )) ≃ B md C ( f ∗ ( E ) , A ).Let us take C = H M (the category of modules over a Hopf algebra), D = Vec (thecategory of vector spaces), and f ∗ the fiber functor. Take A = 1 and E = 1, then we have B md Vec (1 ,
1) = Vec while B md H M ( H ∗ ,
1) = H ∗ H M -mod. They are almost equivalent, i.e.,if H is finite dimensional, but not otherwise. Another example is considered in Section9.1.On the other hand there are examples where it is possible to define M ∗ E , but it is not f ∗ E . Namely, take ǫ ∗ : Vec → H M to be the monoidal unit, with right adjoint V V H .We observe that for any E ∈ Alg ( H M ) the right H M -module E H M -mod is equivalent, asa right Vec-module to E ⋊ H -mod, where ( a ⊗ x )( b ⊗ y ) = a ( x b ) ⊗ x y , and not in generalto E H -mod; thus ǫ ∗ E = E ⋊ H. (2.3)This case considers H M as an admissible (Vec , H M )-bimodule category. Note that wecan also consider it as an admissible ( H M , Vec)-bimodule category since H M = H Vec -modand the resulting map H M → B md H sending V to V ⊗ H with x ( v ⊗ h ) y = x v ⊗ x hy has a right adjoint S ad ( S ) where the action on the latter is via h · s = h sS ( h ).Recall that HH ( M ⊠ D N ) ∗ ≃ HH ( N ) ∗ HH ( M ) ∗ , i.e., compatibility of HH ( − ) withcomposition, implies that HH ( M ) ∗ ch ( A ) ≃ ch ( M ∗ A ) so that ch ( − ) is compatible with5ullbacks. If we assume the compatibility of HH ( − ) with internal Homs when they exist,i.e., HH ( F un ( N , T ) E ) ∗ ≃ HH ( N ) ∗ HH ( T ) ∗ then we obtain the compatibility of ch ( − ) with pushforwards. More precisely, defin-ing M ∗ E by ( M ∗ E ) C -mod = F un ( M , E D -mod) D , if possible, we have ch ( M ∗ E ) = HH ( F un ( M , E D -mod) D ) ∗ k while the latter is equivalent to HH ( M ) ∗ HH ( E D -mod) ∗ k and so to HH ( M ) ∗ ch ( E ), i.e., HH ( M ) ∗ ch ( E ) ≃ ch ( M ∗ E ) . (2.4) Here we concern ourselves with understanding the naive Hochschild and cyclic homologiesof monoidal categories. We make certain additional assumptions, satisfied in our intendedsetting, that allow for a description of Hochschild homology as the category of modulesover a monad. In the cyclic case the monad is equipped with a central element thatsupplies the S -action.Let M be a bimodule category over C such that both actions: ∆ ∗ l : C ⊠ M → M and∆ ∗ r : M ⊠ C → M possess right adjoints: ∆ l ∗ and ∆ r ∗ respectively. We also require thatthe unit and the product functors in C possess right adjoints as well. Definition 3.1.
We will say that a C -bimodule M , as above, has right adjoints . Let us begin with its Hochschild homology category HH ( C , M ). In the usual way,using the left and right actions together with the unit and product of C we may view M ⊠ C ⊠ • as a simplicial object in the 2-category of categories with functors possessingright adjoints . All of the subtleties of tensor products of categories themselves are avoided in our case as they all arise as categories of modules over some algebra, see Section 5. Remark 3.2.
We can also consider the setting of categories with functors possessing leftadjoints. This can arise in the situation that we consider in Section 5. Namely, givena map of algebras f : A → B the forgetful functor f ∗ from B -modules to A -modules hasboth a right, f ∗ = Coinduction , and a left, f ! = Induction , adjoint. We mention that theformer case is what we consider, and it leads to anti-Yetter-Drinfeld contramodules via acertain monad. The latter would lead to the case of anti-Yetter-Drinfeld modules throughthe consideration of the co-monad which is the left adjoint to our monad that we will call A below. Let ∆ denote the simplex category (finite linearly ordered sets with order preservingmaps), then HH ( C , M ) = lim −→ ∆ M ⊠ C ⊠ • , M ⊠ C ⊠ • asa co-simplicial object in categories with functors possessing left adjoints so that we have HH ( C , M ) = lim ←− ∆ M ⊠ C ⊠ • ;the latter is simpler to describe explicitly.Generally if we have C • “fibered” over some index category X , then the inverse limitconsists of the following data: for every x ∈ X we are given an M x ∈ C x together withisomorphisms τ f : f ( M x ) → M y for any f : x → y . Furthermore, if g : y → z then τ g g ( τ f ) = τ gf and τ Id x = Id M x .This data in the case of the simplex index category reduces to an M ∈ C togetherwith an isomorphism (the δ ’s and s ’s below are the usual faces and degeneracies) in C : τ : δ ( M ) → δ ( M ) , subject to the unit condition in C : s ( τ ) = Id M , and the associativity condition in C : δ ( τ ) δ ( τ ) = δ ( τ ) . (3.1)Naturally, M i ∈ C i is obtained from M via M i = δ i δ ( i − · · · δ M , i.e., M i = low ( M ) where low : [0] → [ i ] maps the point to the smallest element. In order to realize the Hochschild homology category as the category of modules overa monad we require additional assumptions that necessitate the Definition 3.3. Moreprecisely, we wish to describe the inverse limit (that gives HH ( C , M ) in our intendedsetting) as the category of modules over a monad on M .Recall the general setting of C • , a co-simplicial object in categories (with left adjoints).We will denote the faces adjoint pairs by ( δ ∗ , δ ∗ ) and the degeneracies by ( s ∗ , s ∗ ).We have δ ∗ δ ∗ = δ ∗ δ ∗ : C → C and so obtain by adjunction (see Section 5.1): δ ∗ δ ∗ → δ ∗ δ ∗ . (3.2)While δ ∗ δ ∗ = δ ∗ δ ∗ : C → C yields: δ ∗ δ ∗ → δ ∗ δ ∗ . (3.3)Furthermore, from δ ∗ δ ∗ = δ ∗ δ ∗ : C → C we have: δ ∗ δ ∗ → δ ∗ δ ∗ . (3.4)7 efinition 3.3. We say that C • has property A if (3.2) , (3.3) , and (3.4) are isomor-phisms. We say that a C -bimodule M , with right adjoints, has property A if C • = M ⊠ C ⊠ • has property A . Definition 3.4.
Let A = δ ∗ δ ∗ ∈ End ( C ) . This is our monad. Define the multiplication on A as follows. First observe that we have a composition ofisomorphisms: τ m : δ ∗ δ ∗ δ ∗ ← δ ∗ δ ∗ δ ∗ = δ ∗ δ ∗ δ ∗ → δ ∗ δ ∗ δ ∗ , (3.5)where the first arrow is provided by (3.2) (and the property A assumption), the secondequality is due to δ ∗ δ ∗ = δ ∗ δ ∗ , and the third arrow is (3.3). The multiplication m : A → A is obtained from τ m by adjunction. Lemma 3.5.
Let M ∈ C with τ : δ ∗ M → δ ∗ M . Equip M with a : A ( M ) → M viaadjunction from τ . Then it is an action if and only if δ ∗ ( τ ) δ ∗ ( τ ) = δ ∗ ( τ ) .Proof. Note that δ ∗ δ ∗ δ ∗ δ ∗ M δ ∗ δ ∗ ( a ) / / m (cid:15) (cid:15) δ ∗ δ ∗ M a (cid:15) (cid:15) δ ∗ δ ∗ M a / / M commutes if and only if δ ∗ δ ∗ δ ∗ M δ ∗ ( a ) / / τ m (cid:15) (cid:15) δ ∗ M τ (cid:15) (cid:15) δ ∗ δ ∗ δ ∗ M δ ∗ ( a ) / / δ ∗ M commutes. We can expand the vertical τ m of the latter diagram into δ ∗ δ ∗ δ ∗ M / / = (cid:15) (cid:15) δ ∗ δ ∗ δ ∗ M δ ∗ ( a ) / / δ ∗ M τ (cid:15) (cid:15) δ ∗ δ ∗ δ ∗ M / / δ ∗ δ ∗ δ ∗ M δ ∗ ( a ) / / δ ∗ M which commutes if and only if δ ∗ δ ∗ M / / = (cid:15) (cid:15) δ ∗ δ ∗ δ ∗ δ ∗ M δ ∗ δ ∗ ( a ) / / δ ∗ δ ∗ M δ ∗ ( τ ) (cid:15) (cid:15) δ ∗ δ ∗ M / / δ ∗ δ ∗ δ ∗ δ ∗ M δ ∗ δ ∗ ( a ) / / δ ∗ δ ∗ M commutes. Observe that under the identification δ ∗ δ ∗ = δ ∗ δ ∗ the top row becomes δ ∗ δ ∗ M → δ ∗ δ ∗ δ ∗ δ ∗ M δ ∗ δ ∗ ( a ) −→ δ ∗ δ ∗ M , i.e., δ ∗ ( δ ∗ M → δ ∗ δ ∗ δ ∗ M δ ∗ ( a ) −→ δ ∗ M ),8hich is δ ∗ ( δ ∗ M τ → δ ∗ M ), so δ ∗ ( τ ). Similarly, under the identification δ ∗ δ ∗ = δ ∗ δ ∗ the bottom row becomes δ ∗ ( τ ) and we are done.Consider u : Id → A given by u : Id = δ ∗ s ∗ s ∗ δ ∗ → δ ∗ δ ∗ (3.6)using the evaluation s ∗ s ∗ → Id . We have the following easy lemma (with proof as inLemma 3.5). Lemma 3.6.
Let M ∈ C with τ : δ ∗ M → δ ∗ M . Then s ∗ ( τ ) = Id if and only if M u → A ( M ) a → M is Id . A In order to demonstrate that A is unital and associative it is helpful to consider cospansin the simplex category. The idea is that A is actually defined (is a 1-morphism from [0]to itself) in the 2-category 2∆ obtained from ∆ by freely adding left adjoints see [6] forexample. In particular, when pushouts are available we can compose cospans and obtain2-morphisms in 2∆ via maps of cospans. We observe that for n ≥ n = 1 , n − δ n (cid:15) (cid:15) δ / / [ n ] δ n +1 (cid:15) (cid:15) [ n ] δ / / [ n + 1]which allows us to compose as follows:[0] δ (cid:31) (cid:31) ❄❄❄❄❄❄❄ [0] δ (cid:31) (cid:31) ❄❄❄❄❄❄❄ δ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ [0] δ (cid:31) (cid:31) ❄❄❄❄❄❄❄ δ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ [0] δ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ [1] δ (cid:31) (cid:31) ❄❄❄❄❄❄❄ [1] δ (cid:31) (cid:31) ❄❄❄❄❄❄❄ δ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ [1] δ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ [2] δ (cid:31) (cid:31) ❄❄❄❄❄❄❄ [2] δ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ [3]so that when considering A = δ ∗ δ ∗ = [0] δ → [1] δ ← [0]we have that A = [0] high → [2] low ← [0]9nd the product m : A → A is given by the unique map of the cospans δ : [1] → [2],where one should note the reversal of the arrows specifying the 2-morphisms. Lemma 3.7.
The -loop A at [0] in is associative.Proof. The two 2-morphisms m ◦ Id and Id ◦ m from A to A coincide since there isonly a unique map from [0] high → [3] low ← [0] to [0] δ → [1] δ ← [0], i.e., f : [1] → [3] with(0 , (0 , u is given by s : [1] → [0] from identity represented as [0] = [0] = [0]to A . Lemma 3.8.
The map u is the unit of A in .Proof. Checking that it is a left and a right unit is checking that two cospan maps from[0] δ → [1] δ ← [0] to itself are both identity, yet there is no other map. Corollary 3.9.
Assuming that (3.2) and (3.4) are isomorphisms, the endofunctor A of C is a monad.Proof. The assumptions allow us to interpret A and A as cospans in C • . Remark 3.10.
Note that (3.2) is required in order to define m , while (3.3) is neededto make sure that τ m of (3.5) is an isomorphism. Finally, (3.4) facilitates the proof ofassociativity. Theorem 3.11.
Let C • denote a cosimplicial object in categories with functors possessingleft adjoints. Assume that C • has property A , then HH ( C • ) = lim ←− ∆ C • = A C -mod . Proof.
By Corollary 3.9 we have that
A ∈
End ( C ) is a monad. Recall that the inverselimit consists of M ∈ C with isomorphisms τ M : δ ∗ M → δ ∗ M such that δ ∗ ( τ M ) δ ∗ ( τ M ) = δ ∗ ( τ M ) (associativity) and s ∗ ( τ M ) = Id M (unitality). Furthermore, maps between suchobjects M and N are ϕ ∈ Hom C ( M, N ) such that δ ∗ ( ϕ ) τ M = τ N δ ∗ ( ϕ ).The correspondence between ( M, τ M ) and ( M, a M ), where τ M : δ ∗ M → δ ∗ M (inverselimit) and a M : δ ∗ δ ∗ M → M (action) is via adjunction. Since δ ∗ ( ϕ ) τ M = τ N δ ∗ ( ϕ ) ifand only if ϕa M = a N δ ∗ δ ∗ ( ϕ ), so the maps in the inverse limit coincide with the maps of A -modules. Note that by Lemmas 3.5 and 3.6 the associativity and unitality conditionscoincide as well.The only issue is that τ M is assumed to be an isomorphism and it is not yet obviousthat any such τ obtained from an action a M : A ( M ) → M via adjunction is automaticallyan isomorphism. To that end observe that · · · → A ( M ) → A ( M ) → M → A -modules that is null-homotopic in C and thus we have a commutative diagram ofexact sequences: δ ∗ A ( M ) / / τ m ( A ( M )) (cid:15) (cid:15) δ ∗ A ( M ) / / τ m ( M ) (cid:15) (cid:15) δ ∗ M τ M (cid:15) (cid:15) / / δ ∗ A ( M ) / / δ ∗ A ( M ) / / δ ∗ M / / τ m (see (3.5)) is an isomorphism (this is where we used (3.3) being anisomorphism) and so τ M is as well. Proposition 3.12.
Let M be a C -bimodule category with right adjoints and property A .Then HH ( C , M ) = A M -modand the canonical map T r : M → HH ( C , M ) is realized as part of an adjoint pair offunctors ( F, U ) : F : M ⇆ A M -mod : U where U forgets the A -module structure and F freely creates it via F ( M ) = A ( M ) .Proof. This is immediate from Theorem 3.11 applied to M ⊠ C ⊠ • . In particular, it is clearthat the right adjoint of T r coincides with U . Remark 3.13.
We observe that for formal reasons there is a canonical identification τ C,M : F ( C · M ) ≃ F ( M · C ) in A M -mod for any C ∈ C and M ∈ M . More generally, for T ∈ M ⊠ C we have τ T : F ( δ ∗ T ) ≃ F ( δ ∗ T ) . This identification is unital and associativein the sense of Lemma 9.1, for the same reasons as in its proof. C If in the discussion of Section 3 we replace an arbitrary C -bimodule category M by C itself, then the structure on C ⊠ • +1 of a cosimplicial object in categories (with left adjoints)extends to that of a cocyclic object in the same 2-category. In this case the inverse limitof that diagram is the cyclic homology HC ( C ) of C .In this section we will address the more general version of the above where we consider C • , a co-cyclic object in categories (with left adjoints). We will require that when consid-ered as a co-simplicial object, it satisfy the property A assumption. This will allow us toapply Theorem 3.11 to it. Definition 3.14.
Let C • be a co-cyclic object in categories (with left adjoints). We saythat it has property A if its restriction to a co-simplicial object has property A . An important, though unsurprising, new feature of A in this setting is ς : Id → A obtained from the extra degeneracy s : [1] → [0]. More precisely, ς : Id = δ ∗ s ∗ s ∗ δ ∗ → δ ∗ δ ∗ = A . (3.7)11et HH ( C • ) denote the inverse limit of C • over the simplex category ∆. Recall thatit consists of M ∈ C equipped with an isomorphism τ M : δ ∗ M → δ ∗ M (that satisfiesunitality and associativity). Note that if HC ( C • ) is the inverse limit of C • over the Connes’cyclic category Λ, then the only additional requirement on M is that s ∗ ( τ ) = Id M . (3.8)On the other hand, without insisting on (3.8), we obtain that HH ( C • ) is equippedwith an element ς = s ∗ ( τ ) of its Drinfeld center. The latter assertion follows immediatelyfrom noting that if α : M → N is a map in HH ( C • ) then δ ∗ ( α ) τ M = τ N δ ∗ ( α ) and soapplying s ∗ to everything we get ας M = ς N α . Note that since τ is an isomorphism, so is s ∗ ( τ ), thus ς is an invertible element of the Drinfeld center of HH ( C • ). Remark 3.15.
Recall Remark 3.13 where it was pointed out that
T r : M → HH ( C , M ) has a unital associative τ T : T r ( δ ∗ T ) → T r ( δ ∗ T ) for T ∈ M ⊠ C . Observe that in ourcurrent setting of M = C , we have T r : C → HH ( C , C ) as well, and whereas the unitalitymeant: for X ∈ C we have τ s ∗ X = Id T rX , now we also get that τ s ∗ X = ς T rX . (3.9) This implies, in particular, that for
X, Y ∈ C we have τ Y,X τ X,Y = ς T r ( X ⊗ Y ) . Observe that the usage of ς for both the element of A in (3.7) and the element of theDrinfeld center is not problematic, since it is immediate that they are one and the same.Namely, if a : A ( M ) → M denotes the action incarnation of the structure on M then s ∗ ( τ ) = aς ( M ). Consequently, completely formally (3.7) specifies a central element of A .To summarize: Lemma 3.16.
Let C • be a co-cyclic object in categories (with left adjoints) and property A , then the monad A on C is equipped with a central element ς : Id → A . Furthermore HC ( C • ) consists of those A -modules in C on which ς acts by identity. Corollary 3.17.
Let C be a monoidal category with right adjoints and property A , then HC ( C ) = A ′C -mod, i.e., the full subcategory of HH ( C , C ) that consists of objects on which ς acts by identity. Motivated by the ∞ -category setting and the derived yoga, we find the followingheuristic useful. Given an algebra A with a central element ς , to consider A -modules onwhich ς acts by identity is to consider A/ (1 − ς )-modules. It would be more enlightenedhowever (in the derived sense) to instead view the cone of (1 − ς ) : A → A as a differentialgraded algebra, i.e., ( A [ ǫ ] , d ) where deg ( ǫ ) = − d = (1 − ς ) ι ǫ (with ι denoting contraction).Note that we have an obvious map of differential graded algebras (DGAs) i : A → A [ ǫ ]so that we get an adjoint pair of functors between A -modules (since A is viewed as a DGAthey are actually complexes) and A [ ǫ ]-modules, namely induction and restriction. Fur-thermore, we can explicitly describe the category of A [ ǫ ]-modules as complexes ( M • , d, h )12here ( M • , d ) is a cohomological complex of A -modules, ( M • , h ) is a homological complexof A -modules, and dh + hd = 1 − ς . Of course the cohomology of such M • is a graded A/ (1 − ς )-module. The adjoint pair of functors is then seen to be the cone of 1 − ς (whichacquires the homotopy h automatically) and the functor that forgets the homotopy.We can apply the above heuristic to the monad A and its ς . We do this in section 6.The result is the correct cyclic homology category of C . More precisely, given C as abovewe replace the naive HC ( C ) by complexes ( M • , d, h ) in HH ( C , C ) with dh + hd = 1 − ς ,where we may recall that HH ( C , C ) comes equipped with an element ς of its Drinfeldcenter. We address the adjunction of (2.1) in more detail here. By the Definition 2.1 of anadmissible bimodule M we need to deal with two dissimilar cases. The first is the case of ρ ∗ : C → D a tensor functor with a right adjoint ρ ∗ . The second is the case of B ∈ Alg ( D )and the two monoidal categories D and B md D B . Then we take the D in the first case tobe the B md D B of the second to obtain the adjunctions between HH ( C ) and HH ( D ) asrequired. Applications of this to the Hopf setting will be examined in Section 8. Consider a monoidal functor ρ ∗ : C → D with a right adjoint ρ ∗ . We will relate theHochschild homologies of C and D by an adjoint pair of functors. Observe that we have ρ ∗ ∆ ∗C σ ∆ C∗ ρ ∗ ≃ ∆ ∗D ( ρ ∗ ⊠ ρ ∗ ) σ ( ρ ∗ ⊠ ρ ∗ )∆ D∗ ≃ ∆ ∗D σ ( ρ ∗ ⊠ ρ ∗ )( ρ ∗ ⊠ ρ ∗ )∆ D∗ → ∆ ∗D σ ∆ D∗ and so, by adjunction, A C ρ ∗ → ρ ∗ A D . Thus we can define HH ( ρ ) ∗ : A D -mod → A C -mod (4.1)as T ρ ∗ T with the action: A C ρ ∗ T → ρ ∗ A D T → ρ ∗ T .On the other hand, also by adjunction, we have ρ ∗ A C → A D ρ ∗ . (4.2)We are now ready to construct the left adjoint to (4.1), i.e., for T ∈ A C -mod the object HH ( ρ ) ∗ T is defined to be the coequalizer of the diagram: A D ρ ∗ A C T Id AD ◦ (4.2) ' ' ❖❖❖❖❖❖❖❖❖❖❖ Id AD ρ ∗ ◦ act / / A D ρ ∗ T A D A D ρ ∗ T mult ◦ Id ρ ∗ T qqqqqqqqqqq
13o that, roughly speaking, HH ( ρ ) ∗ is induction from A C to A D and HH ( ρ ) ∗ is the restric-tion from A D to A C .Of course the adjoint pair of functors also exists from formal considerations as amonoidal ρ ∗ induces a map of cyclic objects in categories and its right adjoint ρ ∗ inducesa map of the corresponding cocyclic ones. Remark 4.1.
Observe that HH ( ρ ) ∗ T r ≃ T rρ ∗ or, equivalently, U HH ( ρ ) ∗ ≃ ρ ∗ U . B -bimodules Let B ∈ Alg ( D ), we need to relate the Hochschild homologies of D and B md D B by anadjoint pair of functors. It will be useful to review a concept very much related to ∆ ∗ ,namely that of internal Homs (see Section 9.2). Briefly, a monoidal category is said to bebiclosed if the product has right adjoints (internal Homs), i.e., we have adjunctions: Hom D ( X ⊗ T, M ) ≃ Hom D ( X, T ⊲ M ) and
Hom D ( T ⊗ X, M ) ≃ Hom D ( X, M ⊳ T ) . Note that if ∆ ∗ exists then we may demonstrate the biclosed property with a bit moreeffort (see Remark 9.2).The main ingredient here is the map HH · ( B, − ) : B md D B → HH ( D ) that sendsa bimodule S to its “Hochschild homology” complex T r ( S ⊗ B ⊗−• ). Note that apply-ing T r allows us to bypass the problem that arises if D is not symmetric and so thereis no Hochschild differential on S ⊗ B ⊗−• itself. Since canonically HH · ( B, S ⊗ B T ) ≃ HH · ( B, T ⊗ B S ) so HH · ( B, − ) induces a map from HH ( B md D B ), i.e., we have: B md D B HH · ( B, − ) & & ▲▲▲▲▲▲▲▲▲▲ T r w w ♥♥♥♥♥♥♥♥♥♥♥♥ HH ( B md D B ) / / ❴❴❴❴❴❴❴❴❴❴ HH ( D )where the induced dashed map has a right adjoint − ⊳ B . More precisely, for M ∈ HH ( D )the object M ⊳ B is not only a B -bimodule in D , but is actually an object in HH ( B md D B )[18]. So we have the required pair of adjoint functors between the Hochschild homologycategories of B md D B and D . Remark 4.2.
Returning to Remark 6.12 we observe that, roughly speaking, we have anequivalence of mixed complexes HH ( A C -mod ) ∗ N ≃ Hom B md C A ( A, N ⊳ A ) . A M as an H M -bimodule We now apply the general observations of Section 3 to the case of Hopf algebras. The goalhere is to demonstrate that the well-known anti-Yetter-Drinfeld contramodules [4] ariseas the Hochschild homology category of H M , the monoidal category of left H -modules.Furthermore, the stable anti-Yetter-Drinfeld contramodules are simply objects in the naivecyclic homology category of H M . 14et A be an algebra in M H , then the structure morphism ∆ r : A → A ⊗ H yields anadjoint pair of functors (this is reviewed below):∆ ∗ r : A ⊗ H M ⇆ A M : ∆ r ∗ . If we set C = H M and M = A M (the category of A -modules) then we have adjunctionsbetween M ⊠ C and M giving the latter a structure of a right C -module such that theaction has a right adjoint. Remark 5.1.
The construction above is a special case of the considerations in Section2. Namely, if A ∈ M H is an algebra then it yields a right module A M H -mod over M H .Furthermore, Vec is an admissible ( H M , M H ) -bimodule and F un ( Vec , A M H -mod ) M H isthe right H M -module A M above. Note that it need not, in general, be admissible, so thatdefining its Chern character would be problematic. Similarly we may consider an algebra A now in H M H (in the sense that it has bothleft and right commuting coactions and both are compatible with the algebra structure),that yields a bimodule category M over C with both actions possessing right adjoints. Remark 5.2.
Note that if A is an algebra in M H (or similarly in H M ) then it willalso yield an H M -bimodule category A M by adding the trivial (missing) H -coaction to A .Furthermore, if A ∈ M H and B ∈ H M are algebras, then A M is a right H M -module and B M is a left H M -module, while A M ⊠ B M = A ⊗ B M is an H M -bimodule. Remark 5.3.
If we let A = H then all of the considerations of M = A M apply also to C = H M itself. More precisely, observe that the coaction maps (using Sweedler notation) ∆ r : a a ⊗ a and ∆ l : a a − ⊗ a , together with the counit map ǫ : H → k and the coproduct∆ : h h ⊗ h , endow A ⊗ H ⊗• with the structure of a co-simplicial object in algebras .More precisely, we have face maps: δ ( a ⊗ h ⊗ · · · ⊗ h n ) = a ⊗ a ⊗ h ⊗ · · · ⊗ h n δ ( a ⊗ h ⊗ · · · ⊗ h n ) = a ⊗ h ⊗ h ⊗ · · · ⊗ h n · · · δ n ( a ⊗ h ⊗ · · · ⊗ h n ) = a ⊗ h ⊗ · · · ⊗ h n ⊗ h n δ n +1 ( a ⊗ h ⊗ · · · ⊗ h n ) = a ⊗ h ⊗ · · · ⊗ h n ⊗ a − and for i = 0 , · · · , n − s i ( a ⊗ h ⊗ · · · ⊗ h n ) = ǫ ( h i +1 ) a ⊗ h ⊗ · · · ⊗ d h i +1 ⊗ · · · ⊗ h n . First, a general observation: given a map of algebras f : A → B we have an adjointpair of functors ( f ∗ , f ∗ ) (5.1)15etween their categories of modules as follows. If M is a B -module then for m ∈ f ∗ ( M ) = M we let a · m = f ( a ) m . On the other hand if N is an A -module we get that f ∗ ( N ) = Hom A ( B, N ) where we view B as a left A -module via f , thus b · ϕ = ϕ ( − b ).Applying this to the cosimplicial object A ⊗ H ⊗• in algebras: using ( − ) ∗ we obtainthat M ⊠ C ⊠ • is a simplicial object in categories with functors possessing right adjoints.Again, we may use ( − ) ∗ to view M ⊠ C ⊠ • as a co-simplicial object in categories withfunctors possessing left adjoints, and apply the discussion in Section 3.The following is an easy, but key, lemma that will connect the general discussion tothe case of Hopf algebras. An important fact to note is the essential use of the antipode S and its inverse S − in the proof of Lemma 5.4. Lemma 5.4.
Let V be an H -module. Consider A ⊗ V as a left A -module via ∆ r and V ⊗ A as a left A -module via ∆ l , then we have isomorphisms of A -modules: A ⊗ V ≃ A ⊗ V and V ⊗ A ≃ V ⊗ A, (5.2) where V denotes V as purely a vector space (multiplicity).Proof. Recall that ∆ r ( a ) = a ⊗ a while ∆ l ( a ) = a − ⊗ a . Consider the ∆ r case first. Let ϕ : A ⊗ V → A ⊗ V be such that ϕ ( a ⊗ v ) = a ⊗ a v , then ϕ ( xa ⊗ v ) = ( xa ) ⊗ ( xa ) v = x a ⊗ x a v = x · ( a ⊗ a v ) = x · ϕ ( a ⊗ v ), so ϕ is indeed a map of A -modules. Note that θ : A ⊗ V → A ⊗ V given by θ ( a ⊗ v ) = a ⊗ S ( a ) v is its inverse.The ∆ l case is similar: ϕ : V ⊗ A → V ⊗ A is given by ϕ ( v ⊗ a ) = a − v ⊗ a , while itsinverse is θ ( v ⊗ a ) = S − ( a − ) v ⊗ a . Corollary 5.5. If H is a Hopf algebra with an invertible antipode, A as above, and N isan A -module, then we have an isomorphism of A ⊗ H -modules: ∆ r ∗ N ≃ Hom k ( H, N ) , with a ⊗ h · ϕ = a ϕ ( S ( a ) − h ) , (5.3) and an isomorphism of H ⊗ A -modules: ∆ l ∗ N ≃ Hom k ( H, N ) , with h ⊗ a · ϕ = a ϕ ( S − ( a − ) − h ) . (5.4) Proof.
Let V = H and trace the isomorphisms of Lemma 5.4 between Hom A ( A ⊗ H, N )and
Hom k ( H, N ). Remark 5.6.
Using Corollary 5.5 in the case of A = H we recover, via (9.3) , exactly theusual formulas for the biclosed structure on H M . Note that the cosimplicial object M ⊠ C ⊠ • , even without the property A assumption,yields an A ∈
End ( A M ) and a u : Id → A which are described explicitly in the following: Corollary 5.7.
Let H and A be as above, the endofunctor A of A M obtained from thecosimplicial object M ⊠ C ⊠ • is as follows: if N ∈ A M then A ( N ) = Hom k ( H, N ) withthe A -action given by a · ϕ = a ϕ ( S ( a ) − a − ) . (5.5)16 urthermore, the natural transformation u : Id → A is given by N → Hom k ( H, N ) where n ϕ ( h ) = ǫ ( h ) n. (5.6) Proof.
The claim (5.5) follows immediately from Corollary 5.5. For (5.6), we note that inour case, N = δ ∗ s ∗ s ∗ δ ∗ N → δ ∗ δ ∗ N translates to N = Hom A ⊗ H ( A, Hom A ( A ⊗ H, N )) =
Hom A ( A ⊗ H, N ) H → Hom A ( A ⊗ H, N )with n ϕ ( a ⊗ h ) = ǫ ( h ) an which identifies with (5.6) via Corollary 5.5. Consider a commutative diagram of algebras: DB f > > ⑦⑦⑦⑦⑦⑦⑦⑦ C g ` ` ❅❅❅❅❅❅❅❅ A s > > ⑦⑦⑦⑦⑦⑦⑦⑦ t ` ` ❅❅❅❅❅❅❅❅ so that we have adjoint pairs of functors between the relevant categories of modules as in(5.1). Furthermore, since s ∗ g ∗ = t ∗ f ∗ we get a natural map f ∗ g ∗ → t ∗ s ∗ by adjunction,i.e., f ∗ g ∗ → t ∗ t ∗ f ∗ g ∗ = t ∗ s ∗ g ∗ g ∗ → t ∗ s ∗ . Lemma 5.8.
The map f ∗ g ∗ → t ∗ s ∗ is an isomorphism if and only if the map g · f : C ⊗ A B → D (5.7) with c ⊗ b g ( c ) f ( b ) is an isomorphism.Proof. For N ∈ C M observe that t ∗ s ∗ ( N ) = Hom A ( B, N ) =
Hom C ( C ⊗ A B, N ) while f ∗ g ∗ ( N ) = Hom C ( D, N ) and the natural map is induced by the (5.7).
Lemma 5.9.
Let H be a Hopf algebra (with an invertible antipode) and A an H -bi-comodule algebra (as above). Consider the monoidal category C = H M and its bimodulecategory M = A M . Then, the cosimplicial object in categories with functors possessingleft adjoints, M ⊠ C ⊠ • has property A .Proof. To prove (3.2) it suffices by Lemma 5.8 to demonstrate that the map A ⊗ H ⊗ A A ⊗ H → A ⊗ H ⊗ Ha ⊗ x ⊗ b ⊗ y a b ⊗ a y ⊗ xb −
17s an isomorphism. Note that the action of A on the right copy of A ⊗ H above is via∆ r = δ . Observe that by Lemma 5.4 the map A ⊗ H ⊗ H → A ⊗ H ⊗ A A ⊗ H → A ⊗ H ⊗ A A ⊗ Ha ⊗ x ⊗ y a ⊗ x ⊗ ⊗ y a ⊗ x ⊗ ⊗ y is an isomorphism. The composition of the latter followed by the former is easily seen tobe a ⊗ x ⊗ y a ⊗ a y ⊗ x which is invertible with inverse: a ⊗ x ⊗ y a ⊗ y ⊗ S ( a ) x .The proof of (3.3) is identical, i.e., apply the same argument to A ⊗ H ⊗ A A ⊗ H → A ⊗ H ⊗ Ha ⊗ x ⊗ b ⊗ y ab ⊗ x y ⊗ x b − to obtain a ⊗ x ⊗ y a ⊗ x y ⊗ x which is invertible using S − .Finally (3.4) concerns the map δ · δ : A ⊗ H ⊗ A ⊗ H A ⊗ H → A ⊗ H being an isomorphism. The action of A ⊗ H on the right copy of A ⊗ H is via δ = ∆ r ⊗ Id H .The composition A ⊗ H ⊗ H → A ⊗ H is now mapping a ⊗ x ⊗ y ⊗ z to a ⊗ a z ⊗ x ⊗ y which is invertible using S .With Lemma 5.9 in hand we can complete the Corollary 5.7 to an explicit descriptionof A as a monad in our special case of A and H . Proposition 5.10.
With H and A as in Corollary 5.7 and under the identification of A with Hom k ( H, − ) , the monadic product on A is given by the following map: Hom k ( H, Hom k ( H, − )) → Hom k ( H, − ) ϕ ( x )( y ) ϕ ( h )( h ) . Proof.
Recall that the product on A is given by the composition: δ ∗ δ ∗ δ ∗ δ ∗ ← δ ∗ δ ∗ δ ∗ δ ∗ = δ ∗ δ ∗ δ ∗ δ ∗ → δ ∗ δ ∗ where the first map is an isomorphism by (3.2). Let N ∈ A M . In our case, the firstidentification is a composition: Hom A ( A ⊗ H, Hom A ( A ⊗ H, N )) ≃ Hom A ( A ⊗ H ⊗ A A ⊗ H, N ) ≃ Hom A ( A ⊗ H ⊗ H, N ) , the second is Hom A ( A ⊗ H ⊗ H, N ) ≃ Hom A ⊗ H ( A ⊗ H ⊗ H, Hom A ( A ⊗ H, N )) , and the third is the evaluation at 1 ∈ A ⊗ H ⊗ H : Hom A ⊗ H ( A ⊗ H ⊗ H, Hom A ( A ⊗ H, N )) → Hom A ( A ⊗ H, N ) . After careful tracing through the maps and identifications via Lemma 5.4 we obtain theresult. 18 roposition 5.11.
Let H be a Hopf algebra with an invertible antipode. Then • HH ( H M , H M ) is the category [ a YD of anti-Yetter-Drinfeld contramodules. • HH ( H M , Vec ) is the category of H -contramodules.Proof. We apply the preceding discussion to an appropriate choice of A . For the first case,let A = H with the usual coactions via the coproduct ∆ : H → H ⊗ H . For the second,let A = k , with the trivial coactions, i.e., ∆ l = ∆ r = u : k → H , where u is the unitinclusion. Remark 5.12.
It is also possible to consider A = H but instead of taking the usual H -coactions, to let the left one be trivial. The consideration of the resulting H M -bimodule H M , namely HH ( H M , H M ) yields the contramodule variant of Hopf modules. Since thelatter is just H M ⊠ H M Vec, so Remark 2.4 demonstrates that the Fundamental Theoremof Hopf contramodules also holds.
Thus for a general A as considered in this section, HH ( H M , A M ) yields the category(of generalized anti-Yetter-Drinfeld contramodules) \ A M H that consists of A -modules and H -contramodules such that the two structures are compatible in the following sense: if M is an A -module and α : Hom k ( H, M ) → M is the H -contramodule structure, then for ϕ ∈ Hom k ( H, M ) and a ∈ A we have α ( a ϕ ( S ( a ) − a − )) = aα ( ϕ )which is obtained directly from (5.5). Having addressed the various Hochschild homology categories, it is time to consider thecyclic version. Let A = H so that we are in the setting of Section 3.3. It remains only todescribe the element ς ∈ A , i.e., the map N = δ ∗ s ∗ s ∗ δ ∗ N → Hom k ( H, N ) for N ∈ H M .Recall that δ ∗ N = Hom k ( H, N ) has two H -actions: H by x ϕ ( S ( x ) − ) and H by ϕ ( − x ). We observe that ς : N = Hom k ( H, N ) H → Hom k ( H, N ) via ς : n ( − ) n ( − ) n . So we see that (compare with (5.6)) ς : N → Hom k ( H, N ) n ϕ ( h ) = hn which together with Proposition 5.11 and Corollary 3.17 yields: Theorem 5.13.
Let H be a Hopf algebra with an invertible antipode, then HC ( H M ) isthe category of stable anti-Yetter-Drinfeld contramodules. Mixed anti-Yetter-Drinfeld contramodules and ch ( A ) Here we pursue the heuristic outlined at the end of Section 3.3, applied to the case ofSection 5. One of the consequences of this point of view is an analogue of [5] for Hopf cycliccohomology; this has been previously attempted in [13] via a very different approach. Theidea is to replace stable anti-Yetter-Drinfeld contramodules ( HC ( H M ) by Theorem 5.13)by mixed anti-Yetter-Drinfeld contramodules introduced below. They are a generalizationof mixed complexes, utilizing the ς in the Drinfeld center of HH ( H M ). Definition 6.1.
Let M be a k -linear category with ς ∈ Aut ( Id M ) equipping M ∈ M with ς M ∈ Aut M ( M ) . Then denote by M S the category of cohomological complexes of objectsin M paired with a homotopy annihilating − ς , i.e., it consists of ( M • , d, h ) with d acohomological differential, h a homological differential and dh + hd = Id M i − ς M i for all i . Our main example of an M as above is HH ( C ) = HH ( C , C ). Thus HH ( C ) S is amodification of the naive HC ( C ). Definition 6.2.
We will call the objects of the category HH ( H M ) S mixed anti-Yetter-Drinfeld contramodules. Thus the classical stable anti-Yetter-Drinfeld contramodules cor-respond to objects concentrated in degree . Note that this is not a needless generalization for two reasons. First, as we will seebelow, this category is a natural recipient of the Chern character of an algebra, and second,the homotopy encodes the analogue of the flat structure on a D X -module, for X living inthe algebraic geometry world.For A a unital associative algebra in H M consider T r ( A ⊗• +1 ) which is a paracyclicobject in [ a YD , i.e., a functor from Λ op ∞ to [ a YD in light of Remark 3.13. More precisely,the faces are obtained via multiplication, the degeneracies via the unit, and finally thecyclic generator τ uses the cyclic property of T r , namely Remark 3.13. Since there is noguarantee that τ to the correct power is identity, it is not a cyclic object.Furthermore, as in [16], T r ( A ⊗−• +1 ) has differentials b and B of degree +1 and − bB + Bb ) | T r ( A ⊗ n +1 ) = 1 − τ n +1 n and the latter is 1 − ς T r ( A ⊗ n +1 ) by Remark 3.15. So we have that ( T r ( A ⊗−• +1 ) , b, B ) is a mixed anti-Yetter-Drinfeldcontramodule. Definition 6.3.
Let A ∈ Alg ( H M ) , define the Chern character of A in [ a YD S as follows: ch ( A ) := ( T r ( A ⊗−• +1 ) , b, B ) . Recall that the above is not the first definition of ch ( A ) that appears in this paper. Itis motivated by the initial Definition 2.2 in light of Section 4.20 emark 6.4. We note that for
V, W ∈ H M we have τ V,W : T r ( V ⊗ W ) → T r ( W ⊗ V ) given by τ V,W ϕ ( h ) = 1 ⊗ h ◦ σ V,W ◦ ϕ ( h ) where σ V,W : V ⊗ W → W ⊗ V sends v ⊗ w to w ⊗ v .Similarly, for S ∈ H M we have τ S : T r (∆ ∗ S ) → T r (∆ ∗ σS ) given by ( τ S ϕ )( h ) =( h ⊗ ϕ ( h ) . Let M be as in the Definition 6.1. We have the following easy lemma that follows from adirect computation. The objects of Vec S below are the usual mixed complexes of [12]. Lemma 6.5. If ( M • , d M , h M ) and ( N • , d N , h N ) are in M S then Hom M ( M • , N • ) • ∈ Vec S where Vec is equipped with the trivial S action. More precisely, Hom M ( M • , N • ) i = Q j Hom ( M j , N i + j ) and d = [ d, − ] , h = [ h, − ] , i.e., dϕ = d N ◦ ϕ − ( − deg ( ϕ ) ϕ ◦ d M , and hϕ = h N ◦ ϕ − ( − deg ( ϕ ) ϕ ◦ h M . The lemma above actually defines module category internal Homs [17]. More precisely,we have that M is a Vec-module category and so K ( M ), the category of cohomologicalcomplexes, is a K (Vec)-module, and similarly Vec S acts on M S with( V • , d V , h V ) · ( M • , d M , h M ) = ( V • ⊗ M • , d V + d M , h V + h M )as expected. Furthermore, if we set Hom ( M • , N • ) := Hom M ( M • , N • ) • ∈ Vec S then Hom
Vec S ( V • , Hom ( M • , N • )) ≃ Hom M S ( V • · M • , N • ) . (6.1) Lemma 6.6.
Let ( X • , d, h ) ∈ Vec S and k be the trivial mixed complex then RHom
Vec S ( k, X • ) ≃ X • [[ y ]] with deg ( y ) = +2 and differential d − hy on the latter. The grading on X • [[ y ]] is as follows: ( X • [[ y ]]) i = Q ∞ j =0 X i − j y j .Proof. As in [12] replace k by ( k [ x, ǫ ] , ǫ/x, ǫ ) with deg ( x ) = − deg ( ǫ ) = − Remark 6.7.
More generally, for X • , Y • ∈ Vec S we have RHom
Vec S ( X • , Y • ) ≃ Hom k ( X • , Y • )[[ y ]] with deg ( y ) = +2 and differential D = [ d, − ] − [ h, − ] y , by (6.1) and Lemma 6.6. roposition 6.8. Let ( M • , d, h ) ∈ M S be such that M i> = 0 and M i is projective in M ; if N • ∈ M S as well then RHom M S ( M • , N • ) ≃ RHom
Vec S ( k, Hom ( M • , N • )) . Proof.
Let M − S denote the subcategory of M S with complexes concentrated in non-positive degrees. Similarly, K − ( M ) denotes cohomological complexes in M in non-positivedegrees. Equip the latter with the projective model structure. We recommend [8] for anoverview.Consider U : M − S → K − ( M ) the forgetful functor (forgetting the homotopy h ) andlet F : K − ( M ) → M − S be Cone (1 − ς ), i.e., for ( A • , d ) ∈ K − ( M ) we have F ( A • , d ) =( A • [ ǫ ] , d + (1 − ς ) ι ǫ , ǫ ) where deg ( ǫ ) = − ι ǫ denotes contraction with ǫ . We note that( F, U ) is an adjoint pair of functors and the model structure on M − S is transferred fromthat of K − ( M ).Observe that F U ( M • ) = ( M • [ ǫ ] , d + (1 − ς ) ι ǫ , ǫ ) is isomorphic to ( M • [ ǫ ] , d, ǫ + h ) via m + ǫm m + hm + ǫm . (6.2)Furthermore we have a commutative diagram: F U ( M • ) (6.2) (cid:15) (cid:15) eval / / M • Id (cid:15) (cid:15) M • [ ǫ ] / / M • where the top arrow is the evaluation map: m + ǫm m + hm and the bottom arrowis m + ǫm m . So ( M • [ ǫ ] , d, ǫ + h ) is cofibrant and thus ( M • [ x, ǫ ] , d + ǫ/x, ǫ + h ) where deg ( x ) = − M • , d, h ).It then follows that RHom M S ( M • , N • ) ≃ Hom M S (( M • [ x, ǫ ] , d + ǫ/x, ǫ + h ) , N • )= Hom M S (( k [ x, ǫ ] , ǫ/x, ǫ ) · M • , N • ) ≃ Hom
Vec S (( k [ x, ǫ ] , ǫ/x, ǫ ) , Hom ( M • , N • )) ≃ RHom
Vec S ( k, Hom ( M • , N • ))We obtain the following as a corollary of the above. Theorem 6.9. If A is an algebra in H M that is projective as an object and M is a stableanti-Yetter-Drinfeld contramodule, then we can reinterpret the old Hopf-cyclic cohomology: HC old ( A, M ) ≃ RHom [ a YD S ( ch ( A ) , M ) . roof. Since A is projective then so is A ⊗ n as the monoidal category H M has exact internalHoms (see [15] for example), i.e., Hom H ( A ⊗ n , − ) ≃ Hom H ( A, Hom l ( A ⊗ n − , − )). Notethat T r ( A ⊗ n ) is also projective since ( T r, U ) is an adjoint pair by Proposition 3.12. ByProposition 6.8 we have
RHom [ a YD S ( ch ( A ) , M ) ≃ RHom
Vec S ( k, Hom [ a YD ( ch ( A ) , M ))and as mixed complexes Hom [ a YD ( ch ( A ) , M ) ≃ Hom H ( A ⊗−• +1 , M ) (the identification isvia Proposition 3.12 again) where the latter obtains the mixed complex structure from thecocyclic object structure on Hom H ( A ⊗ n +1 , M ) defined as usual [4]. Since HC old ( A, M ) isdefined as the cyclic cohomology of this cocyclic object, which is isomorphic to the cycliccohomology [12] of the associated mixed complex, which in turn is exactly the cohomologyof
Hom H ( A ⊗• +1 , M )[[ y ]] with D = b − By , we are done by Lemma 6.6.And so we are prompted to make the following definition: Definition 6.10.
Let A ∈ Alg ( H M ) and M • ∈ [ a YD S then HC • ( A, M • ) := RHom [ a YD S ( ch ( A ) , M • ) . This generalizes the old definition to algebras that are not projective as objects and togeneral mixed anti-Yetter-Drinfeld contramodule coefficients.
Remark 6.11.
Let ( M • , d, h ) ∈ [ a YD S then note that (( M • ) H , d, h ) ∈ Vec S . Fromthe proofs of Theorem 6.9, Proposition 6.8, and the observation that ch (1) ≃ T r (1) weimmediately obtain that if is projective in H M (and thus so is everything) then RHom [ a YD S ( ch (1) , M • ) ≃ RHom
Vec S ( k, ( M • ) H ) so that it is computed by ( M • ) H )[[ y ]] with deg ( y ) = +2 and the differential d − hy . More generally, taking
RHom HH ( C ) S ( ch ( A ) , M ) as the definition of cyclic cohomologyof an algebra A in a monoidal category C with coefficients in M ∈ HH ( C ) S , i.e., of HC • ( A, M ), we get:
Remark 6.12.
The adjunction for cyclic cohomology discussed in [18] can be obtainedfrom (2.2) since for A ∈ Alg ( C ) and M ∈ HH ( D ) S we have: HC • ( M ∗ A, M ) =
RHom HH ( D ) S ( ch ( M ∗ A ) , M ) ≃ RHom HH ( D ) S ( HH ( M ) ∗ ch ( A ) , M ) ≃ RHom HH ( D ) S ( ch ( A ) , HH ( M ) ∗ M ) = HC • ( A, HH ( M ) ∗ M ) . In particular, for N ∈ HH ( D ) S we have HC • ( A, N ) ≃ RHom mixed ( k, HH ( A C -mod ) ∗ N ) so that the generalized cyclic cohomology reduces to computing Ext nmixed ( k, T ) for somemixed complex T . .2 A generalization of the cyclic double complex In this section we take a more hands on approach to defining cyclic cohomology withcoefficients in a mixed anti-Yetter-Drinfeld contramodule by generalizing the original Tsy-gan’s construction of a double complex associated to a precocyclic object. We make noclaims about the compatibility of this method with the derived approach taken above inDefinition 6.10, though no doubt with some mild assumptions they agree.Let A be an associative, not necessarily unital, algebra in H M and take ( M • , d, h ) ∈ [ a YD S that is bounded as a complex. Consider Hom H ( A ⊗• +1 , M j ) for a fixed j , and notethat we don’t have degeneracies given by 1 ∈ A anymore and τ n +1 n is no longer Id as M j is not assumed to be stable, however τ n +1 n is induced by ς M j ∈ Aut ( M j ). Thus it is apreparacocyclic vector space.Define a trigraded object Hom H ( A ⊗• +1 , M • )[ x, ǫ ] with deg ( x ) = +2, deg ( ǫ ) = +1adjoined graded commutative variables. To be backward compatible, the exponent of A gives the first index, total degree of x, ǫ provides the second index, and exponent of M thethird. Thus the total degree of Hom H ( A ⊗ i +1 , M j ) x k ǫ l is i + j + 2 k + l .With the notation as in [16] we have operators: δ = N xι ǫ + (1 − t ) ǫ with δ = (1 − ς ) xδ = bι ǫ − b ′ ǫι ǫ δ = dδ = xh and [ δ , δ ] = δ . Each operator is of total degree 1 and except for the relations indicated above, ev-erything else graded-commutes; so in particular squares to 0. We can thus take the totalcomplex with the differential D = δ + δ + δ − δ yielding cohomology groups as D = 0, i.e., δ fixes the δ = 0 defect caused by theinstability of M j s. Remark 6.13.
It is immediate that if ( M • , d, h ) = M , a single stable anti-Yetter-Drinfeldcontramodule, then the above reduces to the Tsygan’s double complex associated to a pre-cocyclic object Hom H ( A ⊗• +1 , M ) , i.e., it calculates HC old ( A, M ) . H -comodules We mentioned in Remark 3.2 that one may obtain anti-Yetter-Drinfeld modules, a YD , byconsidering ∆ ! instead of ∆ ∗ in the case of H M . However there is a more natural settingin which the module version of coefficients appears, namely that of the case: C = M H , themonoidal category of H -comodules. More precisely, the general methods in Section 3 canbe applied here as they were to the H M case. One may consider H • +1 as a cyclic object24n coalgebras, and thus ( M H ) • +1 is a cyclic object in categories (with right adjoints).For example, the multiplication m : H → H in H induces an adjoint pair of functors: m ! : M H ⇆ M H : m ! , where m ! is the monoidal multiplication in M H and m ! itsright adjoint. Again, by considering ( M H ) • +1 as a cocyclic object in categories (withleft adjoints) we can examine HH ( M H ) as an inverse limit, and so as a module categoryover a monad on M H . To be consistent with Section 3 we let ∆ ∗ = m ! and ∆ ∗ = m ! .Unravelling the general definitions, we get: Definition 7.1.
The adjoint pair of functors (∆ ∗ , ∆ ∗ ) is given by ∆ ∗ : M H ⇆ M H : ∆ ∗ • for S ∈ M H , given by S → S ⊗ H with s s ⊗ s ⊗ s , we have ∆ ∗ S = S with S → S ⊗ H , given by s s ⊗ s s , • for T ∈ M H , given by T → T ⊗ H with t → t ⊗ t , we have ∆ ∗ T = H ⊗ T with H ⊗ T → H ⊗ T ⊗ H , given by x ⊗ t x ⊗ t ⊗ t S ( x ) ⊗ x . Thus for T ∈ M H : A ( T ) = ∆ ∗ σ ∆ ∗ T = H ⊗ T with the coaction H ⊗ T → H ⊗ T ⊗ H : x ⊗ t x ⊗ t ⊗ x t S ( x ) . (7.1)Furthermore, the map A ( T ) → A ( T ) is m ⊗ Id T : H ⊗ T → H ⊗ T where m is themultiplication in H . The identity of A is Id : t ⊗ t while the central element is ς : t t ⊗ t . If T is an A -module then the action map H ⊗ T → T defines an action of H on T .So T is a module over ∆ ∗ σ ∆ ∗ if and only if T ∈ a YD , i.e., T is both an H -comoduleand an H -module with the two structures compatible as dictated by the above (definitionswere given by explicit formulas in [11] and [9] independently). Furthermore, as expectedwe have the free module functor T r : M H → a YD sending T ∈ M H to H ⊗ T with the comodule structure as in (7.1) and the obvious H -action on the left on the first factor. Naturally the functor U the other way, that forgetsthe H -action, is right adjoint to T r .For any M ∈ a YD the S action is given by ς M : m m m so that ς T r ( T ) ( x ⊗ t ) = xt ⊗ t . 25 emark 7.2. We have an analogue of Proposition 5.11: HH ( M H ) = a YD , HC ( M H ) = sa YD , and HH ( M H , Vec ) = H M . Note that the latter is very different from the H M case, due to the fact that while Vec isadmissible as a right M H -module, it is not (in general) as a right H M -module. Definition 7.3.
We will call the objects of a YD S the mixed anti-Yetter-Drinfeld modules . And so forth, everything can be repeated here almost verbatim to define ch ( A ), amixed anti-Yetter-Drinfeld module associated to a unital associative algebra A in M H .The discussion of Section 6.1 can be repeated here as well. Let us examine the Hochschild homology adjunctions of Section 4 in some examples in thesetting where the monoidal categories arise as modules or comodules over Hopf algebras.In particular, let us apply the constructions of Section 4.1 to the case of ρ : K → H a map of Hopf algebras which results in the obvious monoidal functor ρ ∗ : H M → K M with a right adjoint being coinduction, i.e., ρ ∗ M = Hom K ( H, M ).We see that if M ∈ K [ a YD with K -contraaction given by α : Hom ( K, M ) → M then HH ( ρ ) ∗ M = ρ ∗ M = Hom K ( H, M ) with the H -contraaction obtained as the compositionof two maps: Hom ( H, Hom K ( H, M )) → Hom K ( H, Hom ( K, M )) with f ( x ⊗ y ) g f ( h ⊗ k )where g f ( h ⊗ k ) = f ( S ( h ) ρ ( k ) h ⊗ ρ ( k ) h ) and α ◦ − : Hom K ( H, Hom ( K, M )) → Hom K ( H, M ) . If N ∈ H [ a YD with H -contraaction again denoted by α then HH ( ρ ) ∗ N is given by thecoequalizer of the diagram of free objects in K [ a YD : Hom ( K, Hom ( H, N )) α ◦− / / ( −◦ ρ ) ◦− (cid:15) (cid:15) Hom ( K, N ) Hom ( K, Hom ( K, N )) ≃ / / Hom ( K ⊗ K, N ) −◦ ∆ K O O where ∆ K ( k ) = k ⊗ k .Let us return to the ǫ ∗ : Vec → H M example and see the adjunctions between mixedcomplexes and [ a YD S explicitly. For ( W • , d, h ) a mixed complex: HH ( ǫ ) ∗ W • = ( Hom ( H, W • ) , d ◦ − , h ◦ − )26ith action x · f = f ( S ( x ) − x ) and contraaction − ◦ ∆ H : Hom ( H, Hom ( H, W • )) → Hom ( H, W • ). Note that the individual Hom ( H, W i ) are stable. Conversely, if ( M • , d, h ) ∈ [ a YD S then HH ( ǫ ) ∗ M • = ( M • ) H (8.1)is a mixed complex. Remark 8.1.
The same ρ induces ρ ∗ : M K → M H with a right adjoint. Then we obtainan adjunction between K a YD and H a YD that already appeared in [11], i.e., HH ( ρ ) ∗ = Ind HK and HH ( ρ ) ∗ = Res HK . Furthermore, an example of the constructions in Section 4.2 is obtained by consideringthe identification of H M with B md M H H via the fundamental theorem of Hopf modules.Then we get the adjoint pair (( − ) ′ , d ( − )) of functors:( − ) ′ : [ a YD ⇆ a YD : d ( − ) (8.2)from [19]. More precisely, we have Vec, an admissible ( H M , M H )-bimodule. Let A be an H -module algebra and M a mixed anti-Yetter-Drinfeld module. We obtain a generalizationof a result in [19] relating cyclic homologies between H -module algebras and H -comodulealgebras. Namely, by Remark 6.12 we have: HC • ,H ( A ⋊ H, M ) = HC • ,H (Vec ∗ A, M ) ≃ HC • H ( A, HH (Vec) ∗ M ) = HC • H ( A, c M ) . We collect here some material that is not absolutely necessary for the paper, yet withoutwhich it would, we feel, not be complete. QC ( X ) The discussion contained in this section is non-rigorous and is provided for perspectiveonly. Let us briefly illustrate some of the considerations of this paper with the case of C = QC ( X ), quasicoherent sheaves on X , done properly in [1, 2, 3]. For simplicity, assumethat X is an affine scheme. The diagonals and projections endow X • +1 with the structureof a cocyclic object, as usual. Let ∆ : X → X × X be the diagonal which induces theadjoint pair (∆ ∗ , ∆ ∗ ) between QC ( X ) and QC ( X ). Similarly, C ⊠ • +1 = QC ( X • +1 ) is acocyclic object in categories with left adjoints.For T ∈ QC ( X ) we have ∆ ∗ σ ∆ ∗ ( T ) ≃ Ω −• X ⊗ O X T . The multiplication is given bythat of the graded commutative algebra Ω −• X , i.e., functions on LX , the loop space of X ,also known as T X [ − X . To describe the unit 1 and centralelement ς we need the following discussion.Let C −• ( O X , T ) denote the Hochschild homology complex (in negative degrees) of O X with coefficients in T viewed as a diagonal bimodule. It is classical that Ω −• X ⊗ O X and C −• ( O X , T ) are equivalent as complexes in QC ( X ), where the action of O X on C −• ( O X , T ) is via the left action on T . Consider C −• ( O X , T ⊠ O X ) and C −• ( O X , O X ⊠ T )with the obvious non-diagonal bimodule structures on the coefficients, but take care tonote that the action of O X on the complexes is via the underlined components of thecoefficients. Both T ⊠ O X and O X ⊠ T map to T via the O X action on T and these mapsinduce morphisms of the corresponding complexes. Thus T q − iso ←− C −• ( O X , T ⊠ O X ) → C −• ( O X , T )is the identity of ∆ ∗ σ ∆ ∗ while T q − iso ←− C −• ( O X , O X ⊠ T ) → C −• ( O X , T )is the ς of ∆ ∗ σ ∆ ∗ .Note that the identity map is easy to describe more explicitly as O X = Ω X → Ω −• X ,while ς is closely related to J O X ∈ Hom O X ( O X , Ω X [1]), the first order jets of O X . Asis explained in [2, 3] we have that HH ( QC ( X )) consists of modules over the gradedcommutative algebra Ω −• X , i.e., is equivalent to QC ( LX ) while HC ( QC ( X )) is formedby the modules over the differential graded commutative algebra (Ω −• X , d DeRham ) and isclosely related to D X -modules. In particular, the trivialization of the S -action on aparticular object pulled back from QC ( X ) to the Hochschild homology category is relatedto the flat structure on it, i.e., upgrading it to a D X -module. More precisely, a D X -module M , i.e., an O X -module with a compatible action of vector fields, is viewed as Ω −• X ⊗ O X M with the obvious Ω −• X -module structure and homotopy given by d M .Given a map f : X → Y , we have the monoidal functor f ∗ : O Y -mod → O X -mod withthe right adjoint f ∗ . The adjunction of Hochschild homology categories then comes fromthe pullback of forms: f ∗ : Ω Y → Ω X via induction and restriction. Also note that thepair ( f ∗ , f ∗ ) gives us the correct, in the sense of Section 2.2, adjoints on algebras. Moreprecisely, we get that the functors f ∗ : O Y -alg ⇆ O X -alg : f ∗ induce an equivalence F un ( A ⊗ O Y O X -mod , B -mod) O X -mod ≃ F un ( A -mod , B -mod) O Y -mod , since as a right O Y -mod module category B O X -mod -mod = B O Y -mod -mod = B -mod, as B ⊗ O Y M ≃ B ⊗ O X O X ⊗ O Y M .We observe that the case when we replace QC ( X ) by modules over a Hopf algebra H , though the monoidal category H M is no longer symmetric nor braided, is much lesstechnologically sophisticated since ∆ ∗ is exact there. All of the structures involved arevery explicitly describable with none of the subtlety of morphisms in a derived categoryobscuring the picture. In this section we relate the present context to the one considered in [15] where biclosedmonoidal categories (i.e., possessing both left and right internal Homs) were the starting28oint for the development of cyclic cohomology with coefficients. Recall that internalHoms are right adjoints to the monoidal products, i.e.,
Hom C ( X ⊗ Y, Z ) =
Hom C ( X, Hom l ( Y, Z )) =
Hom C ( Y, Hom r ( X, Z )) . A minor generalization of what is considered in [15] is the category Z C ( M op ). Moreprecisely, given a C -bimodule category M we assume that there exist “internal Homs” forthe actions, i.e., Hom M ( M · C, N ) =
Hom M ( M, C ⊲ N ) and similarly for the left action.Then Z C ( M op ) is the center, in the usual sense, of the C -bimodule category M op wherethe action is given by ⊲ and ⊳ . Recall that it consists of objects M together with naturalisomorphisms for every C ∈ C : τ C : C ⊲ M → M ⊳ C, (9.1)subject to the unit condition: τ = Id M , and the associativity condition (for C, D ∈ C ):( τ C ⊳ Id ) ◦ ( Id ⊲ τ D ) = τ C ⊗ D . The maps in Z C ( M op ) are just morphisms in M that are compatible with the τ C ’s.It is perhaps easier to understand the conditions above when equivalently reformulatedin terms of the contravariant functor from M to vector spaces (that M represents). Moreprecisely, consider the C -bimodule category of contravariant functors from M to Vec, withthe usual action: C · G ( − ) = G ( − · C ) and G ( − ) · C = G ( C · − ) . Note that due to the “biclosed” property of the actions on M , the representable functorsform a sub-bimodule category, and it is M op . If we set F ( − ) = Hom M ( − , M ) then weget isomorphisms τ C : C · F → F · C such that τ = Id , but more importantly, to pass C ⊗ D through F via τ we may pass D first, followed by C .Observe that if the C -bimodule M is such that both actions have right adjoints (in thesense of Section 3) then we can do the following. Let σ : C ⊠ M → M ⊠ C be the flip. Inour case we have C • = M ⊠ C ⊠ • fibered over the simplex category ∆ so that the inverselimit consists of M ∈ M together with an isomorphism : τ : ∆ r ∗ M → σ ∆ l ∗ M (9.2)in M ⊠ C subject to a unit and an associativity condition that proves the Lemma 9.1below.For M, N ∈ M and C ∈ C we get Hom M ( N · C, M ) ≃ Hom M ⊠ C ( N ⊠ C, ∆ r ∗ M ) ≃ Hom M ( N, C ◮ ∆ r ∗ M )29nd so we see that C ⊲ M = C ◮ ∆ r ∗ M (9.3)demonstrates the existence of the left adjoint action of C on M op . The right action on M op is obtained from ∆ l ∗ in an identical manner. Thus for such M as we consider in thispaper we have that the C -actions on them are also “biclosed”.The definition of C ◮ T ∈ M for C ∈ C and T ∈ M ⊠ C needs a few words ofexplanation. In the setting of Section 5 that is of primary interest to us it is completelystraightforward. Namely, for N ∈ A M , C ∈ H M , and T ∈ A ⊗ H M we have Hom A ⊗ H ( N ⊗ C, T ) =
Hom A ( N, Hom H ( C, T ))where C ◮ T = Hom H ( C, T ) has an obvious A -action obtained from that on T . Notethat H ◮ T = T as both A -modules and H -modules, where the action of H on the LHSis obtained from the action of H on itself on the right. This allows for the recovery of∆ r ∗ M → σ ∆ l ∗ M of (9.2) from the collection of C ⊲ M → M ⊳ C . Lemma 9.1.
Under the assumption that C and M are as in Section 5 we have that HH ( C , M ) = Z C ( M op ) . Proof.
This is almost a tautology since for M ∈ HH ( C , M ) the τ of (9.2) equips therepresentable F ( − ) = Hom M ( − , M ) with the necessary τ C ’s for C ∈ C via C · F = Hom M ( − · C, M ) ≃ Hom M ⊠ C ( − ⊠ C, ∆ r ∗ M ) ≃ Hom M ⊠ C ( − ⊠ C, σ ∆ l ∗ M ) ≃ Hom M ( C · − , M ) = F · C, while the unit and associativity conditions ensure the same for τ C ’s.The above argument can almost be reversed (via Yoneda) except that not every ob-ject of M ⊠ C is of the form N ⊠ C . This is easily rectified under the assumptions ofSection 5 since the τ C of (9.1) when used with C = H and its naturality, actually equals(recovers/defines) that of (9.2), i.e., τ = τ H .Let us also sketch the equivalence of the centrality associativity to that of (3.1). Theunit condition is similar. Observe that δ ∗ ( τ ) identifies Hom M ( N · ( C ⊗ D ) , M ) with Hom M (( C ⊗ D ) · N, M ) via
Hom M ( N · ( C ⊗ D ) , M ) = Hom M ( δ ∗ δ ∗ ( N ⊠ C ⊠ D ) , M )= Hom M ⊠ C ( N ⊠ C ⊠ D, δ ∗ δ ∗ M ) δ ∗ ( τ ) → Hom M ⊠ C ( N ⊠ C ⊠ D, δ ∗ δ ∗ M )= Hom M (( C ⊗ D ) · N, M ) . Similarly δ ∗ ( τ ) identifies Hom M ( N · ( C ⊗ D ) , M ) with Hom M ( D · N · C, M ), and δ ∗ ( τ )identifies the latter with Hom M (( C ⊗ D ) · N, M ).Note that Lemma 9.1 holds even in the case when H is a bialgebra since the presenceof the antipode S was never used. 30 emark 9.2. Even though a correct definition of C ◮ T can be given (in either the H -module, or H -comodule case), and so the biclosed property would follow from the existenceof right adjoints to the actions, it is more difficult to define than the adjoints. On the otherhand the monadic approach bypasses the biclosed property necessary to speak of Z C ( C op ) ,while dealing with the action adjoints directly. The monadic approach yields immediatelythe usual descriptions of anti-Yetter-Drinfeld modules and contramodules in the comoduleand module case respectively. References [1] David Ben-Zvi, John Francis, and David Nadler. Integral transforms and Drinfeldcenters in derived algebraic geometry.
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E-mail address : [email protected]@uwindsor.ca