aa r X i v : . [ m a t h . QA ] D ec CENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY
NIELS KOWALZIGA
BSTRACT . We study the biclosedness of the monoidal categories of modulesand comodules over a (left or right) Hopf algebroid, along with the bimodulecategory centres of the respective opposite categories and a corresponding cat-egorical equivalence to anti Yetter-Drinfel’d contramodules and anti Yetter-Drinfel’d modules, respectively. This is directly connected to the existence ofa trace functor on the monoidal categories of modules and comodules in ques-tion, which in turn allows to recover (or define) cyclic operators enabling cycliccohomology. C ONTENTS
1. Introduction 12. Categorical preliminaries 42.1. Bimodule categories and centres 42.2. Trace functors 63. Centres and anti Yetter-Drinfel’d contramodules 73.1. Left and right closedness of U - Mod U - Mod as a bimodule category 83.3. Contramodules over bialgebroids 103.4. The bimodule centre in the bialgebroid module category 123.5. Traces on U - Mod
Ext and cyclic cohomology 174. Centres and anti Yetter-Drinfel’d modules 194.1. Comodules over bialgebroids 194.2. Left and right closedness of U - Comod U - Comod as a bimodule category 264.4. The bimodule centre in the bialgebroid comodule category 314.5. Traces on U - Comod
NTRODUCTION
Introducing potential coefficients in cyclic homology or cohomology typicallyasks for more than one algebraic structure in order to obtain from the under-lying chain or cochain complex a paracyclic (or duplicial) object in the sense ofConnes [Co]. For example, in those cyclic theories induced by a Hopf structureon the underlying ring or coring, coefficients might be simultaneously modulesand comodules or simultaneously modules and contramodules, whereas theunderlying (simplicial) complex usually only needs one of them. However, the
Mathematics Subject Classification.
Key words and phrases.
Closed monoidal categories, bimodule categories, centres, cyclic coho-mology, contramodules, Hopf algebroids. presence of two structures instead of one may not always be immediately recog-nised as one of them may be trivial and therefore invisible. This, for example,sometimes happens for bialgebras or bialgebroids with special properties, suchas commutativity or cocommutativity.Whereas up to this point no compatibility between these two algebraic struc-tures is required, passing from paracyclic to cyclic objects, i.e. , those in whichthe cyclic operator powers to the identity, in general asks for some sort of com-patibility condition, which leads to the notion of (stable anti) Yetter-Drinfel’dmodules resp. stable anti Yetter-Drinfel’d contramodules in the two cases ofmodule-comodule resp. module-contramodule mentioned above, which expresswhat happens if action is followed by coaction, and vice versa, resp. con-traaction followed by action, and vice versa again; see, just to name a few,[BePeW, B ¸S, Br, BuCaP, Dr, J ¸S, HKhRS, Kay, Ko1, PSt, RT, Ye] for thesenotions in various contexts. For example, as explained in [Ko2], without spec-ifying the technical details here, if U is a left Hopf algebroid (for example, aHopf algebra or the enveloping algebra A e of an associative algebra or stillthe enveloping algebra of a Lie algebroid) with respect to which N is a Yetter-Drinfel’d module, M a stable anti Yetter-Drinfel’d module, and P a stable antiYetter-Drinfel’d contramodule, then (under suitable projectivity resp. flatnessassumptions), the (co)chain complexes computing the various derived func-tors Tor U ‚ p N, M q , Ext ‚ U p N, P q Cotor U ‚ p N, M q , and Coext U ‚ p N, P q can be made intocyclic modules, which, in particular, implies the existence of (co)cyclic differen-tials of degree ˘ : B : Tor U ‚ p N, M q Ñ
Tor U ‚ ` p N, M q , B : Cotor ‚ U p N, M q Ñ
Cotor ‚ ´ U p N, M q ,B : Ext ‚ U p N, P q Ñ
Ext ‚ ´ U p N, P q , B : Coext U ‚ p N, P q Ñ
Cotor U ‚ ` p N, P q , by abuse of notation all denoted by the same symbol B here, that is, the (in-duced) Connes-Rinehart-Tsygan (co)boundary in its various guises.1.1.
Aims and objectives.
In contrast to Yetter-Drinfel’d kind of objects be-ing interpreted as monoidal centres [Sch1], a categorical understanding of anti
Yetter-Drinfel’d objects is only beginning to emerge. The main objective of thisarticle is to embed the two cases of anti Yetter-Drinfel’d objects mentionedabove in a more categorical setting, inspired by and generalising the ideas in[Sh, KobSh] to the realm of left resp. right Hopf algebroids, which, as alreadyhinted at, allow for the simultaneous generalisation of various (co)homologytheories such as Hopf algebras, associative algebras, Lie algebroids as well as full
Hopf algebroids, that is, those with an antipode in the sense of [BSz].More precisely, whereas it is, as just mentioned, well-known that the cat-egory of Yetter-Drinfel’d modules over a bialgebroid U is equivalent to the(weak) monoidal centre of the category of left U -modules [Sch2, Prop. 4.4] as isthe case for bialgebras, we are going to show in the following that anti Yetter-Drinfel’d modules and anti Yetter-Drinfel’d contramodules correspond to the bimodule category centre of (the opposite of) the category of left U -comodulesand left U -modules, respectively. The main difficulty in dealing here with leftresp. right Hopf algebroids is, apart from the noncommutativity of the basering, the absence of an antipode map which leads to nontrivial associativityconstraints in the bimodule categories in question and hence to considerablymore laborious computations, in striking contrast to the case of Hopf algebras(or even full Hopf algebroids for that matter); this even has implications whenit comes to discuss the relationship between stability and centrality, whichdoes not seem to exactly parallel the Hopf algebraic situation.On the other hand, the sort of disheartening abundance of possibilities fordefining, for example, anti Yetter-Drinfel’d modules in the Hopf algebra case ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 3 (left-left, left-right, and so on) in the left Hopf algebroid case is instantly lim-ited to one (all other possible definitions not being well-defined) and no furtherequivalences need to be established (nor discussed).1.2.
Main results.
Corresponding to the general idea just outlined, assem-bling Lemmata 3.1 & 3.4 with Theorem 3.8, in §3 we essentially show (see themain text for all details, notation, and the precise statements):
Theorem 1.1.
Let a left bialgebroid p U, A q in addition be left Hopf. Then thecategory U - Mod of left U -modules is biclosed, which by adjunction induces thestructure of a bimodule category on its opposite category. The category of stableanti Yetter-Drinfel’d contramodules over U is equivalent to a full subcategory ofthe centre of this bimodule category. In particular, any stable anti Yetter-Drinfel’d contramodule over U can beseen as an object in the centre of U - Mod op . By virtue of this result, in Theo-rems 3.10 & 3.12, we can not only define a so-called trace functor in the senseof Kaledin [Ka2] on the category of left U -modules, but also explicitly constructa cyclic operator in the sense of Connes [Co], that is: Theorem 1.2.
If a left bialgebroid p U, A q is left Hopf and M a stable anti Yetter-Drinfel’d contramodule over U with contraaction γ , then Hom U p´ , M q yields atrace functor U - Mod Ñ k - Mod , which, in particular, implies an isomorphism
Hom U p X b A Y, M q »
Hom U p Y b A X, M q for any X, Y P U - Mod . Its explicit form induces the cyclic operator p τ f qp u , . . . , u q q “ γ ` pp u p q ¨ ¨ ¨ u q ´ p q u q q ➢ f qp´ , u p q , . . . , u q ´ p q q ˘ on the cochain complex C ‚ p U, M q “
Hom A op p U b A op ‚ , M q , which (under suitableprojectivity assumptions) computes Ext ‚ U p A, M q . For details of the precise construction and all notation we refer to §3.6. Withone more (mild) technical assumptions mentioned in Remark 3.13, one caneven replace A by a Yetter-Drinfel’d module N in the Ext -groups above.Dually, passing in §4 to the monoidal category U - Comod of left U -comodulesin relationship to anti Yetter-Drinfel’d modules, assembling the statements ofLemmata 4.10 & 4.13 with Theorem 4.14, we can summarise: Theorem 1.3.
Let a left bialgebroid p U, A q be simultaneously left and rightHopf. Then, under suitable projectivity assumptions, the category U - Comod isbiclosed, which by adjunction induces the structure of a bimodule category onits opposite category. The category of anti Yetter-Drinfel’d modules over U isequivalent to the centre of this bimodule category. Again, asking for stability of the anti Yetter-Drinfel’d modules establishesa categorical equivalence to a full subcategory of this centre. Likewise, if M is now a stable anti Yetter-Drinfel’d module, this allows for the constructionof a trace functor Hom U p´ , M q : U - Comod Ñ k - Mod obeying an analogouscommutation property as above, that is
Hom U p X b A Y, M q »
Hom U p Y b A X, M q for any X, Y P U - Comod .Observe the somewhat unexpected asymmetry between the module and co-module case in Theorems 1.1 and 1.3, both with respect to stability as well asthe number of Hopf structures needed; see Remarks 4.8 and 4.11 for a possibleexplanation.
NIELS KOWALZIG
Notation and conventions.
A very brief exposition on bialgebroids and(left and right) Hopf algebroids as well as the respective relevant notation isgiven in Appendix A at the end of the main text. At this point, we only wantto recall that a left bialgebroid p U, A q is called left resp. right Hopf algebroid ifthe corresponding Hopf-Galois map α ℓ resp. α r is invertible, where α ℓ : § U b A op U Ž Ñ U Ž b A Ż U , u b A op v ÞÑ u p q b A u p q v,α r : U đ b A Ż U Ñ U Ž b A Ż U , u b A v ÞÑ u p q v b A u p q . The Sweedler-type shorthand notations u ` b A op u ´ : “ α ´ ℓ p u b A q ,u r`s b A u r´s : “ α ´ r p b A u q , with summation understood, will be used throughout the entire text. Recallmoreover from Eq. (A.1) the various triangle notations Ż , Ž , § , đ that denote thefour A -module structures on the total space U of a bialgebroid, and occasionallyeven on a U -module: sometimes we decorate U or a U -module by one of thesesymbols to indicate the relevant A -module structure in a specific situation, e.g. ,in a tensor product. The symbol k always denotes a commutative ring, usuallyof characteristic zero. 2. C ATEGORICAL PRELIMINARIES
In this preliminary section, we gather some notions from category theorysuch as module categories and centres of bimodule categories that generalisethe corresponding ideas from algebra and are at the base of our subsequentconsiderations.2.1.
Bimodule categories and centres.
Let p C , b , , α, l, r q be a monoidalcategory, where α is the associativity constraint, and l and r the left resp. rightunit constraint. The following definitions can be found in [EtGeNi, §7.1]. Definition 2.1. A left module category over C is a category M equipped with abifunctor ➤ : C ˆ M Ñ M and natural isomorphisms, again called associativity and unit constraint , φ X,Y,M : p X b Y q ➤ M » ÝÑ X ➤ p Y ➤ M q , M : ➤ M » ÝÑ M (2.1)for all X, Y P C and M P M , such that the customary pentagon and trianglediagrams pp X b Y q b Z q ➤ M α X,Y,Z b id M t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ φ X b Y,Z,M * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ p X b p Y b Z qq ➤ M φ X,Y b Z,M (cid:15) (cid:15) p X b Y q ➤ p Z ➤ M q φ X,Y,Z ➤ M (cid:15) (cid:15) X ➤ pp Y b Z q ➤ M q id X b φ Y,Z,M / / X ➤ p Y ➤ p Z ➤ M qq (2.2) and p X b q ➤ M φ X, ,M / / r X b id M & & ▼▼▼▼▼▼▼▼▼▼ X ➤ p ➤ M q id X b l M x x qqqqqqqqqq X ➤ M (2.3) commute. ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 5
This clearly generalises the idea of a module over a ring. A right modulecategory over C is defined analogously and is the same as a left C op -modulecategory. In this case, we use the notation ➤ : M ˆ C Ñ M , ψ M,X,Y : M ➤ p X b Y q » ÝÑ p M ➤ X q ➤ Y (2.4)for the bifunctor and the associativity constraint. Definition 2.2. A bimodule category over two monoidal categories C and D isa category M that is simultaneously a left C -module and right D -module cate-gory with respective associative constraints φ and ψ , plus middle associativityconstraints given by natural transformations ϑ X,M,Z : p X ➤ M q ➤ Z » ÝÑ X ➤ p M ➤ Z q (2.5)for M P M , X P C , and Z P D , such that the two pentagon diagrams pp X b Y q ➤ M q ➤ Z φ X,Y,M b id Z t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ϑ X b Y,M,Z * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ p X ➤ p Y ➤ M qq ➤ Z ϑ X,Y ➤ M,Z (cid:15) (cid:15) p X b Y q ➤ p M ➤ Z q φ X,Y,M ➤ Z (cid:15) (cid:15) X ➤ pp Y ➤ M q ➤ Z q id X b ϑ Y,M,Z / / X ➤ p Y ➤ p M ➤ Z qq (2.6) and X ➤ p M ➤ p W b Z qq id X b ψ M,W,Z t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ j j ϑ X,M,W b Z ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ X ➤ pp M ➤ W q ➤ Z q O O ϑ X,M ➤ W,Z p X ➤ M q ➤ p W b Z q ψ X ➤ M,W,Z (cid:15) (cid:15) p X ➤ p M ➤ W qq ➤ Z o o ϑ X,M,W b id Z pp X ➤ M q ➤ W q ➤ Z (2.7) commute for all M P M , X, Y P C , and Z, W P D . Remark 2.3.
Note that whereas several relevant examples of monoidal cat-egories are strict, i.e. , where the associative constraint α , along with the leftand right unit constraint l resp. r are the identity transformations such thatthe diagrams (2.2) and (2.3) somewhat simplify, this cannot be said for typicalexamples of (bi)module categories. Here, even for underlying strict monoidalcategories, the left, right, and middle associative constraints φ , ψ , and ϑ from(2.1), (2.4), and (2.5) are not necessarily an easy guess, see Eqs. (4.27) and(4.32) for concrete nontrivial examples. This is mainly due to our dealing with(left or right) Hopf algebroids instead of Hopf algebras and therefore the ab-sence of (the notion of) an antipode resp. its inverse.The definition of the centre of a bimodule category was formulated in thecontext of fusion categories in [GeNaNi, Def. 2.1]; we relax it here to monoidalcategories which is most likely already present in the literature somewhere. Definition 2.4.
The centre of a p C , C q -bimodule category M is a category Z C p M q the objects of which are given by pairs p M, τ q , where M is an objectin M and τ X : M ➤ X » ÝÑ X ➤ M NIELS KOWALZIG are isomorphisms natural in X such that the hexagon diagram X ➤ p M ➤ Z q o o ϑ X,M,Z id X b τ Z w w ♦♦♦♦♦♦♦♦♦♦♦ p X ➤ M q ➤ Z g g τ X b id Z ❖❖❖❖❖❖❖❖❖❖❖ X ➤ p Z ➤ M q g g φ X,Z,M ❖❖❖❖❖❖❖❖❖❖❖ p M ➤ X q ➤ Z ψ M,X,Z ♦♦♦♦♦♦♦♦♦♦♦ p X b Z q ➤ M o o τ X b Z M ➤ p X b Z q (2.8) commutes for all M P M and X, Z P C .The natural transformation τ is called a central structure with respect to M .This definition clearly lifts the idea of the center of a bimodule over a ring to acategorical realm. Remark and Example 2.5.
Of course, a monoidal category is a bimodule cat-egory over itself by means of the monoidal product, but this is often not the onlypossibility and indeed not what we are going to consider in the next sections.If C is biclosed, by means of the left and right internal Homs we can defineadditional right and left C -actions on C itself, that is, we have adjunctions Hom C p X b Y, Z q »
Hom C p Y, Z ➤ X q , Hom C p X b Y, Z q »
Hom C p X, Y ➤ Z q , (2.9)for objects X, Y, Z P C , which flips a left action into a right resp. a right into aleft one. Following [EtNiOs, §2.9], we denote by C op the category opposite to C ,but equipped with the C -bimodule structure given by the adjoint actions ➤ and ➤ , and its centre will be correspondingly denoted by Z C p C op q . Following [Sh,Eq. (2.11)], and similar to [KobSh, Def. 2.3 & Lem. 2.4], we denote by Z C p C op q its full subcategory consisting of objects M such that the identity morphism id M P Hom C p M, M q is mapped to itself via the chain of isomorphisms Hom C p M b , M q » Hom C p , M ➤ M q » Hom C p , M ➤ M q » Hom C p b M, M q , (2.10) given by the adjunctions (2.9) along with the central structure (and suppress-ing the left and right unit constraints).2.2. Trace functors.
We will need one more piece of categorical machinery,the so-called trace functors, introduced by Kaledin [Ka2, Def. 2.1] in an ap-proach to cyclic homology with coefficients and towards a possible understand-ing of cyclic homology as a derived functor [Ka1]:
Definition 2.6. A trace functor consists of a functor T : C Ñ E between a(unital, associative) monoidal category p C , b , q and a category E , together witha family of isomorphisms tr X,Y : T p X b Y q » T p Y b X q for all X, Y P C that is unital (by which we mean tr ,Y “ id ), functorial in X and Y , as well as fulfils the property tr Z,X b Y ˝ tr Y,Z b X ˝ tr X,Y b Z “ id (2.11)for all X, Y, Z P C . Example 2.7.
In the setting we are going to deal with, typical examples oftrace functors of interest for us turn out to be closely related to bimodule cat-egory centres and internal Homs, that is, they arise via the adjunctions (2.9)
ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 7 in connection with Z C p C op q (or rather Z C p C op q ). The study of trace functors ofthe form T “ Hom C p´ , Z C p C op qq will be the goal of the next sections; in §3.6we concretely show how to re-obtain the cyclic operator on the cochain complexcomputing certain Ext groups from a trace functor.3. C
ENTRES AND ANTI Y ETTER -D RINFEL ’ D CONTRAMODULES
As mentioned in Remark 2.5, the main idea in what follows is to define (orfind) the internal Homs of a biclosed monoidal category of our interest, whichthen allows for a left and a right adjoint action, a corresponding bimodulecategory and finally its centre inducing a trace functor.3.1.
Left and right closedness of U - Mod . Let p U, A q be a left bialgebroid(see §A.1). As in the bialgebra case, the monoidal structure on the (strict)monoidal category U - Mod of left U -modules is reflected by the diagonal U -action on the tensor product N b A M of two left U -modules N, M : u p n b A m q : “ ∆ p u qp n b A m q “ u p q n b A u p q m (3.1)for n P N , m P M , and u P U .With respect to the obvious forgetful functor U - Mod Ñ A e - Mod , we some-times denote the induced A -bimodule structure on a left U -module M by a Ż m Ž b : “ s p a q t p b q m, @ m P M, a, b P A. (3.2) Lemma 3.1.
Let p U, A q be a left bialgebroid. ( i ) The category U - Mod of left U -modules is left closed monoidal, that is,has left internal Hom functors: hom ℓ p N, M q : “ Hom U p N b A Ż U , M q , for all N, M P U - Mod , equipped with the left U -action p v ! f qp n b A u q : “ f p n b A uv q (3.3) for every u, v P U and n P N . ( ii ) If the left bialgebroid is on top left Hopf (see §A.2), the category U - Mod is right closed monoidal, that is, has right internal Hom functors: hom r p N, M q : “ Hom A op p N, M q (3.4) for all N, M P U - Mod , equipped with the left U -action p u ➢ g qp n q : “ u ` g p u ´ n q (3.5) for every u P U and n P N . ( iii ) Consequently, for a left Hopf algebroid p U, A q over an underlying left bial-gebroid, the category U - Mod is biclosed monoidal, that is, has both leftand right internal Hom functors.Proof.
This is a well-known result and has already been proven in, for example,[Ko1, Lem. 4.16], see there for all technicalities adapted to our setting here.For later use, we give the adjunction morphisms. As for part (i), this would be ζ : Hom U p N b A P, M q Ñ
Hom U p P, hom ℓ p N, M qq ,f ÞÑ p ÞÑ t n b A u ÞÑ f p n b A up qu ( , ˜ f p p qp n b A q Ð [ n b A p ( Ð [ ˜ f, (3.6) and in part (ii), the claimed adjunction ξ : Hom U p P b A N, M q Ñ
Hom U p P, hom r p N, M qq ,g ÞÑ t p ÞÑ g p p b A ´qu (3.7)is simply the Hom-tensor adjunction. (cid:3) NIELS KOWALZIG
Notation 3.2.
As the left and right internal Homs we use are quite differentin nature and it sometimes turns out to be necessary to remember the explicit U - or A -linearity in question, we shall not always use the sort of concealingnotation hom r and hom ℓ but often write Hom A op and Hom U p´ b A U, ´q instead,even if the internal Homs with their U -module structure are meant. Remark 3.3.
The preceding lemma precisely establishes the setting adaptedto our needs; nevertheless, even without any left Hopf structure, symmetricallyto the case of the left internal Homs, the category U - Mod over a left bialgebroidhas right internal Homs as well (see [Sch2, Prop. 3.3]). Put hom r p N, M q : “ Hom U p U Ž b A N, M q , (3.8)being a left U -module by right multiplication on U in the argument. The orig-inal definition of a left Hopf algebroid [Sch2, Thm. 3.5] then states that a leftbialgebroid p U, A q is called left Hopf if the forgetful functor U - Mod Ñ A e - Mod preserves internal Homs (which is shown to be equivalent to the catchier def-inition mentioned in §A.2). In this case, its right internal Homs (3.8) are iso-morphic (as U -modules) to the ones given in (3.4), with isomorphism given by Hom A op p N, M q Ñ
Hom U p U Ž b A N, M q , g ÞÑ p¨q ➢ g, and inverse f ÞÑ f p b A ´q . On the contrary, the left internal Homs can be sim-plified (or complicated, depending on the point of view) in case more (or rathera different) structure is present. More precisely, in case the left bialgebroid p U, A q in addition is right Hopf, one can set hom ℓ p N, M q : “ Hom A p N, M q withleft U -module structure given by p u ➣ g qp n q : “ u r`s g p u r´s n q , g P Hom A p N, M q , n P N, (3.9)and the same comments apply as above. In the Hopf algebra case, the conditionof being right Hopf corresponds to the antipode being invertible, see Eq. (A.24).We are, however, more interested in the more general approach in Lemma 3.1with only one Hopf structure present, i.e. , the left one.3.2. U - Mod as a bimodule category.
The internal Homs allow to define thestructure of a bimodule category on the category of left U -modules resp. itsopposite in the sense mentioned in Remark 2.5. More precisely, we have: Lemma 3.4.
Let p U, A q be a left bialgebroid. ( i ) Then the operation U - Mod op ˆ U - Mod Ñ U - Mod op , p M, N q ÞÑ M ➤ N : “ hom ℓ p N, M q (3.10) defines on U - Mod op the structure of a right module category over themonoidal category U - Mod . ( ii ) If p U, A q is in addition left Hopf, the operation U - Mod ˆ U - Mod op Ñ U - Mod op , p N, M q ÞÑ N ➤ M : “ hom r p N, M q (3.11) defines on U - Mod op the structure of a left module category over themonoidal category U - Mod . ( iii ) The left and the right action from Eqs. (4.22) and (4.23) define on U - Mod op the structure of a bimodule category over the monoidal cate-gory U - Mod if the left bialgebroid p U, A q is in addition left Hopf. ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 9
Proof. (i): We have to prove that for three U -modules M, N, P P U - Mod thereis a left U -module isomorphism p M ➤ P q ➤ N » M ➤ p P b A N q , which amounts toshow that the k -module isomorphism ψ M,P,N : hom ℓ p P b A N, M q Ñ hom ℓ p N, hom ℓ p P, M qq , (3.12)which on the level of k -modules translates into a map ψ M,P,N : Hom U p P b A N b A U, M q Ñ
Hom U p P b A U, Hom U p N b A U, M qq ,f ÞÑ p b A u ÞÑ t n b A v ÞÑ f p n b A v p q p b A v p q u qu ( , t g p p b A u qp n b A q Ð [ n b A p b A u u Ð [ g, (3.13) where it is straightforward to see that both maps are well-defined and mutualinverses, is in fact an isomorphism of left U -modules. This directly follows from(3.3) by p w ! ψ M,P,N f qp p b A u qp n b A v q “ p ψ M,P,N f qp p b A uw qp n b A v q“ f p n b A v p q p b A v p q uw q “ p w ! f qp n b A v p q p b A v p q uw q“ p ψ M,P,N p w ! f qqp p b A u qp n b A v q for w P U . The truly straightforward but laborious checking of (the analogousright module versions of) the two diagrams (2.2) and (2.3) is omitted.(ii): As for the left action, to analogously fulfil the requirements in Def-inition 2.1, we have to first prove that (2.1) is true, that is, for three U -modules M, N, P P U - Mod there is a left U -module isomorphism P ➤ p N ➤ M q »p P b A N q ➤ M , which amounts to show that the k -module isomorphism φ P,N,M : hom r p P b A N, M q Ñ hom r p P, hom r p N, M qq , (3.14)which results into a map Hom A op p P b A N, M q Ñ
Hom A op p P, Hom A op p N, M qq given by the Hom-tensor adjunction, is an isomorphism of left U -modules. Thatthis is an isomorphism (of k -modules) is obvious, whereas using the left U -action (3.5) on Hom A op p N, M q , along with Eq. (A.7) we immediately see that for f P Hom A op p P b A N, M q one has, abbreviating φ “ φ P,N,M , p u ➢ p φf qqp p qp n q “ ` u ` ➢ p φf qp u ´ p q ˘ p n q “ u `` p φf qp u ´ p qp u `´ n q“ u ` f p u ´p q p b A u ´p q n q “ p u ➢ f qp p b A n q “ φ p u ➢ f qp p qp n q for any u P U , hence u ➢ p φf q “ φ p u ➢ f q as desired, and therefore we obtainan isomorphism of left U -modules as well. In order to effectively obtain a leftmodule category in the sense of Definition 2.1, we still have to verify the pen-tagon resp. triangle axiom (2.2) resp. (2.3), which, however, follow easily fromthe properties of the standard Hom-tensor adjunction, U - Mod being strict.(iii): In this part, we claim that for any
M, N, P P U - Mod , there is an iso-morphism of left U -modules ϑ P,M,N : p P ➤ M q ➤ N » ÝÑ P ➤ p M ➤ N q , the middle associativity constraint from Definition 2.2, that is, an isomorphism hom ℓ p N, hom r p P, M qq » hom r p P, hom ℓ p N, M qq , subject to the two pentagon ax-ioms (2.6) and (2.7). To start with, define the k -module isomorphism ϑ P,M,N : Hom U p N b A Ż U ,
Hom A op p P, M qq Ñ
Hom A op p P, Hom U p N b A Ż U , M qq ,f ÞÑ p ÞÑ t n b A u ÞÑ f p n b A u p q qp u p q p qu ( , t g p u ´ p qp n b A u ` q Ð [ p u Ð [ n b A u ( Ð [ g. (3.15) Verifying that these maps are well-defined and in fact mutual inverses is easyand omitted again. Let us rather show that ϑ is in particular a map (and hence an isomorphism) of left U -modules: we have p v ➢ ϑf qp p qp n b A u q “ ` v ` ! p ϑf qp v ´ p q ˘ p n b A u q “ p ϑf qp v ´ p qp n b A uv ` q“ f p n b A u p q v `p q qp u p q v `p q v ´ p q “ f p n b A u p q v qp u p q p q“ ` ϑ p v ! f q ˘ p p qp n b A u q , abbreviating ϑ “ ϑ P,M,N , where we used the left U -actions (3.5) and (3.3) inthe first step and Eq. (A.4) in the fourth.To conclude the proof of this part, we still have to check the two pentagonaxioms (2.6) and (2.7). We limit ourselves to the second one, being more dif-ficult due to the notably different complexity of the maps ψ and φ from (3.12)and (3.14), respectively.So, let M, N, P, Q P U - Mod . Then diagram (2.7) in this context explicitlyreads:
Hom A op p N, Hom U p P b A Q b A Ż U , M qq Hom A op p N,ψ
M,P,Q q t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ O O ϑ N,M,P b AQ Hom A op p N, Hom U p Q b A Ż U ,
Hom U p P b A Ż U , M qqq O O ϑ N, Hom U p P b AU,M q ,Q Hom U p P b A Q b A Ż U ,
Hom A op p N, M qq ψ Hom A op p N,M q ,P,Q (cid:15) (cid:15) Hom U p Q b A Ż U ,
Hom A op p N, Hom U p P b A Ż U , M qqq j j Hom U p Q b A U, ϑ
N,M,P q ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ Hom U p Q b A Ż U ,
Hom U p P b A Ż U ,
Hom A op p N, M qqq (3.16)
For any f P Hom U p P b A Q b A Ż U ,
Hom A op p N, M qq , going the two steps alongthe top part of this figure amounts to the same as going along the three stepsalong the bottom, which is seen as follows: indeed, for any n P N, q P Q , p P P ,and u, v P U , abbreviating ψ “ ψ M,P,Q and likewise for ϑ , we have ` Hom A op p N, ψ q ˝ ϑ ˝ f ˘ p n qp q b A u qp p b A v q (3.13) “ ` ϑ ˝ f ˘ p n qpp p b A v p q q q b A v p q u q (3.15) “ f p p b A v p q q b A v p q u p q qp v p q u p q n q (3.12) “ p ψ ˝ f qp q b A u p q qp p b A v p q qp v p q u p q n q (3.15) “ p Hom U p Q b A U, ϑ q ˝ ψ ˝ f qp q b A u p q qp u p q n qp p b A v q (3.15) “ p ϑ ˝ Hom U p Q, ϑ b A U q ˝ ψ ˝ f qp n qp q b A u qp p b A v q , that is, the diagram (3.16) commutes. This ends the proof of this part andhence of the entire lemma. (cid:3) * * *The preceding lemma allows to investigate the centre Z U - Mod p U - Mod op q inthe sense of Definition 2.4 of the bimodule category U - Mod op over U - Mod ; butbefore doing so, we need to introduce more algebraic structure to get mean-ingful statements, i.e , that of contramodules resp. anti Yetter-Drinfel’d con-tramodules as already hinted at in the Introduction.3.3.
Contramodules over bialgebroids.
Contramodules in the sense of[EiMo] over coalgebras or corings are a not too wide-spread notion, which issomehow surprising as they turn out to be as natural as comodules (see, e.g. , ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 11 [BBrWi, Br, Po]): as a first approach, they can be thought of as an infinite di-mensional version of modules over the dual of the coring in question. They areof interest for us since not only related to the centre of the bimodule category U - Mod op under investigation but (as a consequence) also appear as naturalcoefficients in the cyclic theory of Ext groups (and as such implicitly used rightfrom the beginning, as detailed in [Ko1, §6], in Connes’ classical cyclic coho-mology theory with its values in the k -linear dual of an associative algebra). Definition 3.5. A right contramodule over a left bialgebroid p U, A q is a right A -module M together with a right A -module map γ : Hom A op p U Ž , M q Ñ M, usually termed the contraaction , subject to the diagram Hom A op p U, Hom A op p U, M qq Hom A op p U,γ q / / » (cid:15) (cid:15) Hom A op p U, M q γ (cid:15) (cid:15) Hom A op p U Ž b A Ż U , M q Hom A op p ∆ ℓ ,M q / / Hom A op p U, M q γ / / M which we will refer to as contraassociativity , as well as Hom A op p A, M q Hom A op p ε,M q / / » & & ▼▼▼▼▼▼▼▼▼▼▼ Hom A op p U, M q γ x x qqqqqqqqqqq M to which we refer as contraunitality. The adjunction of the leftmost vertical arrow in the first diagram is to beunderstood with respect to the right A -action f a : “ f p a Ż ´q on Hom A op p U Ž , M q ;the required right A -linearity of γ then reads γ ` f p a Ż ´q ˘ “ γ p f q a, (3.17)usually excluding the well-definedness of a trivial right contraaction f ÞÑ f p q .Any contramodule M moreover has an induced left A -action am : “ γ ` mε p´ đ a q ˘ “ γ ` mε p a § ´q ˘ , (3.18)which turns M into an A -bimodule and γ into an A -bimodule map, γ ` f p´ đ a q ˘ “ aγ ` f p´q ˘ , (3.19)see [Ko1, Eq. (2.37)]. This yields a a forgetful functor Contramod - U Ñ A e - Mod (3.20)from the category of right U -contramodules to that of A -bimodules.For f P Hom A op p U, M q we may (non-consistently, depending on readabilityin long computations) write both γ p f p´q q as well as γ p f p¨q q or simply γ p f q tounderline where the U -dependency is located: this way, the contraassociativitymay be more compactly expressed as γ ` : γ p g p¨ b A ¨¨qq ˘ “ γ ` g p´ p q b A ´ p q q ˘ , (3.21)for g P Hom A op p U Ž b A Ż U , M q , where the number of dots match the map γ withthe respective argument, and contraunitality as γ p mε p´q q “ m (3.22)for m P M . Finally, a morphism ϕ : M Ñ M of contramodules is a map of right A -modules commuting with the contraaction, that is, ϕ ` γ p f q ˘ “ γ ` ϕ ˝ f ˘ . Anti Yetter-Drinfel’d contramodules.
As already mentioned, coefficientsin cyclic (co)homology theories typically have more than one algebraic struc-ture, like actions, coactions, contraactions, and so forth. A compatibility be-tween these is in general not required as long as one does not impose the con-dition that the cyclic operator powers to the identity. On the contrary, if onedoes, one is led to the notion of anti Yetter-Drinfel’d kind of objects:
Definition 3.6. An anti Yetter-Drinfel’d (aYD) contramodule M over a leftHopf algebroid p U, A q is a left U -module (with action denoted by juxtaposition)being at the same time a right U -contramodule (with contraaction γ ) such thatboth underlying A -bimodule structures (3.2) and (3.20) coincide, i.e. , a Ż m Ž b “ amb, m P M, a, b P A, (3.23)and such that contraaction followed by action results in u p γ p f qq “ γ ` u `p q f p u ´p´q u `p q q ˘ , @ u P U, f P Hom A op p U, M q . (3.24)If action followed by contraaction results in the identity, i.e. , for all m P Mγ p p´q m q “ m (3.25)holds, then M is called stable , where p´q m : u ÞÑ um as a map in Hom A op p U, M q .In [Ko1, p. 1093] one can find additional information about the (not so ob-vious) well-definedness of Eq. (3.24) and further implications: for example, if(3.23) holds, then γ p a Ż f p´q q “ γ ` f p a § ´q ˘ (3.26)is true as well, where on the left hand side the left A -action on M is meant. Remark 3.7.
The category
Contramod - U of right U -contramodules is, in gen-eral, not monoidal and therefore neither are so U aYD contra ´ U , the category ofanti Yetter-Drinfel’d contramodules nor U saYD contra ´ U , the category of stableones. However, in [Ko2, Prop. 3.3] it is shown that Contramod - U is a leftmodule category over U - Comod , the monoidal category of left U -comodules( cf. §4.1), which restricts to the structure UU YD ˆ U aYD contra ´ U Ñ U aYD contra ´ U , p N, M q ÞÑ hom r p N, M q of a left module category on U aYD contra ´ U over the monoidal category UU YD ofYetter-Drinfel’d modules (these are A -bimodules with compatible left U -actionand left U -coaction, which form the monoidal centre of U - Mod , see [Sch2,§4]), which is precisely induced by the action (3.11) defining the right inter-nal Homs.3.4.
The bimodule centre in the bialgebroid module category.
Havingintroduced contramodules, we can now come back to examine the centre of U - Mod op with respect to its adjoint actions. Recall from Definition 2.4 that thecentre Z U - Mod p U - Mod op q is formed by all pairs p M, τ q of objects M P U - Mod op for which there is a family of isomorphisms τ N : N ➤ M » ÝÑ N ➤ M natural in N P U - Mod . With respect to its full subcategory Z U - Mod p U - Mod op q defined by the condition that the identity map id M P Hom U p M, M q is mappedto itself by the chain of isomorphisms (2.10), we have the following result: ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 13
Theorem 3.8.
Let a left bialgebroid p U, A q in addition be left Hopf. ( i ) Then any stable aYD contramodule M induces a central structure τ N : hom ℓ p N, M q Ñ hom r p N, M q , explicitly given on the level of k -modules by τ N : Hom U p N b A Ż U , M q Ñ
Hom A op p N, M q ,f ÞÑ n ÞÑ γ p f p n b A ´qq ( , γ ` p u ➢ g qp p¨q n q ˘ Ð [ n b A u ( Ð [ g. (3.27) ( ii ) Vice versa, for a pair p M, τ q in the centre Z U - Mod p U - Mod op q , the right U -contraaction on M defined by means of γ p g q : “ p τ ´ U g qp b A q , (3.28) for every g P Hom A op p U Ž , M q , induces the structure of an anti Yetter-Drin-fel’d contramodule on M , which is stable if p M, τ q P Z U - Mod p U - Mod op q . ( iii ) Both preceding parts together imply an equivalence U saYD contra ´ U » Z U - Mod p U - Mod op q of categories.Proof. (i): That τ N is well-defined and a morphism of left U -modules if M is an aYD contramodule, and invertible in the given sense if M is stablehas already been proven in [Ko1, Thm. 4.15] (where the rôles of τ and τ ´ are interchanged). We only explicitly show here that τ N is a U -module mor-phism to illustrate where the aYD condition (3.24) is precisely needed: for f P Hom U p N b A U, M q and n P N , we have p u ➢ τ U f qp n q (3.5) “ u ` p τ U f qp u ´ n q (3.27) “ u ` ` γ p f p u ´ n b A ´q ˘ (3.24) “ γ ` u ``p q f p u ´ n b A u `´p´q u ``p q q ˘ (A.7) “ γ ` u `p q u ´ f p n b A p´q u `p q q ˘ (A.4) “ γ ` f p n b A p´q u q ˘ (3.3) “ τ U p u ! f qp n q , where in the fourth step we used the U -linearity of f . Hence, u ➢ τ U p f q “ τ U p u ! f q , (3.29)as claimed. Let us moreover show that τ , or rather τ ´ , is natural in N : for anyleft U -module morphism σ : N Ñ N we want to see that τ ´ N ˝ hom r p σ, M q “ hom ℓ p σ, M q ˝ τ ´ N . Indeed, by the U -linearity of σ , one obtains τ ´ N p g ˝ σ qp n b A u q “ γ ` p u ➢ p g ˝ σ qqp p¨q n q ˘ “ γ ` u ` g p u ´p¨q σ p n qq ˘ “ τ ´ N p g qp σ p n q b A u q , (3.30)for any g P Hom A op p N , M q and n P N .On top, we need to prove that the hexagon axiom (2.8) commutes, whichhere takes the following explicit form: Hom A op p P, Hom U p N b A U, M qq o o ϑ P,N,M
Hom A op p P,τ N q (cid:15) (cid:15) Hom U p N b A U, Hom A op p P, M qq O O Hom U p N b A U,τ P q Hom A op p P, Hom A op p N, M qq O O φ P,N,M
Hom U p N b A U, Hom U p P b A U, M qq O O ψ M,P,N
Hom A op p P b A N, M q o o τ P b N Hom U p P b A N b A U, M q (3.31) Verifying that this diagram in fact commutes with respect to the central struc-ture (3.27) is done as follows. First, for better readability, by abuse of notation let us again abbreviate ϑ “ ϑ P,N,M , and likewise for φ and ψ . For p b A n P P b A N and f P Hom U p P b A N b A U, M q , one then directly computes: p φ ´ ˝ Hom A op p P, τ N q ˝ ϑ ˝ Hom U p P b A U, τ P q ˝ ψ ˝ f qp p b A n q (3.14) “ p Hom A op p P, τ N q ˝ ϑ ˝ Hom U p N b A U, τ P q ˝ ψ ˝ f qp p qp n q (3.27) “ γ ` p ϑ ˝ Hom U p N b A U, τ P q ˝ ψ ˝ f qp p qp n b A p¨q q ˘ (3.15) “ γ ` p Hom U p N b A U, τ P q ˝ ψ ˝ f qp n b A p¨qp q qp p¨qp q p q ˘ (3.27) “ γ ´ : γ ` ` ψ ˝ f ˘` n b A p¨qp q ˘` p¨qp q p b A p¨¨q ˘ ˘ ¯ (3.13) “ γ ´ : γ ` f ` p¨qp q p b A p¨¨qp q n b A p¨¨qp qp¨qp q ˘ ˘ ¯ (3.21) , (3.1) “ γ ` f ` p¨qp q p p b A n b A p¨qp q q ˘ ˘ “ γ ` p¨qp q f ` p b A n b A p¨qp q ˘ ˘ (3.21) “ γ ´ : γ ` p¨¨q f ` p b A n b A p¨q ˘ ˘ ¯ (3.25) “ γ ` f ` p b A n b A p¨q ˘ ˘ (3.27) “ τ P b AN f p p b A n q , which, as required, proves the commutativity of diagram (3.31). Here, in theseventh step we used the U -linearity of f and the stability (3.25) of the aYDcontramodule M in the penultimate. Note that the fourth and fifth line frombottom, despite any appearance, are well-defined by taking Eq. (3.26) into con-sideration.(ii): In this part, we have to show first that (3.28) indeed defines a contraac-tion in the sense of Definition 3.5. To start with, the U -linearity (3.29) of τ U resp. of its inverse implies that γ ` g p a Ż ´q ˘ (3.5) , (A.11) “ ` τ ´ U p t p a q ➢ g q ˘ p b A q (3.29) “ ` t p a q ! τ ´ U g ˘ p b A q (3.3) “ ` τ ´ U g ˘ p b A t p a qq (3.1) “ ` τ ´ U g ˘ p b A q a (3.28) “ γ p g q a for any a P A , which is the required right A -linearity (3.17).As for contraassociativity, observe first that the coproduct ∆ : U Ñ U b A U is a morphism in U - Mod as implied by the diagonal action (3.1). We thereforehave, by means of the naturality (3.30) of the central structure, that τ ´ U p g ˝ ∆ q “ p τ ´ U b AU g q ˝ p ∆ b A id q for g P Hom A op p U b A U, M q , and using this in the firststep below, together with the hexagon axiom for τ ´ in the third, we obtain: γ ` g ˝ ∆ q (3.28) “ ` τ ´ U b AU g q ˝ p ∆ b A id q ˘ p b A q“ p τ ´ U b AU g qp b A b A q (3.31) “ ` ψ ´ ˝ Hom U p N b A U, τ ´ P q ˝ ϑ ´ ˝ Hom A op p P, τ ´ N q ˝ φ ˝ g ˘ p b A b A q (3.13) “ ` Hom U p N b A U, τ ´ P q ˝ ϑ ´ ˝ Hom A op p P, τ ´ N q ˝ φ ˝ g ˘ p b A qp b A q (3.28) “ γ ` p ϑ ´ ˝ Hom A op p P, τ ´ N q ˝ φ ˝ g qp b A q p¨q ˘ (3.15) “ γ ` p Hom A op p P, τ ´ N q ˝ φ ˝ g q p¨q p b A q ˘ (3.28) “ γ ` : γ pp φ ˝ g q p¨qp¨¨q q ˘ (3.14) “ γ ` : γ p g p¨ b A ¨¨qq ˘ , which proves the contraassociativity (3.21). Contraunitality is once moreproven with the help of the naturality of τ ´ : the bialgebroid counit U Ñ A defines an U -action on A by means of u a : “ ε p u đ a q and, by ε p uv q “ ε p u đ ε p v qq ,this yields a morphism in U - Mod . Considering then that for N “ A the central ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 15 structure τ A : hom ℓ p A, M q » M Ñ M » hom r p A, M q is the identity map, we have γ p mε p¨q q “ τ ´ U p L m ˝ ε qp U b A U q “ τ ´ A p L m qp ε p U q b A U q “ L m p A q “ m, which is the contraunitality (3.22), where we defined L m : A Ñ M, a ÞÑ ma asan element in Hom A op p A, M q » M .That the so-defined right U -contraaction (3.28) together with the left U -action (3.5) defines on M an aYD structure is seen as follows: for u P U and f P Hom A op p U, M q , we have u ` γ p g q ˘ “ u p τ ´ U g qp b A q“ p τ ´ U g qp u p q b A u p q q“ p u p q ! τ ´ U g qp u p q b A q“ τ ´ U p u p q ➢ g qp u p q b A q“ τ ´ U ` p u p q ➢ g qp p¨q u p q q ˘ p b A q “ γ ` p u p q ➢ g qp p¨q u p q q ˘ , where in the second step we used the U -linearity of τ ´ U g together with (3.1),moreover Eq. (3.3) in the third step, in the fourth that τ ´ U is a left U -modulemorphism, see Eq. (3.29), and in the fifth the naturality of τ ´ U along with thefact that right multiplication R u : U Ñ U, v ÞÑ vu with an element u P U is amorphism in U - Mod . By (3.2), this simultaneously proves (3.23) and (3.24).Finally, stability follows by the assumption p M, τ q P Z U - Mod p U - Mod op q , thatis, those objects in the centre for which id M P Hom U p M, M q is mapped to itselfby the chain of isomorphisms in (2.10). As before, the map R m : u ÞÑ um in Hom A op p U, M q is a morphism in U - Mod for any m P M , and therefore γ p p´q m q “ p τ ´ U p R m qqp b A q (3.30) “ p τ ´ M id M qp R m p q b A q “ p τ ´ M id M qp m b A q “ m, by naturality again, which signifies the stability of M . Here, in the last stepwe used the defining property of Z U - Mod p U - Mod op q as it explicitly results fromthe inverses of the adjunctions (3.6) and (3.7) in case P “ A .(iii): In this third part, we have to show three things: first, that the object p M, τ q constructed in (i) actually lies in the full subcategory Z U - Mod p U - Mod op q of the centre, the proof of which will be postponed to Remark 3.11; second, thatany morphism M Ñ M of aYD contramodules over U induces a morphism p M, τ q Ñ p ˜ M , ˜ τ q between the corresponding objects in the bimodule centre (andvice versa); third, that the two procedures of how to obtain a central structurefrom a right U -contraaction and a right U -contraaction from a central structureare mutually inverse.As for the second issue, if ϕ : M Ñ ˜ M is a morphism of aYD contramodules,we have to show that for any N P U - Mod the diagram
Hom U p N b A U, M q τ N / / Hom U p N b A U,ϕ q (cid:15) (cid:15) Hom A op p N, M q Hom A op p N,ϕ q (cid:15) (cid:15) Hom U p N b A U, ˜ M q ˜ τ N / / Hom A op p N, ˜ M q (3.32) commutes, and this is obvious since ϕ is both a morphism of right U -contramodules and left U -modules: therefore, for f P Hom U p N b A U, M q , ϕ ` τ N f p n q ˘ “ ϕ ` γ p f p n b A ´qq ˘ “ γ ` p ϕ ˝ f qp n b A ´q ˘ “ ˜ τ N p ϕ ˝ f qp n q . The other way round, let ϕ : p M, τ q Ñ p ˜ M , ˜ τ q be a morphism of objects inthe centre Z U - Mod p U - Mod op q , which means that ϕ is a left U -module mapand that diagram (3.32) commutes. In order to define a morphism of aYD contramodules, we only need to prove that ϕ is also a right U -contramodulemorphism as well. Indeed, ϕ ` γ p g q ˘ “ ϕ ` τ ´ U g p b A q ˘ “ τ ´ U ` p ϕ ˝ g qp b A q ˘ “ γ p ϕ ˝ g q for g P Hom A op p N, M q .Third, and finally, we have to show that obtaining a central structure froma right U -contraaction and a right U -contraaction from a central structure aremutually inverse. As a matter of fact, if a right U -contraaction γ on M is givenand a corresponding central structure τ (and its inverse) is defined by meansof Eq. (3.27), which, in turn, defines a right U -contraaction as in Eq. (3.28),denoted by ˜ γ for the moment, we have for g P Hom A op p U, M q ˜ γ p g q “ τ ´ U g p b A q “ γ ` p ➢ g qp p¨q q ˘ “ γ p g q , which is precisely the right U -contraaction we started with.Vice versa, given a central structure τ that defines a right U -contraaction asin Eq. (3.28) that, in turn, defines a central structure as in Eq. (3.27), denotedby σ for the moment, equally reproduces the central structure τ we startedwith. Indeed, by Eqs. (3.29) and (3.3), we have σ ´ N g p n b A u q “ γ ` p u ➢ g qp p¨q n q ˘ “ ` u ! p τ ´ U g p p¨q n qq ˘ p b A q“ p τ ´ U g p p¨q n qqp b A u q “ τ ´ U g p n b A u q , for g P Hom A op p N, M q , where in the last step we once again used the naturalityof τ p¨q with respect to the map R n : U Ñ N, u ÞÑ un as above. (cid:3) Remark 3.9.
If one desires more structural symmetry and decides to workwith the left and right internal Homs that already exist on the bialgebroidlevel in the spirit of Remark 3.3, then the central structure comes out as τ N : Hom U p N b A Ż U , M q Ñ
Hom U p U Ž b A N, M q ,f ÞÑ u b A n ÞÑ γ ` p u p q ➢ f qp n b A p¨q n q ˘( , γ ` ˜ f p u b A p¨q n q ˘ Ð [ n b A u ( Ð [ ˜ f for any stable aYD contramodule M . Quite on the contrary, if not only aleft Hopf structure but also a right one were present, as also already brieflytouched on in Remark 3.3, one obtained τ N : Hom A p N, M q Ñ
Hom A op p N, M q ,f ÞÑ n ÞÑ γ ` p p¨q ➢ f qp n q ˘( , γ ` g p p¨q n q ˘ Ð [ n ( Ð [ g for the central structure. However, we will be going on with the more generalapproach presented in Theorem 3.8.3.5. Traces on U - Mod . In the spirit of Example 2.7, the bimodule categorycentre just discussed now almost tautologically leads to a trace functor on U - Mod , which, in turn, allows for a cyclic operator on the cochain complexdefining a cyclic cohomology theory for
Ext -groups.
Theorem 3.10.
If the left bialgebroid p U, A q is left Hopf and p M, γ q a stableanti Yetter-Drinfel’d contramodule, then T : “ Hom U p´ , M q yields a trace functor U - Mod Ñ k - Mod , that is, we have an isomorphism tr N,P : Hom U p N b A P, M q » ÝÑ Hom U p P b A N, M q being unital and functorial in N, P P U - Mod . Explicitly, this trace map reads p tr N,P f qp p b A n q : “ γ ` f p n b A p¨q p q ˘ , (3.33) for n P N, p P P . ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 17
Proof.
By Theorem 3.8, Lemma 3.1, and Lemma 3.4, the diagram
Hom U p N b A P, M q ζ / / (cid:15) (cid:15) Hom U p P, hom ℓ p N, M qq Hom U p P,τ N q (cid:15) (cid:15) Hom U p P b A N, M q Hom U p P, hom r p N, M qq , ξ ´ o o commutes if we only showed that tr N,P fits into it at the dotted arrow, that is, tr N,P “ ξ ´ ˝ Hom U p P, τ N q ˝ ζ . Indeed, for f P Hom U p N b A P, M q , we have p ξ ´ ˝ Hom U p P, τ N q ˝ ζ ˝ f qp p b A n q (3.7) “ p Hom U p P, τ N q ˝ ζ ˝ f qp n qp p b A q (3.27) “ γ `` p ζf qp n q ˘ p p´q p q ˘ (3.6) “ γ ` f p n b A p´q p q ˘ (3.33) “ p tr N,P f qp p b A n q . As for the unitality of the trace functor, setting N “ A we directly see that p tr A,P f qp p q “ γ ` f p p¨q p q ˘ “ γ ` p¨q f p p q ˘ “ f p p q , using the U -linearity of f and the stability of M .All remaining properties of a trace functor in Definition 2.6 now directlyfollow from those of the central structure τ ; for example, Eq. (2.11) can beproven via the hexagon axiom (3.31). (cid:3) Remark 3.11.
We are now in a position to complete, with more ease, the proofof part (iii) of Theorem 3.8, that is, that the central object p M, τ q constructedin its first part actually lives in the subcategory Z C p C op q : in view of Theorem3.10, this is a simple consequence of the unitality tr A,P “ id of the trace.3.6. Cyclic structures on
Ext and cyclic cohomology.
In [Ko2, §3.2], wedefined the structure of a cocyclic k -module on the cochain complex computing Ext ‚ U p A, M q , where M is, to begin with, a left U -module right U -contramodulewith contraaction γ : that is, we added a cocyclic operator τ compatible with thesimplicial structure inducing the cochain complex. This way, if M is a stableaYD contramodule, one obtains a cyclic coboundary B : Ext ‚ U p A, M q Ñ
Ext ‚´ U p A, M q that squares to zero (see, for example, [Lo, §§2.5 & 6.1] or [Ts, §5] for generaldetails on (co)cyclic k -modules).In this subsection, we want to show that the trace functor T from Theorem3.10 resp. the map tr in (3.33) induce the same cocyclic operator that was ob-tained in [Ko2, Eq. (3.10)], hence induce the same cyclic cohomology for thecomplex computing Ext ‚ U p A, M q .Let us assume that U Ž is flat as an A -module. In this case, Ext ‚ U p A, M q “ H p Hom U p Bar ‚ p U q , M q , b q , where Bar ‚ p U q “ p § U Ž q b A op ‚` with differential b is the bar resolution of A (essentially defined by the multiplication in U and with augmentation givenby the counit ε ), and which is a left U -module by left multiplication on the firsttensor factor. Elements in tensor powers over A op will typically be denoted bythe comma notation, that is, an elementary tensor in U b A op q by p u , . . . , u q q .Applying for any q P N the isomorphism θ : Hom U p Bar q p U q , M q Ñ Hom A op p U b A op q , M q ,g ÞÑ g p , ¨q , p¨q f Ð [ f, (3.34) where we denoted the left U -action on M by juxtaposition, we obtain that the Ext -groups can equally be computed by the complex C ‚ p U, M q : “ Hom A op p U b A op ‚ , M q , where the cofaces and codegeneracies in degree q P N are explicitly given as p δ i f qp u , . . . , u q ` q “ $&% u f p u , . . . , u q ` q f p u , . . . , u i u i ` , . . . , u q ` q f p u , . . . , ε p u q ` q § u q q if i “ , if ď i ď q, if i “ q ` , p σ j f qp u , . . . , u q ´ q “ f p u , . . . , u j , , u j ` , . . . , u q ´ q for ď j ď q ´ . By means of the cocyclic operator in the form p τ f qp u , . . . , u q q “ γ ` pp u p q ¨ ¨ ¨ u q ´ p q u q q ➢ f qp´ , u p q , . . . , u q ´ p q q ˘ , (3.35) this becomes a cocyclic k -module in the sense of [Lo, §2.5].To see that this cocyclic operator can indeed be considered as originatingfrom a trace functor, we first have to lift it to Hom U p Bar ‚ p U q , M q by the isomor-phism (3.34) in order to place it in the realm of U -linear maps. Secondly, thetensor products appearing in Bar ‚ p U q are not the monoidal products in U - Mod ,which would be needed in (3.33); hence, another two k -module isomorphisms η and χ are required. More precisely: Theorem 3.12.
Let the left bialgebroid p U, A q be left Hopf and M a stable antiYetter-Drinfel’d contramodule. Then the diagram Hom A op p U b A op ‚ , M q τ / / θ ´ (cid:15) (cid:15) Hom A op p U b A op ‚ , M q O O θ Hom U p Bar ‚ p U q , M q η ´ (cid:15) (cid:15) Hom U p Bar ‚ p U q , M q O O χ Hom U p U b A U b A op ‚ , M q tr / / Hom U p U b A op ‚ b A U, M q (3.36) commutes in any degree.Proof. Explicitly, the two k -module isomorphisms η and χ are given as follows:define for any q P N η : Hom U p U b A p U b A op q q , M q Ñ Hom U p U b A op q ` , M q ,f ÞÑ p v, u , . . . , u q q ÞÑ f p v p q b A p v p q u , u , . . . , u q q ( , (3.37) g p v ` , v ´ u , u , . . . u q q Ð [ ` v b A p u , . . . , u q q ˘( Ð [ g, as well as χ : Hom U pp U b A op q q b A U, M q Ñ
Hom U p U b A op q ` , M q ,f ÞÑ p u , . . . , u q , v q ÞÑ f ` p u p q , . . . , u q p q q b A u p q ¨ ¨ ¨ u q p q v ˘( , (3.38) g p u ` , . . . , u q ` , u q ´ ¨ ¨ ¨ u ´ v q Ð [ pp u , . . . , u q q b A v ˘( Ð [ g, where on the left hand side p U b A op q q b A U is seen as left U -module via w ` p u , . . . , u q q b A v ˘ : “ p w p q u , u , . . . , u q q b A w p q v, which is well-defined if thetensor product over A relates the last tensor factor with the first by left mul-tiplication with the target map. It is a straightforward check that the maps η resp. χ are well-defined and that their asserted inverses invert them, indeed. ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 19
We can then compute, for any f P Hom A op p U b A op q , M q , p θ ˝ χ ˝ tr ˝ η ´ ˝ θ ´ ˝ f qp u , . . . , u q q (3.34) “ p χ ˝ tr ˝ η ´ ˝ θ ´ ˝ f qp , u , . . . , u q q (3.38) “ ` tr ˝ η ´ ˝ θ ´ ˝ f ˘` p , u p q , . . . , u q ´ p q q b A u p q ¨ ¨ ¨ u q ´ p q u q ˘ (3.33) “ γ ` ` η ´ ˝ θ ´ ˝ f ˘` u p q ¨ ¨ ¨ u q ´ p q u q b A p p¨q , u p q , . . . , u q ´ p q q ˘ ˘ (3.37) “ γ ` ` θ ´ ˝ f ˘` p u p q ¨ ¨ ¨ u q ´ p q u q q ` , p u p q ¨ ¨ ¨ u q ´ p q u q q ´p¨q , u p q , . . . , u q ´ p q ˘ ˘ (3.34) “ γ ` p u p q ¨ ¨ ¨ u q ´ p q u q q ` f ` p u p q ¨ ¨ ¨ u q ´ p q u q q ´p¨q , u p q , . . . , u q ´ p q ˘ ˘ (3.5) “ γ ` pp u p q ¨ ¨ ¨ u q ´ p q u q q ➢ f qp´ , u p q , . . . , u q ´ p q q ˘ (3.35) “ p τ f qp u , . . . , u q q , which means that diagram (3.36) commutes and hence implies that the co-cyclic operator (3.35) is induced by the trace functor from Theorem 3.10. (cid:3) Remark 3.13.
As already mentioned in Remark 3.7, if M is an aYD contra-module and, say, Q a Yetter-Drinfel’d module ( i.e. , an element in the centre of U - Mod if seen as a bimodule category over itself via the monoidal product),then hom r p Q, M q “
Hom A op p Q, M q is again an aYD contramodule. Hence, ifthis aYD contramodule is stable (which is not equivalent to M being stable),by once more exploiting the Hom-tensor adjunction ξ : Hom U p P b A N b A Q, M q »
Hom U p P b A N, hom r p Q, M qq , it is possible to construct a trace functor T : “ Hom U p´ b A Q, M q , with M and Q as above, and corresponding trace map tr N,P : Hom U p N b A P b A Q, M q » ÝÑ Hom U p P b A N b A Q, M q , for arbitrary N, P P U - Mod , which in the same way as in Theorem 3.12 leadsto the structure of a cyclic k -module on the complex computing Ext ‚ U p Q, M q if U Ž is A -flat. Since this produces even more unpleasant formulæ than thoseseen so far [Ko2, Prop. 3.5], we refrain from spelling out the details here.4. C ENTRES AND ANTI Y ETTER -D RINFEL ’ D MODULES
We now, in a sense, dualise most of the ideas and results of the precedingsection and dedicate our attention to the category of bialgebroid comodules.4.1.
Comodules over bialgebroids.
A left (and analogously right) comoduleover a left bialgebroid p U, A q is simply a comodule over the appurtenant A -coring, see [BrWi, §3]: that is, a left A -module M equipped with a coassociativeand counital coaction λ : M Ñ U Ž b A M, m ÞÑ m p´ q b A m p q . By defining ma : “ ε p m p´ q đ a q m p q “ ε p a § m p´ q q m p q for all a P A equips M with a right A -action as well, and with respect to the resulting A -bimodule structure thecoaction is A -bilinear in the sense of λ p amb q “ a Ż m p´ q đ b b A m p q , a, b P A. (4.1)On the other hand, by virtue of the bialgebroid properties, we have m p´ q b A m p q a “ m p´ q b A ε p a § m p´ q q m p q “ m p´ q Ž ε p a § m p´ q q b A m p q “ a § m p´ q b A m p q , so that the coaction effectively λ corestricts to a map λ : M Ñ U Ž ˆ A M, (4.2)where the subspace U ˆ A M Ă U b A M is defined as U Ž ˆ A M : “ ř i u i b m i P U Ž b A M | ř i a § u i b m i “ ř i u i b m i a, @ a P A ( , (4.3) see [Ta] for more information on the (lax monoidal) product ˆ A . The category U - Comod of left U -comodules is (strict) monoidal. Analogous considerations hold for the category
Comod - U of right U -comod-ules with respect to which we only explicitly state the A -bilinearity of a rightcoaction ρ : M Ñ M ˆ A Ż U , which reads ρ p amb q “ m p q b A a § m p q Ž b, a, b P A. (4.4)4.1.1. A functor between comodule categories.
The standard Hopf algebraicway of transforming a left U -comodule into a right one via the antipode or itspossible inverse does not apply here (as there is no antipode, not even if U is leftor right Hopf) but nevertheless if the left bialgebroid p U, A q is right Hopf and Ż U is A -projective, there is a strict monoidal functor U - Comod Ñ Comod - U ,as shown originally in [Ph] and later, somewhat enhanced, in [ChGaKo,Thm. 4.1.1]. More concretely, given a left U -comodule M , the map M Ñ M b A Ż U , m ÞÑ m r s b A m r s : “ ε p m p´ qr`s q m p q b A m p´ qr´s (4.5)is a right coaction. We refer to op. cit. for the not entirely obvious verificationthat if Ż U is A -projective, then this is a well-defined operation. We reserve thesquare bracket Sweedler notation m ÞÑ m r s b A m r s throughout the entire textfor this kind of right U -coaction only , starting from a left U -comodule.Vice versa, if the left bialgebroid p U, A q is left Hopf and U Ž is A -projective,then there is a strict monoidal functor Comod - U Ñ U - Comod but we are notgoing to need this fact in the sequel.
Remark 4.1.
In case of a Hopf algebra, as follows from Eqs. (A.24), the abovefunctor U - Comod Ñ Comod - U is precisely the one induced by the inverse S ´ of the antipode, whereas Comod - U Ñ U - Comod is induced by S . However, instriking contrast to the Hopf algebra case where essentially it does not matterwhether one uses S or S ´ for either of the functors, for a left bialgebroid thereis no way of obtaining a functor U - Comod Ñ Comod - U in case p U, A q is left Hopf instead of right
Hopf. This defect will become very visible when definingthe left and right internal Homs in U - Comod .For later and frequent use in technical computations, for a left U -comodule M over a left bialgebroid that is, in addition, right Hopf, one easily verifies by(4.1) and (A.15) that for any m P M the compatibility condition p m r sp´ q b A m r sp q q b A m r s “ p m p´ qr`s b A m p q q b A m p´ qr´s (4.6)holds between left U -coaction and induced right U -coaction (4.5) as tensorproducts in p U Ž b A M q đ b A Ż U , where p U b A M q đ “ U đ b A M . If the left bialgebroid p U, A q is both left and right Hopf, by (A.23) one even has m p´ q b A p m p qr s b A m p qr s q “ m r s´ b A p m r s b A m r s` q (4.7)as tensor products in U Ž b A § p M b A Ż U q , where § p M b A U q “ M b A § U .4.1.2. Anti Yetter-Drinfel’d modules.
In the previous sections, we added to ob-jects in the monoidal category U - Mod an additional structure (of right U -contraaction) compatible with the action, which led to the notion of aYD con-tramodules. Now, the monoidal category of interest is U - Comod and the addi-tional structure will be that of a right U -action. Note that the category Mod - U of right U -modules over a left bialgebroid is not monoidal; nonetheless, one stillhas a forgetful functor Mod - U Ñ A e - Mod , with respect to which we denote theinduced A -bimodule structure on a right U -module M by a § m đ b : “ mt p a q s p b q (4.8) ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 21 for m P M , a, b P A . Moreover, Mod - U can be seen as a right module categoryover U - Mod by means of
Mod - U ˆ U - Mod Ñ Mod - U, p M, N q ÞÑ M b A N, induced by the action on elements p m b A n q ➮ u : “ mu r`s b A u r´s n, (4.9)for m P M, n P N , and u P U .Analogous comments apply to the category of anti Yetter-Drinfel’d modules,which we are going to recall next [B ¸S, J ¸S] and which arise when asking forcompatibility between right U -action and left U -coaction. Definition 4.2. An anti Yetter-Drinfel’d (aYD) module M over a left Hopf alge-broid is simultaneously a left U -comodule and right U -module (with action de-noted by juxtaposition) such that both underlying A -bimodule structures from(4.8) and (4.1) coincide, and such that action followed by coaction results into p mu q p´ q b A p mu q p q “ u ´ m p´ q u `p q b A m p q u `p q , u P U, m P M. (4.10)An anti Yetter-Drinfel’d contramodule is called stable if m “ m p q m p´ q . The category U aYD U of aYD modules (resp. the category U saYD U of stableones) is not monoidal as already the category of right U -modules is not so.For later use, we want to state some alternative compatibility conditions inpresence of more structure: if the left bialgebroid p U, A q is not only left Hopfbut also right Hopf, the aYD condition (4.10) is equivalent to p mu r`s q p´ q u r´s b A p mu r`s q p q “ u ´ m p´ q b A m p q u ` , (4.11)as an easy check using (A.21) and (A.13) reveals. In this case, Eq. (4.10) canalso be reformulated with respect to the right U -coaction (4.5), that is, p mu q r s b A p mu q r s “ m r s u r`sp q b A u r´s m r s u r`sp q , (4.12)as one obtains (after a while) applying to (4.10), in this order, Eqs. (4.5), (A.20),(4.2), (A.15), (4.8), (A.21), (A.22), (A.3), and finally (A.10), along with the prop-erties of a bialgebroid counit. Moreover, if M is stable with respect to its leftcoaction, that is, m p q m p´ q “ m , then it is so with respect to its right coaction(4.5) as well, by which we mean m r s m r s “ m r sp q m r sp´ q m r s “ m p q m p´ qr`s m p´ qr´s “ ε p m p´ q q § m p q “ m, (4.13) as results from (4.6) and (A.18).4.2. Left and right closedness of U - Comod . As said before, for a monoidalcategory being closed or even biclosed essentially implies the existence of inter-nal Homs. In case of comodules, this leads to the notion of rational morphismsas introduced by Ulbrich [Ulb], see also [CaGu, ¸StOy] for more information onthe subject in the realm of Hopf algebras. We adapt the idea to the bialgebroidcase here.4.2.1.
Right internal Homs in U - Comod . Let p U, A q be a left algebroid, P be aright U -comodule with right coaction p ÞÑ p p q b A p p q and M a left U -comodulewith left coaction m ÞÑ m p´ q b A m p q , in the sense of §4.1. On Hom A op p P, M q ,consider the following customary A -bimodule structure p a Ż f đ b qp p q “ af p bp q , a, b P A, p P P. (4.14)Define then the map λ r : Hom A op p P, M q Ñ
Hom A op p P, U Ž b A M q ,f ÞÑ p ÞÑ f p p p q q p´ q p p q b A f p p p q q p q ( . (4.15)Now, the canonical map : U Ž b A Hom A op p P, M q Ñ
Hom A op p P, U Ž b A M q is aninjection if U Ž is A -projective. We can then make the following definition: Definition 4.3.
For a right U -comodule P and a left U -comodule M over a leftbialgebroid p U, A q with U Ž projective over A , the A -bimodule HOM r p P, M q : “ t f P Hom A op p P, M q | λ r f P im p qu is called the space of (right) rational morphisms from P to M .In other words, HOM r p P, M q consists of all f P Hom A op p P, M q for which thereexists an element f p´ q b A f p q P U Ž b A Hom A op p P, M q such that p λ r f qp p q “ f p´ q b A f p q p p q for all p P P . By injectivity of the canonical map , we may simply write λ r f “ f p´ q b A f p q for any (right) rational f . If U Ž is finitely generated projective over A , thenclearly all morphisms in Hom A op p P, M q are (right) rational. Lemma 4.4.
Let p U, A q be a left bialgebroid such that U Ž is projective over A .If P is a right U -comodule and M a left U -comodule, then HOM r p P, M q is a left U -comodule with coaction given by ´ ˝ λ r .Proof. We need to show that λ r f lands in U Ž b A HOM r p P, M q for any f P HOM r p P, M q and to check that λ r is counital and coassociative, and this willbe done along the same line of argumentation as in [Ulb, Lem 2.2]. Counitalityis straightforward using the properties of a bialgebroid counit along with the A -linearity (4.1) of the coaction on M . Furthermore, we have for any p P P ` p id b A λ r q λ r f ˘ p p q “ f p´ q b A p λ r f p q qp p q (4.15) “ f p´ q b A f p q p p p q q p´ q p p q b A f p q p p p q q p q (4.15) “ f p p p q q p´ q p p q b A f p p p q q p´ q p p q b A f p p p q q p q “ ` p ∆ b A id q λ r f ˘ p p q . The so-obtained equation not only shows coassociativity but also that λ r f P U b A HOM r p P, M q : the A -bimodule Hom r p N, M q can be seen as a pull-back for λ r and ; but tensoring with the flat A -module U Ž preserves finite limits andhence U Ž b A Hom r p N, M q is the pullback for id U b A λ r and id U b A . Then, from p id U b A λ r q λ r f “ p ∆ b A id U q λ r f one observes p id U b A λ r q λ r f P im p id U b A q andtherefore λ r f P U Ž b A HOM r p N, M q . (cid:3) For the sake of simplicity, by slight abuse of notation, we will denote thecoaction on
HOM r p N, M q by λ r instead of ´ ˝ λ r .Observe that with respect to the A -bimodule structure (4.14), we have by(4.1) and the the right U -comodule version of (4.3), p λ r p a Ż f đ b qqp p q “ a Ż f p p p q q p´ q p p q đ b b A f p p p q q p q , as one rightly would expect from the property (4.1) of a left U -coaction.Now, if the left bialgebroid p U, A q is right Hopf and Ż U projective over A ,using the monoidal functor U - Comod Ñ Comod - U mentioned in §4.1.1, wecan start from two left U -comodules N and M and transform the former into aright one as in Eq. (4.5). Repeating then an analogous discussion as above, wecan define the left U -comodule HOM r p N, M q : “ t f P Hom A op p N, M q | λ r f P im p qu , where p λ r f qp n q “ f ` ε p n p´ qr`s q n p q ˘ p´ q n p´ qr´s b A f ` ε p n p´ qr`s q n p q ˘ p q . (4.16) ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 23
However, instead of using the explicit expression (4.16) in later intricate com-putations, for better readability it is more convenient to consider the left U -comodule N as a right one as in Eq. (4.5) and to stick to the notation usedthere, that is, we will always write the left coaction (4.16) on HOM r p N, M q as p λ r f qp n q “ f p n r s q p´ q n r s b A f p n r s q p q . (4.17)Lemma 4.4 then becomes: Proposition 4.5.
Let p U, A q be a left bialgebroid such that U Ž and Ż U are pro-jective. If p U, A q in addition is right Hopf and both N, M are left U -comodules,then HOM r p N, M q is a left U -comodule with left coaction induced by Eq. (4.16) . Observe that the projectivity of U Ž is needed to have injective (and U - Comod abelian) whereas the one of Ż U to guarantee well-definedness ofEq. (4.5). We will refer to this situation henceforth as U being A -biprojective .4.2.2. Left internal Homs in U - Comod . Let p U, A q be a left bialgebroid and N, M P U - Comod . With respect to the canonical codiagonal left U -coaction M b A Ż U Ñ U Ž b A p M b A Ż U q , m b A u ÞÑ m p´ q u p q b A p m p q b A u p q q , on M b A Ż U , consider the space Hom U p N, M b A Ż U q of left U -colinear maps: foreach of its elements, we are going to deploy a sort of Sweedler notation withsummation understood, that is, for any g P Hom U p N, M b A Ż U q , write g p n q b A g p n q : “ g p n q , and equip Hom U p N, M b A Ż U q with an A -bimodule structure by means of p a § g Ž b qp n q : “ g p n q b A a § g p n q Ž b (4.18)for all a, b P A , see Eq. (A.1) for notation. If p U, A q in addition is left Hopf, define λ ℓ : Hom U p N, M b A Ż U q Ñ Hom U p N, U Ž b A p M b A Ż U qq ,g ÞÑ n ÞÑ g p n q ´ b A § p g p n q b A g p n q ` q ( , (4.19)which becomes well-defined if the first tensor factor relates to the third bymeans of multiplying with the target map from the right. Again, the canonicalmap : U Ž b A Hom U p N, M b A Ż U q Ñ Hom U p N, U Ž b A p M b A Ż U qq is injective if U Ž is A -projective, which allows us to define: Definition 4.6.
For two left U -comodules N, M over a left bialgebroid p U, A q which is left Hopf and with U Ž projective over A , the A -bimodule HOM ℓ p N, M q “ t g P Hom U p N, M b A Ż U q | λ ℓ g P im p qu is called the space of (left) rational morphisms from N to M .In other words, HOM ℓ p N, M q consist of all g P Hom U p N, M b A Ż U q for whichthere exists an element g p´ q b A g p q P U Ž b A Hom U p N, M b A Ż U q such that p λ ℓ g qp n q “ g p´ q b A g p q p n q for all n P N . Again, by injectivity of the canonical map , we may simply write λ ℓ g : “ g p´ q b A g p q for any (left) rational g . As before, if U Ž is finitely generated projective over A ,then all morphisms in Hom U p N, M b A Ż U q are (left) rational. Lemma 4.7.
Let p U, A q be a left Hopf algebroid over a left bialgebroid such that U Ž is projective, and N, M P U - Comod . Then
HOM ℓ p N, M q is a left U -comoduleas well, with coaction given by ´ ˝ λ ℓ . Proof.
Here we argue exactly as in §4.2.1 by which essentially the only aspectleft to show is coassociativity (counitality being obvious from Eq. (A.10)), thatis, for any g P HOM ℓ p N, M q , we have ` p id b A λ ℓ q λ ℓ g ˘ p n q “ g p´ q b A p λ ℓ g p q qp n q (4.19) “ g p´ q b A ` g q p n q ´ b A p g q p n q b A g q p n q ` q ˘ (4.19) “ g p n q ´ b A g p n q `´ b A p g p n q b A g p n q `` q (A.7) “ g p n q ´p q b A g p n q ´p q b A p g p n q b A g p n q ` q“ ` p ∆ b A id q λ ℓ g ˘ p n q , which again implies λ ℓ g P U Ž b A HOM ℓ p N, M q as in the proof of Lemma 4.4. (cid:3) As above, to lighten notation, we will write the coaction on
HOM ℓ p N, M q simply as λ ℓ instead of ´ ˝ λ ℓ . Remark 4.8.
The striking asymmetry in defining
HOM r and HOM ℓ and theircoactions is due to the fact ( cf. Remark 4.1) that for a left bialgebroid one hasa functor U - Comod Ñ Comod - U only in presence of a right Hopf structurebut not in presence of a left one (which notably complicates matters in all whatfollows). Even worse, and in strong contrast to the case of U - Mod in §3.1,where the left internal Homs did not require any
Hopf structure at all, fordefining a coaction on
HOM ℓ a left Hopf structure is sufficient but for its beingleft internal Homs, we additionally will have to assume a right Hopf structureas well, see the subsequent Lemma 4.10. Remark 4.9.
Nevertheless, in case p U, A q “ p
H, k q is a Hopf algebra over a field k with invertible antipode S , all these difficulties disappear and a short compu-tation reveals that HOM ℓ p N, M q and HOM r p N, M q reproduce the well-knowninternal Hom functors (see, e.g. , [CaGu, Prop. 1.2]) which use the antipode andits inverse, i.e. , in both cases the k -module Hom k p N, M q with left coactions p λ ℓ f qp n q “ S p n p´ q q f p n p q q p´ q b k f p n p q q p q , p λ r f qp n q “ f p n p q q p´ q S ´ p n p´ q q b k f p n p q q p q , respectively, for all n P N . Lemma 4.10.
Let p U, A q be a left bialgebroid which is biprojective over A . ( i ) If p U, A q is in addition right Hopf, then the category U - Comod of left U -comodules is right closed monoidal, i.e. , has right internal Hom functors. ( ii ) If the left bialgebroid p U, A q is simultaneously left and right Hopf, U - Comod is left closed monoidal as well, that is, has left internal Homfunctors. As a consequence, in this case U - Comod is biclosed monoidal, i.e. , has both left and right internal Hom functors.Proof.
Let
M, N, P P U - Comod be left U -comodules.(i): As the notation suggests, the right internal Homs are given by the HOM r p N, M q from Definition 4.3, where N is seen as a right U -comodule via(4.5), equipped with the left U -coaction (4.16) resp. (4.17), along with the ad-junction (iso)morphism ξ : Hom U p P b A N, M q Ñ
Hom U p P, HOM r p N, M qq ,f ÞÑ t p ÞÑ f p p b A ´qu , t ˜ f p p qp n q Ð [ p b A n u Ð [ ˜ f , (4.20)induced by the customary Hom-tensor adjunction. To see that ξf indeed landsin Hom U p P, HOM r p N, M qq , we have to show that p ξf qp p q P Hom A op p N, M q isa (right) rational morphism from N to M and that ξf is left U -colinear with ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 25 respect to the coactions of P and HOM r p N, M q in (4.17). Both statements areshown in a single computation only: one has ` λ r ξf p p q ˘ p n q (4.17) “ ` ξf p p q ˘ p n r s q p´ q n r s b A ` ξf p p q ˘ p n r s q p q “ f p p b A n r s q p´ q n r s b A f p p b A n r s q p q “ p p´ q n r sp´ q n r s b A f p p b A n r sp q q (4.6) “ p p´ q n p´ qr`s n p´ qr´s b A f p p b A n r s q (A.18) , (4.2) “ p p´ q b A ` ξf p p p q q ˘ p n q , where we used the U -colinearity of f in the third step; this not only showsthat λ r ξf p p q P U Ž b A Hom A op p N, M q and hence ξf p p q P HOM r p N, M q but simul-taneously that ξf is left U -colinear as well. That the asserted inverse indeedinverts ξ is obvious.(ii): Here, in turn, as the notation again suggests, the left internal Homsare given by the HOM ℓ from Definition 4.6 equipped with the left coaction inEq. (4.19), along with the adjunction morphism ζ : Hom U p N b A P, M q Ñ
Hom U p P, HOM ℓ p N, M qq ,f ÞÑ t p ÞÑ f p´ b A p r s q b A p r s u , (4.21)where p ÞÑ p r s b A p r s is the right U -coaction (4.5) on the left U -comodule P . Toverify that ζf indeed lands in Hom U p P, HOM ℓ p N, M qq , we need to check that ζf is U -colinear and also that ζf p p q P HOM ℓ p N, M q for any p P P , hence that ζf p p q is a (left) rational morphism from N to M and as such left U -colinear again. Asfor the first issue, we compute by means of the codiagonal coaction on M b A U : ` ` ζf p p q ˘ p n q ˘ p´ q b A ` ` ζf p p q ˘ p n q ˘ p q (4.21) “ f p n b A p r s q p´ q p r sp q b A ` f p n b A p r s q p q b A p r sp q ˘ “ n p´ q p r sp´ q p r sp q b A ` f p n p q b A p r sp q q b A p r sp q ˘ (4.6) “ n p´ q p p´ qr`s p p´ qr´sp q b A ` f p n p q b A p p q q b A p p´ qr´sp q ˘ (A.16) , (A.18) , (4.2) “ n p´ q b A ` ζf p p q ˘ p n p q q , where we used the colinearity of f in the second step. Secondly, ` λ ℓ ζf p p q ˘ p n q (4.19) “ ` ζf p p q ˘ p n q ´ b A ` ` ζf p p q ˘ p n q b A ` ζf p p q ˘ p n q ` ˘ (4.21) “ p r s´ b A ` f p n b A p r s q b A p r s` ˘ (4.7) “ p p´ q b A ` f p n b A p p qr s q b A p p qr s ˘ (4.21) “ p p´ q b A ` ζf p p p q q ˘ p n q , which not only shows that λ ℓ ζf p p q P U Ž b A Hom U p N, M b A Ż U q for any p P P and hence ζf p p q P HOM ℓ p N, M q but simultaneously also that ζf is U -colinearin the desired sense, that is, p ζf p p qq p´ q b A p ζf p p qq p q “ p p´ q b A ζf p p p q q .The inverse Hom U p P, HOM ℓ p N, M qq Ñ
Hom U p N b A P, M q of ζ will be given by p ζ ´ g qp n b A p q “ p id b ε q g p p qp n q “ g p p q p n q ε ` g p p q p n q ˘ . In turn, to show that ζ ´ g is in fact a left U -colinear map from N b A P to M ,observe first that g P Hom U p P, HOM ℓ p N, M qq implies two identities, namely p p´ q b A ` g p p p q q p n q b A g p p p q q p n q ˘ “ g p p q p n q ´ b A ` g p p q p n q b A g p p q p n q ` ˘ ,n p´ q b A ` g p p q p n p q q b A g p p q p n p q q ˘ “ g p p q p n q p´ q g p p q p n q p q b A ` g p p q p n q p q b A g p p q p n q p q ˘ , for any n P N and p P P . With the help of these two equations, we proceed by n p´ q p p´ q b A p ζ ´ g qp n p q b A p p q q“ n p´ q p p´ q b A g p p p q q p n p q q ε ` g p p p q q p n p q q ˘ “ g p p p q q p n q p´ q g p p p q q p n q p q p p´ q b A g p p p q q p n q p q ε ` g p p p q q p n q p q ˘ (4.2) “ g p p p q q p n q p´ q g p p p q q p n q p p´ q b A g p p p q q p n q p q “ g p p q p n q p´ q g p p q p n q ` g p p q p n q ´ b A g p p q p n q p q (A.9) “ g p p q p n q p´ q đ ε ` g p p q p n q ˘ b A g p p q p n q p q (4.1) “ p ζ ´ g qp n b A p q p´ q b A p ζ ´ g qp n b A p q p q , and therefore ζ ´ g P Hom U p N b A P, M q as claimed. Verifying that ζ ´ effective-ly inverts ζ is shown by similar computations and is therefore skipped. The laststatement is an obvious consequence of (i) and the statements just verified. (cid:3) Remark 4.11.
One might wonder whether one could not, in the spirit of Re-mark 3.3 for the case of U - Mod , simply transport the left U -coaction (4.19) to Hom A p N, M q by means of the k -linear isomorphism ν : Hom U p N, M b A U q Ñ Hom A p N, M q , f ÞÑ p id b A ε q f (with inverse g ÞÑ t n ÞÑ g p n p q q r s b A g p n p q q r s n p´ q u ), so as to work with theseemingly easier Hom A p N, M q instead of Hom U p N, M b A U q . However, this willnot work since ν is not a morphism of A -bimodules when considering the A -bimodule structure (4.18). Apparently, and in clear contrast to what was saidin Remark 3.3, the left internal Homs HOM ℓ p N, M q “
Hom U p N, M b A U q cannotbe simplified, not even in presence of more structure, cf. also Remark 4.8. Notation 4.12.
Again, as the left and right internal Homs are quite differentand it sometimes is convenient to remember the explicit U -colinearity or A -linearity in question, we shall not always use the sort of concealing notation HOM r and HOM ℓ but often write Hom A op and Hom U p´ , ´ b A U q even if theinternal Homs with their left U -comodule structure are meant.4.3. U - Comod as a bimodule category.
Similar to §3.2, the internal Homsallow to define the structure of a bimodule category on the category of left U -comodules resp. its opposite with the help of the adjoint actions, in the senseexplained in Remark 2.5. More precisely, we have: Lemma 4.13.
Let p U, A q be a left bialgebroid with U biprojective over A . ( i ) If p U, A q is in addition right Hopf, then the operation U - Comod ˆ U - Comod op Ñ U - Comod op , p N, M q ÞÑ N ➤ M : “ HOM r p N, M q (4.22) defines on U - Comod op the structure of a left module category over themonoidal category U - Comod . ( ii ) Likewise, if p U, A q is both left and right Hopf, then the operation U - Comod op ˆ U - Comod Ñ U - Comod op , p M, N q ÞÑ N ➤ M : “ HOM ℓ p N, M q (4.23) defines on U - Comod op the structure of a right module category over themonoidal category U - Comod . ( iii ) Hence, if the left bialgebroid p U, A q is simultaneously left and right Hopf,then the left and the right action from Eqs. (4.22) and (4.23) define on U - Comod op the structure of a bimodule category over the monoidal cat-egory U - Comod . ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 27 ( iv ) The operation (4.22) restricts to a left action UU YD ˆ U aYD U Ñ U aYD U if HOM r p N, M q is seen as a right U -module by means of the right U -action on Hom A op p N, M q defined by p f ➢ u qp n q : “ f p u p q n q u p q (4.24) for N P U - Mod and M P Mod - U . Hence, U aYD U is a left module cate-gory over the monoidal category UU YD .Proof. Let
M, N, P P U - Comod .(i): As for the left action, we have to prove that P ➤ p N ➤ M q » p P b A N q ➤ M as left U -comodules, which amounts to show that the k -module isomorphism φ P,N,M : HOM r p P b A N, M q Ñ
HOM r p P, HOM r p N, M qq (4.25)given by the customary Hom-tensor adjunction is an isomorphism of left U -comodules. Indeed, to start with, if P b A N is a left U -comodule with codiagonalcoaction, it is an easy check that its induced right coaction (4.5) is given by p p b A n q r s b A p p b A n q r s : “ p p r s b A n r s q b A n r s p r s . (4.26)Abbreviating φ “ φ P,N,M , one then has for any p P P , n P N : f p´ q b A p φf r s qp p qp n q “ f p´ q b A f p q p p b A n q“ f p p r s b A n r s q p´ q n r s p r s b A f p p r s b A n r s q p q “ p φf qp p r s qp n r s q p´ q n r s p r s b A p φf qp p r s qp n r s q p q “ p φf qp p r s q p´ q p r s b A p φf qp p r s q p q p n q“ ` p φf q p´ q b A p φf q p q p p q ˘ p n q , hence p id b A φ q λ r f “ λ r p φf q , as desired.In order to effectively obtain a left module category in the sense of Definition2.1, we still have to verify the pentagon resp. triangle axiom (2.2) resp. (2.3),which, however, follow easily from the properties of the standard Hom-tensoradjunction, U - Comod being strict.(ii): The second part is slightly more laborious as the standard Hom-tensor adjunction is not the map that will induce the comodule isomorphism M ➤ p N b A P q » p M ➤ N q ➤ P in question. Observe first that M ➤ p N b A P q “ HOM ℓ p N b A P, M q “
Hom U p N b A P, M b A Ż U q on the level of k -modules, along with p M ➤ N q ➤ P “ HOM ℓ p P, HOM ℓ p N, M qq “
Hom U p P, Hom U p N, M b A Ż U q b A Ż U q , where Hom U p N, M b A Ż U q is seen as an A -bimodule as in (4.18) and as a left U -comodule as in (4.19). We then claim that the map ψ M,N,P : Hom U p N b A P, M b A Ż U q Ñ Hom U p P, Hom U p N, M b A Ż U q b A Ż U q ,f ÞÑ p ÞÑ f p´ b A p r s q b A f p´ b A p r s q p q p r s b A f p´ b A p r s q p q ( , (4.27) where we wrote f p n b A p q “ : f p n b A p q b A f p n b A p q , is an isomorphism ofleft U -comodules. Using the same kind of component-wise notation twice forelements in Hom U p P, Hom U p N, M b A Ż U q b A Ż U q , and abbreviating ψ “ ψ M,N,P ,this can be rewritten as p ψf qp p qp n q “ p ψf q p p q p n q b A p ψf q p p q p n q b A p ψf q p p q“ f p n b A p r s q b A f p n b A p r s q p q p r s b A f p n b A p r s q p q , for all n P N and p P P .We have to show four things now: that p ψf qp p q P HOM ℓ p N, M q b A Ż U for any p P P and any f P HOM ℓ p N b A P, M q , that ψf is U -colinear in the given sense, that ψ is a morphism of left U -comodules, and finally that it is bijective. As forthe first issue, observe that from the left U -colinearity f p n b A p q p´ q f p n b A p q p q b A f p n b A p q p q b A f p n b A p q p q “ n p´ q p p´ q b A f p n p q b A p p q q b A f p n p q b A p p q q (4.28) of an f P Hom U p N b A P, M b A U q follows with Eqs. (4.6), (A.18), and (4.2) that f p n b A p r s q p´ q f p n b A p r s q p q p r s b A f p n b A p r s q p q b A f p n b A p r s q p q p r s b A f p n b A p r s q p q “ n p´ q b A f p n p q b A p r s q b A f p n p q b A p r s q p q p r s b A f p n p q b A p r s q p q , (4.29) and therefore directly λ ℓ ´ p ψf q p p q p n q b A p ψf q p p q p n q ¯ b A p ψf q p p q“ p ψf q p p q p n q p´ q p ψf q p p q p n q p q b p ψf q p p q p n q p q b A p ψf q p p q p n q p q b A p ψf q p p q (4.27) “ f p n b A p r s q p´ q f p n b A p r s q p q p r s b A f p n b A p r s q p q b A f p n b A p r s q p q p r s b A f p n b A p r s q p q (4.29) , (4.27) “ n p´ q b A p ψf qp p qp n p q q , hence p ψf qp p q P HOM ℓ p N, M q b A Ż U for any p P P , as claimed. The secondissue above, i.e. , that ψf is U -colinear, is left to the reader. More interesting, p ψf q p´ q b A p ψf q p q p p qp n q (4.19) “ p ψf q p p q ´ b A p ψf q p p q p n q b A p ψf q p p q p n q b A p ψf q p p q ` (4.27) “ f p n b A p r s q p q´ b A f p n b A p r s q b A f p n b A p r s q p q p r s b A f p n b A p r s q p q` (A.6) “ f p n b A p r s q ´ b A f p n b A p r s q b A f p n b A p r s q `p q p r s b A f p n b A p r s q `p q (4.19) “ f p´ q b A f q p n b A p r s q b A f q p n b A p r s q p q p r s b A f q p n b A p r s q p q (4.27) “ f p´ q b A p ψf p q qp p qp n q , hence ψ is in fact a left U -comodule map, which proves the third issue men-tioned above. Finally, we claim that ψ is bijective, the inverse being given by ψ ´ : Hom U p P, Hom U p N, M b A U q b A U q Ñ Hom U p N b A P, M b A U q ,g ÞÑ n b A p ÞÑ p id b A ε b A id q g p p qp n q ( , or, explicitly, p ψ ´ g qp n b A p q : “ g p p q p n q b A ε ` g p p q p n q ˘ Ż g p p q . (4.30)While ψ ´ ˝ ψ “ id follows directly from the counitality of the coproduct, that ψ ˝ ψ ´ yields the identity is slightly more laborious: the left U -colinearity of g P Hom U p P, Hom U p N, M b A U q b A U q explicitly reads p p´ q b A g p p p q qp n q “ g p p q p´ q b A g p p q p q p n q“ g p p q p´ q g p p q p q b A g p p q q p n q b A g p p q q p n q b A g p p q p q “ g p p q p n q ´ g p p q p q b A g p p q p n q b A g p p q p n q ` b A g p p q p q , and therefore with Eqs. (4.6) and (A.18) b A g p p q p n q b A g p p q p n q b A g p p q“ p r sp´ q p r s b A g p p r sp q q p n q b A g p p r sp q q p n q b A g p p r sp q q“ g p p r s q p n q ´ g p p r s q p q p r s b A g p p r s q p n q b A g p p r s q p n q ` b A g p p r s q p q . (4.31) ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 29
With this, p ψψ ´ g qp p qp n q (4.27) “ p ψ ´ g q p n b A p r s q b A p ψ ´ g q p n b A p r s q p q p r s b A p ψ ´ g q p n b A p r s q p q (4.30) “ g p p r s q p n q b A ε ` g p p r s q p n q ˘ Ż g p p r s q p q p r s b A g p p r s q p q (A.9) “ g p p r s q p n q b A g p p r s q p n q ` g p p r s q p n q ´ g p p r s q p q p r s b A g p p r s q p q (4.31) “ g p p q p n q b A g p p q p n q b A g p p q “ g p p qp n q , as desired.To finalise the proof that we effectively obtain a right module category, weneed to verify the analogous right versions of the pentagon and triangle axioms(2.2) resp. (2.3), which again is lengthy but entirely straightforward to writedown, using Eq. (4.26) and the fact that U - Comod is a strict monoidal category.(iii): In this part, we claim that for any
M, N, P P U - Comod , there is anisomorphism of left U -comodules ϑ P,M,N : p P ➤ M q ➤ N » ÝÑ P ➤ p M ➤ N q the middle associativity constraint required in Definition 2.2 subject to the twopentagon axioms (2.6) and (2.7), which amounts to a left U -comodule isomor-phism HOM ℓ p N, HOM r p P, M qq »
HOM r p P, HOM ℓ p N, M qq . To start with, definethe k -module isomorphism ϑ P,M,N : Hom U p N, Hom A op p P, M q b A U q Ñ Hom A op p P, Hom U p N, M b A U qq given by p ϑ P,M,N f qp p qp n q “ p ϑ P,M,N f qp p q p n q b A p ϑ P,M,N f qp p q p n q“ f p n qp p r s q b A p r s f p n q . (4.32) Its inverse will be defined as p ϑ ´ P,M,N g q p n qp p q b A p ϑ ´ P,M,N g q p n q “ g p p r s q p n q ε p p r sr`s q b A p r sr´s g p p r s q p n q , (4.33) the well-definedness of which over the Sweedler presentation of the right Hopfstructure ( i.e. , that is does not depend on the choice of a representative for theformal expression p r s b A p r sr`s b A p r sr´s ) is not immediately visible to thenaked eye but follows from a detailed consideration not unlikely the proof ofthe well-definedness of the coaction (4.5) in [ChGaKo, Thm. 4.1.1] from theproperty ε p u đ a q “ ε p a § u q of a bialgebroid counit, along with Eqs. (A.20), (4.4),and the right A -module structure on Hom A op p P, M q as in (4.14), which impliesthat the tensor product Hom A op p P, M q b A U is to be understood with respect tothe ideal generated by g p a p¨q q b u ´ g p¨q b a Ż u for a P A and g P Hom A op p P, M q .That the two given maps in (4.32) and (4.33) are mutual inverses followsmore or less immediately from Eqs. (A.14), (A.15), and (A.18).Next, let us verify that ϑ is in fact a map (and hence an isomorphism) of left U -comodules. Abbreviating again ϑ “ ϑ P,M,N for better readability, one has byEqs. (4.19), (4.17), and (A.5) for all p P P and n P N : p ϑf q p´ q b A p ϑf q p q p p qp n q“ p ϑf qp p r s q p´ q p r s b A p ϑf qp p r s q p q p n q“ ` p ϑf qp p r s q p n q ˘ ´ p r s b A p ϑf qp p r s q p n q b A ` p ϑf qp p r s q p n q ˘ ` “ f p n q ´ p r sp q´ p r sp q b A f p n qp p r s q b A p r sp q` f p n q ` “ f p n q ´ b A f p n qp p r s q b A p r s f p n q ` “ f p´ q b A f q p n qp p r s q b A p r s f q p n q“ f p´ q b A p ϑf p q qp p qp n q , for any f P HOM ℓ p N, HOM r p P, M qq Ă
Hom U p N, Hom A op p P, M q b A U q and there-fore λ ℓ ˝ ϑ “ p id U b A ϑ q ˝ λ r , as claimed.To conclude the proof of this part, we still have to check the two pentagon di-agrams (2.6) and (2.7). In full detail, we are going to verify only the second onewhich is more challenging due to the notably different complexity of the maps φ and ψ from (4.25) and (4.27), respectively. Nevertheless, let us briefly indicatehow to also show the first diagram (2.6). So, let P, Q, M, N P U - Comod . Usingthe codiagonal right coaction on P b A Q given by p b A q ÞÑ p p r s b A q r s qb A q r s p r s induced by (4.5), it is not too difficult to see that going from the top in diagram(2.6) clockwise to the bottom right results in a map Hom U p N, Hom A op p P b A Q, M q b A U q Ñ Hom A op p P, Hom A op p Q, Hom U p N, M b A U qqq given by g ÞÑ p ÞÑ g p n qp p r s b A q r s q b A q r s p r s g p n q ( , and without too much effort one verifies that this is the same as going coun-terclockwise the other path. As for the second pentagon diagram (2.7), in thiscontext it explicitly turns into the following one: Hom A op p N, Hom U p P b A Q, M b A U qq Hom A op p N,ψ
M,P,Q q t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ O O ϑ N,M,P b AQ Hom A op p N, Hom U p Q, Hom U p P, M b A U q b A U qq O O ϑ N, Hom U p P,M b AU q ,Q Hom U p P b A Q, Hom A op p N, M q b A U q ψ Hom A op p N,M q ,P,Q (cid:15) (cid:15) Hom U p Q, Hom A op p N, Hom U p P, M b A U qq b A U q j j Hom U p Q,ϑ
N,M,P b A U q ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ Hom U p Q, Hom U p P, Hom A op p N, M q b A U q b A U q (4.34) For any f P Hom U p P b A Q, Hom A op p N, M q b A Ż U q , we will show that going thetwo steps along the top part of this figure amounts to the same as going alongthe three steps along the bottom. Indeed, for any n P N, q P Q , and p P P ,abbreviating ψ “ ψ M,P,Q and analogously for ϑ , we have ` Hom A op p N, ψ q ˝ ϑ ˝ f ˘ p n qp q qp p q (4.27) “ ` ϑ ˝ f ˘ p n q p p b A q r s q b A ` ϑ ˝ f ˘ p n q p p b A q r s q p q q r s b A ` ϑ ˝ f ˘ p n q p p b A q r s q p q (4.32) “ f p p b A q r s qp n r s q b A n r s f p p b A q r s q p q q r s b A n r s f p p b A q r s q p q (4.27) “ p ψ ˝ f q p q q p p qp n r s q b A n r s p ψ ˝ f q p q q p p q b A n r s p ψ ˝ f q p q q (4.32) “ p Hom U p Q, ϑ b A U q ˝ ψ ˝ f q p q qp n r s qp p q b A n r s p Hom U p Q, ϑ b A U q ˝ ψ ˝ f q p q q (4.32) “ p ϑ ˝ Hom U p Q, ϑ b A U q ˝ ψ ˝ f qp n qp q qp p q , that is, the diagram (4.34) commutes indeed. This ends the proof of this part.(iv): Finally, let N P UU YD and M P U aYD U . We have to show that in thiscase HOM r p N, M q is an aYD module as well with respect to the right action ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 31 (4.24) and the left coaction (4.17). Indeed, for any f P HOM r p N, M q , one has λ r p f ➢ u qp n q (4.17) “ p f ➢ u qp n r s q p´ q n r s b A p f ➢ u qp n r s q p q (4.24) “ ` f p u p q n r s q u p q ˘ p´ q n r s b A ` f p u p q n r s q u p q ˘ p q (4.10) , (A.6) “ u ´ f p u `p q n r s q p´ q u `p q n r s b A f p u `p q n r s q p q u `p q (A.14) “ u ´ f ` p u `p q n q r s ˘ p´ q p u `p q n q r s u `p q b A f ` p u `p q n q r s ˘ p q u `p q (4.17) “ u ´ f p´ q u `p q b A f p q p u `p q n q u `p q (4.24) “ u ´ f p´ q u `p q b A p f p q ➢ u `p q qp n q for n P N , u P U , where in the third step we used the fact that M P U aYD U andthat N P UU YD in the fourth (see [Sch2, Def. 4.2]). This concludes the proof. (cid:3) The bimodule centre in the bialgebroid comodule category.
Wecan now, thanks to Lemma 4.13, examine the centre of the bimodule cate-gory U - Comod op with respect to its adjoint actions given by all pairs p M, τ q ofobjects M P U - Comod op for which there is a family of isomorphisms τ N : N ➤ M » ÝÑ N ➤ M of left U -comodules natural in N P U - Comod . With respect to this centre andits full subcategory Z U - Comod p U - Comod op q which we, once more, recall to bedefined by the condition that the identity map id M P Hom U p M, M q is mappedto itself by the chain of isomorphisms (2.10), we can state the following result: Theorem 4.14.
Let an A -biprojective left bialgebroid p U, A q be both left andright Hopf. ( i ) Then any anti Yetter-Drinfel’d module M induces a central structure τ N : HOM ℓ p N, M q Ñ
HOM r p N, M q , explicitly given on the level of k -modules by Hom U p N, M b A U q Ñ Hom A op p N, M q ,f ÞÑ n ÞÑ f p n q p q f p n q p´ q f p n q ( , ` g p n r s q r s b A g p n r s q r s ˘ ➮ n r s Ð [ n ( Ð [ g, (4.35) where the right U -action ➮ is the one defined in (4.9) . ( ii ) Vice versa, for a pair p M, τ q in the centre Z U - Comod p U - Comod op q , theright U -action on M defined by means of mu : “ p τ U f m qp u q , @ u P U, (4.36) where f m P Hom U p U, M b A U q is defined by f m p u q “ m r s b A m r s u for any m P M , induces the structure of an anti Yetter-Drinfel’d module on M . ( iii ) Both preceding parts together induce an equivalence U aYD U » Z U - Comod p U - Comod op q of categories. ( iv ) Imposing stability on anti Yetter-Drinfel’d modules implies U saYD U » Z U - Comod p U - Comod op q as a categorical equivalence. Remark 4.15.
Using that any f P Hom U p N, M b A U q is colinear, we can rewrite τ N f p n q “ ` f p n p q q ε p f p n p q qq ˘ n p´ q (4.37)for the central structure instead of (4.35), which is sometimes more convenientto work with. Proof of Theorem 4.14. (i): We leave it to the reader to check that the two givenmaps in (4.35) are well-defined (checking that τ ´ N g is so is somewhat laboriousbut very similar to the computations that follow below). That they are mutualinverses is in one direction almost immediate, whereas τ ´ N p τ N f qp n q (4.37) “ ` p τ N f qp n r s q r s b A p τ N f qp n r s q r s ˘ ➮ n r s (4.37) “ ´` f p n r sp q q ε p f p n r sp q qq Ż n r sp´ q ˘ r s b A ` f p n r sp q q ε p f p n r sp q qq Ż n r sp´ q ˘ r s ¯ ➮ n r s (4.6) , (4.12) , (A.16) “ ´ f p n p q q r s n p´ qr`sp q b A n p´ qr´sp q tε p f p n p q qq f p n p q q r s n p´ qr`sp q ¯ ➮ n p´ qr´sp q (4.9) , (A.13) , (A.14) “ f p n p q q r s b A tε p f p n p q qq f p n p q q r s n p´ q “ f p n q p qr s b A tε p f p n q p q q f p n q p qr s f p n q p´ q f p n q p q (4.7) , (A.9) “ f p n q r s b A tε p f p n q p q q sε p f p n q r s q f p n q p q “ f p n q b A f p n q “ f p n q for any n P N proves the other direction, using left U -colinearity of f in thefifth step and the aYD condition (4.12) in the third, plus the fact that all four A -actions on U as defined in (A.1) commute.Next, let us check that τ N is natural in N , that is, for any left U -comodulemorphism σ : N Ñ N we want to see that τ N ˝ HOM r p σ, M q “ HOM ℓ p σ, M q˝ τ N .Indeed, by left U -colinearity of σ , τ N p f ˝ σ qp n q “ ` f p σ p n p q qq ε p f p σ p n p q qqq ˘ n p´ q “ ` f p σ p n q p q qq ε p f p σ p n q p q qqq ˘ σ p n q p´ q “ p τ N f qp σ p n qq , (4.38)for any f P HOM ℓ p N , M q , hence the claim.Furthermore, we need to prove that τ N is itself a left U -comodule morphism,that is, λ r τ N “ p id b τ N q λ ℓ . As a matter of fact, one has for any f P HOM ℓ p N, M q : p λ r τ N f qp n q (4.17) “ p τ N f qp n r s q p´ q n r s b A p τ N f qp n r s q p q (4.37) “ ´ f p n r sp q q ε ` f p n r sp q q ˘ n r sp´ q ¯ p´ q n r s b A ´ f p n r sp q q ε ` f p n r sp q q ˘ n r sp´ q ¯ p q (4.6) , (4.10) , (4.1) “ n p´ qr`s´ f p n p q q p´ q ´ ε ` f p n p q q ˘ Ż n p´ qr`s`p q n p´ qr´s ¯ b A f p n p q q p q n p´ qr`s`p q (A.21) , (A.13) “ n p´ q´ f p n p q q p´ q đ ε ` f p n p q q ˘ b A f p n p q q p q n p´ q` “ f p n q p q´ f p n q p´ q´ f p n q p´ q đ ε ` f p n q p q ˘ b A f p n q p q f p n q p´ q` f p n q p q` (A.5) , (A.11) “ f p n q ´ b A f p n q p q f p n q p´ q f p n q ` (A.6) , (4.2) “ f p n q p q´ b A ` f p n q p q ε p f p n q p q` q ˘ f p n q p´ q f p n q p q “ f p n p q q ´ b A ` f p n p q q ε p f p n p q q ` q ˘ n p´ q (4.19) “ f p´ q b A ` f q p n p q q ε p f q p n p q qq ˘ n p´ q (4.37) “ f p´ q b A τ N f p q p n q“ p id b A τ N q λ ℓ f p n q , as desired, where we used the left U -colinearity of f in the fifth and in theeighth step again, along with the aYD condition (4.10) in the third. ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 33
We still need to prove the hexagon axiom (2.8). For better readability, let uswrite down what this means on the level of k -modules: Hom A op p P, Hom U p N, M b A U qq o o ϑ P,N,M
Hom A op p P,τ N q (cid:15) (cid:15) Hom U p N, Hom A op p P, M q b A U q O O Hom U p N,τ P b A U q Hom A op p P, Hom A op p N, M qq O O φ P,N,M
Hom U p N, Hom U p P, M b A U q b A U q O O ψ M,P,N
Hom A op p P b A N, M q o o τ P b N Hom U p P b A N, M b A U q (4.39) Verifying that the above diagram (4.39) commutes with respect to the centralstructure (4.35) is essentially straightforward: by abuse of notation, let usabbreviate ϑ “ ϑ P,N,M and likewise for φ and ψ , along with τ N “ Hom A op p P, τ N q ,and τ P “ Hom U p N, τ P b A U q . We then have for f P Hom U p P b A N, M b A U q : p φ ´ ˝ τ N ˝ ϑ ˝ τ P ˝ ψ ˝ f qp p b A n q (4.25) “ p τ N ˝ ϑ ˝ τ P ˝ ψ ˝ f qp p qp n q (4.35) “ p ϑ ˝ τ P ˝ ψ ˝ f q p p qp n q p q p ϑ ˝ τ P ˝ ψ ˝ f q p p qp n q p´ q p ϑ ˝ τ P ˝ ψ ˝ f q p p qp n q (4.32) “ p τ P ˝ ψ ˝ f q p n qp p r s q p q p τ P ˝ ψ ˝ f q p n qp p r s q p´ q p r s p τ P ˝ ψ ˝ f q p n q (4.35) “ ` p ψ ˝ f q p n q p p r s q p q p ψ ˝ f q p n q p p r s q p´ q p ψ ˝ f q p n q p p r s q ˘ p q ` p ψ ˝ f q p n q p p r s q p q p ψ ˝ f q p n q p p r s q p´ q p ψ ˝ f q p n q p p r s q ˘ p´ q p r s p ψ ˝ f q p n q (4.10) , (A.4) “ p ψ ˝ f q p n q p p r s q p q p ψ ˝ f q p n q p p r s q p´ q p ψ ˝ f q p n q p p r s q p´ q p ψ ˝ f q p n q p p r s q p r s p ψ ˝ f q p n q (4.27) “ f p p r s b A n r s q p q f p p r s b A n r s q p´ q f p p r s b A n r s q p´ q f p p r s b A n r s q p q n r s p r s f p p r s b A n r s q p q (4.28) , (4.2) “ f p p r sp q b A n r sp q q p q f p p r sp q b A n r sp q q p´ q p r sp´ q n r sp´ q n r s p r s f p p r sp q b A n r sp q q (4.6) “ f p p b A n q p q f p p b A n q p´ q f p p b A n q (4.35) “ τ P b AN f p p b A n q for any p b A n P P b A N , which proves the commutativity of diagram (4.39)and concludes the proof of this part.(ii): Let p M, τ q P Z U - Comod p U - Comod op q be an object in the bimodule centre.For any m P M , define f m P Hom U p U, M b A Ż U q by f m p u q : “ m r s b A m r s u, (4.40)where the right coaction on the left comodule M is (as always) the inducedone (4.5). The left U -colinearity of f m is a simple check. However, applying for-mally (4.19), we see that λ ℓ f m p u q “ u ´ m r s´ b A p m r s b A m r s` u ` q “ u ´ m p´ q b A f m p q p u ` q with the help of (4.7), and hence f m is not an element in HOM ℓ p U, M q so that we can not apply the central structure τ U : HOM ℓ p U, M q Ñ
HOM r p U, M q from (4.37) to it. By a standard argument, as in [Sh, p. 479], this problemis circumvented as follows: in general, if N were a finitely A -generated co-module, then obviously HOM ℓ p N, M q “
Hom U p N, M b A Ż U q as comodules. Bywhat is sometimes called the Fundamental Theorem of Comodules [D ˘aN ˘aRa,Thm. 2.1.7], every element of a comodule over a k -coalgebra (where k is afield) is contained in a finite-dimensional subcomodule. This result can beextended to bialgebroids (or general A -corings for that matter) as soon as U Ž is A -projective, which follows from [KaoGoLo, Cor. 2.7 & Prop. 2.8]. Hence, Hom U p N, M b A Ż U q “ lim ÐÝ HOM ℓ p N ι , M q , where the N ι are finitely generated left U -subcomodules, and similarly Hom A op p N, M q “ lim ÐÝ HOM r p N ι , M q . This induces a map
Hom U p N, M b A Ż U q Ñ Hom A op p N, M q with the same prop-erties as τ N and will therefore, by slight abuse of notation, be denoted by thesame symbol. In case N “ U , this is the map used to define the right U -action(4.36) on M , that is, mu : “ p τ U f m qp u q , u P U, m P M. (4.41)Let us show that this, in fact, defines an action with respect to which the left U -comodule M becomes an aYD module: more precisely, we will first prove thatthe what-is-going-to-be action (4.41) is compatible with the left U -coaction on M in the sense of the aYD condition (4.10), or, equivalently, (4.11). To this end,note that considering U as a left U -comodule via the coproduct, the correspond-ing right coaction obtained from Eq. (4.5) reads u r s b A u r s : “ u r`s b A u r´s . (4.42)Moreover, if τ is a central structure, by definition τ U is a left U -comodule iso-morphism Hom U p U, M b A Ż U q Ñ Hom A op p U, M q , and therefore satisfies p τ U f qp u r`s q p´ q u r´s b A p τ U f qp u r`s q p q “ f p´ q b A p τ U f p q qp u q (4.43)with respect to the left U -coaction (4.19) on Hom U p U, M b A Ż U q . Applying thisto f m from (4.40) and considering that p f m q p´ q b A p f m q p q p u q “ u ´ m p´ q b A p m p qr s b A m p qr s u ` q , (4.44)as can be derived from (4.19) and (4.7), we have for the right hand side in (4.43) p τ U f m qp u r`s q p´ q u r´s b A p τ U f m qp u r`s q p q “ p mu r`s q p´ q u r´s b A p mu r`s q p q , (4.45)whereas for the left hand side in (4.43): p f m q p´ q b A τ U p f m q p q p u q “ u ´ m p´ q b A m p qr sp q m p qr sp´ q m p qr s u ` “ u ´ m p´ q b A m p q u ` , with the help of Eq. (4.6). Hence, (4.43) implies (4.11) and therefore the aYDcondition (4.10), as desired.To conclude, let us show that Eq. (4.36) resp. (4.41) effectively defines a right U -action, i.e. , that for any u, v P U p mu q v “ τ U f τ U f m p u q p v q “ p τ U f m qp uv q “ m p uv q (4.46)holds. To this end, first note that the right U -coaction induced by (4.5) on theleft U -comodule Hom U p U, M b A U q explicitly reads for the element f m as follows: p f m q r s p u q b A p f m q r s “ p m r s b A m r sp q u p q q b A m r sp q u p q , (4.47)as seen directly by Eqs. (4.44), (4.5), (4.7), and (A.10), whereas in the samespirit Eq. (4.44) also implies p τ U f m qp u q p´ q b A p τ U f m qp u q p q “ f p´ q u p q b A p τ U f p q qp u p q q . by Eqs. (A.13) and (A.15), and therefrom the expression for the right coaction p τ U f m q r s p u q b A p τ U f m qp u q r s “ τ U p f m q r s p u r`s q b A u r´s p f m q r s “ τ U ` m r s b A m r sp qp¨q ˘ p u r`sp q q b A u r´s m r sp q u r`sp q , (4.48) ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 35 on the element τ U f m in the sense of (4.5) again, where Eqs. (4.47) and (A.15)were used. Proving the associativity (4.46) now essentially hinges on the factthat if τ is a central structure, it makes the diagram (4.39) (resp. (2.8)) com-mute and is natural: the multiplication µ : U đ b A Ż U Ñ U , u b A v ÞÑ uv bythe bialgebroid properties is a morphism in U - Comod , and hence by (4.38) wehave p τ U b AU p f ˝ µ qqp u b A v q “ p τ U f qp uv q for any f P Hom U p U, M b A Ż U q , whichwe are going to exploit in the penultimate step of the following computation: p mu q v (4.41) “ τ U f τ U f m p u q p v q (4.41) “ τ U ` τ U f m p u q r s b A τ U f m p u q r sp¨q ˘ p v q (4.48) “ τ U ` τ U ` m r s b A m r sp qp¨q ˘ p u r`sp q q b A u r´s m r sp q u r`sp q ˘ p v q (4.49) “ τ U ` p Hom U p U, τ U b A U q ˝ ψ ˝ f m q p u r`s qb A u r´s p Hom U p U, τ U b A U q ˝ ψ ˝ f m q ˘ p v q (4.32) “ τ U ` p ϑ ˝ Hom U p U, τ U b A U q ˝ ψ ˝ f m qp u q ˘ p v q (4.25) “ p φ ´ ˝ Hom A op p U, τ U q ˝ ϑ ˝ Hom U p U, τ U b A U q ˝ ψ ˝ f m ˝ µ qp u b A v q (4.39) “ p τ U b AU p f m ˝ µ qqp u b A v q (4.38) “ p τ U f m qp µ p u b A v qq (4.41) “ m p uv q , as claimed. Here, in the fourth step we additionally needed the fact that p ψf m q p v q p u q b A p ψf m q p v q p u q b A p ψf m q p v q“ f m p u b A v r s q b A f m p u b A v r s q p q v r s b A f m p u b A v r s q p q “ m r s b A m r sp q u p q v r`sp q v r´s b A m r sp q u p q v r`sp q “ m r s b A m r sp q u p q b A m r sp q u p q , (4.49)as results from Eqs. (4.42) and (A.13).The unitality of the so-defined action once again follows from the naturality(4.38): for N “ A , the source map s : A Ñ U is a morphism in U - Comod aswell and therefore τ A p f ˝ s qp a q “ p τ U f qp s p a qq for f P HOM ℓ p U, M q . Hence, m U “ p τ U f m qp s p A qq “ p τ A p f m ˝ s qqp A q “ m A “ m, taking into consideration that τ A : HOM ℓ p A, M q » M Ñ HOM r p A, M q » M isthe identity map along with the unitality of the source map, plus the fact that f m ˝ s under the isomorphism Hom U p A, M b A Ż U q » Hom A op p A, M q » M becomesthe map L m : a ÞÑ ma .(iii): Here, we need to verify two things: first, that any morphism M Ñ ˜ M ofaYD modules induces a morphism p M, τ q Ñ p ˜ M , ˜ τ q between the correspondingobjects in the bimodule centre (and vice versa); second, that the two proceduresof how to obtain a central structure from a right U -action and a right U -actionfrom a central structure are mutually inverse.As for the first issue, if ϕ : M Ñ ˜ M is a morphism of aYD modules, we haveto show that for any N P U - Comod the diagram
Hom U p N, M b A U q τ N / / Hom U p N,ϕ b A U q (cid:15) (cid:15) Hom A op p N, M q Hom A op p N,ϕ q (cid:15) (cid:15) Hom U p N, ˜ M b A U q ˜ τ N / / Hom A op p N, ˜ M q (4.50) commutes. Indeed, let n P N and f P Hom U p N, M b A U q . Then ϕ ` τ N f p n q ˘ “ ϕ ` f p n q p q f p n q p´ q f p n q ˘ “ p ϕ ˝ f qp n q p q p ϕ ˝ f qp n q p´ q f p n q“ ˜ τ N ` p ϕ ˝ f q b A f ˘ p n q“ ˜ τ N ` Hom U p N, ϕ b A U q ˝ f ˘ p n q since ϕ is in particular a morphism of right U -modules and left U -comodules.Vice versa, let ϕ : p M, τ q Ñ p ˜ M , ˜ τ q be a morphism of objects in the centre Z U - Comod p U - Comod op q ; this, in particular, means that ϕ is a left U -comodulemap and that the diagram (4.50) commutes. In order to define a morphism ofaYD modules, it suffices to show that ϕ is a right U -module morphism as well.To start with, observe that if ϕ is a left U -comodule map, one has for m P M ϕ p m r s q b A m r s “ ϕ p ε p m p´ qr`s q m p q q b A m p´ qr´s “ ε p m p´ qr`s q ϕ p m p q q b A m p´ qr´s “ ε p ϕ p m q p´ qr`s q ϕ p m q p q b A ϕ p m q p´ qr´s “ ϕ p m q r s b A ϕ p m q r s , that is, it is also a right U -comodule morphism with respect to the right coac-tion (4.5). Applying then diagram (4.50) to the case N “ U , we obtain ϕ p mu q “ ϕ ` τ U f m p u q ˘ “ ˜ τ U ` Hom U p U, ϕ b A U q ˝ f m ˘ p u q“ ˜ τ U ` ϕ p m r s q b A m r sp¨q ˘ p u q“ ˜ τ U ` ϕ p m q r s b A ϕ p m q r sp¨q ˘ p u q“ ˜ τ U f ϕ p m q p u q“ ϕ p m q u for any u P U . Hence, ϕ is a also a morphism of right U -modules.Second, and finally, we have to show that obtaining a central structure froma right U -action and a right U -action from a central structure are mutuallyinverse procedures. Indeed, if a right U -action m b u ÞÑ mu on M P U - Comod isgiven and a corresponding central structure τ is defined by means of Eq. (4.35),which in turn defines a right U -action as in Eq. (4.36), we have p τ U f m qp u q “ m r sp q m r sp´ q m r s u “ mu, with the help of Eq. (4.6) and (A.18), which is just the right U -action that westarted with. Vice versa, given a central structure τ that defines a right U -action as in (4.36) that, in turn, defines a central structure as in (4.35), in asimilar way reproduces the central structure τ we started with. To see this,assume that τ is the central structure defined by the action (4.36); we willshow now that τ “ τ . Indeed, for g P Hom U p N, M b A Ż U q , one has, using (4.35) τ N g p n q “ ` g p n p q q ε p g p n p q qq ˘ n p´ q “ τ U ` f g p n p q q ε p g p n p q qq ˘ p n p´ q q , (4.51)where f m P Hom U p U, M b A Ż U q was, as before, the element defined in Eq. (4.40).Before we continue, note that any left U -coaction λ : N Ñ U Ž b A N on N is a morphism in U - Comod if U Ž b A N is seen as a free left U -comodule, i.e. ,ignoring the coaction on N and only taking the coproduct on U into account.From the naturality of a central structure we obtain τ N p ˜ g ˝ λ q “ τ U b AN p ˜ g q ˝ λ (4.52)for any ˜ g P Hom U p U Ž b A N, M b A Ż U q , along with τ U b AN ˜ g p u b A n q “ τ U f ˜ g p u p q b A n q ε p ˜ g p u p q b A n qq p u p q q . (4.53) ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 37
Set then ˜ g : “ ε b g . Combining (4.52) and (4.53) and comparing the outcomewith Eq. (4.51), we obtain: τ N g p n q “ τ N pp ε b g q ˝ λ qp n q“ τ U b AN p ε b g qp n p´ q b A n p q q“ τ U ` f g p n p q q ε p g p n p q qq ˘ p n p´ q q “ τ N g p n q , as desired.(iv): We only have to show that the chain of isomorphisms in (2.10) mapsthe identity map id M to itself when using the central structure τ N from part (i),as well as, vice versa, that m p q m p´ q “ m for the action from (4.41) if p M, τ q P Z U - Comod p U - Comod op q . The first issue will follow directly from the unitality(4.55) of the trace (4.54) discussed below, and is therefore postponed.Vice versa, note that if id M P Hom U p M, M q is mapped to itself by means of(2.10), then, by virtue of the adjunctions (4.20) and (4.21), τ M p id M b A q “ id M . We can then argue as above Eq. (4.52): the left coaction λ : M Ñ U Ž b A M isa morphism in U - Comod if U Ž b A M is seen as a free left comodule. Define ˜ g P Hom U p U Ž b A M, M b A Ż U q by ˜ g “ ε b A id M b A and apply (4.52) and (4.53)to it, observing that p ˜ g ˝ λ qp m q “ m b A and ˜ g p u b A n q ε p ˜ g p u b A n qq “ ε p u q m ;that is, for any m P M , we have m “ τ M p id M b A qp m q “ τ M p ˜ g ˝ λ qp m q (4.52) “ τ U b AM p ˜ g qp m p´ q b A m p q q (4.53) “ τ U f ε p m p´ q q m p q p m p´ q q “ τ U f m p q p m p´ q q (4.36) “ m p q m p´ q , using f ε p u q m p¨q “ f m p p¨q Ž ε p u qq , which results from (4.40) with (4.4). Hence,the aYD module M defined in (ii) is stable if p M, τ q P Z U - Comod p U - Comod op q ,which concludes the proof. (cid:3) Remark 4.16.
Observe that comparing the situation for U - Mod and aYD con-tramodules resp. U - Comod and aYD modules is less symmetric than expected:whereas U - Mod was biclosed in presence of one Hopf structure only (or actu-ally none), this is apparently not the case for U - Comod , where left and rightHopf structures are needed. On the other hand, for defining a central structurefor U - Comod the stability of an aYD module in Theorem 4.14 was not needed,whereas for U - Mod in Theorem 3.8 the stability of aYD contramodules im-mediately came into play not only when asking the central structure τ to beinvertible (which could be weakened) but already when asking the hexagonaxiom (3.31) to be fulfilled.4.5. Traces on U - Comod . In a spirit analogous to what was done in §3.5, wecan now state a dual version of Theorem 3.10:
Theorem 4.17.
Let an A -biprojective left bialgebroid p U, A q be both left andright Hopf. If M is a stable anti Yetter-Drinfel’d module, then T : “ Hom U p´ , M q yields a trace functor U - Comod Ñ k - Mod , that is, we have a family of isomor-phisms tr N,P : Hom U p N b A P, M q » ÝÑ Hom U p P b A N, M q , functorial in N, P P U - Comod , given by p tr N,P f qp p b A n q : “ f p n b A p r s q p r s , (4.54) for n P N and p P P . Proof.
Analogously to the proof of Theorem 3.10, by Theorem 4.14, Lemma4.10 and Lemma 4.13, it is enough to show that the diagram
Hom U p N b A P, M q ζ / / (cid:15) (cid:15) Hom U p P, HOM ℓ p N, M qq Hom U p P,τ N q (cid:15) (cid:15) Hom U p P b A N, M q Hom U p P, HOM r p N, M qq , ξ ´ o o commutes, that is, that tr N,P fits into it at the dotted arrow. Indeed, for f P Hom U p N b A P, M q , we have p ξ ´ ˝ Hom U p P, τ N q ˝ ζ ˝ f qp p b A n q (4.20) “ p Hom U p P, τ N q ˝ ζ ˝ f qp p qp n q (4.35) “ ` p ζ ˝ f qp p q p n q ˘ p q ` p ζ ˝ f qp p q p n q ˘ p´ q ` p ζ ˝ f qp p q p n q ˘ (4.21) “ f p n b A p r s q p q f p n b A p r s q p´ q p r s “ f p n b A p r s q p r s (4.54) “ p tr N,P f qp p b A n q , where we used the stability of M in the penultimate step. Unitality of thistrace functor, that is, tr A,P “ id , is then immediate: since for N “ A the left U -colinearity of an element f P Hom U p P, M q also implies right U -colinearity inthe sense of f p p r s q b A p r s “ f p p q r s b A f p p q r s , we have p tr A,P f qp p q “ f p p r s q p r s “ f p p q r s f p p q r s (4.13) “ f p p q . (4.55)All remaining properties in Definition 2.6 of a trace functor now follow fromthose of the central structure τ ; for example, Eq. (2.11) can be seen directlyfrom the hexagon axiom (4.39). (cid:3) Remark 4.18.
Dually to Remark 3.13, this trace functor can analogously beenhanced by introducing more coefficients: if M is an aYD module and Q aYetter-Drinfel’d module, then HOM r p Q, M q is again an aYD module as provenin the fourth part of Lemma 4.13. Hence, if this aYD module is stable (which isnot equivalent to M being stable), by ξ : Hom U p P b A N b A Q, M q »
Hom U p P b A N, HOM r p Q, M qq , it is possible to construct a trace functor T : “ Hom U p´ b A Q, M q , with M and Q as above, and corresponding trace map tr N,P : Hom U p N b A P b A Q, M q » ÝÑ Hom U p P b A N b A Q, M q , for arbitrary N, P P U - Comod .A PPENDIX
A. L
EFT AND RIGHT H OPF ALGEBROIDS
A.1.
Bialgebroids.
A left bialgebroid p U, A, ∆ , ε, s, t q , introduced first in [Ta]and rediscovered a couple of times, is a generalisation of a k -bialgebra to abialgebra object over a noncommutative base ring A , consisting of a compatiblealgebra and coalgebra structure over A e resp. over A . In particular, there is aring homomorphism resp. antihomomorphism s, t : A Ñ U ( source resp. target )that induce four commuting A -module structures on U , denoted by a § b Ż u Ž c đ d : “ t p c q s p b q us p d q t p a q (A.1)for u P U, a, b, c, d P A , which we abbreviate by §Ż U Žđ , depending on the relevantaction(s) in question. Moreover, apart from the multiplication, U also carries acomultiplication ∆ : U Ñ U ˆ A U Ă U Ž b A Ż U , u ÞÑ u p q b A u p q and a counit ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 39 ε : U Ñ A subject to certain identities that at some points differ from those inthe bialgebra case, see [BSz, Ta] or elsewhere. To the A e -ring U ˆ A U : “ ř i u i b v i P U Ž b A Ż U | ř i a § u i b v i “ ř i u i b v i đ a, @ a P A ( we usually refer to as Sweedler-Takeuchi product .A.2.
Left and right Hopf algebroids.
Generalising Hopf algebras ( i.e. , bial-gebras with an antipode) to noncommutative base rings is a much more chal-lenging task. If one wants to avoid the abundance of structure maps that ac-company the notion of a full
Hopf algebroid as in [BSz], that is, two bialgebroidstructures (meaning two coproducts, two counits, eight A -actions on the totalspace, etc.) and an antipode map as sort of intertwiner between these, one re-nounces on the idea of an antipode and rather requires a certain Hopf-Galoismap to be invertible [Sch2], which even leads to a more general concept thanthat of full Hopf algebroids. More precisely, if p U, A q is a left bialgebroid, con-sider the maps α ℓ : § U b A op U Ž Ñ U Ž b A Ż U , u b A op v ÞÑ u p q b A u p q v,α r : U đ b A Ż U Ñ U Ž b A Ż U , u b A v ÞÑ u p q v b A u p q , (A.2)of left U -modules. Then the left bialgebroid p U, A q is called a left Hopf algebroid or simply left Hopf if α ℓ is invertible and right Hopf algebroid or right Hopf ifthis is the case for α r . Adopting kind of Sweedler notations u ` b A op u ´ : “ α ´ ℓ p u b A q u r`s b A u r´s : “ α ´ r p b A u q , with, as usual, summation understood, one proves that for a left Hopf algebroid u ` b A op u ´ P U ˆ A op U, (A.3) u `p q b A u `p q u ´ “ u b A P U Ž b A Ż U, (A.4) u p q` b A op u p q´ u p q “ u b A op P § U b A op U Ž , (A.5) u `p q b A u `p q b A op u ´ “ u p q b A u p q` b A op u p q´ , (A.6) u ` b A op u ´p q b A u ´p q “ u `` b A op u ´ b A u `´ , (A.7) p uv q ` b A op p uv q ´ “ u ` v ` b A op v ´ u ´ , (A.8) u ` u ´ “ s p ε p u qq , (A.9) ε p u ´ q § u ` “ u, (A.10) p s p a q t p b qq ` b A op p s p a q t p b qq ´ “ s p a q b A op s p b q (A.11)are true [Sch2], where in (A.3) we mean the Takeuchi-Sweedler product U ˆ A op U : “ ř i u i b v i P § U b A op U Ž | ř i u i Ž a b v i “ ř i u i b a § v i , @ a P A ( , and if the left bialgebroid p U, A q is right Hopf, in the same spirit one verifies u r`s b A u r´s P U ˆ A U, (A.12) u r`sp q u r´s b A u r`sp q “ b A u P U Ž b A Ż U, (A.13) u p qr´s u p q b A u p qr`s “ b A u P U đ b A Ż U , (A.14) u r`sp q b A u r´s b A u r`sp q “ u p qr`s b A u p qr´s b A u p q , (A.15) u r`sr`s b A u r`sr´s b A u r´s “ u r`s b A u r´sp q b A u r´sp q , (A.16) p uv q r`s b A p uv q r´s “ u r`s v r`s b A v r´s u r´s , (A.17) u r`s u r´s “ t p ε p u qq , (A.18) u r`s đ ε p u r´s q “ u, (A.19) p s p a q t p b qq r`s b A p s p a q t p b qq r´s “ t p b q b A t p a q , (A.20) see [BSz, Prop. 4.2], where in (A.12) we denoted U ˆ A U : “ ř i u i b v i P U đ b A Ż U | ř i a Ż u i b v i “ ř i u i b v i đ a, @ a P A ( . If the left bialgebroid p U, A q is simultaneously left and right Hopf, the compat-ibility between the two (inverses of the) Hopf-Galois maps comes out as: u `r`s b A op u ´ b A u `r´s “ u r`s` b A op u r`s´ b A u r´s , (A.21) u ` b A op u ´r`s b A u ´r´s “ u p q` b A op u p q´ b A u p q , (A.22) u r`s b A u r´s ` b A op u r´s´ “ u p qr`s b A u p qr´s b A op u p q , (A.23)see [ChGaKo, Lem. 2.3.4]. A simultaneous left and right Hopf structure on aleft bialgebroid still does not imply the existence of an antipode required inthe definition of a full Hopf algebroid. For example, the universal envelopingalgebra VL of a Lie-Rinehart algebra p A, L q constitutes a left bialgebroid thatis both left and right Hopf but still does not admit an antipode in general.However, in case p U, A q “ p
H, k q is actually a Hopf algebra over a field k , theinvertibility of α ℓ guarantees the existence of the antipode S and the invert-ibility of α r the existence of S ´ . More precisely, in these cases we had h ` b k h ´ “ h p q b S p h p q q h r`s b k h r´s “ h p q b S ´ p h p q q , (A.24)for any h P H . R EFERENCES[BePeW] G. Benkart, M. Pereira, and S. Witherspoon,
Yetter-Drinfeld modules under cocycletwists , J. Algebra (2010), no. 11, 2990–3006.[BBrWi] G. Böhm, T. Brzezi ´nski, and R. Wisbauer,
Monads and comonads on module categories ,J. Algebra (2009), no. 5, 1719–1747.[B ¸S] G. Böhm and D. ¸Stefan, (Co)cyclic (co)homology of bialgebroids: an approach via (co)monads ,Comm. Math. Phys. (2008), no. 1, 239–286.[BSz] G. Böhm and K. Szlachányi,
Hopf algebroids with bijective antipodes: axioms, integrals, andduals , J. Algebra (2004), no. 2, 708–750.[Br] T. Brzezi ´nski,
Hopf-cyclic homology with contramodule coefficients , Quantum groups and non-commutative spaces, Aspects Math., E41, Vieweg + Teubner, Wiesbaden, 2011, pp. 1–8.[BrWi] T. Brzezi ´nski and R. Wisbauer,
Corings and comodules , London Mathematical Society Lec-ture Note Series, vol. 309, Cambridge University Press, Cambridge, 2003.[BuCaP] D. Bulacu, S. Caenepeel, and F. Panaite,
Yetter-Drinfeld categories for quasi-Hopf alge-bras , Comm. Algebra (2006), no. 1, 1–35.[CaGu] S. Caenepeel and T. Guédénon, On the cohomology of relative Hopf modules , Commun.Algebra (2005), no. 11, 4011–4034.[ChGaKo] S. Chemla, F. Gavarini, and N. Kowalzig, Duality features of left Hopf algebroids , Al-gebr. Represent. Theory (2016), no. 4, 913–941.[Co] A. Connes, Cohomologie cyclique et foncteurs
Ext n , C. R. Acad. Sci. Paris Sér. I Math. (1983), no. 23, 953–958.[D˘aN˘aRa] S. D˘asc˘alescu, C. N˘ast˘asescu, and ¸S. Raianu, Hopf algebras , Monographs and Text-books in Pure and Applied Mathematics, vol. 235, Marcel Dekker, Inc., New York, 2001.[Dr] V. Drinfel’d,
Quantum groups , Proceedings of the International Congress of Mathematicians,Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 798–820.[EiMo] S. Eilenberg and J. Moore,
Foundations of relative homological algebra , Mem. Amer. Math.Soc. No. (1965).[EtGeNi] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor categories , vol. 205, Amer. Math.Soc., Providence, RI, 2015.[EtNiOs] P. Etingof, D. Nikshych, and V. Ostrik,
Fusion categories and homotopy theory , QuantumTopol. (2010), no. 3, 209–273.[GeNaNi] S. Gelaki, D. Naidu, and D. Nikshych, Centers of graded fusion categories , Algebra &Number Theory (2009), no. 8, 959–990.[HKhRS] P. Hajac, M. Khalkhali, B. Rangipour, and Y. Sommerhäuser, Stable anti-Yetter-Drinfeldmodules , C. R. Math. Acad. Sci. Paris (2004), no. 8, 587–590.
ENTRES, TRACE FUNCTORS, AND CYCLIC COHOMOLOGY 41 [J ¸S] P. Jara and D. ¸Stefan,
Hopf-cyclic homology and relative cyclic homology of Hopf-Galois ex-tensions , Proc. London Math. Soc. (3) (2006), no. 1, 138–174.[Ka1] D. Kaledin, Cyclic homology with coefficients. , Algebra, arithmetic, and geometry. In honorof Y. I. Manin on the occasion of his 70th birthday. Vol. II, Boston, MA: Birkhäuser, 2009,pp. 23–47.[Ka2] D. Kaledin,
Trace theories and localization , Stacks and categories in geometry, topology, andalgebra, Contemp. Math., vol. 643, Amer. Math. Soc., Providence, RI, 2015, pp. 227–262.[KaoGoLo] L. El Kaoutit, J. Gómez-Torrecillas, and F. J. Lobillo,
Semisimple corings , AlgebraColloq. (2004), no. 4, 427–442.[Kay] A. Kaygun, Bialgebra cyclic homology with coefficients , K -Theory (2005), no. 2, 151–194.[KobSh] I. Kobyzev and I. Shapiro, A categorical approach to cyclic cohomology of quasi-Hopfalgebras and Hopf algebroids , Appl. Categ. Structures (2019), no. 1, 85–109.[Ko1] N. Kowalzig, When
Ext is a Batalin-Vilkovisky algebra , J. Noncommut. Geom. A noncommutative calculus on the cyclic dual of
Ext , preprint (2019), arXiv:1912.08145 , to appear in Ann. Sc. Norm. Super. Pisa, Cl. Sci.[Lo] J.-L. Loday,
Cyclic homology , second ed., Grundlehren Math. Wiss., vol. 301, Springer-Verlag,Berlin, 1998.[PSt] F. Panaite and M. Staic,
Generalized (anti) Yetter-Drinfeld modules as components of abraided T -category , Israel J. Math. (2007), 349–365.[Ph] H. H. Phùng, Tannaka-Krein duality for Hopf algebroids , Israel J. Math. (2008), 193–225.[Po] L. Positselski,
Contramodules , preprint (2015), arXiv:1503.00991 .[RT] D. Radford and J. Towber,
Yetter-Drinfel’d categories associated to an arbitrary bialgebra , J.Pure Appl. Algebra (1993), no. 3, 259–279.[Sch1] P. Schauenburg, Hopf bimodules over Hopf-Galois-extensions, Miyashita-Ulbrich actions,and monoidal center constructions , Comm. Algebra (1996), no. 1, 143–163.[Sch2] P. Schauenburg, Duals and doubles of quantum groupoids ( ˆ R -Hopf algebras) , New trendsin Hopf algebra theory (La Falda, 1999), Contemp. Math., vol. 267, Amer. Math. Soc., Provi-dence, RI, 2000, pp. 273–299.[Sh] I. Shapiro, On the anti-Yetter-Drinfeld module-contramodule correspondence , J. Noncommut.Geom. (2019), no. 2, 473–497.[ ¸StOy] D. ¸Stefan and F. Van Oystaeyen, The Wedderburn-Malcev theorem for comodule algebras ,Commun. Algebra (1999), no. 8, 3569–3581.[Ta] M. Takeuchi, Groups of algebras over A b A , J. Math. Soc. Japan (1977), no. 3, 459–492.[Ts] B. Tsygan, Cyclic homology , Cyclic homology in non-commutative geometry, EncyclopaediaMath. Sci., vol. 121, Springer, Berlin, 2004, pp. 73–113.[Ulb] K.-H. Ulbrich,
Smash products and comodules of linear maps , Tsukuba J. Math. (1990),no. 2, 371–378.[Ye] D. Yetter, Quantum groups and representations of monoidal categories , Math. Proc. Cam-bridge Philos. Soc. (1990), no. 2, 261–290.D
IPARTIMENTO DI M ATEMATICA , U
NIVERSITÀ DI N APOLI F EDERICO
II, V IA C INTIA , 80126N
APOLI , I
TALY
Email address ::