A noncommutative calculus on the cyclic dual of Ext
aa r X i v : . [ m a t h . K T ] D ec A NONCOMMUTATIVE CALCULUS ON THE CYCLIC DUAL OF
Ext
NIELS KOWALZIGA
BSTRACT . We show that if the cochain complex computing
Ext groups (in the categoryof modules over Hopf algebroids) admits a cocyclic structure, then the noncommutativeCartan calculus structure on
Tor over
Ext dualises in a cyclic sense to a calculus on
Coext over
Cotor . More precisely, the cyclic duals of the chain resp. cochain spacescomputing the two classical derived functors lead to complexes that compute the more ex-otic ones, giving a cyclic opposite module over an operad with multiplication that induceoperations such as a Lie derivative, a cap product (or contraction), and a (cyclic) differen-tial, along with higher homotopy operators defining a noncommutative Cartan calculus upto homotopy. In particular, this allows to recover the classical Cartan calculus from dif-ferential geometry or the Chevalley-Eilenberg calculus for Lie(-Rinehart) algebras withoutany finiteness conditions or the use of topological tensor products. C ONTENTS
Introduction 11. Contramodules over bialgebroids 52. The derived functors
Cotor and
Coext
63. The complex computing
Ext as a cocyclic module 94. The complex computing
Coext as a cyclic module 125. The noncommutative calculus structure on
Coext over
Cotor
NTRODUCTION
Aims and objectives.
Higher structures on cohomology or homology, such as brack-ets, products, and differentials, are typically only part of a richer structure on pairs ofcohomology and homology groups, where one acts on the other in various ways, as agraded module or graded Lie algebra module, for example. Usually, these operationscan already be observed on a cochain resp. chain level, often encoded in the action ofan operad on a module or opposite module, fulfilling certain axioms only up to ho-motopy and accordingly involving more or less explicit higher homotopy operators aswell. The probably most basic example here is given by the pair of multivector fieldsand differential forms, seen as cohomology and homology groups with zero differen-tials, where the former acts on the latter by contraction and Lie derivative, and both areequipped with differentials that give, depending on the precise context, rise to de Rhamor Lie algebra cohomology resp. homology. Algebraically, this idea was formalised in
Mathematics Subject Classification.
Key words and phrases.
Noncommutative calculi, cyclic homology, Hopf algebroids, operads, contramod-ules, Lie-Rinehart algebras. [GeDaTs, NeTs, TaTs, Ts1] by the notion of noncommutative differential calculus , whichalso runs under the name
Batalin-Vilkoviski˘ı (BV) module , and has been an active researchtopic since [DoTaTs2, DoTaTs3, La, Ts2, ArKe, He, Tam], finding its possibly highestdegree of abstraction so far in the definition of the
Kontsevich-Soibelman operad (as in-troduced in [KS1, KS2], see also [DoTaTs1, §4]) that essentially encodes calculi. Laterwork, for example in [KoKr], resulted in a homotopy calculus structure on the cochainand chain complexes that compute
Ext groups and
Tor groups over quite general rings,more precisely over so-called Hopf algebroids, which enlarged the Hochschild case from[NeTs] and also allowed for more general coefficients, from which one can deduce, as anexample, that the Hochschild cohomology of twisted Calabi-Yau algebras forms a Batalin-Vilkoviski˘ı (BV) algebra . The approach in [KoKr] was formalised in an operadic languagein [Ko1] by determining the minimal ingredients required in order to obtain a (homotopy)noncommutative calculus.The main objective in the article at hand is to investigate what happens to a (homo-topy) noncommutative calculus when applying to it what is called cyclic duality , whichtransforms cocyclic objects in cyclic ones and vice versa, see [Co1] and §A.2. More pre-cisely, by using the operadic formalism developed in [Ko1] and the cyclic structure on thecochain complex computing
Ext groups obtained in [Ko3], we use cyclic duality both onthe cochain resp. chain complexes that eventually yield the noncommutative calculus on
Tor over
Ext in [KoKr] to a obtain a homotopy noncommutative calculus on the chainresp. cochain complexes that leads to a natural calculus of
Coext over
Cotor when de-scending to (co)homology. This approach turns out to be versatile enough to include theclassical Cartan calculus in differential geometry as an example.The pattern behind our construction is quite striking: starting from a cyclic unital op-posite module M over an operad O with multiplication (the chain space that computes Tor over the cochain space that computes
Ext ), one obtains a noncommutative calculuson p H ‚ p O q , H ‚ p M qq . Adding the assumption that the operad O is cyclic, one can passto the cyclic duals both for O ‚ and M ‚ with the result that now rôles are exchanged and O ‚ is a cyclic unital opposite module over M ‚ (the chain space that computes Coext overthe cochain space that computes
Cotor ), which means that now p H ‚ p M q , H ‚ p O qq yieldsa noncommutative calculus. As a side remark, both H ‚ p O q and H ‚ p M q even becomeBatalin-Vilkoviski˘ı algebras here, that is, a Gerstenhaber algebra whose bracket measuresthe failure of the cyclic differential to be a (graded) derivation of the cup product. We won-der whether one could observe this sort of dual behaviour on a much more general levelonly involving, say, two cyclic operads with a mutual action, but were at present not ableto make this idea more precise.0.2. Main results.
In §3.1, we improve earlier work [Ko3, Prop. 4.8] by observing thateven in the non-finite case the category U aYD contra ´ U of anti Yetter-Drinfel’d (aYD) con-tramodules over a left bialgebroid p U, A q , while not being monoidal, is a module categoryover UU YD , the monoidal category of Yetter-Drinfel’d (YD) modules over U , which re-lies essentially on the fact that this is already the case for right U -contramodules over themonoidal category of left U -comodules. Expressed in simpler terms, in Proposition 3.2 weprove that if M P U aYD contra ´ U and N P UU YD , then Hom A op p M, N q is an aYD con-tramodule over U again. This observation allows to generalise [Ko3, Cor. 4.13] to moregeneral coefficients in Proposition 3.4, see the main text for all details: Proposition 0.1. If M is an aYD contramodule and N a YD module over a left bialgebroid p U, A q such that Hom A op p N, M q is stable, then (when U Ž is A -flat) the cochain complexcomputing Ext ‚ U p N, M q is a cyclic k -module. In a standard way, as briefly explained in Eq. (A.2), this yields a degree ´ differentialon the cochain complex that induces a differential B : Ext ‚ U p N, M q Ñ
Ext ‚´ U p N, M q on cohomology. NONCOMMUTATIVE CALCULUS ON THE CYCLIC DUAL OF
Ext One of the main feature of Connes’ cyclic category is its self-duality, as mentioned in§A.2. This allows to construct, as in Eq. (A.3), from any cocyclic k -module a cyclic k -module essentially by treating cofaces as degeneracies and codegeneracies as faces, exceptfor one of them the definition of which involves the cocyclic operator (they are infinitelymany ways for such a procedure due to the infinite number of autoequivalences of thecyclic category). While it is known that in case the Hochschild cochain complex is cyclic(which, as a side remark, is usually not the case) the result is trivial, in general for Hopfalgebroids the situation is richer. In Lemma 4.2 and Theorem 4.4, we show: Theorem 0.2. If p U, A q is both a left and a right Hopf algebroid and M a stable aYD con-tramodule over U , then the cyclic dual of the cochain complex that computes Ext ‚ U p A, M q yields a chain complex computing Coext U ‚ p A, M q , along with a degree ` differential B : Coext U ‚ p A, M q Ñ
Coext U ‚` p A, M q . More precisely, if γ denotes the right U -contraaction on M and if we indicate by u r`s b A u r´s for u P U a Sweedler-type notation for the inverse of one of the canon-ical Hopf-Galois maps, that is, the right Hopf structure, we obtain on the chain spaces C ‚ p U, M q : “ Hom A p U b A ‚ , M q the following structure maps of a cyclic k -module: p d i f qp u | . . . | u n ´ q “ $&% γ ` p´q r`s f p p´q r´s | u | . . . | u n ´ q ˘ f p u | . . . | ∆ u i | . . . | u n ´ q f p u | . . . | u n ´ | q if i “ , if ď i ď n ´ , if i “ n, p s j f qp u | . . . | u n ` q “ f p u | . . . | ε p u j ` q| . . . | u n ` q for ď j ď n, p tf qp u | . . . | u n q “ γ ´ pp p´q u q ➣ f q ` u | . . . | u n | ˘¯ , where ➣ denotes the left U -action on Hom A p U b A n , M q as in Eq. (0.3) and the verticalbars denote a certain tensor product over A , see Eq. (0.4). From a broader perspective, thiscyclic k -module and the corresponding differential B are part of what is called a homotopynoncommutative or homotopy Cartan calculus , also known as homotopy BV module , seeAppendix C.1. Such a differential calculus typically arises from a so-called cyclic oppo-site module over an operad with multiplication, as quoted in Theorem C.1; the operad inquestion here arises from the complex computing the derived functor Cotor ‚ U p A, A q . Inthis spirit, in Theorem 5.2, we prove: Theorem 0.3. If M is a stable aYD contramodule over a left bialgebroid p U, A q whichis both left and right Hopf, the chain complex computing Coext U ‚ p A, M q can be seen asa cyclic unital opposite module over the cochain complex computing Cotor ‚ U p A, A q , seenas an operad with multiplication, such that the underlying cyclic k -module structure is theone listed right above. This, as already mentioned, has Corollary 5.3 as an immediate consequence:
Corollary 0.4.
The couple consisting of the cochain complex computing
Cotor ‚ U p A, A q and the chain complex computing Coext U ‚ p A, M q can be equipped with the structure of ahomotopy noncommutative calculus if M is a stable aYD contramodule over U . In partic-ular, this induces the structure of a BV module on Coext U ‚ p A, M q over Cotor ‚ U p A, A q . NIELS KOWALZIG
Explicitly, along with the homotopy or higher B -operators S and T , see Eqs. (C.8), thecalculus operators of cyclic differential, contraction, and Lie derivative read as follows: p Bf qp v | . . . | v n q “ n ` ř i “ p´ q p i ´ q n γ ´ p¨q r`s p v i ➣ f q ` p¨q r´s p v i ` | . . . | v n ` q| v | . . . | v i ´ ˘¯ , p ι w f qp v | . . . | v n ´ p q “ γ ` p¨q r`s f p p¨q r´s p u | . . . | u p q| v | . . . | v n ´ p q ˘ , p L w f qp v | . . . | v n ´ p ` q “ n ´ p ` ř i “ p´ q p p ´ qp i ´ q f ` v | . . . | v i ´ | v i p u | . . . | u p q| v i ` | . . . | v n ´ p ` ˘ ` p ř i “ p´ q n p i ´ q` p ´ γ ´ p¨q r`s p u i ➣ f q ` p¨q r´s p u i ` | . . . | u p q| v | . . . | v n ´ p ` | u | . . . | u i ´ ˘¯ , for w : “ p u | . . . | u p q P U b A p and f P M p n q , where denotes the diagonal action in themonoidal category U - Mod of left U -modules.Our main application of this machinery consists in showing in §6 that the noncommuta-tive calculus on Coext and
Cotor provides a natural framework for including the classicalCartan calculus in differential geometry as an example: in Theorem 6.2, we show:
Theorem 0.5.
Let p A, L q be a Lie-Rinehart algebra, where L is projective over A ofpossibly infinite dimension, and VL its universal enveloping algebra. Then the anti-symmetrisation map induces an isomorphism of BV modules (or Cartan calculi) between `Ź nA L, Hom A p Ź nA L, M q ˘ and ` Cotor ‚ VL p A, A q , Coext VL ‚ p A, M q ˘ . Here, by isomorphism of BV modules we mean a pair of isomorphisms of the respectiveunderlying k -modules that commute with all calculus operators B, ι, L , S , T , and alsoinduce an isomorphism of Gerstenhaber algebras, see Lemma 6.1 and Eqs. (6.10)–(6.12)for details.This, in particular, contains the Chevalley-Eilenberg calculus for Lie algebras and thecalculus known for Lie algebroids as vector bundles over smooth manifolds which, in turn,includes the classical Cartan calculus if the vector bundle in question is the tangent bundle.A related but more restrictive result was already obtained in [KoKr] by developing acalculus on Tor over
Ext . There, however, finiteness of L as an A -module was necessaryto assume since the construction not only hinges on jet spaces JL as a bialgebroid dualto VL but also passes through a sort of double dual that plays the rôle of the space ofmultivector fields; in particular, one has to make use of topological tensor products alongwith completions. Here, none of all this is required and the result can be obtained bypurely algebraic operations. Finally, it appears (to us) more natural to regard VL as thespace of differential operators on a manifold (in case L arises from a Lie algebroid) insteadof Hom A p JL, A q .0.3. Notation and conventions.
All notation for bialgebroids, cyclic modules, operadsetc. is explained in the respective appendices at the end of the main text. Here, we onlyintroduce some basic notation globally used.The symbol k always denotes a commutative ring, usually of characteristic zero. For aleft bialgebroid p U, A q and a left U -module M , we most of the time denote the U -actionjust by juxtaposition, except for a few cases: for example, the monoidal structure on thecategory U - Mod of left U -modules is reflected by the diagonal U -action on the tensorproducts of two left U -modules N, M , that is, 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 (0.1)for n P N, m P M , and u P U . If U is on top a left resp. right Hopf algebroid (see Appen-dix D), one obtains a left U -module structure on Hom A op p N, M q resp. on Hom A p N, M q :in the first case, for all f P Hom A op p N, M q , set p u ➢ f qp n q : “ u ` f p u ´ n q , (0.2) NONCOMMUTATIVE CALCULUS ON THE CYCLIC DUAL OF
Ext and in the second case, for all g P Hom A p N, M q , put p u ➣ g qp n q : “ u r`s g p u r´s n q . (0.3)Recall from Eq. (D.1) the various triangle notations Ż , Ž , § , đ that denote the four A -modulestructures on the total space U of a bialgebroid, and occasionally even on a U -module. Weabbreviate tensor products U Ž b A Ż U with a vertical bar and tensor products in § U b A op U Ž with a comma, that is, write p u | . . . | u n q : “ u b A ¨ ¨ ¨ b A u n P Ż U Ž b A n , (0.4)as well as p u , . . . , u n q : “ u b A op ¨ ¨ ¨ b A op u n P § U Ž b A op n . (0.5)This would somehow make more sense the other way round as the analogue of the bar resolution is defined on U b A op n , while U b A n is the right space for the cobar resolution,but for notational consistency with the predecessor [Ko3] of this article, we decided to stickto the comma notation with respect to the tensor powers over A op .Finally, to keep things simple in homological considerations, we always assume (andsometimes even repeat this explicitly) that U Ž is flat as an A -module.1. C ONTRAMODULES OVER BIALGEBROIDS
Contramodules over coalgebras were introduced in [EiMo] not too long after the notionof comodules but are, in striking contrast to the latter, basically unknown to most of themathematical community.They are dealt with, for example, in [BöBrzWi, Brz] and gained the attention they de-serve in particular in [Po1, Po2]. For finite dimensional bialgebras (or bialgebroids), acontramodule should be thought of as a module over the dual. Contramodules also pop upas natural coefficients in the cyclic theory of
Ext groups and were implicitly used in theclassical cyclic cohomology theory by Connes [Co2] by choosing coefficients in the lineardual of an algebra, as explained in [Ko3, §6].
Definition 1.1. A right contramodule over a left bialgebroid p U, A q is a right A -module M along with a right A -module map γ : Hom A op p U Ž , M q Ñ M, called the contraaction , subject to contraassociativity , 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, as well as contraunitality , Hom A op p A, M q Hom A op p ε,M q / / » * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ Hom A op p U, M q γ (cid:15) (cid:15) M. The adjunction of the leftmost vertical arrow in the first diagram is to be understood withrespect to the right A -module structure on Hom A op p U Ž , M q defined by f a : “ f p a Ż ´q for a P A ; the right A -linearity of γ in the definition then means γ ` f p a Ż ´q ˘ “ γ p f q a. (1.1)Observe that there also is an induced left A -action on M given by am : “ γ ` mε p´ đ a q ˘ “ γ ` mε p a § ´q ˘ , (1.2) NIELS KOWALZIG turning M into an A -bimodule, and with respect to which γ becomes an A -bimodule map: γ ` f p´ đ a q ˘ “ aγ ` f p´q ˘ , (1.3)as shown in [Ko3, Eq. (2.37)]. In particular, we obtain a forgetful functor Contramod - U Ñ A e - Mod (1.4)from the category of right U -contramodules to that of A -bimodules.In general, we denote the “free entry” in the contraaction γ by hyphens or dots: for f P Hom A op p U, M q we may write both γ p f p´q q as well as γ p f p¨q q or simply γ p f q , dependingon readability in long computations: this way, the contraassociativity may be compactlyexpressed as γ ` : γ p g p¨ b A ¨¨qq ˘ “ γ ` g p´ p q b A ´ p q q ˘ , (1.5)for g P Hom A op p U Ž b A Ż U , M q , where the dots match the map γ with the respectiveargument, and contraunitality as γ p mε p´q q “ m (1.6)for m P M . Finally, do not confuse the operation of contraaction with that of contraction dealt with from §5 onwards. Example 1.2.
Eq. (1.1) in general excludes the existence of a trivial right contraaction f ÞÑ f p q , in full analogy to the fact that for bialgebroids in general there is no trivial(left or right) coaction. However, if A is commutative and source and target map happento coincide, then such a trivial contraaction is possible. This is, for example, the case forbialgebras or cocommutative bialgebroids, which we will explicitly exploit in §6. Example 1.3.
Essentially for the same reasons, again in contrast to coalgebra theory, du-alising bialgebroid comodules generally does not furnish examples of contramodules: incase of a coalgebra C , that is, for A “ k , and N a left C -comodule with coaction λ N , thelinear dual Hom k p N, k q is a right C -contramodule with contraaction γ : “ Hom p λ N , k q .Trying to generalise this to a bialgebroid p U, A q , for N P U - Comod neither the rightdual
Hom A op p N, A q nor the left dual Hom A p N, A q make this formula well-defined sincelinearity of the left coaction reads λ N p anb q “ a Ż n p´ q đ b b A n p q for a, b P A and n P N .2. T HE DERIVED FUNCTORS
Cotor
AND
Coext
The functors
Cotor and
Coext are, in a sense we will briefly explain in this section,dual to the well-known
Tor and
Ext and might appear an exotic and possibly not toourgent extension of the theory; on the other hand, as we are going to see in the examplesection, they yield a direct algebraic and natural approach if one wants to embed the Cartancalculus in differential geometry into a more abstract framework.2.1.
Cotor.
In this subsection, we will describe the derived functor of the cotensor prod-uct, which is called
Cotor in analogy to the derived functor
Tor of the ordinary tensorproduct; see [EiMo, Do] for classical information on the subject in the realm of customarycoalgebras, or [BrzWi] for general corings, or still [Ra, App. A] for commutative bial-gebroids. For comodules over bialgebroids and all involved technical features, see, forexample, [Tak].For a general bialgebroid p U, A q , the categories U - Comod and
Comod - U of leftresp. right comodules are not necessarily abelian, but are so if we assume that Ż U resp. U Ž are flat over A ; hence, to avoid all problems in this direction as this is not our mainfocus, let us directly assume that U Ž is flat, as mentioned in §0.3. To shorten terminology,we shall not use a “relative” language, that is, we call a right U -comodule P injective if itis a direct summand in one of the form X b A U for a right A -module X , which, in turn,we term free . NONCOMMUTATIVE CALCULUS ON THE CYCLIC DUAL OF
Ext Definition 2.1.
Let p U, A q be a left bialgebroid, P P Comod - U with right coaction ̺ P ,and M P U - Comod with left coaction λ M . The cotensor product P U M is defined asthe equaliser of the pair of maps p ̺ P b A M, P b A λ M q : P b A M Ñ P b A Ż U Ž b A M, that is, by the kernel of the difference map.More explicitly, the cotensor product is given by the subspace P U M “ p b A m P P b A M | p p q b A p p q b A m “ p b A m p´ q b A m p q u , where we wrote ̺ P p p q “ p p q b A p p q and λ M p m q “ m p´ q b A m p q for the right resp.left U -coaction. For any M P U - Comod , there is a natural isomorphism U U M Ñ M given by u b A m ÞÑ ε p u q m with inverse the left coaction λ M . More generally, for anyright A -module X (that is, for any free right U -comodule of the form X b A U q , we havean isomorphism φ : p X b A U q U M Ñ X b A M, p x b A u q U M ÞÑ x b A ε p u q m, (2.1)with inverse induced by the coaction of M as above. The functor of taking cotensor prod-ucts is left exact in the first variable if M is flat as a left A -module; the same holds in thesecond variable if P is flat as a right A -module. As a consequence, we can define its rightderived functors Cotor U : more precisely, considering that under the flatness assumptionson U the category Comod - U has enough injectives, for any cochain complex p I ‚ , B q ofinjective right U -comodules, the resulting cochain complex I ‚ U M is acyclic. Hence, if P Ñ I ‚ is a resolution of the right U -comodule P by injectives, we define Cotor ‚ U p P, M q : “ H ` I ‚ U M, B b A M ˘ . The standard way of resolving the right U -comodule P is by the well-known cobar cochaincomplex: set Cob n p P, U q : “ P b A Ż U Ž b A n ` for any n P N , and define the differential B “ ř n ` i “ p´ q i B i : Cob n p P, U q Ñ
Cob n ` p P, U q , where B i p p | u | . . . | u n ` q “ " p ̺ P p p q| u | . . . | u n ` qp p | u | . . . | ∆ p u i q| . . . | u n ` q if i “ , if ď i ď n ` , (2.2) using the notation introduced in (0.4). As a consequence, Cotor ‚ U p P, M q can be computedby the cochain complex Cob ‚ p P, U q U M with differential B . Now, the right coaction on Cob n p P, U q is simply defined by the coproduct on the rightmost tensor factor of U , whichtherefore yields a free (hence injective) resolution of P . Applying the isomorphism φ from (2.1), we see that Cotor ‚ U p P, M q can effectively be computed by the chain complex P b A U b A ‚ b A M with differential B “ ř n ` i “ p´ q i B i in degree n , which is typicallymore convenient to consider. Here, the cofaces B i : “ φ ˝ B i ˝ φ ´ come out as: B i p p | u | . . . | u n | m q “ $&% p ̺ P p p q| u | . . . | u n | m qp p | u | . . . | ∆ p u i q| . . . | u n | m qp p | u | . . . | u n | λ M p m qq if i “ , if ď i ď n, if i “ n ` . (2.3) In case P “ A , we will denote the resulting chain complex complex by C ‚ co p U, M q : “ U b A ‚ b A M, (2.4)with differential B as above and right U -coaction on A given by the target map.2.2. Coext.
In this subsection, another not too well-known derived functor is introduced,the so-called
Coext , the definition of which for coalgebras over a commutative ring appearsin [Po1, §0.2] and possibly (much) earlier elsewhere. As above, we have to adapt theconstruction given in op. cit. to a relative setting as we are dealing with corings over a(possibly noncommutative) base algebra A . NIELS KOWALZIG
Definition 2.2.
Let p U, A q be a left bialgebroid, p P, ̺ P q a right U -comodule and p M, γ q aright U -contramodule. The space of cohomomorphisms Cohom U p P, M q is defined as thecoequaliser of the pair of maps ` Hom A op p ̺ P , M q , Hom A op p P, γ q ˘ :Hom A op p P b A Ż U , M q »
Hom A op p P, Hom A op p U, M qq Ñ Hom A op p P, M q , that is, the cokernel of the difference map.Here, the A op -linearity on both sides of the adjunction refers to U Ž . More explicitly,the space of cohomomorphisms can be described as the quotient Cohom U p P, M q “
Hom A op p P, M q{ I, where I is the k -module generated by g ˝ ̺ P ´ γ p g p´ b A ¨qq | g P Hom A op p P b A Ż U , M q ( . That this yields a well-defined construction with respect to the right A -action follows from g p ̺ P p pa qq ´ γ ` g p pa b A ¨q ˘ “ g p p p q b A p p q Ž a q ´ γ ` g p p b A a Ż p¨q q ˘ “ g p p p q b A p p q q a ´ γ ` g p p b A ¨q ˘ a for a P A, p P P , using right linearity of the right coaction ̺ P : p ÞÑ p p q b A p p q alongwith (1.1).For any right U -contramodule M , there is a natural isomorphism Cohom U p U, M q Ñ
M, f ÞÑ γ p f q with inverse m ÞÑ mε p¨q . More generally, for any right A -module X (thatis, for any free right U -comodule of the form X b A U ), we have an isomorphism ϑ : Cohom U p X b A U, M q Ñ
Hom A op p X, M q ,f ÞÑ x ÞÑ γ ` f p x b A ¨q ˘( , (2.5) g p xε p u qq Ð [ x b A u ( Ð [ g. (2.6)As for coalgebras [Po1, §0.2.5], the functor of cohomomorphisms over a bialgebroid p U, A q is left exact in the first variable (and right exact in the second), and hence we candefine in a standard way its right derived functors Coext U : by the flatness assumption on U Ž and similarly to the preceding subsection, for any cochain complex p I ‚ , B q of injectiveright U -comodules, the resulting chain complex Cohom U p I ‚ , M q is acyclic. Hence, asabove, for any resolution P Ñ I ‚ by injectives, define Coext U ‚ p P, M q : “ H ` Cohom U p I ‚ , M q , Hom A op pB , M q ˘ . Using the cobar cochain complex from (2.2) again,
Coext U ‚ p P, M q can be computed bythe chain complex Cohom U p Cob ‚ p P, U q , M q with differential b “ ř n ` i “ p´ q i b i , where b i f : “ f ˝ B i for any f P Cohom U p Cob n p P, U q , M q . Again, considering the comodulestructure of the cobar complex and applying the isomorphism ϑ from (2.5), this time wesee that Coext U ‚ p P, M q can effectively be computed by the chain complex Hom A op p P b A U b A ‚ , M q with differential b “ ř ni “ p´ q i b i in degree n P N , which usually is morepractical, again. Here, the faces b i : “ ϑ ˝ b i ˝ ϑ ´ by a quick computation using Eqs. (2.5),(2.6), (1.1), and (1.6), result into p b i f qp p | u | . . . | u n ´ q “ $&% f p ̺ P p p q| u | . . . | u n ´ q f p p | u | . . . | ∆ p u i q| . . . | u n ´ q γ p f p p | u | . . . | u n ´ | p¨q qq if i “ , if ď i ď n ´ , if i “ n, (2.7) for any f P Hom A op p P b A U b A n , M q . In case P “ A , we will denote the resulting chaincomplex as D ‚ p U, M q : “ Hom A op p U b A ‚ , M q (2.8)with differential b as above and right U -coaction on A again given by the target map. NONCOMMUTATIVE CALCULUS ON THE CYCLIC DUAL OF
Ext Of course, one could equally resolve M by (relative) projective contramodules (see[Po1, §0.2] again) to compute Coext U ‚ p P, M q but we are not going to pursue this possibil-ity here. 3. T HE COMPLEX COMPUTING
Ext
AS A COCYCLIC MODULE
Anti Yetter-Drinfel’d contramodules.
In most cyclic theories, not only the ones in-cluding a Hopf structure on the underlying ring or coring, to obtain a para-(co)cyclic objectof any kind the possible coefficients typically exhibit more than one algebraic structure, forexample, they need to be both modules and comodules or both modules and contramod-ules. In many cases, these double structures are not immediately recognized as such sinceone of them might be trivial as happens in §6.1, for example. In any case, to pass frompara-(co)cyclic to truly cyclic ones, that is, the (co)cyclic operator powers to the identity,a compatibility condition between these two algebraic structures is required. In the case athand, we are interested in the following definition from [Ko3, Def. 4.3]:
Definition 3.1. An anti Yetter-Drinfel’d (aYD) contramodule M over a left Hopf algebroid p U, A q is simultaneously a left U -module (with action simply denoted by juxtaposition)and a right U -contramodule (with contraaction denoted by γ ) such that both underlying A -bimodule structures from (D.1) and (1.4) coincide, that is, a Ż m Ž b “ amb, m P M, a, b P A, (3.1)and such that action and contraaction are compatible in the sense that 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 . (3.2)An anti Yetter-Drinfel’d contramodule is called stable if γ p p´q m q “ m (3.3)for all m P M , where we denote p´q m : u ÞÑ um as a map in Hom A op p U, M q .A similar definition in the realm of Hopf algebras appeared first in [Brz], whereas forHopf algebroids to our knowledge first in [Ko3]. We refer to op. cit. , p. 1093, for more in-formation about the (not so obvious) well-definedness of Eq. (3.2) and further implications.In particular, one can show that γ p a Ż f p´q q “ γ ` f p a § ´q ˘ , (3.4)where on the left hand side the left A -action on M is meant.The category U aYD contra ´ U of right aYD contramodules over a left bialgebroid U isnot known to be monoidal, and it is also not likely to be the case, considering the fact thatin finite dimensions this category is equivalent to that of left modules over the (right) dual U ˚ , see [Ko3, Lem. 4.6], which in general is not monoidal. However, similar to the case ofaYD modules , the category U aYD contra ´ U is a module category over UU YD , the categoryof Yetter-Drinfel’d (YD) modules over U , see [Sch, Def. 4.2]. More precisely, with thefollowing we improve Proposition 4.8 in [Ko3] by removing the finiteness condition: Proposition 3.2.
Let p U, A q be a left bialgebroid. ( i ) The operation U - Comod ˆ Contramod - U Ñ Contramod - U, p N, M q ÞÑ N M : “ Hom A op p N, M q (3.5) defines on Contramod - U the structure of a module category over the monoidalcategory U - Comod . ( ii ) The operation ( ) restricts to a left action UU YD ˆ U aYD contra ´ U Ñ U aYD contra ´ U . Hence, U aYD contra ´ U is a module category over the monoidal category UU YD . Proof.
As for the first part, we have to show that for a left U -comodule N and M a right U -contramodule, Hom A op p N, M q can be endowed with a right contraaction aswell. From the adjunction Hom A op p N b A N, M q »
Hom A op p N , Hom A op p N, M qq for N, N P U - Comod , one then obtains p N b A N q M “ N p N M q and hence, b A being the monoidal product in U - Comod , the claim.To this end, let λ N : n ÞÑ n p´ q b A n p q denote the left U -coaction on N whereas γ M the U -contraaction on M , and consider Hom A op p N, M q as a right A -module by p ha qp n q : “ h p an q for a P A and h P Hom A op p N, M q . The following then defines a U -contraaction on Hom A op p N, M q : γ : Hom A op p U, Hom A op p N, M qq Ñ
Hom A op p N, M q ,f ÞÑ n ÞÑ γ M ` f p n p´ qp´q b A n p q q ˘ , (3.6)with the adjunction Hom A op p U, Hom A op p N, M qq »
Hom A op p U Ž b A N, M q implicitlyunderstood. To show that this indeed defines a contraaction, we will make use of thefact that γ M is already a contraaction, i.e. , that Eqs. (1.1)–(1.6) hold for γ M . The right A -linearity (1.1) follows for γ by simply observing λ N p na q “ n p´ q đ a b A n p q alongwith the right A -module structure on Hom A op p N, M q as above. Furthermore, for g P Hom A op p U b A U, Hom A op p N, M qq and n P N , γ ` : γ p g p¨ b A ¨¨qq ˘ p n q “ γ M ` : γ M p g p n p´ qp¨q b A n p´ qp¨¨q b A n p q qq ˘ “ γ M ` g p n p´ qp´qp q b A n p´ qp´qp q b A n p q q ˘ “ γ ` g p p´qp q b A p´qp q q ˘ p n q , which is (1.5) for the map γ from (3.6). In the same spirit one proves (1.6) and therefore, γ indeed constitutes a right U -contraaction on Hom A op p N, M q .As for the second part, assume now that N P UU YD and M P U aYD contra ´ U , thatis, both N, M in particular are left U -modules. Then Hom A op p N, M q becomes a left U -module as well by Eq. (0.2), and in order to prove that Hom A op p N, M q even turns intoa stable aYD contramodule over U , we need to show that this left action is compatiblewith the right contraaction in the sense of Eqs. (3.1)–(3.2). Let h P Hom A op p N, M q and a, b P A . That h Ž b “ hb follows immediately from (0.2) and (D.11). On the other hand, p ah qp n q (1.2) “ γ ` hε p p´q đ a q ˘ p n q (3.6) “ γ M ` h p ε p n p´ qp´q đ a q n p q ˘ (1.3) “ a Ż γ M ` h p ε p n p´ qp´q q n p q ˘ “ a Ż γ M ` h p nε p p´q qq ˘ “ a Ż γ M ` h p n q ε p p´q q ˘ (1.6) “ a Ż h p n q , where in the fourth step we used the properties of a bialgebroid counit, counitality andthe fact that the coaction maps into a Takeuchi subspace similar to the coproduct as inAppendix D. This proves (3.1). Moreover, using the fact that N is a YD module and hencethe compatibility p u p q n q p´ q u p q b A p u p q n q p q “ u p q n p´ q b A u p q n p q (3.7)holds between left U -action and left U -coaction (see [Sch, Def. 4.2]), one computes for f P Hom A op p U, Hom A op p N, M qq that p u ➢ γ p f qqp n q (0.2) “ u ` p γ p f qp u ´ n qq (3.6) “ u ` ` γ M ` f pp u ´ n q p´ qp´q b A p u ´ n q p q q ˘˘ (3.2) “ γ M ` u ``p q f pp u ´ n q p´ q u `´p´q u ``p q b A p u ´ n q p q q ˘ (D.7) “ γ M ` u `p q f pp u ´p q n q p´ q u ´p qp´q u `p q b A p u ´p q n q p q q ˘ (3.7) “ γ M ` u `p q f p u ´p q n p´ qp´q u `p q b A u ´p q n p q q ˘ (D.6) , (0.2) “ γ M ` p u p q ➢ f qp n p´ qp´q u p q b A n p q q ˘ (3.6) “ γ ` p u p q ➢ f qp p´q u p q q ˘ p n q , NONCOMMUTATIVE CALCULUS ON THE CYCLIC DUAL OF
Ext which by (D.6) again is (3.2) for Hom A op p N, M q with U -action (0.2) and U -contraaction(3.6). (cid:3) Remark 3.3.
The possible stability of the aYD contramodule
Hom A op p N, M q does not automatically follow from the possible stability of the aYD contramodule M : the stabilitycondition for Hom A op p N, M q explicitly reads γ p p´q ➢ h qp n q “ γ M ` p n p´ qp´q q ➢ h p n p q q ˘ “ h p n q (3.8)for h P Hom A op p N, M q . Even in case of a Hopf algebra over a commutative ring k withinvolutive antipode S , considering M “ k as a stable aYD contramodule with trivial actionand trivial contraaction (Example 1.2), the left hand side in (3.8) reads γ p p´q ➢ h qp n q “ h p S p n p´ q q n p q q , which in general is different from the right hand side h p n q .3.2. The cocyclic module.
In [Ko3, §4.2], for a left Hopf algebroid p U, A q and a left U -module right U -contramodule M , we defined a para-cocyclic k -module structure on C ‚ p U, M q : “ Hom A op p U b A op ‚ , M q , (3.9)where the tensor products are taken with respect to the A -bimodule structure § U Ž . Explic-itly, in degree q P N the structure maps are given by 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 ´ , 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 ˘ , (3.10) the cosimplicial part of which computes the Ext functor in case U Ž is flat as an A -module,that is, H p C ‚ p U, M q , δ q » Ext ‚ U p A, M q , where as always δ : “ ř n ` i “ p´ q i δ i . Thispara-cocyclic k -module becomes cyclic if M is a stable aYD contramodule.In view of §3.1, we can now fill in more general coefficients in the first entry: Proposition 3.4.
Let p U, A q be a left Hopf algebroid, N a left U -module left U -comodule,and M a left U -module right U -contramodule. Then C ‚ p U b A op N, M q : “ Hom A op p U b A op ‚ b A op N, M q can be given the structure of a para-cocyclic k -module (the cohomology of which computes Ext ‚ U p N, M q if U Ž is A -flat), which is cyclic if N is a YD module, M an aYD contramod-ule, and Hom A op p N, M q is stable as in ( ) . In particular, the cyclic coboundary inducesan operator B : Ext ‚ U p N, M q Ñ
Ext ‚´ U p N, M q , which squares to zero.Proof. We simply have to transport the structure maps (3.10) on C ‚ p U, Hom A op p N, M qq for the contramodule Hom A op p N, M q to the space C ‚ p U b A op N, M q by thecorrect isomorphism of k -modules that produces the correct underlying cosimpli-cial k -module: this isomorphism between Hom A op p U b A op ‚ , Hom A op p N, M qq and Hom A op p U b A op ‚ b A op N, M q is not a simple adjunction but rather the adjunction Hom A op p U b A op ‚ , Hom A op p N, M qq »
Hom A op pp U b A op ‚ q b A N, M q , where on the righthand side the A op -linearity refers to the right A -module structure on N , followed by the k -module isomorphism χ : Hom A op pp U b A op q q b A N, M q Ñ
Hom A op p U b A op q b A op N, M q ,f ÞÑ p u , . . . , u q , n q ÞÑ f pp u p q , . . . , u q p q q| u p q ¨ ¨ ¨ u q p q n q ( , g p u ` , . . . , u q ` , u q ´ ¨ ¨ ¨ u ´ n q Ð [ pp u , . . . , u q q| n q ( Ð [ g, in degree q P N , where on the right hand side the A op -linearity now refers to the right A -module structure on the first tensor factor of U . Defining then δ i : “ χ ˝ δ i ˝ χ ´ and σ j : “ χ ˝ σ j ˝ χ ´ by means of the cofaces and codegeneracies from (3.10), a quickcomputation reveals p δ i g qp u , . . . , u q ` , n q “ $&% u g p u , . . . , u q ` , n q g p u , . . . , u i u i ` , . . . , u q ` , n q g p u , . . . , u q , u q ` n q if i “ , if ď i ď q, if i “ q ` , p σ j g qp u , . . . , u q ´ , n q “ f p u , . . . , u j , , u j ` , . . . , u q ´ , n q for ď j ď q ´ , for the cosimplicial k -module structure on C ‚ p U b A op N, M q , which clearly yields acochain complex that computes Ext ‚ U p N, M q if U Ž is A -flat. Likewise, by putting τ : “ χ ˝ τ ˝ χ ´ we can promote this cosimplicial module to a para-cocyclic one whichis cyclic if N is a YD module, M an aYD module, and Hom A op p N, M q stable as followsdirectly from the respective property of the structure maps (3.10), Proposition 3.2, and thefact that χ is a k -module isomorphism. Explicitly, the cocyclic operator is given by p τ g qp u , . . . , u q , n q “ γ M ´ u p q` ¨ ¨ ¨ u q ´ p q` u q ` g ` p n p´ q u q ´ u q ´ p q´ ¨ ¨ ¨ u p q´p¨q q ` , u p q` ,. . . , u q ´ p q` , u p q´ ¨ ¨ ¨ u q ´ p q´ p n p´ q u q ´ u q ´ p q´ ¨ ¨ ¨ u p q´p¨q q ´ n p q ˘¯ , where γ M denotes the contraaction on M and which, however, is neither nice nor reallyhelpful to stare at but at least reduces to τ f in (3.10) again if N “ A .The statement about the cyclic coboundary follows by a standard argument involvingan SBI sequence, see [Lo, §2.2]. (cid:3)
4. T
HE COMPLEX COMPUTING
Coext
AS A CYCLIC MODULE
In the rest of this article, for the mere sake of simplicity to avoid too messy formulæ, werestrict ourselves to the case in which N equals the base algebra A itself, with left U -actiongiven by ua : “ ε p u đ a q for u P U, a P A , and left U -coaction given by the source map.The aim of this section is to compute the cyclic dual in the sense of §A.2 of the cocyclicmodule C ‚ p U, M q from (3.9)–(3.10), where M is a stable aYD contramodule. Merelyapplying the formula for cyclic duality in (A.3) does not quite yield the desired result aswe are interested in obtaining a cyclic structure on the complex Cohom U p U b A ‚` , M q » D ‚ p U, M q as in (2.8) that computes Coext by means of the cobar resolution using coprod-ucts, which, as a k -module, is quite different from Hom U p U b A op ‚` , M q » C ‚ p U, M q which computes Ext by means of the bar resolution using products. To circumvent thisproblem, one uses k -linear isomorphisms which transform one complex into the other andwhich are basically higher order Hopf-Galois maps. From a more abstract point of view,the cyclic operator arises from a distributive law between two monads, and the isomor-phism from the following Lemma maps one monad into the other. Since our main goal in§5 is to obtain a chain complex on which the cochain complex C ‚ co p U, A q acts in a naturalway, it turns out to be more constructive to detect the cyclic structure on C ‚ p U, M q : “ Hom A p U b A ‚ , M q , where the A -linearity refers to the A -module structure Ż U on the first tensor factor, and toconnect it to the chain complex D ‚ p U, M q afterwards.The subsequent lemma is a straightforward verification relying on Hopf-Galois yoga,that is, on the identities (D.3)–(D.19) for left resp. right Hopf algebroids and in particularon the mixed ones in Eqs. (D.21)–(D.23). Lemma 4.1.
Let p U, A q be both a left and a right Hopf algebroid and M a left U -module.Then for each n P N there is a k -linear isomorphism ξ : C n p U, M q Ñ C n p U, M q , (4.1) g ÞÑ p u | . . . | u n q ÞÑ u ` g p u ´ u ` , . . . , u n ´ ´ u n ` , u n ´ q ( , NONCOMMUTATIVE CALCULUS ON THE CYCLIC DUAL OF
Ext the inverse of which being given by ξ ´ : C n p U, M q Ñ C n p U, M q , (4.2) f ÞÑ p u , . . . , u n q ÞÑ u r`s ¨ ¨ ¨ u n r`s f ` u n r´s p¨ ¨ ¨ p u r´s p u r´s | q| q ¨ ¨ ¨ | q ˘( , where denotes the diagonal U -action ( ) on U b A n . These isomorphisms allow to obtain the structure maps of a cyclic k -module on C ‚ p U, M q as calculated in the next lemma, which again is achieved by computation only. Lemma 4.2.
Let p U, A q be both a left and a right Hopf algebroid and M a stable aYDcontramodule over U . Then the cyclic dual as defined in Eqs. ( A.3 ) intertwined by theisomorphism ( ) obtained from the cocyclic k -module C ‚ p U, M q with structure maps ( ) produces in degree n P N the following morphisms p d i f qp u | . . . | u n ´ q “ $&% γ ` p´q r`s f p p´q r´s | u | . . . | u n ´ q ˘ f p u | . . . | ∆ u i | . . . | u n ´ q f p u | . . . | u n ´ | q if i “ , if ď i ď n ´ , if i “ n, (4.3) p s j f qp u | . . . | u n ` q “ f p u | . . . | ε p u j ` q| . . . | u n ` q for ď j ď n, (4.4) p tf qp u | . . . | u n q “ γ ´ pp p´q u q ➣ f q ` u | . . . | u n | ˘¯ (4.5) on C n p U, M q , where ➣ denotes the left U -action on Hom A p U b A n , M q as in p . q , con-sidering U b A n as a left U -module via the diagonal action ( ) . That these structural maps are well-defined and in particular have the correct left A -linearity is in case of d and t not obvious but follows from (D.17) together with (1.3). Proof of Lemma 4.2.
Following the mapping rule in (A.3), the claim explicitly reads: d “ ξ ˝ σ n ´ τ ˝ ξ ´ , d i “ ξ ˝ σ i ´ ˝ ξ ´ , s j “ ξ ˝ δ j ˝ ξ ´ , t “ ξ ˝ τ ´ ˝ ξ ´ for ď i ď n and ď j ď n with respect to the operators p δ j , σ i , τ q from Eqs. (3.10).For the simplicial part, we only show how d is computed (as this is already fiddly enough)and leave the rest to the reader. Indeed, p d f qp u | . . . | u n q “ p ξ ˝ σ n ´ τ ˝ ξ ´ f qp u | . . . | u n q (4.1) “ u ` p σ n ´ τ ˝ ξ ´ f qp u ´ u ` , . . . , u n ´ ´ u n ` , u n ´ q (3.10) “ u ` p τ ˝ ξ ´ f qp u ´ u ` , . . . , u n ´ ´ u n ` , u n ´ , q (3.10) “ u ` γ ` pp u ´p q u `p q ¨ ¨ ¨ u n ´ ´p q u n `p q u n ´p q q ➢ p ξ ´ f qqp p´q , u ´p q u `p q , . . . , u n ´ ´p q u n `p q , u n ´p q q ˘ (D.7) , (D.4) “ u ` γ ` p u ´p q ➢ p ξ ´ f qqp p´q , u ´p q u ` , . . . , u n ´ ´ u n ` , u n ´ q ˘ (0.3) “ u ` γ ` u ´p q` p ξ ´ f qp u ´p q´p´q , u ´p q u ` , . . . , u n ´ ´ u n ` , u n ´ q ˘ (3.2) “ γ ` u ``p q u ´p q` p ξ ´ f qp u ´p q´ u `´p´q u ``p q , u ´p q u ` , . . . , u n ´ ´ u n ` , u n ´ q ˘ (D.7) , (D.5) “ γ ` u ``p q u `´ p ξ ´ f qp p´q u ``p q , u ´ u ` , . . . , u n ´ ´ u n ` , u n ´ q ˘ (D.4) “ γ ` p ξ ´ f qp p´q u ` , u ´ u ` , . . . , u n ´ ´ u n ` , u n ´ q ˘ (4.2) “ γ ´ p´qr`s u `r`s u ´r`s ¨ ¨ ¨ u n `r`s u n ´r`s f ` u n ´r´s pp u n `r´s u n ´ ´r´s q p¨ ¨ ¨ pp u `r´s u ´r´s q p u `r´sp´qr´s | q| q ¨ ¨ ¨ | q ˘¯ (D.22) , (D.21) , (D.9) “ γ ´ p´qr`s ε p u p qr`s q ¨ ¨ ¨ ε p u n p qr`s q f ` u n p q pp u n p qr´s u n ´ p q q p¨ ¨ ¨ pp u p qr´s u p q q p u p qr´sp´qr´s | q| q ¨ ¨ ¨ | q ˘¯ (D.15) , (D.12) “ γ ´ p´qr`s f ` u n r`s pp u n r´s u n ´ r`s q p¨ ¨ ¨ pp u r´s u r`s q p u r´sp´qr´s | q| q ¨ ¨ ¨ | q ˘¯ (D.13) “ γ ` p´qr`s f p p´qr´s | u | . . . | u n q ˘ . Observe that the aYD condition (3.2) was used in line seven and left A -linearity of f inthe penultimate line. Finally, the computation of t runs along the same lines taking intoconsideration that the inverse of τ in (3.10) is given by p τ ´ f qp u , . . . , u n q “ γ ` u ` f p u ` , . . . , u n ` , u n ´ ¨ ¨ ¨ u ´p´q q ˘ , see [Ko3], Eq. (4.19), where it is denoted by τ due to the use of an opposite convention. (cid:3) Remark 4.3.
For later use, we also want to mention the inverse of t if M is a stable aYDcontramodule, defined by t ´ : “ ξ ˝ τ ˝ ξ ´ . A direct computation using (3.10), (4.1), and(4.2) yields p t ´ f qp u | . . . | u n q “ γ ´ p¨q r`s p u n ➣ f q ` p¨q r´s | u | . . . | u n ´ ˘¯ (4.6)for f P C n p U, M q . Theorem 4.4. If p U, A q is both a left and a right Hopf algebroid and M a stable aYD con-tramodule over U , then the cyclic dual ` C ‚ p U, M q , d ‚ , s ‚ , t ˘ defines a cyclic k -module thesimplicial part of which induces a chain complex that is isomorphic to the chain complex ` D ‚ p U, M q , b ˘ as in ( ) – ( ) computing Coext U ‚ p A, M q .Proof. The first statement is a tautological consequence of how cyclic duals are constructed(the cyclic dual of a cocyclic module being a cyclic module) along with the fact that themaps (4.1) and (4.2) are isomorphisms of k -modules.As for the second part, we want to show that p C ‚ p U, M q , d q » p D ‚ p U, M q , b q as chaincomplexes, where d “ ř ni “ p´ q n d i in degree n for the faces in Eqs. (4.3). To this end,consider first the following k -linear isomorphism ζ : Hom A p U b A n , M q Ñ Hom A op p U b A n , M q ,f ÞÑ p u | . . . | u n q ÞÑ p u n ➣ f qp | u | . . . | u n ´ q ( , p u ➢ g qp u | . . . | u n | q Ð [ p u | . . . | u n qu Ð [ g. (4.7) Coupling then this isomorphism with the cyclic operator t , which is an isomorphism aswell if M is a stable aYD module — with inverse quoted in Eq. (4.6) — does the job;that is, defining η : “ ζ ˝ t , we obtain an isomorphism C ‚ p U, M q » D ‚ p U, M q with theproperty that η ˝ d i “ b i ˝ η for all faces, that is, for all ď i ď n in degree n . We onlyshow this for i “ which is the most intricate case, and leave the rest to the reader. On theother hand, it turns out to be more convenient working with the inverse, and we thereforecompute η ´ first: for g P Hom A op p U b A n , M q , we have p η ´ g qp u | . . . | u n q “ p t ´ ζ ´ g qp u | . . . | u n q (4.6) “ γ ´ p¨q r`s p u n ➣ p ζ ´ g qqp p¨q r´s | u | . . . | u n ´ q ¯ (D.16) “ γ ´ p¨q r`s u n r`s ` ζ ´ g ˘` u n r`sr´sp¨q r´s | u n r´s p u | . . . | u n ´ q ˘¯ (4.7) “ γ ´ p¨q r`s u n r`sr`s ` p u n r`sr´sp¨q r´s q ➢ g ˘` u n r´s p u | . . . | u n ´ q| ˘¯ (0.2) “ γ ´ p¨q r`s u n r`sr`s u n r`sr´s` p p¨q r´s ➢ g ˘` p u n r`sr´s´p q u n r´s q p u | . . . | u n ´ q| u n r`sr´s´p q ˘¯ (D.23) , (D.18) “ γ ´ p¨q r`s p p¨q r´s ➢ g ˘` p u n r`sp q u n r´s q p u | . . . | u n ´ q| u n r`sp q ˘¯ (D.13) , (0.3) “ γ ´ p¨q r`s p¨q r´s` g ` p¨q r´s´ p u | . . . | u n q ˘¯ (D.23) , (D.18) “ γ ´ g ` p¨q p u | . . . | u n q ˘¯ , using right A -linearity of g in line six and eight. With this, we compute on one side p η ´ b g qp u | . . . | u n ´ q “ γ ´ p b g q ` p¨q p u | . . . | u n ´ q ˘¯ “ γ ´ g ` | p¨q p u | . . . | u n ´ q ˘¯ , NONCOMMUTATIVE CALCULUS ON THE CYCLIC DUAL OF
Ext using b from (2.7) for N “ A , and on the other side, by means of d from (4.3), p d η ´ g qp u | . . . | u n ´ q (4.3) “ γ ´ p¨q r`s p η ´ g qp p¨q r´s | u | . . . | u n ´ q ¯ “ γ ´ p¨q r`s : γ ` g p p¨¨q p p¨q r´s | u | . . . | u n ´ qq ˘¯ (3.2) “ γ ´ : γ ` p¨q r`s`p q g pp p¨q r`s´ p¨¨qp¨q r`s`p q q p p¨q r´s | u | . . . | u n ´ qq ˘¯ (1.5) , (D.21) “ γ ´ p¨q p q`r`sp q g ` p p¨q p q´ p¨q p q p¨q p q`r`sp q q p p¨q p q`r´s | u | . . . | u n ´ q ˘¯ (D.5) “ γ ´ p¨q r`sp q g ` p¨q r`sp q p p¨q r´s | u | . . . | u n ´ q ˘¯ (D.13) “ γ ´ p¨q p q g ` | p¨q p q p u | . . . | u n ´ q ˘¯ (1.5) “ γ ´ : γ ` p¨¨q g p | p¨q p u | . . . | u n ´ qq ˘¯ (3.3) “ γ ´ g ` | p¨q p u | . . . | u n ´ q ˘¯ “ p η ´ b g qp u | . . . | u n ´ q , where we used the aYD property in line four and stability in line nine. Verifying analogousidentities for all faces, we altogether obtain η ˝ d “ b ˝ η and hence, η : p C ‚ p U, M q , d q » ÝÑp D ‚ p U, M q , b q gives the desired isomorphism of chain complexes. (cid:3)
5. T
HE NONCOMMUTATIVE CALCULUS STRUCTURE ON
Coext
OVER
Cotor
In this section, we advance to the core of the article by defining the structure of a cyclicunital opposite module on the chain complex computing
Coext over the operad with mul-tiplication given by the cochain complex computing
Cotor , which induces a noncommu-tative calculus up to homotopy, see Theorem C.1.Since we would like to see from now on the cochain resp. chain spaces just mentionedfrom a more operadic point of view, we change the notation and set for n, p P N O p p q : “ C p co p U, A q , M p n q : “ C n p U, M q , where p U, A q for the time being is only a left bialgebroid and M a right U -contramodule.The operadic structure of O was explicitly described in [Ko2, Eqs. (3.3)–(3.5)] for generalcoefficients (more precisely, with coefficients being (braided) commutative monoids in thebraided category of Yetter-Drinfel’d modules, see there). Here, we will only deal withthe case of coefficients in the base algebra A but a more general approach would also bepossible without too much additional effort.The operadic structure on O is defined by the partial composition maps ˝ i : O p p q b O p q q Ñ O p p ` q ´ q given as p u | . . . | u p q ˝ i p v | . . . | v q q : “ p u | . . . | u i ´ | u i p v | . . . | v q q| u i ` | . . . | u p q (5.1)for all ď i ď p , where as always denotes the diagonal action (0.1) given by the p q ´ q -fold iterated coproduct ∆ q ´ on elements of degree q . For q “ , set ∆ “ id U and becomes the multiplication in U , whereas for q “ , that is, an element in O p q “ A , set ∆ ´ “ ε , the counit of U . In particular, O is an operad with multiplication (see §B.1), themultiplication, the identity, and the unit being given by p µ, , e q : “ ` p U | U q , U , A ˘ . (5.2)As far as the opposite O -module structure on M is concerned, define for all i “ , . . . , n ´ p ` and ď p ď n , the operation ‚ i : O p p q b M p n q Ñ M p n ´ p ` q ,w b f ÞÑ f ` p´q | . . . | p´q i ´ | p´q i w | p´q i ` | . . . | p´q n ˘ , (5.3) declared to be zero if p ą n . Explicitly, this has to be read as follows: let w “p u | . . . | u p q P U b A p and f P Hom A p U b A n , M q . Then ` p u | . . . | u p q ‚ i f ˘ p v | . . . | v n ´ p ` q : “ f ` v | . . . | v i ´ | v i p q u | . . . | v i p p q u p | v i ` | . . . | v n ´ p ` ˘ for all i “ , . . . , n ´ p ` and p v | . . . | v n ´ p ` q P U b A n ´ p ` , and where p v i p q | . . . | v i p p q q denotes the p p ´ q -fold iterated coproduct ∆ p ´ p v i q . Again, for an element in O p q “ A acting on f , one sets ∆ ´ “ ε as above, and hence ` a ‚ i f ˘ p v | . . . | v n ` q : “ f ` v | . . . | v i ´ | ε p v i đ a q| v i ` | . . . | v n ` ˘ (5.4)for a P A and i “ , . . . , n ` . Lemma 5.1.
The operations ( ) induce on M the structure of a unital opposite O -module.Proof. By definition, we have to check the identities (C.4) for the operations (5.3) and theoperadic structure on O spelled out in (5.1) above. This is an obvious verification thathinges essentially on coassociativity along with the compatibility between product andcoproduct, which is why we omit it. (cid:3) A possible Hopf structure on a left bialgebroid p U, A q comes into play when one wantsto promote this opposite module into a cyclic one, in particular when adding the extraoperation ‚ . To this end, let U be both a left and right Hopf algebroid, and set p u | . . . | u p q ‚ f : “ γ ` p¨q r`s p u ➣ f qp p¨q r´s p u | . . . | u p q| p´q | . . . | p´q n ´ p ` q ˘ , (5.5)for f P M p n q , declared to be zero this time if p ą n ` . That this is a well-definedexpression, indeed, is possibly not obvious at first sight, but follows from A -linearity ofthe coproduct as well as Eqs. (3.4), (D.12), (D.17), and (D.20). Explicitly on an element p v | . . . | v n ´ p ` q P U b A n ´ p ` and all short-hand notations written out, this reads as ` p u | . . . | u p q ‚ f ˘ p v | . . . | v n ´ p ` q“ γ ` p¨qr`s u r`s f p u r´sp qp¨qr´sp q u | . . . | u r´sp p ´ qp¨qr´sp p ´ q u p | u r´sp p q v | . . . | u r´sp n q v n ´ p ` q ˘ . Then, along with the cyclic operator p tf qp v | . . . | v n q “ γ ´ pp p¨q v q ➣ f q ` v | . . . | v n | ˘¯ from (4.5), we can turn the opposite O -module M into a cyclic one: Theorem 5.2. If M is a stable aYD contramodule over a left bialgebroid p U, A q whichis both left and right Hopf, the extra operation ( ) turns p M , t q into a cyclic unitalopposite module over the operad with multiplication p O , µ, e q , the underlying cyclic k -module structure of which coincides with the one of the cyclic dual of C ‚ p U, M q given inEqs. ( ) – ( ) .Proof. We first prove the second statement regarding the cyclic k -module structure. Asthe cyclic operator coincides by construction, we only need to check that the simplicialstructure defined in Eqs. (4.3)–(4.4) coincides with the one originating from being a cyclicopposite O -module by the general construction in Eqs. (C.6): with µ “ p U | U q , oneimmediately sees from (5.5) and (5.3) that p µ ‚ f qp v | . . . | v n ´ q “ γ ` p¨q r`s f p p¨q r´s | v | . . . | v n ´ q ˘ (5.6) p µ ‚ i f qp v | . . . | v n ´ q “ f p v | . . . | ∆ v i | . . . | v n ´ q , NONCOMMUTATIVE CALCULUS ON THE CYCLIC DUAL OF
Ext for i “ , . . . , n , which are the first two lines in Eqs. (4.3); as for the last face, compute p µ ‚ tf qp v | . . . | v n ´ q (5.6) “ γ ` p¨q r`s tf p p¨q r´s | v | . . . | v n ´ q ˘ (4.5) “ γ ´ p¨q r`s : γ ` p¨¨q r`s p¨q r´sr`s f pp p¨q r´sr´s p¨¨q r´s q p v | . . . | v n ´ | qq ˘¯ (3.2) , (D.17) “ γ ´ : γ ` p¨q r`s`p q p¨q r`s´r`s p¨¨q r`s p¨q r`s`p qr`s p¨q r´sr`s f pp p¨q r´sr´s p¨q r`s`p qr´s p¨¨q r´s p¨q r`s´r´s q p v | . . . | v n ´ | qq ˘¯ (D.22) , (D.4) “ γ ´ : γ ` p¨¨q r`s p¨q r`sp qr`s p¨q r´sr`s f pp p¨q r´sr´s p¨q r`sp qr´s p¨¨q r´s p¨q r`sp q q p v | . . . | v n ´ | qq ˘¯ (D.17) , (D.13) “ γ ´ : γ ` p p¨¨q r`s f pp p¨¨q r´s p¨q q p v | . . . | v n ´ | qq ˘¯ (1.5) “ γ ` p¨q p qr`s f pp p¨q p qr´s p¨q p q q p v | . . . | v n ´ | qq ˘ (D.14) “ γ ` p¨q f p v | . . . | v n ´ | q ˘ (3.3) “ f p v | . . . | v n ´ | q , which is the last line in (4.3). For the degeneracies we obtain for all j “ , . . . , n bysimply staring at (5.4) along with (5.2) p e ‚ j ` f qp v | . . . | v n ` q “ f ` v | . . . | ε p v j ` q| . . . | v n ` ˘ , which is (4.4).To conclude the proof, we have to check Eq. (C.5) in this situation, that is t p w ‚ i f q “ w ‚ i ` f for ď i ď n ´ p and w P O p p q , f P M p n q but we are going to do this only for i “ as this is the most difficult case; the verification for ď i ď n ´ p will be left to thereader. Indeed, for w “ p u | . . . | u p q P O p p q , we have p t p w ‚ f qqp v | . . . | v n ´ p ` q (4.5) “ γ ´` p p¨q v q ➣ p w ‚ f q ˘` v | . . . | v n ´ p ` | ˘¯ (5.5) “ γ ´ p p¨q v q r`s : γ ` p¨¨qr`s p u ➣ f q ` p¨¨qr´s p u | . . . | u p q|p p¨q v q r´s p v | . . . | v n ´ p ` | q ˘˘¯ (3.2) , (D.17) “ γ ´ : γ ` p p¨q v q r`s`p q p p¨q v q r`s´r`sp¨¨qr`s p p¨q v q r`s`p qr`s p u ➣ f q ` p p¨q v q r`s`p qr´sp¨¨qr´s p p¨q v q r`s´r´s p u | . . . | u p q|p p¨q v q r´s p v | . . . | v n ´ p ` | q ˘˘¯ (D.22) , (D.4) “ γ ´ : γ ` p¨¨qr`s p p¨q v q r`sp qr`s p u ➣ f q ` p p¨q v q r`sp qr´sp¨¨qr´s p p¨q v q r`sp q p u | . . . | u p q|p p¨q v q r´s p v | . . . | v n ´ p ` | q ˘˘¯ (1.5) , (D.17) “ γ ` p¨qp qr`sp¨qp qr`sp qr`s v r`sp qr`s p u ➣ f q ` v r`sp qr´sp¨qp qr`sp qr´sp¨qp qr´sp¨qp qr`sp q v r`sp q p u | . . . | u p q|p p¨q v q r´s p v | . . . | v n ´ p ` | q ˘˘ (D.15) , (D.14) “ γ ` p¨qr`sp qp¨qr`sp qr`s v r`sp qr`s p u ➣ f q ` v r`sp qr´sp¨qr`sp qr´s v r`sp q p u | . . . | u p q|p v r´sp¨qp qr´s q p v | . . . | v n ´ p ` | q ˘˘ (D.15) “ γ ` p¨qp qp¨qp qr`sr`s v r`sp qr`s p u ➣ f q ` v r`sp qr´sp¨qp qr`sr´s v r`sp q p u | . . . | u p q|p v r´sp¨qp qr´s q p v | . . . | v n ´ p ` | q ˘˘ (1.5) “ γ ´ : γ ` p¨¨qp¨qr`sr`s v r`sp qr`s p u ➣ f q ` v r`sp qr´sp¨qr`sr´s v r`sp q p u | . . . | u p q|p v r´sp¨qr´s q p v | . . . | v n ´ p ` | q ˘˘¯ (3.3) “ γ ` p¨qr`sr`s v r`sp qr`s p u ➣ f q ` v r`sp qr´sp¨qr`sr´s v r`sp q p u | . . . | u p q|p v r´sp¨qr´s q p v | . . . | v n ´ p ` | q ˘˘ (D.15) , (D.16) “ γ ` p p¨q v q r`s p u ➣ f q ` pp p¨q v p q q r´sp q v p q q p u | . . . | u p q|p p¨q v p q q r´sp q p v | . . . | v n ´ p ` | q ˘˘ “ γ ´ pp p¨q v p q u q ➣ f q ` v p q p u | . . . | u p q| v | . . . | v n ´ p ` | ˘¯ (4.5) “ tf ` v p q u | v p q p u | . . . | u p q| v | . . . | v n ´ p ` ˘ (5.3) “ p w ‚ tf qp v | . . . | v n ` p ´ q , which finishes the proof. (cid:3) From [Ko1, Thm. 5.4], one then obtains at once
Corollary 5.3.
The couple p O , M q as defined above can be equipped with the structureof a homotopy Cartan calculus if M is a stable aYD contramodule over U . In particular,this induces the structure of a BV module on Coext U ‚ p A, M q over Cotor ‚ U p A, A q . With the help of Eqs. (C.7)–(C.8), we can then explicitly obtain the operations that de-fine the calculus structure, cf. §C.1. For example, for the cap product ι , the cyclic cobound-ary B , and the Lie derivative L , a not really quick computation using Eqs. (3.2), (4.5), (5.1),(5.2), (5.3), (5.5), and basically all of the identities (D.3)–(D.23) yields: p Bf qp v | . . . | v n q “ n ` ř i “ p´ q p i ´ q n γ ´ p¨q r`s p v i ➣ f q ` p¨q r´s p v i ` | . . . | v n ` q| v | . . . | v i ´ ˘¯ p ι w f qp v | . . . | v n ´ p q “ γ ` p¨q r`s f p p¨q r´s p u | . . . | u p q| v | . . . | v n ´ p q ˘ (5.7) p L w f qp v | . . . | v n ´ p ` q “ n ´ p ` ř i “ p´ q p p ´ qp i ´ q f ` v | . . . | v i ´ | v i p u | . . . | u p q| v i ` | . . . | v n ´ p ` ˘ ` p ř i “ p´ q n p i ´ q` p ´ γ ´ p¨q r`s p u i ➣ f q ` p¨q r´s p u i ` | . . . | u p q| v | . . . | v n ´ p ` | u | . . . | u i ´ ˘¯ for w : “ p u | . . . | u p q P O p p q , f P M p n q , and p v | . . . | v k q P U b A k . Here, if i ă j appears in a sum, an element p u j | . . . | u i q has to be read as A : for example, in the cyclicboundary B the first and the last term have to be read as γ ` p p´q v ➣ f q ` v | . . . | v n ` ˘˘ resp. γ ` p´q p v n ` ➣ f q ` v | . . . | v n ˘˘ , and similarly in the expression for the Lie derivative.We spare the reader at this point to be confronted with the explicit expressions of thehomotopy operators S and T . Remark 5.4.
As already mentioned, the restriction to trivial coefficients (that is, the basealgebra A ) in the operadic structure of C p co p U, A q is not necessary and has only been madeto avoid too cumbersome formulæ that might obscure the general idea. Replacing A by a(braided) commutative monoid in the braided category of Yetter-Drinfel’d modules wouldalso work, see [Ko2, Thm. 1.3].6. E XAMPLE : C
ARTAN CALCULI IN DIFFERENTIAL GEOMETRY
We already briefly mentioned that the noncommutative calculus on
Coext and
Cotor contains the classical Cartan calculus known from differential geometry as an examplein a natural way. In a more restricted context, this was already achieved in [KoKr] by acalculus on
Ext and
Tor which, however, passes through a sort of double dual, and as aconsequence requires a certain finiteness condition, the use of topological tensor productsas well as completions. As we will explain now, the calculus structure obtained in the pre-vious section applied to the special case of differential geometry does not ask for anythingof all that and therefore yields a much more direct and even more general approach as onecan start from Lie-Rinehart algebras of infinite dimension.6.1.
The homotopy calculus structure for cocommutative bialgebroids.
In a cocom-mutative bialgebroid p U, A q , the base algebra A is necessarily commutative and the sourcemap equals the target one. This, in turn, implies that there exceptionally exists a trivial con-traaction as discussed in Example 1.2: any right A -module M is a right U -contramoduleby means of Hom A p U, M q Ñ M , f ÞÑ f p q ; if the right A -module M also happens to NONCOMMUTATIVE CALCULUS ON THE CYCLIC DUAL OF
Ext be a left U -module, it automatically becomes a stable aYD contramodule over U , that is,fulfils Eqs. (3.1)–(3.3) as one quickly verifies by Eqs. (D.6) and (D.4). Using the triv-ial contraaction notably simplifies the structure maps of the cocyclic k -module C ‚ p U, M q from Lemma 4.4, which for any f P C n p U, M q now become p d i f qp u | . . . | u n ´ q “ $&% f p | u | . . . | u n ´ q f p u | . . . | ∆ u i | . . . | u n ´ q f p u | . . . | u n ´ | q if i “ , if ď i ď n ´ , if i “ n, (6.1) p s j f qp u | . . . | u n ` q “ f p u | . . . | ε p u j ` q| . . . | u n ` q for ď j ď n, p tf qp u | . . . | u n q “ p u ➣ f qp u | . . . | u n | q . Observe that in this situation D ‚ p U, M q and C ‚ p U, M q are not only isomorphic as com-plexes but equal, that is, b “ d , as seen from Eq. (2.7) and Eq. (4.3). Also note that now d “ Hom A pB , M q (6.2)in case the left and right U -comodules in (2.3) are given by A itself, where B is the differ-ential of the cochain complex C ‚ co p U, A q computing Cotor ‚ U p A, A q .On top, the trivial contraaction notably entangles the calculus operators from Eqs. (5.7),which reduce to p Bf qp v | . . . | v n ` q “ n ` ř i “ p´ q p i ´ q n p v i ➣ f qp v i ` | . . . | v n ` | v | . . . | v i ´ q (6.3) p ι w f qp v | . . . | v n ´ p q “ f p u | . . . | u p | v | . . . | v n ´ p q (6.4) p L w f qp v | . . . | v n ´ p ` q “ n ´ p ` ř i “ p´ q p p ´ qp i ´ q f ` v | . . . | v i ´ | v i p u | . . . | u p q| v i ` | . . . | v n ´ p ` ˘ (6.5) ` p ř i “ p´ q n p i ´ q` p ´ p u i ➣ f qp u i ` | . . . | u p | v | . . . | v n ´ p ` | u | . . . | u i ´ q , for w “ p u | . . . | u p q . In particular, ι now becomes a simple insertion of w into f resp.literally a contraction of f by the element w . Furthermore, observe that in a cocommutativeleft bialgebroid there is no distinction between left and right Hopf algebroid structure, thatis u ` b A op u ´ “ u r`s b A u r´s for any u P U , and therefore also u ➢ f “ u ➣ f .6.2. Lie-Rinehart algebras and classical Cartan calculus.
Let p A, L q be a Lie-Rinehartalgebra (see [Ri] for details), with L not necessarily finitely generated as a module over thecommutative k -algebra A , and write the anchor map L Ñ Der k p A q as X ÞÑ t a ÞÑ X p a qu .We call the elements of the exterior algebra Ź ‚ A L over A multivector fields . Then the triple p Ź ‚ A L, , r¨ , ¨s SN q defines a dg-Lie algebra with respect to the zero differential along withthe Schouten-Nijenhuis bracket r¨ , ¨s SN over A , and a Gerstenhaber algebra if we add thewedge product. Let M be both an A -module and a left L -module, where the two actionsdo not commute but rather reflect the presence of the anchor map, which is equivalent tosaying that M is a VL -module. The dual space Hom A p Ź ‚ A L, M q of alternating M -valued A -multilinear forms constitutes a mixed complex ` Hom A p Ź ‚ A L, M q , , d dR ˘ , where d dR : Hom A p Ź nA L, M q Ñ
Hom A p Ź n ` A L, M q is the de Rham-Chevalley-Eilenberg differential d dR ω p X , . . . , X n ` q : “ n ` ř i “ p´ q i ´ X i ` ω p X , . . . , ˆ X i , . . . , X n q ˘ ` ř i ă j p´ q i ` j ´ ω pr X i , X j s , X , . . . , ˆ X i , . . . , ˆ X j , . . . , X n ` q , (6.6) where as usual ˆ X i means omission. Moreover, recall from [Ri] the cap product (or con-traction , not to be confused with contraaction) and Lie derivative for a multivector field Y : “ Y ^ ¨ ¨ ¨ ^ Y p , that is, ι : Ź pA L b Hom A p Ź nA L, M q Ñ
Hom A p Ź n ´ pA L, M q ,ι Y ω p X , . . . , X n ´ p q : “ ω p Y , . . . , Y p , X , . . . , X n ´ p q , L : Ź pA L b Hom A p Ź nA L, M q Ñ
Hom A p Ź n ´ p ` A L, M q , L Y ω p X , . . . , X n ´ p ` q : “ p ř i “ p´ q i ´ Y i ` ω p Y , . . . , ˆ Y i , . . . , Y p , X , . . . , X n ´ p ` q ˘ ` p ř j “ n ´ p ` ř i “ p´ q j ω p Y , . . . , ˆ Y j , . . . , Y p , X , . . . , r Y j , X i s , . . . , X n ´ p ` q` ř i ă j p´ q i ` j ´ ω pr Y i , Y j s , Y , . . . , ˆ Y i , . . . , ˆ Y j , . . . , Y p , X , . . . , X n ´ p ` q . If we additionally choose the homotopy operators S “ T “ to be zero, this yields a ho-motopy calculus on the pair `Ź nA L, Hom A p Ź nA L, M q ˘ . More precisely, the (co)simplicialdifferentials being zero, this even furnishes a calculus fulfilling the customary identities(C.3). In case p A, L q “ p C p Q q , Γ p E qq arises from a Lie algebroid E Ñ Q over a smoothmanifold Q as introduced in [Pr], the above calculus is the one given by E -differentialforms and E -multivector fields (as for example detailed in [CaWe, §18]), and if the Liealgebroid is given by the tangent bundle T Q , this yields the well-known Cartan calculus indifferential geometry [C].6.3.
The bialgebroid of differential operators.
The universal enveloping algebra VL (as introduced in [Ri]) of a Lie-Rinehart algebra p A, L q is not only a left bialgebroid butalso a left and right Hopf algebroid over this left bialgebroid structure; this still does notgive a (full) Hopf algebroid in the sense of [Bö] as in general an antipode does not exist[KoPo, Prop. 3.11]. The algebra VL is generated by elements a P A and X P L , andthe left and right Hopf algebroid structure on VL comes out as follows: source and targetmaps are equal and equal the canonical injection A Ñ VL , henceforth suppressed fromnotation. We therefore modify the notation for the tensor products in the Hopf-Galoismaps (D.2) by indicating the position of the elements in A in the quotient, that is, write VL b ll VL : “ VL Ž b A Ż VL and VL b lr VL : “ § VL b A op VL Ž , which in this casecoincides with VL đ b A Ż VL . On generators, the structure maps then read as ∆ p X q “ X b ll ` b ll X, X ` b rl X ´ “ X r`s b rl X r´s “ X b rl ´ b rl X, ∆ p a q “ a b ll , a ` b rl a ´ “ a r`s b rl a r´s “ a b rl , (6.7) along with ε p X q “ and ε p a q “ a . If p A, L q arises from a Lie algebroid as above, onemight want to consider VL as the space of differential operators on a smooth manifold.The bialgebroid VL is, in particular, cocommutative and hence the considerations madein the preceding section §6.1 apply.6.4. The Hochschild-Kostant-Rosenberg map.
From this section onwards, let us as-sume that Q Ď k and that L is A -flat. For simplicity, as mentioned before, let us put P “ A for the complexes defined in §2.1 and §2.2, to keep the formulæ on a reasonable level ofcomplexity. The Hochschild-Kostant-Rosenberg (HKR) map of antisymmetrisation in thiscontext reads: Alt : Ź nA L Ñ VL b llA n X ^ ¨ ¨ ¨ ^ X n ÞÑ { n ! ř σ P S p n q p´ q σ p X σ p q | . . . | X σ p n q q . (6.8)The first statement of the following lemma is well-known in its various guises, see, e.g. ,[Ko2, Thm. 3.13] for the statement in precisely the same context as here. The secondstatement simply follows from the observation made in (6.2) and the comments at thebeginning of §6.1. Lemma 6.1.
The HKR map is a quasi-isomorphism of cochain complexes which inducesan isomorphism Ź ‚ A L Ñ Cotor ‚ U p A, A q of Gerstenhaber algebras. For a left VL -module NONCOMMUTATIVE CALCULUS ON THE CYCLIC DUAL OF
Ext M (seen also as a trivial right VL -contramodule), the pull-back Hom A p Alt , M q yieldsa quasi-isomorphism ` C ‚ p U, M q , d ˘ Ñ ` Hom A p Ź ‚ A L, M q , ˘ of chain complexes. Inparticular, in homology Coext VL ‚ p A, M q »
Hom A p Ź ‚ A L, M q (6.9) holds. Observe that we only assume A -flatness of L here, but not necessarily finite dimensions.That Alt is a quasi-isomorphism even in case L is infinite dimensional follows from anargument as in the proof of [Lo, Thm. 3.2.2], see also [CE, §XIII.7] for the free case. Ontop, the HKR map induces a map of mixed complexes, see, e.g. , [Ko2, Thm. 3.13] again aswell as the subsequent theorem. With the help of this lemma, one can now prove the mainresult in this example section, which tells us that the classical Cartan calculus is containedin the more general approach presented here: Theorem 6.2.
Let p A, L q be a Lie-Rinehart algebra, with L projective but notnecessarily finitely generated as an A -module. Then the HKR map Alt inducesan isomorphism of BV modules (or calculi) between `Ź nA L, Hom A p Ź nA L, M q ˘ and ` Cotor ‚ VL p A, A q , Coext VL ‚ p A, M q ˘ . In particular, this means that Alt (resp. its pull-back)commutes with all possible calculus operators in the sense of
Hom A p Alt , M q ˝ B “ d dR ˝ Hom A p Alt , M q , (6.10) Hom A p Alt , M q ˝ L Alt p Y q “ L Y ˝ Hom A p Alt , M q , (6.11) Hom A p Alt , M q ˝ ι Alt p Y q “ ι Y ˝ Hom A p Alt , M q , (6.12) for a multivector field Y P Ź ‚ A L , where the last two identities hold on homology only.Proof. The statement essentially follows from the preceding Lemma 6.1 along withEqs. (6.3)–(6.5). Indeed, for any f P C n ´ p VL, A q , we have using (6.3), (0.3), and (6.7) pp Bf q ˝ Alt qp X ^ ¨ ¨ ¨ ^ X n q“ n ! ř σ P S p n q p´ q σ n ř i “ p´ q p i ´ q n p X σ p i q ➣ f qp X σ p i ` q | . . . | X σ p n q | X σ p q | . . . | X σ p i ´ q q“ nn ! ř σ P S p n q p´ q σ p X σ p q ➣ f qp X σ p q | . . . | X σ p n q q“ p n ´ q ! ř σ P S p n q p´ q σ X σ p q f p X σ p q | . . . | X σ p n q q´ p n ´ q ! ř σ P S p n q p´ q σ n ´ ř i “ f p X σ p q | . . . | X σ p i q X σ p i ` q | . . . | X σ p n q q“ p n ´ q ! ř τ P S p n ´ q p´ q τ n ř i “ p´ q i ´ X i ` f p X σ p q | . . . | X σ p i ´ q | X σ p i ` q | . . . | X σ p n q q ˘ ´ p n ´ q ! ř σ P S p n q p´ q σ n ´ ř i “ f p X σ p q | . . . | X σ p i q X σ p i ` q | . . . | X σ p n q q“ n ř i “ p´ q i ´ X i ` p f ˝ Alt qp X , . . . , ˆ X i , . . . , X n q ˘ ` ř i ă j p´ q i ` j ´ p f ˝ Alt qpr X i , X j s , X , . . . , ˆ X i , . . . , ˆ X j , . . . , X n q“ d dR p f ˝ Alt qp X , . . . , X n q , using the fact that Alt is a map of Gerstenhaber algebras and hence
Alt pr X, Y sq “ XY ´ Y X , where on the right hand side
X, Y are seen as elements in VL . Hence, Eq. (6.10) isproven. As for Eq. (6.11), as in [Ka, Prop. XVIII.7.6] we rather consider the left inverse of Alt given by P : “ pr ^ ¨ ¨ ¨ ^ pr , where pr : VL Ñ L denotes the natural projection, and which is a quasi-isomorphism as well. One immediately sees from (6.4) that ι w p θ ˝ P qp v | . . . | v n ´ p q “ p θ ˝ P qp w | v | . . . | v n ´ p q“ θ p pr p u q ^ ¨ ¨ ¨ ^ pr p u p q ^ pr p v q ^ ¨ ¨ ¨ ^ pr p v n ´ p qq“ ` p ι P p w q θ q ˝ P ˘ p v | . . . | v n ´ p q for w “ p u | . . . | u p q and θ P Hom p Ź nA L, M q . By functoriality, in homology the map P also becomes a right inverse of Alt and (6.11) follows. The last equation follows from thepreceding two along with (C.3) when descending to (co)homology. (cid:3)
Lie-Rinehart cohomology and cyclic homology.
We conclude this example sectionby a few words on the relation between Lie-Rinehart cohomology and cyclic cohomology.Ignoring the zero differential in the mixed complex ` Hom A p Ź ‚ A L, M q , , d dR ˘ , one ob-tains a Chevalley-Eilenberg cochain complex that generalises the classical complex com-puting Lie algebra cohomology: Definition 6.3. [Ri] The cohomology of the cochain complex ` Hom A p Ź ‚ A L, M q , d dR ˘ denoted by H ‚ p L, M q is called the Lie-Rinehart cohomology (with values in M ) of p A, L q .Dually to Lemma 6.1, it is a well-known fact that if L is A -projective, then the Lie-Rinehart cohomology is an Ext -group again, see [Ri, §4]. More precisely, together withEq. (6.9) we have
Ext ‚ VL p A, M q » H ‚ p L, M q , Coext VL ‚ p A, M q »
Hom A p Ź ‚ A L, M q , which allows us to state: Proposition 6.4.
Let p A, L q be a Lie-Rinehart algebra, where L is projective but not nec-essarily finite as an A -module, and M a left VL -module. Then the HKR map ( ) inducesthe isomorphisms HC n p VL, M q »
Hom A p Ź nA L, M q{ d dR ` Hom A p Ź n ´ A L, M q ˘ ‘ H n ´ p L, M q ‘ H n ´ p L, M q ‘ ¨ ¨ ¨ ,HC n p VL, M q » H n p L, M q ‘ H n ´ p L, M q ‘ ¨ ¨ ¨ , where HC ‚ denotes the cyclic homology defined by the cyclic module ( ) , and HC ‚ thecyclic cohomology with respect to the cocyclic module ( ) for the trivial contraaction.Proof. The first isomorphism follows from Lemma 6.1 together with Eq. (6.10) by com-puting the total homology of the trivial mixed complex ` Hom A p Ź ‚ A L, M q , , d dR ˘ . Thesecond isomorphism follows from the fact that the cyclic boundary B associated to thecocyclic module (3.10) for the trivial contraaction in case of a cocommutative bialgebroidinduces the zero map in the cohomology of the columns of the respective mixed complex,which can be either computed directly or obtained by simply A -linearly dualising [KoPo,Thm. 2.16 & 3.14]. (cid:3) A PPENDIX
A. T
HE CYCLIC CATEGORY
A.1.
Cocyclic and para-cocyclic modules.
Recall from, e.g. , [Lo, §6.1] that a cyclic k -module is a simplicial k -module p C ‚ , d ‚ , s ‚ q resp. a cocyclic k -module is a cosimplicial k -module p C ‚ , δ ‚ , σ ‚ q together with k -linear maps t : C n Ñ C n resp. τ : C n Ñ C n indegree n , satisfying, respectively d i ˝ t “ " t ˝ d i ´ if ď i ď n,d n if i “ ,s i ˝ t “ " t ˝ s i ´ if ď i ď n,t ˝ s n if i “ ,t n ` “ id C n , τ ˝ δ i “ " δ i ´ ˝ τδ n if ď i ď n, if i “ ,τ ˝ σ i “ " σ i ´ ˝ τσ n ˝ τ if ď i ď n, if i “ ,τ n ` “ id C n . (A.1) NONCOMMUTATIVE CALCULUS ON THE CYCLIC DUAL OF
Ext In the definition of a para-cyclic resp. para-cocyclic k -module one drops the last iden-tity, that is, the cyclic resp. cocyclic operator does not power to the identity any more.More conceptually, cyclic k -modules resp. cocyclic ones can be viewed as functors Λ op Ñ k - Mod resp. Λ Ñ k - Mod , where Λ is Connes’ cyclic category, see loc. cit. for a detailed description. A cyclic k -module allows to introduce the cyclic (or Connes-Rinehart-Tsygan ) boundary B : “ p ´ p´ q n t q s ´ N , (A.2)where s ´ : “ ts n is the extra degeneracy and N : “ ř ni “ p´ q i ` n t in the norm operator ;an analogous construction leads to the cyclic co boundary in case of a cocyclic k -module. Inboth cases, together with the respective (co)simplicial (co)boundary summing all (co)faceswith alternating sign, this leads to a mixed complex, the total (co)homology of whichdefines cyclic (co)homology.A.2. The cyclic dual.
It is a well-known fact (see [Co1] or [Lo, Prop. 6.1.11]) that thecyclic category Λ is self-dual, which allows to identify cocyclic k -modules and cyclic k -modules, even in infinitely many ways due to the autoequivalences of the cyclic category[Lo, §6.1.14]. The standard choice to pass from a cocyclic module p X ‚ , δ ‚ , σ ‚ , τ q to acyclic module p X ‚ , d ‚ , s ‚ , t q is given by setting X n : “ X n for all n P N along with d : “ σ n ´ τ, d i : “ σ i ´ , s j : “ δ j , t : “ τ ´ (A.3)for ď i ď n and ď j ď n . Observe that in this convention the last coface δ n ` is notused. A PPENDIX
B. A
LGEBRAIC OPERADS
B.1.
Operads and Gerstenhaber algebras. A non- Σ operad O in the category k - Mod of k -modules is a sequence t O p n qu n ě of k -modules endowed with k -bilinear operations ˝ i : O p p q b O p q q Ñ O p p ` q ´ q for i “ , . . . , p subject to ϕ ˝ i ψ “ if p ă i or p “ , p ϕ ˝ i ψ q ˝ j χ “ $’&’% p ϕ ˝ j χ q ˝ i ` r ´ ψ if j ă i,ϕ ˝ i p ψ ˝ j ´ i ` χ q if i ď j ă q ` i, p ϕ ˝ j ´ q ` χ q ˝ i ψ if j ě q ` i. (B.1)Call the operad unital if there is an identity P O p q such that ϕ ˝ i “ ˝ ϕ “ ϕ for all ϕ P O p p q and i ď p , and call the operad with multiplication if there exist a multiplication µ P O p q and a unit e P O p q such that µ ˝ µ “ µ ˝ µ and µ ˝ e “ µ ˝ e “ . Anoperad with multiplication will be denoted p O , µ, e q . Such an object naturally defines acosimplicial k -module given by O p : “ O p p q with faces and degeneracies for ϕ P O p p q given by δ ϕ : “ µ ˝ ϕ , δ i ϕ : “ ϕ ˝ i µ for i “ , . . . , p , and δ p ` ϕ : “ µ ˝ ϕ , alongwith σ j p ϕ q : “ ϕ ˝ j ` e for j “ , . . . , p ´ . One obtains a cochain complex denotedby the same symbol O , with O p n q in degree n , differential O p n q Ñ O p n ` q given by δ : “ ř n ` i “ p´ q i δ i , and cohomology H ‚ p O q : “ H p O , δ q . Define then the cup product ψ ` ϕ : “ p µ ˝ ψ q ˝ ϕ P O p p ` q q , (B.2)for ϕ P O p p q and ψ P O p q q . As a consequence, p O , ` , δ q determines a dg algebra. Onefurthermore defines the Gerstenhaber bracket as t ϕ, ψ u : “ ϕ t ψ u ´ p´ q p p ´ qp q ´ q ψ t ϕ u , (B.3)where ϕ t ψ u : “ ř pi “ p´ q p q ´ qp i ´ q ϕ ˝ i ψ P O p p ` q ´ q is the sum over all possiblepartial compositions. Observe that t µ, µ u “ as well as δϕ “ p´ q p ` t µ, ϕ u . (B.4)It is well-known that in cohomology p H ‚ p O q , ` , t¨ , ¨uq constitutes a Gerstenhaber algebra. A PPENDIX
C. N
ONCOMMUTATIVE CALCULI AND OPPOSITE OPERAD MODULES
C.1.
Noncommutative differential calculi [GeDaTs] . Let p X ‚ , b, B q be a mixed com-plex, and let p G ‚ , δ, t¨ , ¨u , ` q be both a dg associative algebra and a dg Lie algebra (withdegree shifted by one) such that its cohomology H ‚ p G , δ q is a Gerstenhaber algebra (whichone may refer to as homotopy Gerstenhaber algebra). The mixed complex X is called a homotopy Gerstenhaber module over G if p X ´‚ , b q is both a dg module over p G ‚ , ` , ι q anda dg Lie algebra module over p G ‚ r s , t¨ , ¨u , L q by means of two respective actions ι : G p b X n Ñ X n ´ p , L : G p b X n Ñ X n ´ p ` , called cap product (or contraction ) and Lie derivative , respectively, such that, writing ι ϕ : “ ι p ϕ b ¨q for ϕ P G and similarly for all operators in the sequel, the Gelfan’d-Daletski˘ı-Tsygan homotopy formula r ι ϕ , L ψ s ´ ι t ϕ,ψ u “ r b, T ϕ,ψ s ´ T δϕ,ψ ´ p´ q ϕ T ϕ,δψ (C.1)holds, where T is an operator T : G p b G q b X n Ñ X n ` p ` q ´ . A homotopy Gerstenhabermodule X is called homotopy Batalin-Vilkoviski˘ı (BV) module over G if there is an addi-tional operator S : G p b X n Ñ X n ` p ´ such that the Cartan-Rinehart homotopy formulæ L ϕ “ r B, ι ϕ s ` r b, S ϕ s ` S δϕ , r S ϕ , L ψ s ´ S t ϕ,ψ u “ r B, T ϕ,ψ s (C.2)are verified. A Gerstenhaber resp. BV module is then defined by analogous relations thatwould hold on homology H ‚ p M, b q and cohomology H ‚ p G , δ q (which then becomes a trueGerstenhaber algebra) setting all homotopy terms to zero. For example, in case of a BVmodule one has the following relations: ι ϕ ` ψ “ ι ϕ ι ψ , L t ϕ,ψ u “ r L ϕ , L ψ s , r ι ϕ , L ψ s “ ι t ϕ,ψ u , L ϕ “ r B, ι ϕ s . (C.3)Inspired by the obvious resemblance of these identities with the well-known ones in dif-ferential geometry, a BV module structure is also called a noncommutative differentialcalculus in [Ts1] and a noncommutative Cartan calculus in [FiKo]; one might also want tocall this a Tamarkin-Tsygan calculus since these structures are analysed in detail in [TaTs].In this spirit, one may equally speak of a homotopy noncommutative differential/Cartancalculus or simply a homotopy calculus on the pair p G , X q instead of a homotopy BVmodule X over G .C.2. (Cyclic) opposite O -modules [Ko1] . Let O be an operad with partial composi-tion denoted by ˝ i , as above. A (left) opposite O -module is a sequence of k -modules t M p n qu n ě together with k -linear operations, ‚ i : O p p q b M p n q Ñ M p n ´ p ` q for i “ , . . . , n ´ p ` , declared to be zero if p ą n , and subject to ϕ ‚ i p ψ ‚ j x q “ $’&’% ψ ‚ j p ϕ ‚ i ` q ´ x q if j ă i, p ϕ ˝ j ´ i ` ψ q ‚ i x if j ´ p ă i ď j,ψ ‚ j ´ p ` p ϕ ‚ i x q if ď i ď j ´ p, (C.4)for ϕ P O p p q , ψ P O p q q , and x P M p n q , where p ą , q ě , n ě (in case p “ deletethe middle relation). An opposite O -module is called unital if ‚ i x “ x for i “ , . . . , n and all x P M p n q .A cyclic (unital, left) opposite O -module is a (unital, left) opposite O -module M en-dowed with two additional structures: an extra ( k -linear) composition map ‚ : O p p q b M p n q Ñ M p n ´ p ` q , ď p ď n ` , declared to be zero if p ą n ` such that the relations (C.4) and unitality are fulfilled for i “ as well; moreover, a degree-preserving morphism t : M p n q Ñ M p n q for all n ě with the property t n ` “ id M p n q and such that t p ϕ ‚ i x q “ ϕ ‚ i ` t p x q , i “ , . . . , n ´ p, (C.5) NONCOMMUTATIVE CALCULUS ON THE CYCLIC DUAL OF
Ext holds for ϕ P O p p q and x P M p n q .See [Ko1] or [FiKo] for more information, examples, and illustrations on (cyclic) oppo-site O -modules (termed “comp modules” in the former).A cyclic unital opposite module p M , t q over an operad with multiplication p O , µ, e q car-ries the structure of a cyclic k -module [Ko1, Prop. 3.5]: the faces d i : M p n q Ñ M p n ´ q and degeneracies s j : M p n q Ñ M p n ` q of the underlying simplicial object given by d i p x q “ µ ‚ i x, i “ , . . . , n ´ ,d n p x q “ µ ‚ t p x q ,s j p x q “ e ‚ j ` x, j “ , . . . , n, (C.6)where x P M p n q , can be easily shown to be compatible with the cyclic operator t in the sense of Eqs. (A.1). Defining the differential b : M p n q Ñ M p n ´ q by b “ ř ni “ p´ q i d i , the pair p M , b q becomes a chain complex, and by means of B : M p n q Ñ M p n ` q defined as in Eq. (A.2), the triple p M , b, B q becomes a mixed (chain) complex.Here, the extra degeneracy turns out as s ´ : “ t s n “ e ‚ ´ , explaining the terminol-ogy extra operation for ‚ . To simplify matters, we will usually work on the normalisedcomplex Ď M , the quotient of M by the (acyclic) subcomplex spanned by the images of thedegeneracy maps. For example, on Ď M the cyclic coboundary simplifies to s ´ N , whichin this case becomes explicitly B p x q “ n ÿ i “ p´ q in e ‚ t i p x q . (C.7)Likewise, s O denotes the intersection of the kernels of the codegeneracies in the cosimpli-cial k -module obtained from the operad with multiplication p O , µ, e q .The nice feature of cyclic opposite O -modules is that they automatically turn into ho-motopy BV modules (see [Ko1, Thm. 5.4]): Theorem C.1.
The structure of a cyclic unital opposite module p M , t q over an operad withmultiplication p O , µ, e q induces a homotopy calculus on the pair p O , M q of k -modules. For use in the main text, we will give some explicit formulæ, see [Ko1] and also [FiKo,§6]. For ϕ P O p p q , ψ P O p q q , and x P M p n q ι ϕ x “ p µ ˝ ϕ q ‚ x, L ϕ x “ n ´ p ` ř i “ p´ q p p ´ qp i ´ q ϕ ‚ i x ` p ř i “ p´ q n p i ´ q` p ´ ϕ ‚ t i ´ p x q , S ϕ x “ n ´ p ` ř j “ n ´ p ` ř i “ j p´ q n p j ´ q`p p ´ qp i ´ q e ‚ ` ϕ ‚ i t j ´ p x q ˘ , T ϕ,ψ p x q “ p ´ ř j “ p ´ ř i “ j p´ q n p j ´ q`p q ´ qp i ´ j q` p p ϕ ˝ p ´ i ` j ψ q ‚ t j ´ p x q . (C.8)Observe the formal analogy between the cap product ι ϕ x “ : ϕ a x and the cup product ϕ ` ψ “ p µ ˝ ψ q ˝ ϕ in the operad O . With these explicit expressions, it is an essen-tially direct (but not-so-straightforward) check that on the normalised complex Ď M and forelements in s O , the homotopy formulæ (C.1) and (C.2) hold.A PPENDIX
D. L
EFT AND RIGHT H OPF ALGEBROIDS
D.1.
Bialgebroids [Tak] . A left bialgebroid p U, A, ∆ , ε, s, t q is a generalisation of a k -bialgebra over a noncommutative base ring A ; more precisely, it consists of a compati-ble algebra and coalgebra structure over A e resp. over A ; see, for example, [Bö] for alltechnical details. In particular, it comes along with a ring homomorphism resp. antihomo-morphism s, t : A Ñ U (called source resp. target) that equip U with four commuting A -module structures, denoted a § b Ż u Ž c đ d : “ t p c q s p b q us p d q t p a q (D.1) for u P U, a, b, c, d P A , and this situation will be abbreviated by §Ż U Žđ or any variationthereof, depending on the action considered in a specific construction. In the same spirit,there is an obvious forgetful functor U - Mod Ñ A e - Mod and therefore, for a left U -module M , we sometimes denote the induced A -bimodule structure by a Ż m Ž b : “ s p a q t p b q m for m P M , a, b P A . Furthermore, as mentioned, along with a product in U ,one has a coproduct ∆ : U Ñ U Ž ˆ A Ż U Ă U Ž b A Ż U , u ÞÑ u p q b A u p q and a counit ε : U Ñ A subject to certain technicalities which we are not going to explain here butrefer to [Tak] or elsewhere. Here, 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 ( is sometimes called Sweedler-Takeuchi product .D.2.
Left and right Hopf algebroids [Sch] . Generalising Hopf algebras (bialgebras withan antipode) to noncommutative base rings is less straightforward and instead of askingfor an antipode to exist, one rather wants a certain Hopf-Galois map to be invertible. Moreprecisely, for a left bialgebroid p U, A q , consider the U -module morphisms 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 , (D.2)and call the left bialgebroid p U, A q a left Hopf algebroid if α ℓ is a bijection and right Hopfalgebroid if α r is so. With the shorthand notation u ` b A op u ´ : “ α ´ ℓ p u b A q and u r`s b A u r´s : “ α ´ r p b A u q , one easily verifies that for a left Hopf algebroid u ` b A op u ´ P U ˆ A op U, (D.3) u `p q b A u `p q u ´ “ u b A P U Ž b A Ż U, (D.4) u p q` b A op u p q´ u p q “ u b A op P § U b A op U Ž , (D.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´ , (D.6) u ` b A op u ´p q b A u ´p q “ u `` b A op u ´ b A u `´ , (D.7) p uv q ` b A op p uv q ´ “ u ` v ` b A op v ´ u ´ , (D.8) u ` u ´ “ s p ε p u qq , (D.9) ε p u ´ q § u ` “ u, (D.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 (D.11)holds, where in (D.3) we mean 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 ( . If the left bialgebroid p U, A q is a right Hopf algebroid instead, one analogously obtains: u r`s b A u r´s P U ˆ A U, (D.12) u r`sp q u r´s b A u r`sp q “ b A u P U Ž b A Ż U, (D.13) u p qr´s u p q b A u p qr`s “ b A u P U đ b A Ż U , (D.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 , (D.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 , (D.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 , (D.17) u r`s u r´s “ t p ε p u qq , (D.18) u r`s đ ε p u r´s q “ u, (D.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 , (D.20)where in (D.12) we mean 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 ( . NONCOMMUTATIVE CALCULUS ON THE CYCLIC DUAL OF
Ext If the left bialgebroid p U, A q happens to be simultaneously a left and a right Hopf algebroid,it is an easy check that on top the mixed compatibility relations 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 , (D.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 , (D.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 . (D.23)hold between left and right Hopf structures. Let us conclude by remarking that a left bial-gebroid which is both left and right Hopf still does not imply the existence of an antipoderequired in the definition of a (full) Hopf algebroid in [Bö]: for example, the universalenveloping algebra VL from §6 in general does not admit an antipode [KoPo, Prop. 3.11].R EFERENCES[ArKe] M. Armenta and B. Keller,
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