(Co)cyclic (Co)homology of Bialgebroids: An Approach via (Co)monads
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- 作者：Gabriella B?hm and Drago? ?tefan
- 刊名：Communications in Mathematical Physics
- 出版年：2008
- 出版时间：August, 2008
- 年：2008
- 卷：282
- 期：1
- 页码：239-286
- 全文大小：947.5 KB
文摘
For a (co)monad T l on a category M{\mathcal{M}}, an object X in M{\mathcal{M}} , and a functor \varvec P:M ? C{{\varvec {\Pi}}:\mathcal{M} \to \mathcal{C}} , there is a (co)simplex Z*:=\varvec P Tl*+1 X{Z^\ast:={\varvec {\Pi} {T_l}}^{\ast +1} X} in C{\mathcal{C}} . The aim of this paper is to find criteria for para-(co)cyclicity of Z *. Our construction is built on a distributive law of T l with a second (co)monad T r on M{\mathcal{M}} , a natural transformation i:\varvec P Tl ? P Tr{i:{\varvec {\Pi} {T_l} \to {\bf \Pi} {T_r}}} , and a morphism w:\varvec TrX ? \varvec TlX{w:{\varvec {T_r}}X \to {\varvec {T_l}}X} in M{\mathcal{M}} . The (symmetrical) relations i and w need to satisfy are categorical versions of Kaygun’s axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads \varvec Tl=T ?R (-){{\varvec {T_l}}=T \otimes_R (-)} and \varvec Tr=(-)?R T{{\varvec {T_r}}=(-)\otimes_R T} on the category of R-bimodules. The functor Π can be chosen such that Zn=T[^(?)]R?[^(?)]R T [^(?)]RX{Z^n=T\widehat{\otimes}_R\cdots \widehat{\otimes}_R T \widehat{\otimes}_RX} is the cyclic R-module tensor product. A natural transformation i:T [^(?)]R (-) ? (-) [^(?)]R T{{i}:T \widehat{\otimes}_R (-) \to (-) \widehat{\otimes}_R T} is given by the flip map and a morphism w: X ?R T ? T?R X{w: X \otimes_R T \to T\otimes_R X} is constructed whenever T is a (co)module algebra or coring of an R-bialgebroid. The notion of a stable anti-Yetter-Drinfel’d module over certain bialgebroids, the so-called ¡Á R -Hopf algebras, is introduced. In the particular example when T is a module coring of a ¡Á R -Hopf algebra B{\mathcal{B}} and X is a stable anti-Yetter-Drinfel’d B{\mathcal{B}} -module, the para-cyclic object Z * is shown to project to a cyclic structure on T?R *+1 ?B X{T^{\otimes_R\, \ast+1} \otimes_{\mathcal{B}} X} . For a B{\mathcal{B}} -Galois extension S ¨ª T{S \subseteq T} , a stable anti-Yetter-Drinfel’d B{\mathcal{B}} -module T S is constructed, such that the cyclic objects B?R *+1 ?B TS{\mathcal{B}^{\otimes_R\, \ast+1} \otimes_{\mathcal{B}} T_S} and T[^(?)]S *+1{T^{\widehat{\otimes}_S\, \ast+1}} are isomorphic. This extends a theorem by Jara and Ştefan for Hopf Galois extensions. As an application, we compute Hochschild and cyclic homologies of a groupoid with coefficients in a stable anti-Yetter-Drinfel’d module, by tracing it back to the group case. In particular, we obtain explicit expressions for (coinciding relative and ordinary) Hochschild and cyclic homologies of a groupoid. The latter extends results of Burghelea on cyclic homology of groups.