(Co)cyclic (Co)homology of Bialgebroids: An Approach via (Co)monads

Abstract

For a (co)monad T _l on a category ${\mathcal{M}}$, an object X in ${\mathcal{M}}$ , and a functor ${{\varvec {\Pi}}:\mathcal{M} \to \mathcal{C}}$ , there is a (co)simplex ${Z^\ast:={\varvec {\Pi} {T_l}}^{\ast +1} X}$ in ${\mathcal{C}}$ . 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 ${\mathcal{M}}$ , a natural transformation ${i:{\varvec {\Pi} {T_l} \to {\bf \Pi} {T_r}}}$ , and a morphism ${w:{\varvec {T_r}}X \to {\varvec {T_l}}X}$ in ${\mathcal{M}}$ . 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 ${{\varvec {T_l}}=T \otimes_R (-)}$ and ${{\varvec {T_r}}=(-)\otimes_R T}$ on the category of R-bimodules. The functor Π can be chosen such that ${Z^n=T\widehat{\otimes}_R\cdots \widehat{\otimes}_R T \widehat{\otimes}_RX}$ is the cyclic R-module tensor product. A natural transformation ${{i}:T \widehat{\otimes}_R (-) \to (-) \widehat{\otimes}_R T}$ is given by the flip map and a morphism ${w: X \otimes_R T \to T\otimes_R X}$ 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  ×  _R -Hopf algebras, is introduced. In the particular example when T is a module coring of a  ×  _R -Hopf algebra ${\mathcal{B}}$ and X is a stable anti-Yetter-Drinfel’d ${\mathcal{B}}$ -module, the para-cyclic object Z _* is shown to project to a cyclic structure on ${T^{\otimes_R\, \ast+1} \otimes_{\mathcal{B}} X}$ . For a ${\mathcal{B}}$ -Galois extension ${S \subseteq T}$ , a stable anti-Yetter-Drinfel’d ${\mathcal{B}}$ -module T _S is constructed, such that the cyclic objects ${\mathcal{B}^{\otimes_R\, \ast+1} \otimes_{\mathcal{B}} T_S}$ and ${T^{\widehat{\otimes}_S\, \ast+1}}$ 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.