(Co)cyclic (co)homology of bialgebroids: An approach via (co)monads

摘要

For a (co) monad T-l on a category M, an object X in M, and a functor Pi : M --> C, there is a (co) simplex Z* := Pi T*(+1)(l) X in 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 Tl with a second (co) monad T-r on M, a natural transformation i : Pi T-l --> Pi T-r, and a morphism w : TrX --> TlX in 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 T-l = T circle times(R) (-) and T-r = (-) circle times(R) T on the category of R-bimodules. The functor Pi can be chosen such that Z(n) = T (circle times) over cap (R) ...(circle times) over cap T-R circle times(circle times) over cap X-R is the cyclic R-module tensor product. A natural transformation i : T (circle times) over cap circle times(R)(-) --> (-)(circle times) over cap T-R is given by the flip map and a morphism w : X circle times T-R --> T circle times(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'dmodule over certain bialgebroids, the so-called x(R)-Hopf algebras, is introduced. In the particular example when T is a module coring of a x(R)-Hopf algebra B and X is a stable anti-Yetter-Drinfel'd B-module, the para-cyclic object Z* is shown to project to a cyclic structure on T*((circle times) over capR+ 1)circle times(B) X. For a B-Galois extension S subset of T, a stable anti-Yetter-Drinfel'd B-module T-S is constructed, such that the cyclic objects B*(circle times R+ 1) circle times(B) T-S and T*((circle times) over capS+1) are isomorphic. This extends a theorem by Jara and Stefan 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.