(co)cyclic (co)homology of Bialgebroids: an Approach via (co)monads

For a (co)monad Tl on a category M, an object X in M, and a functor Π : M → C, there is a (co)simplex Z := ΠTl ∗+1 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 Tr on M, a natural transformation i : ΠTl → ΠTr , 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 Tl = T ⊗R (−) and Tr = (−) ⊗R T on the category of R-bimodules. The functor Π can be chosen such that Z = T b ⊗R . . . b ⊗RT b ⊗RX is the cyclic R-module tensor product. A natural transformation i : T b ⊗R(−) → (−)b ⊗RT is given by the flip map and a morphism w : X ⊗R T → T ⊗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, so called ×R-Hopf algebras, is introduced. In the particular example when T is a module coring of a ×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 TR ∗+1 ⊗B X. For a B-Galois extension S ⊆ T , a stable anti Yetter-Drinfel’d B-module TS is constructed, such that the cyclic objects BR ∗+1 ⊗B TS and T b ⊗S ∗+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. Latter extends results of Burghelea on cyclic homology of groups.