Bialgebra Cyclic Homology with Coefficients Part II

This is the second part of the article [3]. In the first paper we developed a cyclic homology theory for B–module coalgebras with coefficients in stable B–module/comodules where B was just a bialgebra. The construction we gave for the cyclic homology theory for B–module coalgebras used mainly the coalgebra structure on B. In the first part of this paper, we present the dual picture. Namely, a cyclic homology theory for B–comodule algebras with coefficients in a stable B–module/comodule where B is just a bialgebra. Our theory is an extension of the theory developed in [2] by lifting two restrictions: (i) our theory uses bialgebras as opposed to Hopf algebras (ii) the coefficient module/comodules are just stable as opposed to stable antiYetter-Drinfeld. In the second part of this paper, we recover the main result of [4]. Namely, these two cyclic theories are dual in the sense of (co)cyclic objects, whenever the input pair (H,X) has the property that H is a Hopf algebra and X is a stable anti-Yetter-Drinfeld module.