A model structure and Hopf-cyclic theory on the category of coequivariant modules over a comodule algebra

Abstract

Let H be a coFrobenius Hopf algebra over a field k. Let A be a right H-comodule algebra over k.

We recall that the category $M^H$ of right H-comodules admits a certain model structure whose homotopy category is equivalent to the stable category of right H-comodules given in [5]. In the first part of this paper, we show that the category ${\mathrm{LMod}}_A\left(M^H\right)$ of left A-module objects in $M^H$ admits a model structure, which becomes a model subcategory of the category of $A\#H^{⁎}$-modules endowed with a model structure given in [14] if H is finite dimensional with a certain assumption. Note that ${\mathrm{LMod}}_A\left(M^H\right)$ is not a Frobenius category in general. We also construct a functorial cofibrant replacement by proceeding the similar argument as in [16].

Hopf-cyclic theory is refered as a theory of cyclic homology of (co)module (co)algebra over a Hopf algebra H whose coefficients in Hopf H-modules. In the latter half of this paper, we see that cyclic H-comodules which give Hopf-cyclic (co)homology with coefficients in Hopf H-modules are contructible in the homotopy category of right H-comodules, and we investigate a Hopf-cyclic (co)homology in slightly modified setting by assuming A a right H-comodule k-Hopf algebra with H-colinear bijective antipode in stable category of right H-comodules and give an analogue of the characteristic map.