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

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 of right H-comodules admits a certain model structure whose homotopy category is equivalent to the stable category of right H-comodules given in Farina's paper. In the first part of this paper, we show that the category of left A-module objects in the category of right H-comodules admits a model structure, which becomes a model subcategory of the category of H*-equivariant A-modules endowed with a model structure given in the author's previous paper if H is finite dimensional with a certain assumption. Note that this category is not a Frobenius category in general. We also construct a functorial cofibrant replacement by proceeding the similar argument as in Qi's paper. 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. We remark that, as an expansion of an idea of taking trivial comodule k as the coefficients, if we take an A-coinvariant part of M assuming M a Hopf A-module in the category of right H-comodules, we have the degree shift of cyclic modules.