Cohomology of Modules Over -categories and Co--categories

Abstract

AbstractLet$H$be a Hopf algebra. We consider$H$-equivariant modules over a Hopf module category${\mathcal{C}}$as modules over the smash extension${\mathcal{C}}\#H$. We construct Grothendieck spectral sequences for the cohomologies as well as the$H$-locally finite cohomologies of these objects. We also introduce relative$({\mathcal{D}},H)$-Hopf modules over a Hopf comodule category${\mathcal{D}}$. These generalize relative$(A,H)$-Hopf modules over an$H$-comodule algebra$A$. We construct Grothendieck spectral sequences for their cohomologies by using their rational$\text{Hom}$objects and higher derived functors of coinvariants.