Cohomology of modules over $H$-categories and co-$H$-categories

Abstract

Let $H$ 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 $Hom$ objects and higher derived functors of coinvariants.