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

## TL;DR

This paper develops spectral sequences for cohomology theories of modules over Hopf algebra categories, generalizing classical Hopf module concepts and providing tools for their cohomological analysis.

## Contribution

It introduces a framework for cohomology of modules over $H$-categories and co-$H$-categories, including spectral sequences and generalizations of Hopf modules.

## Key findings

- Constructed Grothendieck spectral sequences for cohomologies.
- Extended the concept of Hopf modules to relative $( ext{D},H)$-Hopf modules.
- Provided methods to compute cohomologies using rational $Hom$ objects.

## 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.