Comodule theories in Grothendieck categories and relative Hopf objects

TL;DR

This paper develops the theory of comodules and Hopf modules in Grothendieck categories, exploring their properties, cohomology, and injective resolutions within noncommutative base change frameworks.

Contribution

It introduces a categorical framework for comodules over noncommutative bases, including relative Hopf modules, and analyzes their cohomology and injective resolutions.

Findings

Characterization of when comodules are locally finitely generated or noetherian

Development of spectral sequences for cohomology in Hopf module categories

Decomposition of injective resolutions using support and associated primes

Abstract

We develop the categorical algebra of the noncommutative base change of a comodule category by means of a Grothendieck category $\mathfrak S$. We describe when the resulting category of comodules is locally finitely generated, locally noetherian or may be recovered as a coreflective subcategory of the noncommutative base change of a module category. We also introduce the category ${_A}\mathfrak S^H$ of relative $(A,H)$-Hopf modules in $\mathfrak S$, where $H$ is a Hopf algebra and $A$ is a right $H$-comodule algebra. We study the cohomological theory in ${_A}\mathfrak S^H$ by means of spectral sequences. Using coinduction functors and functors of coinvariants, we study torsion theories and how they relate to injective resolutions in ${_A}\mathfrak S^H$. Finally, we use the theory of associated primes and support in noncommutative base change of module categories to give direct sum decompositions of minimal injective resolutions in ${_A}\mathfrak S^H$.