Fibrewise stratification of group representations

TL;DR

This paper develops a fibrewise stratification framework for the stable categories of Gorenstein projective modules over finite cocommutative Hopf algebras, linking local and global representation theory.

Contribution

It introduces a novel stratification approach for stable categories of Gorenstein projective modules over Hopf algebras, generalizing existing theories to new algebraic contexts.

Findings

Describes the lattice of localising tensor ideals via fibres over the spectrum of R.

Establishes stratification of the stable category under natural cohomological conditions.

Applies to group algebras and exterior algebras over regular rings.

Abstract

Given a finite cocommutative Hopf algebra $A$ over a commutative regular ring $R$, the lattice of localising tensor ideals of the stable category of Gorenstein projective $A$-modules is described in terms of the corresponding lattices for the fibres of $A$ over the spectrum of $R$. Under certain natural conditions on the cohomology of $A$ over $R$, this yields a stratification of the stable category. These results apply when $A$ is the group algebra over $R$ of a finite group, and also when $A$ is the exterior algebra on a finite free $R$-module.