Permutation modules and cohomological singularity

TL;DR

This paper introduces a new invariant for finitely generated group representations over noetherian rings, using cohomology and singularity categories to identify permutation module-controlled representations.

Contribution

It defines a novel invariant that links group cohomology with singularity categories, advancing the understanding of representation control by permutation modules.

Findings

Invariant effectively detects permutation module-controlled representations

Connects group cohomology with singularity categories

Provides new tools for analyzing representations over noetherian rings

Abstract

We define a new invariant of finitely generated representations of a finite group, with coefficients in a commutative noetherian ring. This invariant uses group cohomology and takes values in the singularity category of the coefficient ring. It detects which representations are controlled by permutation modules.