On cohomological characterizations of endotrivial modules

TL;DR

This paper explores the cohomological properties of endotrivial modules over finite groups, revealing connections between different cohomology theories and providing new characterizations through topos-theoretic methods.

Contribution

It demonstrates the link between ech and category cohomology approaches to endotrivial modules, offering new insights and characterizations based on topos theory.

Findings

Unified understanding of cohomology approaches to endotrivial modules

New characterizations derived from topos-theoretic perspective

Enhanced conceptual framework for studying endotrivial modules

Abstract

Given a general finite group $G$, there are various finite categories whose cohomology theories are of great interests. Recently Balmer and Grodal gave some new characterizations of the groups of endotrivial modules, via \v{C}ech cohomology and category cohomology, respectively, defined on certain orbit categories. These two seemingly different approaches share a common root in topos theory. We shall demonstrate the connection, which leads to a better understanding as well as new characterizations of the group of endotrivial modules.