The Module Structure of a Group Action on a Ring

TL;DR

This paper explores the structure of modules arising from finite group actions on graded rings, linking homological algebra and geometric properties of the spectrum to understand module multiplicities.

Contribution

It introduces a homological algebra framework to analyze module multiplicities in group actions on graded rings, connecting algebraic and geometric perspectives.

Findings

Describes module multiplicities via homological algebra.

Links module structure to the geometry of the group action.

Provides a method to analyze indecomposable summands in graded modules.

Abstract

Consider a finite group $G$ acting on a graded Noetherian $k$-algebra $S$, for some field $k$ of characteristic $p$; for example $S$ might be a polynomial ring. Regard $S$ as a $kG$-module and consider the multiplicity of a particular indecomposable module as a summand in each degree. We show how this can be described in terms of homological algebra and how it is linked to the geometry of the group action on the spectrum of $S$.