The prime spectra of relative stable module categories

TL;DR

This paper investigates the prime spectra of relative stable module categories over a discrete valuation ring, revealing a structure that decomposes into multiple copies of the spectrum over the residue field, enhancing understanding of modular representation theory.

Contribution

It characterizes the prime ideal spectrum of the relative stable module category over a DVR as a disjoint union of multiple copies of the spectrum over the residue field, extending previous spectral analyses.

Findings

Prime spectrum decomposes into n disjoint copies.

Spectrum over DVR relates to spectrum over residue field.

Provides structural insight into relative stable categories.

Abstract

For a finite group $G$ and an arbitrary commutative ring $R$, Brou\'e has placed a Frobenius exact structure on the category of finitely generated $RG$-modules by taking the exact sequences to be those that split upon restriction to the trivial subgroup. The corresponding stable category is then tensor triangulated. In this paper we examine the case $R=S/t^n$, where $S$ is a discrete valuation ring having uniformising parameter $t$. We prove that the prime ideal spectrum (in the sense of Balmer) of this `relative' version of the stable module category of $RG$ is a disjoint union of $n$ copies of that for $kG$, where $k$ is the residue field of $S$.