The endomorphism ring of the trivial module in a localized category

TL;DR

This paper investigates the structure of the endomorphism ring of the trivial module in a localized stable module category for a finite group, revealing it is a local ring with an infinitely generated nilpotent maximal ideal under certain conditions.

Contribution

It characterizes the endomorphism ring in a localized category for specific subvarieties and group conditions, highlighting its local and infinitely generated nilpotent maximal ideal.

Findings

Endomorphism ring is a local ring with an infinitely generated nilpotent maximal ideal.

The structure depends on the support variety and the p-rank of elementary abelian p-subgroups.

An example shows the endomorphism ring of a compact object may not be finitely presented.

Abstract

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p >0$. Let $\mathcal{M}$ be the thick tensor ideal of finitely generated modules whose support variety is in a fixed subvariety $V$ of the projectivized prime ideal spectrum $\operatorname{Proj} \operatorname{H}^*(G,k)$. Let $\mathcal{C}$ denote the Verdier localization of the stable module category $\operatorname{stmod}(kG)$ at $\mathcal{M}$. We show that if $V$ is a finite collection of closed points and if the $p$-rank every maximal elementary abelian $p$-subgroups of $G$ is at least 3, then the endomorphism ring of the trivial module in $\mathcal{C}$ is a local ring whose unique maximal ideal is infinitely generated and nilpotent. In addition, we show an example where the endomorphism ring in $\mathcal{C}$ of a compact object is not finitely presented as a module over the endomorphism ring of the trivial module.