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.
Categories: math.RT