  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.
