The endomorphism ring of the trivial module in a localized category

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