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. Let 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). Let C denote the Verdier localization of the stable module category stmod(kG) at 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 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 of a compact object is not finitely presented as a module over the endomorphism ring of the trivial module.