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Let $k$ be a field of characteristic $2$ and let $H$ be a finite group orgroup scheme. We show that the negative Tate cohomology ring$\widehat{\text{H}}^{\leq 0}(H,k)$ can be realized as the endomorphism ring ofthe trivial module in a Verdier localization of the stable category of$kG$-modules for $G$ an extension of $H$. This means in some cases that theendomorphism of the trivial module is a local ring with infinitely generatedradical with square zero. This stands in stark contrast to some knowncalculations in which the endomorphism ring of the trivial module is the degreezero component of a localization of the cohomology ring of the group.

Negative cohomology and the endomorphism ring of the trivial module