Negative cohomology and the endomorphism ring of the trivial module

TL;DR

This paper demonstrates that negative Tate cohomology rings can be realized as endomorphism rings of trivial modules within localized stable categories, revealing new algebraic structures and contrasts with previous cohomology calculations.

Contribution

It introduces a novel realization of negative Tate cohomology as endomorphism rings in localized stable categories, highlighting cases with local rings and infinitely generated radicals.

Findings

Negative Tate cohomology rings are realizable as endomorphism rings.

Endomorphism rings can be local with infinitely generated radicals.

Contrasts with known cohomology ring localizations.

Abstract

Let $k$ be a field of characteristic $2$ and let $H$ be a finite group or group scheme. We show that the negative Tate cohomology ring $\widehat{\text{H}}^{\leq 0}(H,k)$ can be realized as the endomorphism ring of the 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 the endomorphism of the trivial module is a local ring with infinitely generated radical with square zero. This stands in stark contrast to some known calculations in which the endomorphism ring of the trivial module is the degree zero component of a localization of the cohomology ring of the group.