Cohomological finite-generation for finite supergroup schemes

TL;DR

This paper computes extension groups in superfunctor categories to identify universal extension classes, demonstrating that the cohomology ring of finite supergroup schemes is finitely generated, with implications for supergroup cohomology.

Contribution

It introduces new universal extension classes for supergroups and proves finite generation of their cohomology rings, extending classical results to superalgebra contexts.

Findings

Cohomology rings of finite supergroup schemes are finitely generated.

Universal extension classes for supergroups are constructed and analyzed.

Connections to classical cohomology of linear groups are established.

Abstract

In this paper we compute extension groups in the category of strict polynomial superfunctors and thereby exhibit certain "universal extension classes" for the general linear supergroup. Some of these classes restrict to the universal extension classes for the general linear group exhibited by Friedlander and Suslin, while others arise from purely super phenomena. We then use these extension classes to show that the cohomology ring of a finite supergroup scheme---equivalently of a finite-dimensional cocommutative Hopf superalgebra---over a field is a finitely-generated algebra. Implications for the rational cohomology of the general linear supergroup are also discussed.