Cohomological finite generation for restricted Lie superalgebras and finite supergroup schemes

TL;DR

This paper proves that the cohomology ring of finite-dimensional restricted Lie superalgebras over fields of characteristic p > 2 is finitely generated, using explicit projective resolutions and relating the problem to universal extension classes.

Contribution

It establishes the finite generation of cohomology rings for restricted Lie superalgebras and connects the problem for supergroup schemes to conjectured universal extension classes.

Findings

Cohomology ring of restricted Lie superalgebras is finitely generated.

Reduction of supergroup scheme cohomology finite generation to universal extension classes.

Use of explicit projective resolutions for proof.

Abstract

We prove that the cohomology ring of a finite-dimensional restricted Lie superalgebra over a field of characteristic $p > 2$ is a finitely-generated algebra. Our proof makes essential use of the explicit projective resolution of the trivial module constructed by J. Peter May for any graded restricted Lie algebra. We then prove that the cohomological finite generation problem for finite supergroup schemes over fields of odd characteristic reduces to the existence of certain conjectured universal extension classes for the general linear supergroup $GL(m|n)$ that are similar to the universal extension classes for $GL_n$ exhibited by Friedlander and Suslin.