Bounding the dimensions of rational cohomology groups

TL;DR

This paper establishes bounds on the dimensions of rational cohomology groups for algebraic groups over fields of positive characteristic, with applications to symmetric and Chevalley groups.

Contribution

It provides new bounds on cohomology group dimensions for algebraic groups and applies these results to improve estimates for symmetric and Chevalley groups.

Findings

Bounded the dimension of rational cohomology groups by a constant times the module dimension.

Derived effective bounds on the first cohomology of symmetric groups.

Significantly improved estimates for the second cohomology of finite Chevalley groups.

Abstract

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple simply-connected algebraic group over $k$ that is defined and split over the prime field $\mathbb{F}_p$. In this paper we investigate situations where the dimension of a rational cohomology group for $G$ can be bounded by a constant times the dimension of the coefficient module. We then demonstrate how our results can be applied to obtain effective bounds on the first cohomology of the symmetric group. We also show how, for finite Chevalley groups, our methods permit significant improvements over previous estimates for the dimensions of second cohomology groups.