Extensions for Finite Chevalley Groups III: Rational and Generic Cohomology

TL;DR

This paper investigates the existence and stability of generic cohomology for finite Chevalley groups, providing new vanishing results and extending previous stability and vanishing range findings.

Contribution

It introduces new vanishing theorems for G and B-cohomology, and extends stability and vanishing range results for finite group cohomology of Chevalley groups.

Findings

Vanishing of G and B-cohomology groups established.

New stability ranges for generic cohomology derived.

Extended vanishing ranges for finite group cohomology of G(F_q).

Abstract

Let $G$ be a connected reductive algebraic group and $B$ be a Borel subgroup defined over an algebraically closed field of characteristic $p>0$. In this paper, the authors study the existence of generic $G$-cohomology and its stability with rational $G$-cohomology groups via the use of methods from the authors' earlier work. New results on the vanishing of $G$ and $B$-cohomology groups are presented. Furthermore, vanishing ranges for the associated finite group cohomology of $G({\mathbb F}_{q})$ are established which generalizes earlier work of Hiller, in addition to stability ranges for generic cohomology which improves on seminal work of Cline, Parshall, Scott and van der Kallen.