On unipotent algebraic G-groups and 1-cohomology

TL;DR

This paper advances the understanding of non-abelian 1-cohomology for unipotent algebraic groups acted upon by connected algebraic groups, establishing new isomorphisms, filtrations, and stability properties that impact subgroup structure analysis.

Contribution

It introduces new results on the structure and cohomology of unipotent algebraic G-groups, extending classical theorems and establishing rational stability and generic cohomology.

Findings

The restriction map H^1(G,Q) to H^1(B,Q) is an isomorphism.

Connected unipotent algebraic G-groups admit filtrations with G-module quotients.

The paper establishes rational stability and generic cohomology for these groups.

Abstract

In this paper we consider non-abelian 1-cohomology for groups with coefficients in other groups. We prove versions of the `five lemma' arising from this situation. We go on to show that a connected unipotent algebraic group Q acted on morphically by a connected algebraic group G admits a filtration with successive quotients having the structure of G-modules. From these results we deduce extensions to results due to Cline, Parshall, Scott and van der Kallen. Firstly, if G is a connected, reductive algebraic group with Borel subgroup B and Q a unipotent algebraic G-group, we show the restriction map H^1(G,Q)\to H^1(B,Q) is an isomorphism. We also show that this situation admits a notion of rational stability and generic cohomology. We use these results to obtain corollaries about complete reducibility and subgroup structure.