On unipotent algebraic G-groups and 1-cohomology

Details On unipotent algebraic G-groups and 1-cohomology On unipotent algebraic G-groups and 1-cohomology

2010-11-04

arxiv.org/abs/1011.1183v4

Mathematics

Group Theory

25 pages; v4; to appear in Transactions of the AMS

Scientific paper

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 unipotent algebraic group $Q$ acted on morphically by an 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.

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