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. First, 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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- David I. Stewart
- Affiliation: Department of Mathematics and Statistics, New College, Oxford, OX1 3BN, United Kingdom
- MR Author ID: 884527
- Received by editor(s): September 26, 2011
- Received by editor(s) in revised form: March 17, 2012, April 2, 2012, and April 9, 2012
- Published electronically: July 10, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: Trans. Amer. Math. Soc. 365 (2013), 6343-6365
- MSC (2010): Primary 20G07, 20G10; Secondary 18G50
- DOI: https://doi.org/10.1090/S0002-9947-2013-05853-9
- MathSciNet review: 3105754