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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: David I. Stewart
Journal: Trans. Amer. Math. Soc. 365 (2013), 6343-6365
MSC (2010): Primary 20G07, 20G10; Secondary 18G50
Published electronically: July 10, 2013
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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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Additional Information

David I. Stewart
Affiliation: Department of Mathematics and Statistics, New College, Oxford, OX1 3BN, United Kingdom

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05853-9
PII: S 0002-9947(2013)05853-9
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.