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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On unipotent algebraic $G$-groups and $1$-cohomology
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by David I. Stewart PDF
Trans. Amer. Math. Soc. 365 (2013), 6343-6365 Request permission

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
  • 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