Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Remarks on non-abelian cohomology of proalgebraic groups

Author: Richard Hain
Journal: J. Algebraic Geom. 22 (2013), 581-598
Published electronically: March 21, 2013
MathSciNet review: 3048547
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Abstract: In this paper we develop a theory of non-abelian cohomology for proalgebraic groups which is used in J. Amer. Math. Soc. 24 (2011), 709-769 to study the unipotent section conjecture. The non-abelian cohomology $ H^1_{\mathrm {nab}}(\mathcal {G},\mathcal {P})$ is a scheme. The argument $ \mathcal {G}$ is a proalgebraic group; the coefficient group $ \mathcal {P}$ is prounipotent with trivial center and endowed with an outer action of $ \mathcal {G}$. This outer action uniquely determines an extension $ \widehat {\mathcal {G}}$ of $ \mathcal {G}$ by $ \mathcal {P}$. With suitable hypotheses, the scheme $ H^1_{\mathrm {nab}} (\mathcal {G},\mathcal {P})$ parametrizes the $ \mathcal {P}$ conjugacy classes of sections of $ \widehat {\mathcal {G}} \to \mathcal {G}$.

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Richard Hain
Affiliation: Department of Mathematics, Duke University, Durham, North Carolina 27708-0320

Received by editor(s): October 13, 2010
Received by editor(s) in revised form: April 11, 2011
Published electronically: March 21, 2013
Additional Notes: Supported in part by National Science Foundation grant DMS-1005675.
Article copyright: © Copyright 2013 University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.