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Grothendieck’s theorem on non-abelian $H^2$ and local-global principles

Authors: Yuval Z. Flicker, Claus Scheiderer and R. Sujatha
Journal: J. Amer. Math. Soc. 11 (1998), 731-750
MSC (1991): Primary 14L30, 11R34, 12G05
MathSciNet review: 1608617
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Abstract: A theorem of Grothendieck asserts that over a perfect field $k$ of cohomological dimension one, all non-abelian $H^{2}$-cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of this theorem. The generalization — to the context of perfect fields of virtual cohomological dimension one — takes the form of a local-global principle for the $H^{2}$-sets with respect to the orderings of the field. This principle asserts in particular that an element in $H^{2}$ is neutral precisely when it is neutral in the real closure with respect to every ordering in a dense subset of the real spectrum of $k$. Our techniques provide a new proof of Grothendieck’s original theorem. An application to homogeneous spaces over $k$ is also given.

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Additional Information

Yuval Z. Flicker
Affiliation: Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, Ohio 43210-1174

Claus Scheiderer
Affiliation: Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
MR Author ID: 212893

R. Sujatha
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Bombay 400 005, India
MR Author ID: 293023
ORCID: 0000-0003-1221-0710

Received by editor(s): September 2, 1997
Received by editor(s) in revised form: March 16, 1998
Article copyright: © Copyright 1998 American Mathematical Society