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
DOI:
https://doi.org/10.1090/S0894-0347-98-00271-9
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
Email:
flicker@math.ohio-state.edu
Claus Scheiderer
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
MR Author ID:
212893
Email:
claus.scheiderer@mathematik.uni-regensburg.de
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
Email:
sujatha@math.tifr.res.in
Received by editor(s):
September 2, 1997
Received by editor(s) in revised form:
March 16, 1998
Article copyright:
© Copyright 1998
American Mathematical Society