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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Grothendieck’s theorem on non-abelian $H^2$ and local-global principles
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by Yuval Z. Flicker, Claus Scheiderer and R. Sujatha
J. Amer. Math. Soc. 11 (1998), 731-750


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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Bibliographic Information
  • Yuval Z. Flicker
  • Affiliation: Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, Ohio 43210-1174
  • Email:
  • Claus Scheiderer
  • Affiliation: Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
  • MR Author ID: 212893
  • Email:
  • 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:
  • Received by editor(s): September 2, 1997
  • Received by editor(s) in revised form: March 16, 1998
  • © Copyright 1998 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 11 (1998), 731-750
  • MSC (1991): Primary 14L30, 11R34, 12G05
  • DOI:
  • MathSciNet review: 1608617