Grothendieck’s theorem on non-abelian 𝐻² and local-global principles

Flicker, Yuval¹ ; Scheiderer, Claus² ; Sujatha, R.³

¹ Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, Ohio 43210-1174
² Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
³ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Bombay 400005, India

Journal of the American Mathematical Society, Tome 11 (1998) no. 3, pp. 731-750

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-98-00271-9

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