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

Flicker, Yuval; Scheiderer, Claus; Sujatha, R.

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

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

DOI: https://doi.org/10.1090/S0894-0347-98-00271-9

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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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