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

Abstract

A theorem of Grothendieck asserts that over a perfect field k k of cohomological dimension one, all non-abelian H 2 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 H^{2} -sets with respect to the orderings of the field. This principle asserts in particular that an element in H 2 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 k . Our techniques provide a new proof of Grothendieck’s original theorem. An application to homogeneous spaces over k k is also given.