2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76738A 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.22 pages, AMS-TeX; accepted for publication by the Journal of the AMSAlgebraic GeometryNumber Theory14L30; 11R34; 12G05text