Principal bundles of reductive groups over affine schemes

TL;DR

This paper proves that principal G-bundles over certain affine schemes are trivial when extended from a semi-local regular domain to its field of fractions, under specific conditions on the group scheme G and the base field.

Contribution

It establishes the triviality of the kernel of the etale cohomology map for reductive semi-simple simply connected group schemes over affine schemes, extending previous results to broader contexts.

Findings

Kernel of H^1 map is trivial under given conditions

Reductive semi-simple simply connected groups are trivial over affine schemes

Results hold for semi-local rings with infinite base fields

Abstract

Let R be a semi-local regular domain containing an infinite perfect field k, and let K be the field of fractions of R. Let G be a reductive semi-simple simply connected R-group scheme such that each of its R-indecomposable factors is isotropic. We prove that for any Noetherian affine scheme A over k, the kernel of the map of etale cohomology sets H^1(A\times_k R,G)-> H^1(A\times_ k K,G), induced by the inclusion of R into K, is trivial. If R is the semi-local ring of several points on a k-smooth scheme, then it suffices to require that k is infinite and keep the same assumption concerning G.