On the Grothendieck--Serre conjecture concerning principal G-bundles over semi-local Dedekind domains

TL;DR

This paper proves that for certain reductive group schemes over semi-local Dedekind domains, principal G-bundles trivial over the field of fractions are already trivial over the base, extending previous results in algebraic geometry.

Contribution

It establishes the triviality of the kernel of the cohomology map for a class of reductive groups, extending Nisnevich's theorem to broader cases.

Findings

Kernel of H^1_et(R,G) to H^1_et(K,G) is trivial for specified G

Extends Nisnevich's theorem to new classes of group schemes

Supports the Grothendieck--Serre conjecture in this context

Abstract

Let R be a semi-local Dedekind domain and let K be the field of fractions of R. Let G be a reductive semisimple simply connected R-group scheme such that every semisimple normal R-subgroup scheme of G contains a split R-torus G_m. We prove that the kernel of the map H^1_et(R,G)-> H^1_et(K,G) induced by the inclusion of R into K, is trivial. This result partially extends a theorem of Nisnevich.