On the Grothendieck–Serre Conjecture Concerning Principal G-Bundles Over Semilocal Dedekind Domains

Abstract

Let R be a semilocal 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 \( {\mathbb{G}}_{m,R} \). It is proved that the kernel of the map

\( {H}_{\overset{\prime }{e}t}^1\left(R,\kern0.5em G\right)\to {H}_{\overset{\prime }{e}t}^1\left(K,\kern0.5em G\right) \)

induced by the inclusion of R into K is trivial. This result partially extends the Nisnevich theorem.