The Grothendieck-Serre Conjecture over Semilocal Dedekind Rings

Abstract

For a reductive group scheme $G$ over a semilocal Dedekind ring $R$ with total ring of fractions $K$, we prove that no nontrivial $G$-torsor trivializes over $K$. This generalizes a result of Nisnevich-Tits, who settled the case when $R$ is local. Their result, in turn, is a special case of a conjecture of Grothendieck-Serre that predicts the same over any regular local ring. With a patching technique and weak approximation in the style of Harder, we reduce to the case when $R$ is a complete discrete valuation ring. Afterwards, we consider Levi subgroups to reduce to the case when $G$ is semisimple and anisotropic, in which case we take advantage of Bruhat-Tits theory to conclude. Finally, we show that the Grothendieck-Serre conjecture implies that any reductive group over the total ring of fractions of a regular semilocal ring $S$ has at most one reductive $S$-model.