The Grothendieck-Serre Conjecture over Semilocal Dedekind Rings

TL;DR

This paper proves the Grothendieck-Serre conjecture for reductive group schemes over semilocal Dedekind rings, showing that nontrivial torsors do not trivialize over the total ring of fractions, extending previous results.

Contribution

It generalizes the Grothendieck-Serre conjecture to semilocal Dedekind rings using patching, weak approximation, and Bruhat-Tits theory, and relates to uniqueness of reductive models.

Findings

No nontrivial G-torsor trivializes over the total ring of fractions.

Reduction to semisimple anisotropic case using Bruhat-Tits theory.

Implication that reductive groups over regular semilocal rings have at most one reductive model.

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.