2019/02/06 by Ning Guo, Guo, Ning · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1902.02315
openalex publication_date 2019/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.