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.

对于在分数为K的总环的半局部Dedekind环R上的归约群方案G，我们证明没有非平凡的G -torsor在K上平凡。这归纳了Nisnevich-Tits的结果，他解决了R是局部的情况。反过来，他们的结果是格罗腾迪克-塞勒猜想的一个特例，该猜想在任何规则的本地环上都可以预测出同样的结果。借助修补技术和Harder风格的弱近似，我们可以简化为R是完整的离散估值环的情况。之后，我们考虑将李维斯子群归结为G是半简单且各向异性的，在这种情况下，我们利用Bruhat-Tits理论得出结论。最后，我们证明了Grothendieck-Serre猜想表明，规则半局部环S的分数的总环上的任何还原基团最多具有一个还原S模型。