The Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.

中文翻译：

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Grothendieck-Serre 在准分裂无分支案例中

Grothendieck-Serre 猜想预测，在规约群方案G下，规则局部环R上的每个一般平凡的躯干都是平凡的。我们在G是准分裂且R是未分支的情况下解决它。一些使我们能够克服迄今为止使混合特征案例遥不可及的障碍的一些技术包括离散估值环上的 Noether 归一化版本，以及有助于通过的混合特征中平滑相对曲线的合适表示引理通过切除和修补到相对仿射线。