论文标题
在准清分未受到的案例中的grothendieck-serre
Grothendieck-Serre in the quasi-split unramified case
论文作者
论文摘要
Grothendieck-Serre的猜想预测，在常规本地环$ r $上，每个逐渐琐碎的托尔索尔$ g $都是微不足道的。我们在$ g $是准清单而$ r $的情况下解决了这一问题。迄今为止，使我们无法克服的障碍物的某些技术使混合特征的情况无法触及，包括在离散估值环上进行的Noether归一化，以及在混合特征中平滑相对曲线的合适呈现引理，以促进通过切除和斑点通过相对仿射线传递到相对效果线。
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.
