论文标题
Grothendieck-关于估值环的猜想
The Grothendieck--Serre conjecture over valuation rings
论文作者
论文摘要
在本文中，我们建立了关于估值环的Grothendieck-serre猜想：对于估值戒指$ v $的还原集团方案$ g $，而分数$ k $，如果$ k $ a $ g $ -torsor，则是$ v $的$ g $ -torsor，如果它比$ k $的微不足道。原始的Grothendieck-Serre猜想和奇异性的分辨率预测了这一结果。我们的证明的新颖性在于克服了普通非差异评估环带来的微妙之处。通过使用Flasque的分辨率并使用当地的同谋，我们证明了Colliot-Thélène-Sansuc的Tori案例的非非核心对应物。然后，利用代数化技术的优势，我们获得了Henselian等级的段落。最后，我们引入LEVI亚组，并使用各向异性群体的理性点的完整性减少到半型各向异性案例中，在这种情况下，我们呼吁在Bruhat-tim-tits理论中诉诸于律法亚组的特性来得出结论。在最后一部分中，通过使用反射式滑轮的属性，我们还证明了Nisnevich的纯度猜想的变体。
In this article, we establish the Grothendieck-Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$, a $G$-torsor over $V$ is trivial if it is trivial over $K$. This result is predicted by the original Grothendieck-Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène-Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat-Tits theory to conclude. In the last section, by using properties of reflexive sheaves, we also prove a variant of Nisnevich's purity conjecture.
