论文标题
Grothendieck本地化问题的统一处理
A uniform treatment of Grothendieck's localization problem
论文作者
论文摘要
令$ f \ colon y \ to x $成为当地noetherian方案的适当平坦形态。然后，$ x $以$ f $在生成下是稳定的$ x $中的基因座。我们证明，在$ x $的正式纤维上的适当假设下，即使$ f $仅关闭且平坦，同一属性也适用于其他局部属性。我们证明这一说法的证明简化为一个纯粹的本地问题，称为Grothendieck的本地化问题。为了解决Grothendieck的问题，我们提供了一个通用框架，可以统一地处理以前已知的该问题的情况，并且在新情况下也解决了此问题，即较弱的正态性，拟态性，$ f $ - 理性性和财产“ Cohen-Macaulay和$ f $ f $ impentive）。对于较弱的正态性陈述，我们证明弱态性总是从卡地亚分隔线中提升。我们还解决了Grothendieck的本地化问题，以等于零的终端，规范和合理的概念。
Let $f\colon Y \to X$ be a proper flat morphism of locally noetherian schemes. Then, the locus in $X$ over which $f$ is smooth is stable under generization. We prove that under suitable assumptions on the formal fibers of $X$, the same property holds for other local properties of morphisms, even if $f$ is only closed and flat. Our proof of this statement reduces to a purely local question known as Grothendieck's localization problem. To answer Grothendieck's problem, we provide a general framework that gives a uniform treatment of previously known cases of this problem, and also solves this problem in new cases, namely for weak normality, seminormality, $F$-rationality, and the property "Cohen-Macaulay and $F$-injective." For the weak normality statement, we prove that weak normality always lifts from Cartier divisors. We also solve Grothendieck's localization problem for terminal, canonical, and rational singularities in equal characteristic zero.
