Abstract

Let $\underset{¯}{x}=x_1,\dots ,x_k$ denote an ordered sequence of elements of a commutative ring R. Let M be an R-module. We recall the two notions that $\underset{¯}{x}$ is M-proregular given by Greenlees and May and Lipman and show that both notions are equivalent. As a main result we prove a cohomological characterization for $\underset{¯}{x}$ to be M-proregular in terms of Čech cohomology. This implies also that $\underset{¯}{x}$ is M-weakly proregular if it is M-proregular. A local-global principle for proregularity and weakly proregularity is proved. This is used for a result about prisms as introduced by Bhatt and Scholze.

中文翻译：

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关于正则序列和棱镜的应用

摘要

让 $\underset{¯}{X}=X_1,\dots ,X_{克}$表示交换环R的有序元素序列。令M为R模。我们记得这两个概念$\underset{¯}{X}$是M -proregular 由格林利斯和梅和利普曼给出，并表明这两个概念是等价的。作为主要结果，我们证明了$\underset{¯}{X}$在 Čech 上同调方面是M -proregular。这也意味着$\underset{¯}{X}$是中号-weakly proregular如果是中号-proregular。证明了前正则性和弱前正则性的局部全局原理。这用于 Bhatt 和 Scholze 介绍的关于棱镜的结果。