Lemma 31.20.3 (063E)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.20: Regular ideal sheaves
Lemma 31.20.3 (cite)

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Lemma 31.20.3. Let $X$ be a ringed space. Let $\mathcal{J}$ be a sheaf of ideals. We have the following implications: $\mathcal{J}$ is regular $\Rightarrow$ $\mathcal{J}$ is Koszul-regular $\Rightarrow$ $\mathcal{J}$ is $H_1$ -regular $\Rightarrow$ $\mathcal{J}$ is quasi-regular.

Proof. The lemma immediately reduces to Lemma 31.20.1. $\square$

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