Lemma 31.5.8 (05AR)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.5: Weakly associated points
Lemma 31.5.8 (cite)

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Lemma 31.5.8. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. If $\mathfrak m_ x$ is a finitely generated ideal of $\mathcal{O}_{X, x}$ , then

$x \in \text{Ass}(\mathcal{F}) \Leftrightarrow x \in \text{WeakAss}(\mathcal{F}).$

In particular, if $X$ is locally Noetherian, then $\text{Ass}(\mathcal{F}) = \text{WeakAss}(\mathcal{F})$ .

Proof. See Algebra, Lemma 10.66.9. $\square$

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