The Stacks project

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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