The Stacks project

Lemma 71.2.8. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then $\text{Ass}(\mathcal{F}) \cap W$ is finite for every quasi-compact open $W \subset |X|$.

Proof. Choose a quasi-compact scheme $U$ and an étale morphism $U \to X$ such that $W$ is the image of $|U| \to |X|$. Then $U$ is a Noetherian scheme and we may apply Divisors, Lemma 31.2.5 to conclude. $\square$

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