Lemma 31.2.6. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then

**Proof.**
If $\mathcal{F} = 0$, then $\text{Ass}(\mathcal{F}) = \emptyset $ by definition. Conversely, if $\text{Ass}(\mathcal{F}) = \emptyset $, then $\mathcal{F} = 0$ by Algebra, Lemma 10.63.7. To translate from schemes to algebra, restrict to any affine and use Lemma 31.2.2.
$\square$

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