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

$\mathcal{F} = 0 \Leftrightarrow \text{Ass}(\mathcal{F}) = \emptyset .$

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. $\square$

There are also:

• 2 comment(s) on Section 31.2: Associated points

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).