The Stacks project

Lemma 31.4.2. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then

  1. the generic points of irreducible components of $\text{Supp}(\mathcal{F})$ are associated points of $\mathcal{F}$, and

  2. an associated point of $\mathcal{F}$ is embedded if and only if it is not a generic point of an irreducible component of $\text{Supp}(\mathcal{F})$.

In particular an embedded point of $X$ is an associated point of $X$ which is not a generic point of an irreducible component of $X$.

Proof. Recall that in this case $Z = \text{Supp}(\mathcal{F})$ is closed, see Morphisms, Lemma 29.5.3 and that the generic points of irreducible components of $Z$ are associated points of $\mathcal{F}$, see Lemma 31.2.9. Finally, we have $\text{Ass}(\mathcal{F}) \subset Z$, by Lemma 31.2.3. These results, combined with the fact that $Z$ is a sober topological space and hence every point of $Z$ is a specialization of a generic point of $Z$, imply (1) and (2). $\square$

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