Lemma 31.2.9. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $x \in \text{Supp}(\mathcal{F})$ be a point in the support of $\mathcal{F}$ which is not a specialization of another point of $\text{Supp}(\mathcal{F})$. Then $x \in \text{Ass}(\mathcal{F})$. In particular, any generic point of an irreducible component of $X$ is an associated point of $X$.

Proof. Since $x \in \text{Supp}(\mathcal{F})$ the module $\mathcal{F}_ x$ is not zero. Hence $\text{Ass}(\mathcal{F}_ x) \subset \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ is nonempty by Algebra, Lemma 10.63.7. On the other hand, by assumption $\text{Supp}(\mathcal{F}_ x) = \{ \mathfrak m_ x\}$. Since $\text{Ass}(\mathcal{F}_ x) \subset \text{Supp}(\mathcal{F}_ x)$ (Algebra, Lemma 10.63.2) we see that $\mathfrak m_ x$ is associated to $\mathcal{F}_ x$ and we win. $\square$

Comment #2326 by Jarek on

Perhaps add a corollary that for coherent sheaf $\mathcal{F}$, we have $\operatorname{Supp}(\mathcal{F}) = \overline{\operatorname{Ass}(\mathcal{F})}$.

Comment #2398 by on

Please edit the latex file adding the lemma and email me the new version. Thanks!

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