Lemma 31.5.7 (05AQ)—The Stacks project

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Lemma 31.5.7. Let $X$ be a 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 $x \in \text{WeakAss}(\mathcal{F})$. In particular, any generic point of an irreducible component of $X$ is weakly associated to $\mathcal{O}_ X$.

Proof. Since $x \in \text{Supp}(\mathcal{F})$ the module $\mathcal{F}_ x$ is not zero. Hence $\text{WeakAss}(\mathcal{F}_ x) \subset \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ is nonempty by Algebra, Lemma 10.66.5. On the other hand, by assumption $\text{Supp}(\mathcal{F}_ x) = \{ \mathfrak m_ x\} $. Since $\text{WeakAss}(\mathcal{F}_ x) \subset \text{Supp}(\mathcal{F}_ x)$ (Algebra, Lemma 10.66.6) we see that $\mathfrak m_ x$ is weakly associated to $\mathcal{F}_ x$ and we win. $\square$

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