The Stacks project

Lemma 29.5.2. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. The support of $\mathcal{F}$ is closed under specialization.

Proof. If $x' \leadsto x$ is a specialization and $\mathcal{F}_ x = 0$ then $\mathcal{F}_{x'}$ is zero, as $\mathcal{F}_{x'}$ is a localization of the module $\mathcal{F}_ x$. Hence the complement of $\text{Supp}(\mathcal{F})$ is closed under generalization. $\square$

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