The Stacks project

Lemma 30.9.7. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then $\text{Supp}(\mathcal{F})$ is closed, and $\mathcal{F}$ comes from a coherent sheaf on the scheme theoretic support of $\mathcal{F}$, see Morphisms, Definition 29.5.5.

Proof. Let $i : Z \to X$ be the scheme theoretic support of $\mathcal{F}$ and let $\mathcal{G}$ be the finite type quasi-coherent sheaf on $Z$ such that $i_*\mathcal{G} \cong \mathcal{F}$. Since $Z = \text{Supp}(\mathcal{F})$ we see that the support is closed. The scheme $Z$ is locally Noetherian by Morphisms, Lemmas 29.15.5 and 29.15.6. Finally, $\mathcal{G}$ is a coherent $\mathcal{O}_ Z$-module by Lemma 30.9.1 $\square$

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