Lemma 30.26.7. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ X$-module. The following are equivalent

the support of $\mathcal{F}$ is proper over $S$,

the scheme theoretic support of $\mathcal{F}$ (Morphisms, Definition 29.5.5) is proper over $S$, and

there exists a closed subscheme $Z \subset X$ and a finite type, quasi-coherent $\mathcal{O}_ Z$-module $\mathcal{G}$ such that (a) $Z \to S$ is proper, and (b) $(Z \to X)_*\mathcal{G} = \mathcal{F}$.

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