Lemma 30.26.7 (0CYS)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 30: Cohomology of Schemes
Section 30.26: Being proper over a base
Lemma 30.26.7 (cite)

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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) $(Z \to X)_*\mathcal{G} = \mathcal{F}$ .

Proof. The support $\text{Supp}(\mathcal{F})$ of $\mathcal{F}$ is a closed subset of $X$ , see Morphisms, Lemma 29.5.3. Hence we can apply Definition 30.26.2. Since the scheme theoretic support of $\mathcal{F}$ is a closed subscheme whose underlying closed subset is $\text{Supp}(\mathcal{F})$ we see that (1) and (2) are equivalent by Definition 30.26.2. It is clear that (2) implies (3). Conversely, if (3) is true, then $\text{Supp}(\mathcal{F}) \subset Z$ (an inclusion of closed subsets of $X$ ) and hence $\text{Supp}(\mathcal{F})$ is proper over $S$ for example by Lemma 30.26.3. $\square$

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