Lemma 75.7.7 (0CZI)—The Stacks project

Lemma 75.7.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ 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 $Y$,
the scheme theoretic support of $\mathcal{F}$ (Morphisms of Spaces, Definition 67.15.4) is proper over $Y$, and
there exists a closed subspace $Z \subset X$ and a finite type, quasi-coherent $\mathcal{O}_ Z$-module $\mathcal{G}$ such that (a) $Z \to Y$ 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 of Spaces, Lemma 67.15.2. Hence we can apply Definition 75.7.2. Since the scheme theoretic support of $\mathcal{F}$ is a closed subspace whose underlying closed subset is $\text{Supp}(\mathcal{F})$ we see that (1) and (2) are equivalent by Definition 75.7.2. It is clear that (2) implies (3). Conversely, if (3) is true, then $\text{Supp}(\mathcal{F}) \subset |Z|$ and hence $\text{Supp}(\mathcal{F})$ is proper over $Y$ for example by Lemma 75.7.3. $\square$

Comments (2)

Comment #2304 by Matthew Emerton on November 18, 2016 at 12:14

I think that the citation to Def. 28.5.5 in the statement of the lemma is to a definition in the context of schemes, although this lemma is about algebraic spaces.

Comment #2382 by Johan on February 16, 2017 at 23:37

Thanks, fixed here.

Comments (2)

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