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

## Comments (2)

Comment #2304 by Matthew Emerton on

Comment #2382 by Johan on