Lemma 66.8.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

$f$ is quasi-compact,

for every scheme $Z$ and any morphism $Z \to Y$ the morphism of algebraic spaces $Z \times _ Y X \to Z$ is quasi-compact,

for every affine scheme $Z$ and any morphism $Z \to Y$ the algebraic space $Z \times _ Y X$ is quasi-compact,

there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that $V \times _ Y X \to V$ is a quasi-compact morphism of algebraic spaces, and

there exists a surjective étale morphism $Y' \to Y$ of algebraic spaces such that $Y' \times _ Y X \to Y'$ is a quasi-compact morphism of algebraic spaces, and

there exists a Zariski covering $Y = \bigcup Y_ i$ such that each of the morphisms $f^{-1}(Y_ i) \to Y_ i$ is quasi-compact.

## Comments (1)

Comment #1109 by Evan Warner on