Lemma 66.40.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent

$f$ is proper,

for every scheme $Z$ and every morphism $Z \to Y$ the projection $Z \times _ Y X \to Z$ is proper,

for every affine scheme $Z$ and every morphism $Z \to Y$ the projection $Z \times _ Y X \to Z$ is proper,

there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that $V \times _ Y X \to V$ is proper, and

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

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