The Stacks project

Lemma 73.8.3. Let $S$ be a scheme. Let $f : Y \to X$ be a surjective proper morphism of algebraic spaces over $S$. Then $\{ Y \to X\} $ is a ph covering.

Proof. Let $U \to X$ be a morphism with $U$ affine. By Chow's lemma (in the weak form given as Cohomology of Spaces, Lemma 69.18.1) we see that there is a surjective proper morphism of schemes $V \to U$ which factors through $Y \times _ X U \to U$. Taking any finite affine open cover of $V$ we obtain a standard ph covering of $U$ refining $\{ X \times _ Y U \to U\} $ as desired. $\square$

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