Lemma 81.14.6. Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U$ be an algebraic space of finite type and separated over $B$. Let $V \to U$ be an étale morphism. If $V$ has a compactification $V \subset Y$ over $B$, then there exists a $V$-admissible blowing up $Y' \to Y$ and an open $V \subset V' \subset Y'$ such that $V \to U$ extends to a proper morphism $V' \to U$.
Proof. Consider the scheme theoretic image $Z \subset Y \times _ B U$ of the “diagonal” morphism $V \to Y \times _ B U$. If we replace $Y$ by a $V$-admissible blowing up, then $Z$ is replaced by the strict transform with respect to this blowing up, see Lemma 81.14.5. Hence by More on Morphisms of Spaces, Lemma 76.39.4 we may assume $Z \to Y$ is an open immersion. If $V' \subset Y$ denotes the image, then we see that the induced morphism $V' \to U$ is proper because the projection $Y \times _ B U \to U$ is proper and $V' \cong Z$ is a closed subspace of $Y \times _ B U$. $\square$
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