The Stacks project

Lemma 37.47.4. Let $S$ be a quasi-compact and quasi-separated scheme. If $U \subset S$ is a quasi-compact open and $V \to U$ is a surjective finite morphism, then there exists a surjective finite morphism $\overline{V} \to S$ with $\overline{V} \times _ S U$ isomorphic to $V$ as schemes over $U$.

Proof. By Zariski's Main Theorem (Lemma 37.42.3) we can assume $V$ is a quasi-compact open in a scheme $V'$ finite over $S$. After replacing $V'$ by the scheme theoretic image of $V$ we may assume that $V$ is dense in $V'$. It follows that $V' \times _ S U = V$ because $V \to V' \times _ S U$ is closed as $V$ is finite over $U$. Let $Z$ be the reduced induced scheme structure on $S \setminus U$. Then $\overline{V} = V' \amalg Z$ works. $\square$

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