The Stacks project

Lemma 38.33.6. Let $S$ be a quasi-compact and quasi-separated scheme. Let $U$ be a scheme of finite type and separated over $S$. Let $V \subset U$ be a quasi-compact open. If $V$ has a compactification $V \subset Y$ over $S$, 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 $Z \subset Y \times _ S U$ of the “diagonal” morphism $V \to Y \times _ S U$. 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 38.33.5. Hence by Lemma 38.31.3 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 $Y \times _ S U \to U$ is proper and $V' \cong Z$ is a closed subscheme of $Y \times _ S U$. $\square$

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