The Stacks project

Lemma 38.31.2. Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme over $S$. Let $U \subset S$ be a quasi-compact open. Assume

  1. $X \to S$ is proper, and

  2. $X_ U \to U$ is finite locally free.

Then there exists a $U$-admissible blowup $S' \to S$ such that the strict transform of $X$ is finite locally free over $S'$.

Proof. By Lemma 38.31.1 we may assume that $X \to S$ is flat and of finite presentation. After replacing $S$ by a $U$-admissible blowup if necessary, we may assume that $U \subset S$ is scheme theoretically dense. Then $f$ is finite by Lemma 38.11.4. Hence $f$ is finite locally free by Morphisms, Lemma 29.48.2. $\square$

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