The Stacks project

Lemma 76.39.2. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $U \subset B$ be an open subspace. Assume

  1. $B$ is quasi-compact and quasi-separated,

  2. $U$ is quasi-compact,

  3. $f : X \to B$ is proper, and

  4. $f^{-1}(U) \to U$ is finite locally free.

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

Proof. By Lemma 76.39.1 we may assume that $X \to B$ is flat and of finite presentation. After replacing $B$ by a $U$-admissible blowup if necessary, we may assume that $U \subset B$ is scheme theoretically dense. Then $f$ is finite by Lemma 76.37.4. Hence $f$ is finite locally free by Morphisms of Spaces, Lemma 67.46.6. $\square$

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