The Stacks project

Lemma 38.31.3. Let $\varphi : X \to S$ be a separated morphism of finite type with $S$ quasi-compact and quasi-separated. Let $U \subset S$ be a quasi-compact open such that $\varphi ^{-1}U \to U$ is an isomorphism. Then there exists a $U$-admissible blowup $S' \to S$ such that the strict transform $X'$ of $X$ is isomorphic to an open subscheme of $S'$.

Proof. The discussion in Remark 38.30.1 applies. Thus we may do a first $U$-admissible blowup and assume the complement $S \setminus U$ is the support of an effective Cartier divisor $D$. In particular $U$ is scheme theoretically dense in $S$. Next, we do another $U$-admissible blowup to get to the situation where $X \to S$ is flat and of finite presentation, see Lemma 38.31.1. In this case the result follows from Lemma 38.11.5. $\square$

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