The Stacks project

Lemma 76.39.4. 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. $B$ quasi-compact and quasi-separated,

  2. $U$ is quasi-compact,

  3. $f$ is of finite type

  4. $f^{-1}(U) \to U$ is an isomorphism.

Then there exists a $U$-admissible blowup $B' \to B$ such that $U$ is scheme theoretically dense in $B'$ and such that the strict transform $X'$ of $X$ maps isomorphically to an open subspace of $B'$.

Proof. This lemma is a generalization of Lemma 76.39.3. As the composition of $U$-admissible blowups is $U$-admissible (Divisors on Spaces, Lemma 71.19.2) we can proceed in stages. Pick a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ B$ with $|B| \setminus |U| = |V(\mathcal{I})|$. Replace $B$ by the blowup of $B$ in $\mathcal{I}$ and $X$ by the strict transform of $X$. After this replacement $B \setminus U$ is the support of an effective Cartier divisor $D$ (Divisors on Spaces, Lemma 71.17.4). In particular $U$ is scheme theoretically dense in $B$ (Divisors on Spaces, Lemma 71.6.4). Next, we do another $U$-admissible blowup to get to the situation where $X \to B$ is flat and of finite presentation, see Lemma 76.39.1. Note that $U$ is still scheme theoretically dense in $B$. Hence $X \to B$ is an open immersion by Lemma 76.37.5. $\square$

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