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The Stacks project

Lemma 71.19.4. Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let U \subset X be a quasi-compact open subspace. Let b_ i : X_ i \to X, i = 1, \ldots , n be U-admissible blowups. There exists a U-admissible blowup b : X' \to X such that (a) b factors as X' \to X_ i \to X for i = 1, \ldots , n and (b) each of the morphisms X' \to X_ i is a U-admissible blowup.

Proof. Let \mathcal{I}_ i \subset \mathcal{O}_ X be the finite type quasi-coherent sheaf of ideals such that V(\mathcal{I}_ i) is disjoint from U and such that X_ i is isomorphic to the blowup of X in \mathcal{I}_ i. Set \mathcal{I} = \mathcal{I}_1 \cdot \ldots \cdot \mathcal{I}_ n and let X' be the blowup of X in \mathcal{I}. Then X' \to X factors through b_ i by Lemma 71.17.10. \square


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