Lemma 31.34.3. Let $X$ be a quasi-compact and quasi-separated scheme. Let $U, V \subset X$ be quasi-compact open subschemes. Let $b : V' \to V$ be a $U \cap V$-admissible blowup. Then there exists a $U$-admissible blowup $X' \to X$ whose restriction to $V$ is $V'$.
Proof. Let $\mathcal{I} \subset \mathcal{O}_ V$ be the finite type quasi-coherent sheaf of ideals such that $V(\mathcal{I})$ is disjoint from $U \cap V$ and such that $V'$ is isomorphic to the blowup of $V$ in $\mathcal{I}$. Let $\mathcal{I}' \subset \mathcal{O}_{U \cup V}$ be the quasi-coherent sheaf of ideals whose restriction to $U$ is $\mathcal{O}_ U$ and whose restriction to $V$ is $\mathcal{I}$ (see Sheaves, Section 6.33). By Properties, Lemma 28.22.2 there exists a finite type quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_ X$ whose restriction to $U \cup V$ is $\mathcal{I}'$. The lemma follows. $\square$
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