The Stacks project

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 $\mathcal{I}$. By Limits of Spaces, Lemma 70.9.8 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. $\square$