The Stacks project

Proof. Choose a finite type quasi-coherent sheaf of ideals $\mathcal{I}$, resp. $\mathcal{J}$ such that $X \setminus U = V(\mathcal{I})$, resp. $X \setminus V = V(\mathcal{J})$, see Limits of Spaces, Lemma 70.14.1. Then $|V(\mathcal{I}\mathcal{J})| = |X|$. Hence $\mathcal{I}\mathcal{J}$ is a locally nilpotent sheaf of ideals. Since $\mathcal{I}$ and $\mathcal{J}$ are of finite type and $X$ is quasi-compact there exists an $n > 0$ such that $\mathcal{I}^ n \mathcal{J}^ n = 0$. We may and do replace $\mathcal{I}$ by $\mathcal{I}^ n$ and $\mathcal{J}$ by $\mathcal{J}^ n$. Whence $\mathcal{I} \mathcal{J} = 0$. Let $b : X' \to X$ be the blowing up in $\mathcal{I} + \mathcal{J}$. This is $U \cup V$-admissible as $|V(\mathcal{I} + \mathcal{J})| = |X| \setminus |U| \cup |V|$. We will show that $X'$ is a disjoint union of open subspaces $X' = X'_1 \amalg X'_2$ as in the statement of the lemma.

Since $|V(\mathcal{I} + \mathcal{J})|$ is the complement of $|U \cup V|$ we conclude that $V \cup U$ is scheme theoretically dense in $X'$, see Lemmas 71.17.4 and 71.6.4. Thus if such a decomposition $X' = X'_1 \amalg X'_2$ into open and closed subspaces exists, then $X'_1$ is the scheme theoretic closure of $U$ in $X'$ and similarly $X'_2$ is the scheme theoretic closure of $V$ in $X'$. Since $U \to X'$ and $V \to X'$ are quasi-compact taking scheme theoretic closures commutes with étale localization (Morphisms of Spaces, Lemma 67.16.3). Hence to verify the existence of $X'_1$ and $X'_2$ we may work étale locally on $X$. This reduces us to the case of schemes which is treated in the proof of Divisors, Lemma 31.34.5. $\square$