The Stacks project

Lemma 38.33.2. Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open.

  1. If $Z_1, Z_2 \subset X$ are closed subschemes of finite presentation such that $Z_1 \cap Z_2 \cap U = \emptyset $, then there exists a $U$-admissible blowing up $X' \to X$ such that the strict transforms of $Z_1$ and $Z_2$ are disjoint.

  2. If $T_1, T_2 \subset U$ are disjoint constructible closed subsets, then there is a $U$-admissible blowing up $X' \to X$ such that the closures of $T_1$ and $T_2$ are disjoint.

Proof. Proof of (1). The assumption that $Z_ i \to X$ is of finite presentation signifies that the quasi-coherent ideal sheaf $\mathcal{I}_ i$ of $Z_ i$ is of finite type, see Morphisms, Lemma 29.21.7. Denote $Z \subset X$ the closed subscheme cut out by the product $\mathcal{I}_1 \mathcal{I}_2$. Observe that $Z \cap U$ is the disjoint union of $Z_1 \cap U$ and $Z_2 \cap U$. By Divisors, Lemma 31.34.5 there is a $U \cap Z$-admissible blowup $Z' \to Z$ such that the strict transforms of $Z_1$ and $Z_2$ are disjoint. Denote $Y \subset Z$ the center of this blowing up. Then $Y \to X$ is a closed immersion of finite presentation as the composition of $Y \to Z$ and $Z \to X$ (Divisors, Definition 31.34.1 and Morphisms, Lemma 29.21.3). Thus the blowing up $X' \to X$ of $Y$ is a $U$-admissible blowing up. By general properties of strict transforms, the strict transform of $Z_1, Z_2$ with respect to $X' \to X$ is the same as the strict transform of $Z_1, Z_2$ with respect to $Z' \to Z$, see Divisors, Lemma 31.33.2. Thus (1) is proved.

Proof of (2). By Properties, Lemma 28.24.1 there exists a finite type quasi-coherent sheaf of ideals $\mathcal{J}_ i \subset \mathcal{O}_ U$ such that $T_ i = V(\mathcal{J}_ i)$ (set theoretically). By Properties, Lemma 28.22.2 there exists a finite type quasi-coherent sheaf of ideals $\mathcal{I}_ i \subset \mathcal{O}_ X$ whose restriction to $U$ is $\mathcal{J}_ i$. Apply the result of part (1) to the closed subschemes $Z_ i = V(\mathcal{I}_ i)$ to conclude. $\square$


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