The Stacks project

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 of Spaces, Lemma 67.28.12. Denote $Z \subset X$ the closed subspace 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 on Spaces, Lemma 71.19.5 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 on Spaces, Definition 71.19.1 and Morphisms of Spaces, Lemma 67.28.2). 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 on Spaces, Lemma 71.18.3. Thus (1) is proved.

Proof of (2). By Limits of Spaces, Lemma 70.14.1 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 Limits of Spaces, Lemma 70.9.8 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 subspaces $Z_ i = V(\mathcal{I}_ i)$ to conclude. $\square$