Lemma 38.33.4. Let $S$ be a quasi-compact and quasi-separated scheme. Let $U \to X_1$ and $U \to X_2$ be open immersions of schemes over $S$ and assume $U$, $X_1$, $X_2$ of finite type and separated over $S$. Then there exists a commutative diagram
\[ \xymatrix{ X_1' \ar[d] \ar[r] & X & X_2' \ar[l] \ar[d] \\ X_1 & U \ar[l] \ar[lu] \ar[u] \ar[ru] \ar[r] & X_2 } \]
of schemes over $S$ where $X_ i' \to X_ i$ is a $U$-admissible blowup, $X_ i' \to X$ is an open immersion, and $X$ is separated and finite type over $S$.
Proof.
Throughout the proof all schemes will be separated of finite type over $S$. This in particular implies these schemes are quasi-compact and quasi-separated and the morphisms between them are quasi-compact and separated. See Schemes, Sections 26.19 and 26.21. We will use that if $U \to W$ is an immersion of such schemes over $S$, then the scheme theoretic image $Z$ of $U$ in $W$ is a closed subscheme of $W$ and $U \to Z$ is an open immersion, $U \subset Z$ is scheme theoretically dense, and $U \subset Z$ is dense topologically. See Morphisms, Lemma 29.7.7.
Let $X_{12} \subset X_1 \times _ S X_2$ be the scheme theoretic image of $U \to X_1 \times _ S X_2$. The projections $p_ i : X_{12} \to X_ i$ induce isomorphisms $p_ i^{-1}(U) \to U$ by Morphisms, Lemma 29.6.8. Choose a $U$-admissible blowup $X_ i^ i \to X_ i$ such that the strict transform $X_{12}^ i$ of $X_{12}$ is isomorphic to an open subscheme of $X_ i^ i$, see Lemma 38.31.3. Let $\mathcal{I}_ i \subset \mathcal{O}_{X_ i}$ be the corresponding finite type quasi-coherent sheaf of ideals. Recall that $X_{12}^ i \to X_{12}$ is the blowup in $p_ i^{-1}\mathcal{I}_ i \mathcal{O}_{X_{12}}$, see Divisors, Lemma 31.33.2. Let $X_{12}'$ be the blowup of $X_{12}$ in $p_1^{-1}\mathcal{I}_1 p_2^{-1}\mathcal{I}_2 \mathcal{O}_{X_{12}}$, see Divisors, Lemma 31.32.12 for what this entails. We obtain in particular a commutative diagram
\[ \xymatrix{ X_{12}' \ar[d] \ar[r] & X_{12}^2 \ar[d] \\ X_{12}^1 \ar[r] & X_{12} } \]
where all the morphisms are $U$-admissible blowing ups. Since $X_{12}^ i \subset X_ i^ i$ is an open we may choose a $U$-admissible blowup $X_ i' \to X_ i^ i$ restricting to $X_{12}' \to X_{12}^ i$, see Divisors, Lemma 31.34.3. Then $X_{12}' \subset X_ i'$ is an open subscheme and the diagram
\[ \xymatrix{ X_{12}' \ar[d] \ar[r] & X_ i' \ar[d] \\ X_{12}^ i \ar[r] & X_ i^ i } \]
is commutative with vertical arrows blowing ups and horizontal arrows open immersions. Note that $X'_{12} \to X_1' \times _ S X_2'$ is an immersion and proper (use that $X'_{12} \to X_{12}$ is proper and $X_{12} \to X_1 \times _ S X_2$ is closed and $X_1' \times _ S X_2' \to X_1 \times _ S X_2$ is separated and apply Morphisms, Lemma 29.41.7). Thus $X'_{12} \to X_1' \times _ S X_2'$ is a closed immersion. It follows that if we define $X$ by glueing $X_1'$ and $X_2'$ along the common open subscheme $X_{12}'$, then $X \to S$ is of finite type and separated (Lemma 38.33.1). As compositions of $U$-admissible blowups are $U$-admissible blowups (Divisors, Lemma 31.34.2) the lemma is proved.
$\square$
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