Lemma 76.40.1. Let $S$ be a scheme. Let $Y$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \to X_1$ and $U \to X_2$ be open immersions of algebraic spaces over $Y$ and assume $U$, $X_1$, $X_2$ of finite type and separated over $Y$. 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 algebraic spaces over $Y$ 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 $Y$.
Proof.
Throughout the proof all the algebraic spaces will be separated of finite type over $Y$. This in particular implies these algebraic spaces are quasi-compact and quasi-separated and that the morphisms between them will be quasi-compact and separated. See Morphisms of Spaces, Sections 67.4 and 67.8. We will use that if $U \to W$ is an immersion of such spaces over $Y$, then the scheme theoretic image $Z$ of $U$ in $W$ is a closed subspace of $W$ and $U \to Z$ is an open immersion, $U \subset Z$ is scheme theoretically dense, and $|U| \subset |Z|$ is dense. See Morphisms of Spaces, Lemma 67.17.7.
Let $X_{12} \subset X_1 \times _ Y X_2$ be the scheme theoretic image of $U \to X_1 \times _ Y X_2$. The projections $p_ i : X_{12} \to X_ i$ induce isomorphisms $p_ i^{-1}(U) \to U$ by Morphisms of Spaces, Lemma 67.16.7. 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 subspace of $X_ i^ i$, see Lemma 76.39.4. 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 on Spaces, Lemma 71.18.3. 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 on Spaces, Lemma 71.17.10 for what this entails. We obtain 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 on Spaces, Lemma 71.19.3. Then $X_{12}' \subset X_ i'$ is an open subspace 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 $X'_{12} \to X_1' \times _ Y X_2'$ is an immersion and proper (use that $X'_{12} \to X_{12}$ is proper and $X_{12} \to X_1 \times _ Y X_2$ is closed and $X_1' \times _ Y X_2' \to X_1 \times _ Y X_2$ is separated and apply Morphisms of Spaces, Lemma 67.40.6). Thus $X'_{12} \to X_1' \times _ Y X_2'$ is a closed immersion. If we define $X$ by glueing $X_1'$ and $X_2'$ along the common open subspace $X_{12}'$, then $X \to Y$ is of finite type and separated1. As compositions of $U$-admissible blowups are $U$-admissible blowups (Divisors on Spaces, Lemma 71.19.2) the lemma is proved.
$\square$
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