The Stacks project

Proof. Choose a compactification $U_ i \subset X_ i$ for $i = 1, 2$. We may assume $U_ i$ is scheme theoretically dense in $X_ i$. We may assume there is an open $V_ i \subset X_ i$ and a proper morphism $\psi _ i : V_ i \to U$ extending $\text{id} : U_ i \to U_ i$, see Lemma 38.33.6. Picture

If $\{ i, j\} = \{ 1, 2\} $ denote $Z_ i = U \setminus U_ j = U_ i \setminus (U_1 \cap U_2)$ and $Z_ j = U \setminus U_ i = U_ j \setminus (U_1 \cap U_2)$. Thus we have

Denote $Z_{i, i} \subset V_ i$ the inverse image of $Z_ i$ under $\psi _ i$. Observe that $\psi _ i$ is an isomorphism over an open neighbourhood of $Z_ i$. Denote $Z_{i, j} \subset V_ i$ the inverse image of $Z_ j$ under $\psi _ i$. Observe that $\psi _ i : Z_{i, j} \to Z_ j$ is a proper morphism. Since $Z_ i$ and $Z_ j$ are disjoint closed subsets of $U$, we see that $Z_{i, i}$ and $Z_{i, j}$ are disjoint closed subsets of $V_ i$.

Denote $\overline{Z}_{i, i}$ and $\overline{Z}_{i, j}$ the closures of $Z_{i, i}$ and $Z_{i, j}$ in $X_ i$. After replacing $X_ i$ by a $V_ i$-admissible blowup we may assume that $\overline{Z}_{i, i}$ and $\overline{Z}_{i, j}$ are disjoint, see Lemma 38.33.2. We assume this holds for both $X_1$ and $X_2$. Observe that this property is preserved if we replace $X_ i$ by a further $V_ i$-admissible blowup.

Set $V_{12} = V_1 \times _ U V_2$. We have an immersion $V_{12} \to X_1 \times _ S X_2$ which is the composition of the closed immersion $V_{12} = V_1 \times _ U V_2 \to V_1 \times _ S V_2$ (Schemes, Lemma 26.21.9) and the open immersion $V_1 \times _ S V_2 \to X_1 \times _ S X_2$. Let $X_{12} \subset X_1 \times _ S X_2$ be the scheme theoretic image of $V_{12} \to X_1 \times _ S X_2$. The projection morphisms

are proper as $X_1$ and $X_2$ are proper over $S$. If we replace $X_1$ by a $V_1$-admissible blowing up, then $X_{12}$ is replaced by the strict transform with respect to this blowing up, see Lemma 38.33.5.

Denote $\psi : V_{12} \to U$ the compositions $\psi = \psi _1 \circ p_1|_{V_{12}} = \psi _2 \circ p_2|_{V_{12}}$. Consider the closed subscheme

The morphism $p_1|_{V_{12}} : V_{12} \to V_1$ is an isomorphism over an open neighbourhood of $Z_{1, 2}$ because $\psi _2 : V_2 \to U$ is an isomorphism over an open neighbourhood of $Z_2$ and $V_{12} = V_1 \times _ U V_2$. By Lemma 38.33.3 there exists a $V_1$-admissible blowing up $X_1' \to X_1$ such that the strict tranform $p'_1 : X'_{12} \to X'_1$ of $p_1$ is an isomorphism over an open neighbourhood of the closure of $Z_{1, 2}$ in $X'_1$. After replacing $X_1$ by $X'_1$ and $X_{12}$ by $X'_{12}$ we may assume that $p_1$ is an isomorphism over an open neighbourhood of $\overline{Z}_{1, 2}$.

The reduction of the previous paragraph tells us that

where the intersection taken in $X_1 \times _ S X_2$. Namely, the inverse image $p_1^{-1}(\overline{Z}_{1, 2})$ in $X_{12}$ maps isomorphically to $\overline{Z}_{1, 2}$. In particular, we see that $Z_{12, 2}$ is dense in $p_1^{-1}(\overline{Z}_{1, 2})$. Thus $p_2$ maps $p_1^{-1}(\overline{Z}_{1, 2})$ into $\overline{Z}_{2, 2}$. Since $\overline{Z}_{2, 2} \cap \overline{Z}_{2, 1} = \emptyset $ we conclude.

Consider the schemes

obtained by glueing. Let us apply Lemma 38.33.1 to see that $W_ i \to S$ is separated. First, $U \to S$ and $X_ i \to S$ are separated. The immersion $U_ i \to U \times _ S (X_ i \setminus \overline{Z}_{i, j})$ is closed because any specialization $u_ i \leadsto u$ with $u_ i \in U_ i$ and $u \in U \setminus U_ i$ can be lifted uniquely to a specialization $u_ i \leadsto v_ i$ in $V_ i$ along the proper morphism $\psi _ i : V_ i \to U$ and then $v_ i$ must be in $Z_{i, j}$. Thus the image of the immersion is closed, whence the immersion is a closed immersion.

On the other hand, for any valuation ring $A$ over $S$ with fraction field $K$ and any morphism $\gamma : \mathop{\mathrm{Spec}}(K) \to (U_1 \cap U_2)$ over $S$, there is an $i$ and an extension of $\gamma $ to a morphism $h_ i : \mathop{\mathrm{Spec}}(A) \to W_ i$. Namely, for both $i = 1, 2$ there is a morphism $g_ i : \mathop{\mathrm{Spec}}(A) \to X_ i$ extending $\gamma $ by the valuative criterion of properness for $X_ i$ over $S$, see Morphisms, Lemma 29.42.1. Thus we only are in trouble if $g_ i(\mathfrak m_ A) \in \overline{Z}_{i, j}$ for $i = 1, 2$. This is impossible by the emptyness of the intersection of $X_{12}$ and $\overline{Z}_{1, 2} \times _ S \overline{Z}_{2, 1}$ we proved above.

Consider a diagram

as in Lemma 38.33.4. By the previous paragraph for every solid diagram

where $\mathop{\mathrm{Im}}(\gamma ) \subset U_1 \cap U_2$ there is an $i$ and an extension $h_ i : \mathop{\mathrm{Spec}}(A) \to W_ i$ of $\gamma $. Using the valuative criterion of properness for $W'_ i \to W_ i$, we can then lift $h_ i$ to $h'_ i : \mathop{\mathrm{Spec}}(A) \to W'_ i$. Hence the dotted arrow in the diagram exists. Since $W$ is separated over $S$, we see that the arrow is unique as well. This implies that $W \to S$ is universally closed by Morphisms, Lemma 29.42.2. As $W \to S$ is already of finite type and separated, we win. $\square$