The Stacks project

Proof. Choose a compactification $U_ i \subset X_ i$ over $B$ 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 $U_ i \to U$, see Lemma 81.14.6. Picture

Denote $Z_1 \subset U$ the reduced closed subspace corresponding to the closed subset $|U| \setminus |U_2|$. Recall that $f^{-1}Z_1$ is a closed subspace of $U_1$ mapping isomorphically to $Z_1$. Denote $Z_2 \subset U$ the reduced closed subspace corresponding to the closed subset $|U| \setminus \mathop{\mathrm{Im}}(|f|) = |U_2| \setminus \mathop{\mathrm{Im}}(|U_1 \times _ U U_2| \to |U_2|)$. Thus we have

set theoretically. Denote $Z_{i, i} \subset V_ i$ the inverse image of $Z_ i$ under $\psi _ i$. Observe that $\psi _2$ is an isomorphism over an open neighbourhood of $Z_2$. Observe that $Z_{1, 1} = \psi _1^{-1}Z_1 = f^{-1}Z_1 \amalg T$ for some closed subspace $T \subset V_1$ disjoint from $f^{-1}Z_1$ and furthermore $\psi _1$ is étale along $f^{-1}Z_1$. 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 subspaces of $U$, we see that $Z_{i, i}$ and $Z_{i, j}$ are disjoint closed subspaces of $V_ i$.

Denote $\overline{Z}_{i, i}$ and $\overline{Z}_{i, j}$ the scheme theoretic images of $Z_{i, i}$ and $Z_{i, j}$ in $X_ i$. We recall that $|Z_{i, j}|$ is dense in $|\overline{Z}_{i, j}|$, see Morphisms of Spaces, Lemma 67.17.7. 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 81.14.2. 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. Hence we may replace $X_1$ by another $V_1$-admissible blowup and assume $|\overline{Z}_{1, 1}|$ is the disjoint union of the closures of $|T|$ and $|f^{-1}Z_1|$ in $|X_1|$.

Set $V_{12} = V_1 \times _ U V_2$. We have an immersion $V_{12} \to X_1 \times _ B X_2$ which is the composition of the closed immersion $V_{12} = V_1 \times _ U V_2 \to V_1 \times _ B V_2$ (Morphisms of Spaces, Lemma 67.4.5) and the open immersion $V_1 \times _ B V_2 \to X_1 \times _ B X_2$. Let $X_{12} \subset X_1 \times _ B X_2$ be the scheme theoretic image of $V_{12} \to X_1 \times _ B X_2$. The projection morphisms

are proper as $X_1$ and $X_2$ are proper over $B$. 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 81.14.5.

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 subspace

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 81.14.3 there exists a $V_1$-admissible blowing up $X_1' \to X_1$ such that the strict transform $p'_1 : X'_{12} \to X'_1$ 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 result of the previous paragraph tells us that

where the intersection taken in $X_1 \times _ B X_2$. 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.

It turns out that we need to do one additional blowing up before we can conclude the argument. Namely, let $V_2 \subset W_2 \subset X_2$ be the open subspace with underlying topological space

Since $p_2(p_1^{-1}\overline{Z}_{1, 2})$ is contained in $W_2$ (see above) we see that replacing $X_2$ by a $W_2$-admissible blowup and $X_{21}$ by the corresponding strict transform will preserve the property of $p_1$ being an isomorphism over an open neighbourhood of $\overline{Z}_{1, 2}$. Since $\overline{Z}_{2, 1} \cap W_2 = \overline{Z}_{2, 1} \cap V_2 = Z_{2, 1}$ we see that $Z_{2, 1}$ is a closed subspace of $W_2$ and $V_2$. Observe that $V_{12} = V_1 \times _ U V_2 = p_1^{-1}(V_1) = p_2^{-1}(V_2)$ as open subspaces of $X_{12}$ as it is the largest open subspace of $X_{12}$ over which the morphism $\psi : V_{12} \to U$ extends; details omitted¹. We have the following equalities of closed subspaces of $V_{12}$:

Here and below we use the slight abuse of notation of writing $p_2$ in stead of the restriction of $p_2$ to $V_{12}$, etc. Since $p_2^{-1}(Z_{2, 1})$ is a closed subspace of $p_2^{-1}(W_2)$ as $Z_{2, 1}$ is a closed subspace of $W_2$ we conclude that also $p_1^{-1}f^{-1}Z_1$ is a closed subspace of $p_2^{-1}(W_2)$. Finally, the morphism $p_2 : X_{12} \to X_2$ is étale at points of $p_1^{-1}f^{-1}Z_1$ as $\psi _1$ is étale along $f^{-1}Z_1$ and $V_{12} = V_1 \times _ U V_2$. Thus we may apply Lemma 81.14.4 to the morphism $p_2 : X_{12} \to X_2$, the open $W_2$, the closed subspace $Z_{2, 1} \subset W_2$, and the closed subspace $p_1^{-1}f^{-1}Z_1 \subset p_2^{-1}(W_2)$. Hence after replacing $X_2$ by a $W_2$-admissible blowup and $X_{12}$ by the corresponding strict transform, we obtain for every geometric point $\overline{y}$ of the closure of $|p_1^{-1}f^{-1}Z_1|$ a local ring map $\mathcal{O}_{X_{12}, \overline{y}} \to \mathcal{O}$ such that $\mathcal{O}_{X_2, p_2(\overline{y})} \to \mathcal{O}$ is finite flat.

Consider the algebraic space

and with $T \subset V_1$ as in the first paragraph the algebraic space

obtained by pushout, see Lemma 81.9.2. Let us apply Lemma 81.14.1 to see that $W_ i \to B$ is separated. First, $U \to B$ and $X_ i \to B$ are separated. Let us check the quasi-compact immersion $U_ i \to U \times _ B (X_ i \setminus \overline{Z}_{i, j})$ is closed using the valuative criterion, see Morphisms of Spaces, Lemma 67.42.1. Choose a valuation ring $A$ over $B$ with fraction field $K$ and compatible morphisms $(u, x_ i) : \mathop{\mathrm{Spec}}(A) \to U \times _ B X_ i$ and $u_ i : \mathop{\mathrm{Spec}}(K) \to U_ i$. Since $\psi _ i$ is proper, we can find a unique $v_ i : \mathop{\mathrm{Spec}}(A) \to V_ i$ compatible with $u$ and $u_ i$. Since $X_ i$ is proper over $B$ we see that $x_ i = v_ i$. If $v_ i$ does not factor through $U_ i \subset V_ i$, then we conclude that $x_ i$ maps the closed point of $\mathop{\mathrm{Spec}}(A)$ into $Z_{i, j}$ or $T$ when $i = 1$. This finishes the proof because we removed $\overline{Z}_{i, j}$ and $\overline{T}$ in the construction of $W_ i$.

On the other hand, for any valuation ring $A$ over $B$ with fraction field $K$ and any morphism

over $B$, we claim that after replacing $A$ by an extension of valuation rings, there is an $i$ and an extension of $\gamma$ to a morphism $h_ i : \mathop{\mathrm{Spec}}(A) \to W_ i$. Namely, we first extend $\gamma$ to a morphism $g_2 : \mathop{\mathrm{Spec}}(A) \to X_2$ using the valuative criterion of properness. If the image of $g_2$ does not meet $\overline{Z}_{2, 1}$, then we obtain our morphism into $W_2$. Otherwise, denote $\overline{z} \in \overline{Z}_{2, 1}$ a geometric point lying over the image of the closed point under $g_2$. We may lift this to a geometric point $\overline{y}$ of $X_{12}$ in the closure of $|p_1^{-1}f^{-1}Z_1|$ because the map of spaces $|p_1^{-1}f^{-1}Z_1| \to |\overline{Z}_{2, 1}|$ is closed with image containing the dense open $|Z_{2, 1}|$. After replacing $A$ by its strict henselization (More on Algebra, Lemma 15.123.6) we get the following diagram

where $\mathcal{O}_{X_{12}, \overline{y}} \to \mathcal{O}$ is the map we found in the 5th paragraph of the proof. Since the horizontal composition is finite and flat we can find an extension of valuation rings $A'/A$ and dotted arrow making the diagram commute. After replacing $A$ by $A'$ this means that we obtain a lift $g_{12} : \mathop{\mathrm{Spec}}(A) \to X_{12}$ whose closed point maps into the closure of $|p_1^{-1}f^{-1}Z_1|$. Then $g_1 = p_1 \circ g_{12} : \mathop{\mathrm{Spec}}(A) \to X_1$ is a morphism whose closed point maps into the closure of $|f^{-1}Z_1|$. Since the closure of $|f^{-1}Z_1|$ is disjoint from the closure of $|T|$ and contained in $|\overline{Z}_{1, 1}|$ which is disjoint from $|\overline{Z}_{1, 2}|$ we conclude that $g_1$ defines a morphism $h_1 : \mathop{\mathrm{Spec}}(A) \to W_1$ as desired.

Consider a diagram

as in More on Morphisms of Spaces, Lemma 76.40.1. By the previous paragraph for every solid diagram

where $\mathop{\mathrm{Im}}(\gamma ) \subset \mathop{\mathrm{Im}}(U_1 \times _ U U_2 \to U)$ there is an $i$ and an extension $h_ i : \mathop{\mathrm{Spec}}(A) \to W_ i$ of $\gamma$ after possibly replacing $A$ by an extension of valuation rings. 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 after possibly extending $A$. Since $W$ is separated over $B$, we see that the choice of extension isn't needed and the arrow is unique as well, see Morphisms of Spaces, Lemmas 67.41.5 and 67.43.1. Then finally the existence of the dotted arrow implies that $W \to B$ is universally closed by Morphisms of Spaces, Lemma 67.42.5. As $W \to B$ is already of finite type and separated, we win. $\square$