The Stacks project

Proof. Omitted. Hint: check that $\Delta : X \to X \times _ S X$ is closed by using the open covering of $X \times _ S X$ given by $U \times _ S U$, $U \times _ S V$, $V \times _ S U$, and $V \times _ S V$. $\square$

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, Lemma 29.21.7. Denote $Z \subset X$ the closed subscheme 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, Lemma 31.34.5 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, Definition 31.34.1 and Morphisms, Lemma 29.21.3). 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, Lemma 31.33.2. Thus (1) is proved.

Proof of (2). By Properties, Lemma 28.24.1 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 Properties, Lemma 28.22.2 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 subschemes $Z_ i = V(\mathcal{I}_ i)$ to conclude. $\square$

Proof. Let $T' \subset V$ be the complement of the maximal open over which $f|_ U$ is an isomorphism. Then $T', T$ are closed in $V$ and $T \cap T' = \emptyset $. Since $V$ is a spectral topological space, we can find constructible closed subsets $T_ c, T'_ c$ with $T \subset T_ c$, $T' \subset T'_ c$ such that $T_ c \cap T'_ c = \emptyset $ (choose a quasi-compact open $W$ of $V$ containing $T'$ not meeting $T$ and set $T_ c = V \setminus W$, then choose a quasi-compact open $W'$ of $V$ containing $T_ c$ not meeting $T'$ and set $T'_ c = V \setminus W'$). By Lemma 38.33.2 we may, after replacing $Y$ by a $V$-admissible blowing up, assume that $T_ c$ and $T'_ c$ have disjoint closures in $Y$. Set $Y_0 = Y \setminus \overline{T}'_ c$, $V_0 = V \setminus T'_ c$, $U_0 = U \times _ V V_0$, and $X_0 = X \times _ Y Y_0$. Since $U_0 \to V_0$ is an isomorphism, we can find a $V_0$-admissible blowing up $Y'_0 \to Y_0$ such that the strict transform $X'_0$ of $X_0$ maps isomorphically to $Y'_0$, see Lemma 38.31.3. By Divisors, Lemma 31.34.3 there exists a $V$-admissible blow up $Y' \to Y$ whose restriction to $Y_0$ is $Y'_0 \to Y_0$. If $f' : X' \to Y'$ denotes the strict transform of $f$, then we see what we want is true because $f'$ restricts to an isomorphism over $Y'_0$. $\square$

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

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

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. $\square$

Proof. Denote $Z' \to Z$ the strict transform. The morphism $Z' \to X'$ induces a morphism $Z' \to X' \times _ S Y$ which is a closed immersion (as $Z'$ is a closed subscheme of $X' \times _ X Z$ by definition). Thus to finish the proof it suffices to show that the scheme theoretic image $Z''$ of $V \to Z'$ is $Z'$. Observe that $Z'' \subset Z'$ is a closed subscheme such that $V \to Z'$ factors through $Z''$. Since both $V \to X \times _ S Y$ and $V \to X' \times _ S Y$ are quasi-compact (for the latter this follows from Schemes, Lemma 26.21.14 and the fact that $X' \times _ S Y \to X \times _ S Y$ is separated as a base change of a proper morphism), by Morphisms, Lemma 29.6.3 we see that $Z \cap (U \times _ S Y) = Z'' \cap (U \times _ S Y)$. Thus the inclusion morphism $Z'' \to Z'$ is an isomorphism away from the exceptional divisor $E$ of $Z' \to Z$. However, the structure sheaf of $Z'$ does not have any nonzero sections supported on $E$ (by definition of strict transforms) and we conclude that the surjection $\mathcal{O}_{Z'} \to \mathcal{O}_{Z''}$ must be an isomorphism. $\square$

Proof. Consider the scheme theoretic image $Z \subset Y \times _ S U$ of the “diagonal” morphism $V \to Y \times _ S U$. If we replace $Y$ by a $V$-admissible blowing up, then $Z$ is replaced by the strict transform with respect to this blowing up, see Lemma 38.33.5. Hence by Lemma 38.31.3 we may assume $Z \to Y$ is an open immersion. If $V' \subset Y$ denotes the image, then we see that the induced morphism $V' \to U$ is proper because the projection $Y \times _ S U \to U$ is proper and $V' \cong Z$ is a closed subscheme of $Y \times _ S U$. $\square$

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 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 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. $\square$

Proof. We first reduce to the Noetherian case. We strongly urge the reader to skip this paragraph. There exists a closed immersion $X \to X'$ with $X' \to S$ of finite presentation and separated. See Limits, Proposition 32.9.6. If we find a compactification of $X'$ over $S$, then taking the scheme theoretic image of $X$ in this will give a compactification of $X$ over $S$. Thus we may assume $X \to S$ is separated and of finite presentation. We may write $S = \mathop{\mathrm{lim}}\nolimits S_ i$ as a directed limit of a system of Noetherian schemes with affine transition morphisms. See Limits, Proposition 32.5.4. We can choose an $i$ and a morphism $X_ i \to S_ i$ of finite presentation whose base change to $S$ is $X \to S$, see Limits, Lemma 32.10.1. After increasing $i$ we may assume $X_ i \to S_ i$ is separated, see Limits, Lemma 32.8.6. If we can find a compactification of $X_ i$ over $S_ i$, then the base change of this to $S$ will be a compactification of $X$ over $S$. This reduces us to the case discussed in the next paragraph.

Assume $S$ is Noetherian. We can choose a finite affine open covering $X = \bigcup _{i = 1, \ldots , n} U_ i$ such that $U_1 \cap \ldots \cap U_ n$ is dense in $X$. This follows from Properties, Lemma 28.29.4 and the fact that $X$ is quasi-compact with finitely many irreducible components. For each $i$ we can choose an $n_ i \geq 0$ and an immersion $U_ i \to \mathbf{A}^{n_ i}_ S$ by Morphisms, Lemma 29.39.2. Hence $U_ i$ has a compactification over $S$ for $i = 1, \ldots , n$ by taking the scheme theoretic image in $\mathbf{P}^{n_ i}_ S$. Applying Lemma 38.33.7 $(n - 1)$ times we conclude that the theorem is true. $\square$