The Stacks project

86.28 Completions and morphisms, II

To obtain Artin's theorem on dilatations, we need to match formal modifications with actual modifications in the correspondence given by Theorem 86.27.4. We urge the reader to skip this section.

Lemma 86.28.1. With assumptions and notation as in Theorem 86.27.4 let $f : X' \to X$ correspond to $g : W \to X_{/T}$. Then $f$ is quasi-compact if and only if $g$ is quasi-compact.

Proof. If $f$ is quasi-compact, then $g$ is quasi-compact by Lemma 86.23.5. Conversely, assume $g$ is quasi-compact. Choose an étale covering $\{ X_ i \to X\} $ with $X_ i$ affine. It suffices to prove that the base change $X' \times _ X X_ i \to X_ i$ is quasi-compact, see Morphisms of Spaces, Lemma 65.8.8. By Formal Spaces, Lemma 85.13.3 the base changes $W_ i \times _{X_{/T}} (X_ i)_{/T} \to (X_ i)_{/T}$ are quasi-compact. By Lemma 86.27.1 we reduce to the case described in the next paragraph.

Assume $X$ is affine and $g : W \to X_{/T}$ quasi-compact. We have to show that $X'$ is quasi-compact. Let $V \to X'$ be a surjective étale morphism where $V = \coprod _{j \in J} V_ j$ is a disjoint union of affines. Then $V_{/T} \to X'_{/T} = W$ is a surjective étale morphism. Since $W$ is quasi-compact, then we can find a finite subset $J' \subset J$ such that $\coprod _{j \in J'} (V_ j)_{/T} \to W$ is surjective. Then it follows that

\[ U \amalg \coprod \nolimits _{j \in J'} V_ j \longrightarrow X' \]

is surjective (and hence $X'$ is quasi-compact). Namely, we have $|X'| = |U| \amalg |W_{red}|$ as $X'_{/T} = W$. $\square$

Lemma 86.28.2. With assumptions and notation as in Theorem 86.27.4 let $f : X' \to X$ correspond to $g : W \to X_{/T}$. Then $f$ is quasi-separated if and only if $g$ is so.

Proof. If $f$ is quasi-separated, then $g$ is quasi-separated by Lemma 86.23.7. Conversely, assume $g$ is quasi-separated. We have to show that $f$ is quasi-separated. Exactly as in the proof of Lemma 86.28.1 we may check this over the members of a étale covering of $X$ by affine schemes using Morphisms of Spaces, Lemma 65.4.12 and Formal Spaces, Lemma 85.26.5. Thus we may and do assume $X$ is affine.

Let $V \to X'$ be a surjective étale morphism where $V = \coprod _{j \in J} V_ j$ is a disjoint union of affines. To show that $X'$ is quasi-separated, it suffices to show that $V_ j \times _{X'} V_{j'}$ is quasi-compact for all $j, j' \in J$. Since $W$ is quasi-separated the fibre products $(V_ j \times _ Y V_{j'})_{/T} = (V_ j)_{/T} \times _{X'_{/T}} (V_{j'})_{/T}$ are quasi-compact for all $j, j' \in J$. Since $X$ is Noetherian affine and $U' \to U$ is an isomorphism, we see that

\[ (V_ j \times _{X'} V_{j'}) \times _ X U = (V_ j \times _ X V_{j'}) \times _ X U \]

is quasi-compact. Hence we conclude by the equality

\[ |V_ j \times _{X'} V_{j'}| = |(V_ j \times _{X'} V_{j'}) \times _ X U| \amalg |(V_ j \times _{X'} V_{j'})_{/T, red}| \]

and the fact that a formal algebraic space is quasi-compact if and only if its associated reduced algebraic space is so. $\square$

Lemma 86.28.3. With assumptions and notation as in Theorem 86.27.4 let $f : X' \to X$ correspond to $g : W \to X_{/T}$. Then $f$ is separated $\Leftrightarrow $ $g$ is separated and $\Delta _ g : W \to W \times _{X_{/T}} W$ is rig-surjective.

Proof. If $f$ is separated, then $g$ is separated and $\Delta _ g$ is rig-surjective by Lemmas 86.23.7 and 86.23.11. Assume $g$ is separated and $\Delta _ g$ is rig-surjective. Exactly as in the proof of Lemma 86.28.1 we may check this over the members of a étale covering of $X$ by affine schemes using Morphisms of Spaces, Lemma 65.4.4 (locality on the base of being separated for morphisms of algebraic spaces), Formal Spaces, Lemma 85.26.2 (being separated for morphisms of formal algebraic spaces is preserved by base change), and Lemma 86.21.4 (being rig-surjective is preserved by base change). Thus we may and do assume $X$ is affine. Furthermore, we already know that $f : X' \to X$ is quasi-separated by Lemma 86.28.2.

By Cohomology of Spaces, Lemma 67.19.1 and Remark 67.19.3 it suffices to show that given any commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X' \ar[d] \\ \mathop{\mathrm{Spec}}(R) \ar[r]^ p \ar@{-->}[ru] & X' \times _ X X' } \]

where $R$ is a complete discrete valuation ring with fraction field $K$, there is a dotted arrow making the diagram commute (as this will give the uniqueness part of the valuative criterion). Let $h : \mathop{\mathrm{Spec}}(R) \to X$ be the composition of $p$ with the morphism $Y \times _ X Y \to X$. There are three cases: Case I: $h(\mathop{\mathrm{Spec}}(R)) \subset U$. This case is trivial because $U' = X' \times _ X U \to U$ is an isomorphism. Case II: $h$ maps $\mathop{\mathrm{Spec}}(R)$ into $T$. This case follows from our assumption that $g : W \to X_{/T}$ is separated. Namely, if $Z$ denotes the reduced induced closed subspace structure on $T$, then $h$ factors through $Z$ and

\[ W \times _{X_{/T}} Z = X' \times _ X Z \longrightarrow Z \]

is separated by assumption (and for example Formal Spaces, Lemma 85.26.5) which implies we get the lifting property by Cohomology of Spaces, Lemma 67.19.1 applied to the displayed arrow. Case III: $h(\mathop{\mathrm{Spec}}(K))$ is not in $T$ but $h$ maps the closed point of $\mathop{\mathrm{Spec}}(R)$ into $T$. In this case the corresponding morphism

\[ p_{/T} : \text{Spf}(R) \longrightarrow (X' \times _ X X')_{/T} = W \times _{X_{/T}} W \]

is an adic morphism (by Formal Spaces, Lemma 85.10.4 and Definition 85.19.3). Hence our assumption that $\Delta _ g : W \to W \times _{X_{/T}} W$ is rig-surjective implies we can lift $p_{/T}$ to a morphism $\text{Spf}(R) \to W = X'_{/T}$, see Lemma 86.21.11. Algebraizing the composition $\text{Spf}(R) \to X'$ using Formal Spaces, Lemma 85.29.3 we find a morphism $\mathop{\mathrm{Spec}}(R) \to X'$ lifting $p$ as desired. $\square$

Lemma 86.28.4. With assumptions and notation as in Theorem 86.27.4 let $f : X' \to X$ correspond to $g : W \to X_{/T}$. Then $f$ is proper if and only if $g$ is a formal modification (Definition 86.24.1).

Proof. If $f$ is proper, then $g$ is a formal modification by Lemma 86.24.3. Assume $g$ is a formal modification. By Lemmas 86.28.1 and 86.28.3 we see that $f$ is quasi-compact and separated.

By Cohomology of Spaces, Lemma 67.19.2 and Remark 67.19.3 it suffices to show that given any commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X' \ar[d]^ f \\ \mathop{\mathrm{Spec}}(R) \ar[r]^ p \ar@{-->}[ru] & X } \]

where $R$ is a complete discrete valuation ring with fraction field $K$, there is a dotted arrow making the diagram commute. There are three cases: Case I: $p(\mathop{\mathrm{Spec}}(R)) \subset U$. This case is trivial because $U' \to U$ is an isomorphism. Case II: $p$ maps $\mathop{\mathrm{Spec}}(R)$ into $T$. This case follows from our assumption that $g : W \to X_{/T}$ is proper. Namely, if $Z$ denotes the reduced induced closed subspace structure on $T$, then $p$ factors through $Z$ and

\[ W \times _{X_{/T}} Z = X' \times _ X Z \longrightarrow Z \]

is proper by assumption which implies we get the lifting property by Cohomology of Spaces, Lemma 67.19.2 applied to the displayed arrow. Case III: $p(\mathop{\mathrm{Spec}}(K))$ is not in $T$ but $p$ maps the closed point of $\mathop{\mathrm{Spec}}(R)$ into $T$. In this case the corresponding morphism

\[ p_{/T} : \text{Spf}(R) \longrightarrow X'_{/T} = W \]

is an adic morphism (by Formal Spaces, Lemma 85.10.4 and Definition 85.19.3). Hence our assumption that $g : W \to X_{/T}$ be rig-surjective implies we can lift $g_{/T}$ to a morphism $\text{Spf}(R') \to W = X'_{/T}$ for some extension of complete discrete valuation rings $R \subset R'$. Algebraizing the composition $\text{Spf}(R') \to X'$ using Formal Spaces, Lemma 85.29.3 we find a morphism $\mathop{\mathrm{Spec}}(R') \to X'$ lifting $p$ as desired. $\square$

Lemma 86.28.5. With assumptions and notation as in Theorem 86.27.4 let $f : X' \to X$ correspond to $g : W \to X_{/T}$. Then $f$ is étale if and only if $g$ is étale.

Proof. If $f$ is étale, then $g$ is étale by Lemma 86.23.2. Conversely, assume $g$ is étale. Since $f$ is an isomorphism over $U$ we see that $f$ is étale over $U$. Thus it suffices to prove that $f$ is étale at any point of $X'$ lying over $T$. Denote $Z \subset X$ the reduced closed subspace whose underlying topological space is $|Z| = T \subset |X|$, see Properties of Spaces, Definition 64.12.5. Letting $Z_ n \subset X$ be the $n$th infinitesimal neighbourhood we have $X_{/T} = \mathop{\mathrm{colim}}\nolimits Z_ n$. Since $X'_{/T} = W \to X_{/T}$ we conclude that $f^{-1}(Z_ n) = X' \times _ X Z_ n \to Z_ n$ is étale by the assumed étaleness of $g$. By More on Morphisms of Spaces, Lemma 74.20.3 we conclude that $f$ is étale at points lying over $T$. $\square$


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