The Stacks project

Proof. If $f$ is quasi-compact, then $g$ is quasi-compact by Lemma 88.23.5. 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 67.8.8. By Formal Spaces, Lemma 87.17.3 the base changes $W_ i \times _{X_{/T}} (X_ i)_{/T} \to (X_ i)_{/T}$ are quasi-compact. By Lemma 88.27.1 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

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

Proof. If $f$ is quasi-separated, then $g$ is quasi-separated by Lemma 88.23.7. Conversely, assume $g$ is quasi-separated. We have to show that $f$ is quasi-separated. Exactly as in the proof of Lemma 88.28.1 we may check this over the members of a étale covering of $X$ by affine schemes using Morphisms of Spaces, Lemma 67.4.12 and Formal Spaces, Lemma 87.30.5. 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

is quasi-compact. Hence we conclude by the equality

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

Proof. If $f$ is separated, then $g$ is separated and $\Delta _ g$ is rig-surjective by Lemmas 88.23.7 and 88.23.11. Assume $g$ is separated and $\Delta _ g$ is rig-surjective. Exactly as in the proof of Lemma 88.28.1 we may check this over the members of a étale covering of $X$ by affine schemes using Morphisms of Spaces, Lemma 67.4.4 (locality on the base of being separated for morphisms of algebraic spaces), Formal Spaces, Lemma 87.30.2 (being separated for morphisms of formal algebraic spaces is preserved by base change), and Lemma 88.21.4 (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 88.28.2.

By Cohomology of Spaces, Lemma 69.19.1 and Remark 69.19.3 it suffices to show that given any commutative diagram

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

is separated by assumption (and for example Formal Spaces, Lemma 87.30.5) which implies we get the lifting property by Cohomology of Spaces, Lemma 69.19.1 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

is an adic morphism (by Formal Spaces, Lemma 87.14.4 and Definition 87.23.2). 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 88.21.11. Algebraizing the composition $\text{Spf}(R) \to X'$ using Formal Spaces, Lemma 87.33.3 we find a morphism $\mathop{\mathrm{Spec}}(R) \to X'$ lifting $p$ as desired. $\square$

Proof. If $f$ is proper, then $g$ is a formal modification by Lemma 88.24.3. Assume $g$ is a formal modification. By Lemmas 88.28.1 and 88.28.3 we see that $f$ is quasi-compact and separated.

By Cohomology of Spaces, Lemma 69.19.2 and Remark 69.19.3 it suffices to show that given any commutative diagram

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

is proper by assumption which implies we get the lifting property by Cohomology of Spaces, Lemma 69.19.2 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

is an adic morphism (by Formal Spaces, Lemma 87.14.4 and Definition 87.23.2). 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 87.33.3 we find a morphism $\mathop{\mathrm{Spec}}(R') \to X'$ lifting $p$ as desired. $\square$

Proof. If $f$ is étale, then $g$ is étale by Lemma 88.23.2. 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 66.12.5. 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 76.20.3 we conclude that $f$ is étale at points lying over $T$. $\square$