The Stacks project

Lemma 87.28.3. With assumptions and notation as in Theorem 87.27.4 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 87.23.7 and 87.23.11. Assume $g$ is separated and $\Delta _ g$ is rig-surjective. Exactly as in the proof of Lemma 87.28.1 we may check this over the members of a ├ętale covering of $X$ by affine schemes using Morphisms of Spaces, Lemma 66.4.4 (locality on the base of being separated for morphisms of algebraic spaces), Formal Spaces, Lemma 86.30.2 (being separated for morphisms of formal algebraic spaces is preserved by base change), and Lemma 87.21.4 (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 87.28.2.

By Cohomology of Spaces, Lemma 68.19.1 and Remark 68.19.3 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 86.30.5) which implies we get the lifting property by Cohomology of Spaces, Lemma 68.19.1 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 86.14.4 and Definition 86.23.2). 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 87.21.11. Algebraizing the composition $\text{Spf}(R) \to X'$ using Formal Spaces, Lemma 86.33.3 we find a morphism $\mathop{\mathrm{Spec}}(R) \to X'$ lifting $p$ as desired. $\square$

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