Lemma 88.28.3. With assumptions and notation as in Theorem 88.27.4 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 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

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

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 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

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

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. $\square$

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