The Stacks project

Lemma 86.24.3. With assumptions and notation as in Theorem 86.23.4 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 86.20.6 and 86.20.10. Assume $g$ is separated and $\Delta _ g$ is rig-surjective. Exactly as in the proof of Lemma 86.24.1 we may check this over the members of a ├ętale 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 85.25.2 (being separated for morphisms of formal algebraic spaces is preserved by base change), and Lemma 86.18.4 (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 86.24.2.

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 85.25.5) 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 86.18.10. Algebraizing the composition $\text{Spf}(R) \to X'$ using Formal Spaces, Lemma 85.28.3 we find a morphism $\mathop{\mathrm{Spec}}(R) \to X'$ lifting $p$ as desired. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0ARW. Beware of the difference between the letter 'O' and the digit '0'.