The Stacks project

Lemma 85.10.12. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $T \subset |X|$ be a closed subset. Let $W \to X_{/T}$ be an object of the category $\mathcal{C}_{X_{/T}}$ and let $Y \to X$ be the object corresponding to $W$ via Theorem 85.10.9. Then $Y \to X$ is separated if and only if $W \to X_{/T}$ is separated and $\Delta : W \to W \times _{X_{/T}} W$ is rig-surjective.

Proof. These conditions may be checked after base change to an affine scheme étale over $X$, resp. a formal affine algebraic space étale over $X_{/T}$, see Morphisms of Spaces, Lemma 64.4.12 as well as Formal Spaces, Lemma 84.23.5. If $U \to X$ ranges over étale morphisms with $U$ affine, then the formal completions $U_{/T} \to X_{/T}$ give a family of formal affine coverings as in Formal Spaces, Definition 84.7.1. Thus we may and do assume $X$ is affine. In the proof of both directions we may assume that $Y \to X$ and $W \to X_{/T}$ are quasi-separated by Lemma 85.10.11.

Proof of easy direction. Assume $Y \to X$ is separated. Then $Y \to Y \times _ X Y$ is a closed immersion and it follows that $W \to W \times _{X_{/T}} W$ is a closed immersion too, i.e., we see that $W \to X_{/T}$ is separated. Let

\[ p : \text{Spf}(R) \longrightarrow W \times _{X_{/T}} W = (Y \times _ X Y)_{/T} \]

be an adic morphism where $R$ is a complete discrete valuation ring with fraction field $K$. The composition into $Y \times _ X Y$ corresponds to a morphism $g : \mathop{\mathrm{Spec}}(R) \to Y \times _ X Y$, see Formal Spaces, Lemma 84.26.3. Since $p$ is an adic morphism, so is the composition $\text{Spf}(R) \to X$. Thus we see that $g(\mathop{\mathrm{Spec}}(K))$ is a point of

\[ (Y \times _ X Y) \times _ X (X \setminus T) \cong X \setminus T \cong Y \times _ X (X \setminus T) \]

(small detail omitted). Hence this lifts to a $K$-point of $Y$ and we obtain a commutative diagram

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

Since $Y \to X$ was assumed separated we find the dotted arrow exists (Cohomology of Spaces, Lemma 66.19.1). Applying the functor completion along $T$ we find that $p$ can be lifted to a morphism into $W$, i.e., $W \to W \times _{X_{/T}} W$ is rig-surjective.

Proof of hard direction. Assume $W \to X_{/T}$ separated and $W \to W \times _{X_{/T}} W$ rig-surjective. By Cohomology of Spaces, Lemma 66.19.1 and Remark 66.19.3 it suffices to show that given any commutative diagram

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

where $R$ is a complete discrete valuation ring with fraction field $K$, there is at most one dotted arrow making the diagram commute. Let $h : \mathop{\mathrm{Spec}}(R) \to X$ be the composition of $g$ with the morphism $Y \times _ X Y \to X$. There are three cases: Case I: $h(\mathop{\mathrm{Spec}}(R)) \subset (X \setminus T)$. This case is trivial because $Y \times _ X (X \setminus T) = X \setminus T$. Case II: $h$ maps $\mathop{\mathrm{Spec}}(R)$ into $T$. This case follows from our assumption that $W \to X_{/T}$ is separated. Namely, if $T$ denotes the reduced induced closed subspace structure on $T$, then $h$ factors through $T$ and

\[ W \times _{X_{/T}} T = Y \times _ X T \longrightarrow T \]

is separated by assumption (and for example Formal Spaces, Lemma 84.23.5) which implies we get the lifting property by Cohomology of Spaces, Lemma 66.19.1 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

\[ g_{/T} : \text{Spf}(R) \longrightarrow (Y \times _ X Y)_{/T} = W \times _{X_{/T}} W \]

is an adic morphism (detail omitted). Hence our assumption that $W \to W \times _{X_{/T}} W$ be rig-surjective implies we can lift $g_{/T}$ to a morphism $e : \text{Spf}(R) \to W = Y_{/T}$ (see Lemma 85.9.11 for why we do not need to extend $R$). Algebraizing the composition $\text{Spf}(R) \to Y$ using Formal Spaces, Lemma 84.26.3 we find a morphism $\mathop{\mathrm{Spec}}(R) \to Y$ lifting $g$ 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'.