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. $\square$

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