The Stacks project

Lemma 85.10.13. 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 proper if and only if the following conditions hold

  1. $W \to X_{/T}$ is proper,

  2. $W \to X_{/T}$ is rig-surjective, and

  3. $\Delta : W \to W \times _{X_{/T}} W$ 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 64.40.2 as well as Formal Spaces, Lemma 84.24.2. 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 separated and quasi-compact and that $W \to W \times _{X_{/T}} W$ is rig-surjective by Lemmas 85.10.10 and 85.10.12.

Proof of the easy direction. Assume $Y \to X$ is proper. Then $Y_{/T} = Y \times _ X X_{/T} \to X_{/T}$ is proper too. Let

\[ p : \text{Spf}(R) \longrightarrow X_{/T} \]

be an adic morphism where $R$ is a complete discrete valuation ring with fraction field $K$. Then $p$ corresponds to a morphism $g : \mathop{\mathrm{Spec}}(R) \to X$, see Formal Spaces, Lemma 84.26.3. Since $p$ is an adic morphism, we have $p(\mathop{\mathrm{Spec}}(K)) \not\in T$. Since $Y \to X$ is an isomorphism over $X \setminus T$ we can lift to $X$ and obtain a commutative diagram

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

Since $Y \to X$ was assumed proper we find the dotted arrow exists. (Cohomology of Spaces, Lemma 66.19.2). Applying the functor completion along $T$ we find that $p$ can be lifted to a morphism into $W$, i.e., $W \to X_{/T}$ is rig-surjective.

Proof of hard direction. Assume $W \to X_{/T}$ proper, $W \to W \times _{X_{/T}} W$ rig-surjective, and $W \to X_{/T}$ rig-surjective. By Cohomology of Spaces, Lemma 66.19.2 and Remark 66.19.3 it suffices to show that given any commutative diagram

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

where $R$ is a complete discrete valuation ring with fraction field $K$, there is a 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 proper. 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 proper by assumption which implies we get the lifting property by Cohomology of Spaces, Lemma 66.19.2 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

\[ g_{/T} : \text{Spf}(R) \longrightarrow Y_{/T} = W \]

is an adic morphism (detail omitted). Hence our assumption that $W \to X_{/T}$ be rig-surjective implies we can lift $g_{/T}$ to a morphism $e : \text{Spf}(R') \to W = Y_{/T}$ for some extension of complete discrete valuation rings $R \subset R'$. 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 $g$. By the discussion in Cohomology of Spaces, Remark 66.19.3 this is sufficient to conclude that $Y \to X$ is proper. $\square$


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