The Stacks project

Lemma 70.22.1. Let $S$ be a scheme. Let $f : X \to Y$ and $h : U \to X$ be morphisms of algebraic spaces over $S$. Assume that $Y$ is locally Noetherian, that $f$ and $h$ are of finite type, that $f$ is separated, and that the image of $|h| : |U| \to |X|$ is dense in $|X|$. If given any commutative solid diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & U \ar[r]^ h & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[rr] \ar@{-->}[rru] & & Y } \]

where $A$ is a discrete valuation ring with field of fractions $K$, there exists a dotted arrow making the diagram commute, then $f$ is proper.

Proof. It suffices to prove that $f$ is universally closed. Let $V \to Y$ be an ├ętale morphism where $V$ is an affine scheme. By Morphisms of Spaces, Lemma 67.9.5 it suffices to prove that the base change $X \times _ Y V \to V$ is universally closed. By Properties of Spaces, Lemma 66.4.3 the image $I$ of $|U \times _ Y V| \to |X \times _ Y V|$ is the inverse image of the image of $|h|$. Since $|X \times _ Y V| \to |X|$ is open (Properties of Spaces, Lemma 66.16.7) we conclude that $I$ is dense in $|X \times _ Y V|$. Therefore the assumptions of the lemma are satisfied for the morphisms $U \times _ Y V \to X \times _ Y V \to V$. Hence we may assume $Y$ is an affine scheme.

Assume $Y$ is an affine scheme. Then $U$ is quasi-compact. Choose an affine scheme and a surjective ├ętale morphism $W \to U$. Then we may and do replace $U$ by $W$ and assume that $U$ is affine. By the weak version of Chow's lemma (Cohomology of Spaces, Lemma 69.18.1) we can choose a surjective proper morphism $X' \to X$ where $X'$ is a scheme. Then $U' = X' \times _ X U$ is a scheme and $U' \to X'$ is of finite type. We may replace $X'$ by the scheme theoretic image of $h' : U' \to X'$ and hence $h'(U')$ is dense in $X'$. We claim that for every diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & U' \ar[r]^ h & X' \ar[d]^{f'} \\ \mathop{\mathrm{Spec}}(A) \ar[rr] \ar@{-->}[rru] & & Y } \]

where $A$ is a discrete valuation ring with field of fractions $K$, there exists a dotted arrow making the diagram commute. Namely, we first get an arrow $\mathop{\mathrm{Spec}}(A) \to X$ by the assumption of the lemma and then we lift this to an arrow $\mathop{\mathrm{Spec}}(A) \to X'$ using the valuative criterion for properness (Morphisms of Spaces, Lemma 67.44.1). The morphism $X' \to Y$ is separated as a composition of a proper and a separated morphism. Thus by the case of schemes the morphism $X' \to Y$ is proper (Limits, Lemma 32.16.1). By Morphisms of Spaces, Lemma 67.40.7 we conclude that $X \to Y$ is proper. $\square$

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