The Stacks project

70.22 Refined Noetherian valuative criteria

This section is the analogue of Limits, Section 32.16. One usually does not have to consider all possible diagrams with valuation rings when checking valuative criteria.

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 étale 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 étale 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. $\square$

Lemma 70.22.2. 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 $f$ is locally of finite type and quasi-separated, that $h$ is of finite type, 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 at most one dotted arrow making the diagram commute, then $f$ is separated.

Proof. We will apply Lemma 70.22.1 to the morphisms $U \to X$ and $\Delta : X \to X \times _ Y X$. We check the conditions. Observe that $\Delta $ is quasi-compact because $f$ is quasi-separated. Of course $\Delta $ is locally of finite type and separated (true for any diagonal morphism). Finally, suppose given a commutative solid diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & U \ar[r]^ h & X \ar[d]^\Delta \\ \mathop{\mathrm{Spec}}(A) \ar[rr]^{(a, b)} \ar@{-->}[rru] & & X \times _ Y X } \]

where $A$ is a discrete valuation ring with field of fractions $K$. Then $a$ and $b$ give two dotted arrows in the diagram of the lemma and have to be equal. Hence as dotted arrow we can use $a = b$ which gives existence. This finishes the proof. $\square$

Lemma 70.22.3. 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 quasi-separated, and that $h(U)$ 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 unique dotted arrow making the diagram commute, then $f$ is proper.

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