## 105.11 Noetherian valuative criterion

In this section we will discuss (refined) valuative criteria for morphisms of algebraic stacks using only discrete valuation rings in the Noetherian setting. There are many different variants and we will add more here over time as needed.

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks (or algebraic spaces or schemes). A refined valuative criterion is one where we are given a morphism $\mathcal{U} \to \mathcal{X}$ (with some properties) and we only look at existence or uniqueness of dotted arrows in solid diagrams of the form

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[d] \ar[r] & \mathcal{U} \ar[r] & \mathcal{X} \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[rr] \ar@{..>}[rru] & & \mathcal{Y} }$

We use this terminology below to describe the results we have obtained sofar.

Non-Noetherian valuative criteria for morphisms of algebraic stacks

1. Morphisms of Stacks, Section 100.40 (for separatedness of the diagonal),

2. Morphisms of Stacks, Section 100.41 (for separatedness),

3. Morphisms of Stacks, Section 100.42 (for universal closedness),

4. Morphisms of Stacks, Section 100.43 (for properness).

For algebraic spaces we have the following valuative criteria

1. Morphisms of Spaces, Section 66.42 (for universal closedness),

2. Morphisms of Spaces, Lemma 66.42.5 (refined for universal closedness)

3. Morphisms of Spaces, Section 66.43 (for separatedness),

4. Morphisms of Spaces, Section 66.44 (for properness),

5. Decent Spaces, Section 67.16 (for universal closedness for decent spaces),

6. Decent Spaces, Lemma 67.17.11 (for universal closedness for decent morphisms between algebraic spaces),

7. Cohomology of Spaces, Section 68.19 contains Noetherian valuative criteria

1. Cohomology of Spaces, Lemma 68.19.1 (for separatedness using discrete valuation rings),

2. Cohomology of Spaces, Lemma 68.19.2 (for properness using discrete valuation rings),

3. Cohomology of Spaces, Remark 68.19.3 (discusses how to reduce to complete discrete valuation rings),

8. Limits of Spaces, Section 69.21 discussing Noetherian valuative criteria

1. Limits of Spaces, Lemma 69.21.2 (for separatedness using discrete valuation rings and generic points)

2. Limits of Spaces, Lemma 69.21.3 (for properness using discrete valuation rings and generic points)

3. Limits of Spaces, Lemma 69.21.4 (for universal closedness using discrete valuation rings).

9. Limits of Spaces, Section 69.22 discussing refined Noetherian valuative criteria

1. Limits of Spaces, Lemmas 69.22.1 and 69.22.3 (refined for properness using discrete valuation rings),

2. Limits of Spaces, Lemma 69.22.2 (refined for separatedness using discrete valuation rings),

For schemes we have the following valuative criteria

1. Schemes, Section 26.20 (for universal closedness)

2. Schemes, Section 26.22 (for separatedness),

3. Morphisms, Section 29.42 (for properness)

4. Morphisms, Lemma 29.42.2 (refined for universal closedness),

5. Limits, Section 32.15 discussing Noetherian valuative criteria

1. Limits, Lemma 32.15.2 (for separatedness using discrete valuation rings and generic points)

2. Limits, Lemma 32.15.3 (for properness using discrete valuation rings and generic points)

3. Limits, Lemma 32.15.4 (for universal closedness using discrete valuation rings).

6. Limits, Section 32.16 discussing refined Noetherian valuative criteria

1. Limits, Lemmas 32.16.1 and 32.16.3 (refined for properness using discrete valuation rings),

2. Limits, Lemma 32.16.2 (refined for separatedness using discrete valuation rings),

7. Limits, Section 32.17 discussing valuative criteria over a Noetherian base where one can get discrete valuation rings essentially of finite type over the base.

This ends our list of previous results.

Many of the results in this section can (and perhaps should) be proved by appealing to the following lemma, although we have not always done so.

Lemma 105.11.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ finite type and $\mathcal{Y}$ locally Noetherian. Let $y \in |\mathcal{Y}|$ be a point in the closure of the image of $|f|$. Then there exists a commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r] & \mathcal{Y} }$

of algebraic stacks where $A$ is a discrete valuation ring and $K$ is its field of fractions mapping the closed point of $\mathop{\mathrm{Spec}}(A)$ to $y$.

Proof. Choose an affine scheme $V$, a point $v \in V$ and a smooth morphism $V \to \mathcal{Y}$ mapping $v$ to $y$. The map $|V| \to |\mathcal{Y}|$ is open and by Properties of Stacks, Lemma 99.4.3 the image of $|\mathcal{X} \times _\mathcal {Y} V| \to |V|$ is the inverse image of the image of $|f|$. We conclude that the point $v$ is in the closure of the image of $|\mathcal{X} \times _\mathcal {Y} V| \to |V|$. If we prove the lemma for $\mathcal{X} \times _\mathcal {Y} V \to V$ and the point $v$, then the lemma follows for $f$ and $y$. In this way we reduce to the situation described in the next paragraph.

Assume we have $f : \mathcal{X} \to Y$ and $y \in |Y|$ as in the lemma where $Y$ is a Noetherian affine scheme. Since $f$ is quasi-compact, we conclude that $\mathcal{X}$ is quasi-compact. Hence we can choose an affine scheme $W$ and a surjective smooth morphism $W \to \mathcal{X}$. Then the image of $|f|$ is the same as the image of $|W| \to |Y|$. In this way we reduce to the case of schemes which is Limits, Lemma 32.15.1. $\square$

Lemma 105.11.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume

1. $\mathcal{Y}$ is locally Noetherian,

2. $f$ is locally of finite type and quasi-separated,

3. for every commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_ x \ar[d]_ j & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r]^ y \ar@{-->}[ru] & \mathcal{Y} }$

where $A$ is a discrete valuation ring and $K$ its fraction field and any $2$-arrow $\gamma : y \circ j \to f \circ x$ the category of dotted arrows (Morphisms of Stacks, Definition 100.39.1) is either empty or a setoid with exactly one isomorphism class.

Then $f$ is separated.

Proof. To prove that $f$ is separated we have to show that $\Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ is proper. We already know that $\Delta$ is representable by algebraic spaces, locally of finite type (Morphisms of Stacks, Lemma 100.3.3) and quasi-compact and quasi-separated (by definition of $f$ being quasi-separated). Choose a scheme $U$ and a surjective smooth morphism $U \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$. Set

$V = \mathcal{X} \times _{\Delta , \mathcal{X} \times _\mathcal {Y} \mathcal{X}} U$

It suffices to show that the morphism of algebraic spaces $V \to U$ is proper (Properties of Stacks, Lemma 99.3.3). Observe that $U$ is locally Noetherian (use Morphisms of Stacks, Lemma 100.17.5 and the fact that $U \to \mathcal{Y}$ is locally of finite type) and $V \to U$ is of finite type and quasi-separated (as the base change of $\Delta$ and properties of $\Delta$ listed above). Applying Cohomology of Spaces, Lemma 68.19.2 it suffices to show: Given a commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_ v \ar[d]_ j & V \ar[d]^ g \ar[r] & \mathcal{X} \ar[d]^\Delta \\ \mathop{\mathrm{Spec}}(A) \ar[r]^ u \ar@{-->}[ru] \ar@{..>}[rru] & U \ar[r] & \mathcal{X} \times _\mathcal {Y} \mathcal{X} }$

where $A$ is a discrete valuation ring and $K$ its fraction field, there is a unique dashed arrow making the diagram commute. By Morphisms of Stacks, Lemma 100.39.4 the categories of dashed and dotted arrows are equivalent. Assumption (3) implies there is a unique dotted arrow up to isomorphism, see Morphisms of Stacks, Lemma 100.41.1. We conclude there is a unique dashed arrow as desired. $\square$

Lemma 105.11.3. Let $f : \mathcal{X} \to \mathcal{Y}$ and $h : \mathcal{U} \to \mathcal{X}$ be morphisms of algebraic stacks. Assume that $\mathcal{Y}$ is locally Noetherian, that $f$ and $h$ are of finite type, that $f$ is separated, and that the image of $|h| : |\mathcal{U}| \to |\mathcal{X}|$ is dense in $|\mathcal{X}|$. If given any $2$-commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-u \ar[d]_ j & \mathcal{U} \ar[r]_ h & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[rr]^-y & & \mathcal{Y} }$

where $A$ is a discrete valuation ring with field of fractions $K$ and $\gamma : y \circ j \to f \circ h \circ u$ there exist an extension $K'/K$ of fields, a valuation ring $A' \subset K'$ dominating $A$ such that the category of dotted arrows for the induced diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K') \ar[r]_-{x'} \ar[d]_{j'} & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A') \ar[r]^-{y'} \ar@{..>}[ru] & \mathcal{Y} }$

with induced $2$-arrow $\gamma ' : y' \circ j' \to f \circ x'$ is nonempty (Morphisms of Stacks, Definition 100.39.1), then $f$ is proper.

Proof. It suffices to prove that $f$ is universally closed. Let $V \to \mathcal{Y}$ be a smooth morphism where $V$ is an affine scheme. By Properties of Stacks, Lemma 99.4.3 the image $I$ of $|\mathcal{U} \times _\mathcal {Y} V| \to |\mathcal{X} \times _\mathcal {Y} V|$ is the inverse image of the image of $|h|$. Since $|\mathcal{X} \times _\mathcal {Y} V| \to |\mathcal{X}|$ is open (Morphisms of Stacks, Lemma 100.27.15) we conclude that $I$ is dense in $|\mathcal{X} \times _\mathcal {Y} V|$. Also since the category of dotted arrows behaves well with respect to base change (Morphisms of Stacks, Lemma 100.39.4) the assumption on existence of dotted arrows (after extension) is inherited by the morphisms $\mathcal{U} \times _\mathcal {Y} V \to \mathcal{X} \times _\mathcal {Y} V \to V$. Therefore the assumptions of the lemma are satisfied for the morphisms $\mathcal{U} \times _\mathcal {Y} V \to \mathcal{X} \times _\mathcal {Y} V \to V$. Hence we may assume $\mathcal{Y}$ is an affine scheme.

Assume $\mathcal{Y} = Y$ is an affine scheme. (From now on we no longer have to worry about the $2$-arrows $\gamma$ and $\gamma '$, see Morphisms of Stacks, Lemma 100.39.3.) Then $\mathcal{U}$ is quasi-compact. Choose an affine scheme $U$ and a surjective smooth morphism $U \to \mathcal{U}$. Then we may and do replace $\mathcal{U}$ by $U$. Thus we may assume that $\mathcal{U}$ is an affine scheme.

Assume $\mathcal{Y} = Y$ and $\mathcal{U} = U$ are affine schemes. By Chow's lemma (Theorem 105.10.3) we can choose a surjective proper morphism $X \to \mathcal{X}$ where $X$ is an algebraic space. We will use below that $X \to Y$ is separated as a composition of separated morphisms. Consider the algebraic space $W = X \times _\mathcal {X} U$. The projection morphism $W \to X$ is of finite type. We may replace $X$ by the scheme theoretic image of $W \to X$ and hence we may assume that the image of $|W|$ in $|X|$ is dense in $|X|$ (here we use that the image of $|h|$ is dense in $|\mathcal{X}|$, so after this replacement, the morphism $X \to \mathcal{X}$ is still surjective). We claim that for every solid commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & W \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[rr] \ar@{..>}[rru] & & Y }$

where $A$ is a discrete valuation ring with field of fractions $K$, there exists a dotted arrow making the diagram commute. First, it is enough to prove there exists a dotted arrow after replacing $K$ by an extension and $A$ by a valuation ring in this extension dominating $A$, see Morphisms of Spaces, Lemma 66.41.4. By the assumption of the lemma we get an extension $K'/K$ and a valuation ring $A' \subset K'$ dominating $A$ and an arrow $\mathop{\mathrm{Spec}}(A') \to \mathcal{X}$ lifting the composition $\mathop{\mathrm{Spec}}(A') \to \mathop{\mathrm{Spec}}(A) \to Y$ and compatible with the composition $\mathop{\mathrm{Spec}}(K') \to \mathop{\mathrm{Spec}}(K) \to W \to X$. Because $X \to \mathcal{X}$ is proper, we can use the valuative criterion of properness (Morphisms of Stacks, Lemma 100.43.1) to find an extension $K''/K'$ and a valuation ring $A'' \subset K''$ dominating $A'$ and a morphism $\mathop{\mathrm{Spec}}(A'') \to X$ lifting the composition $\mathop{\mathrm{Spec}}(A'') \to \mathop{\mathrm{Spec}}(A') \to \mathcal{X}$ and compatible with the composition $\mathop{\mathrm{Spec}}(K'') \to \mathop{\mathrm{Spec}}(K') \to \mathop{\mathrm{Spec}}(K) \to X$. Then $K''/K$ and $A'' \subset K''$ and the morphism $\mathop{\mathrm{Spec}}(A'') \to X$ is a solution to the problem posed above and the claim holds.

The claim implies the morphism $X \to Y$ is proper by the case of the lemma for algebraic spaces (Limits of Spaces, Lemma 69.22.1). By Morphisms of Stacks, Lemma 100.37.6 we conclude that $\mathcal{X} \to Y$ is proper as desired. $\square$

Lemma 105.11.4. Let $f : \mathcal{X} \to \mathcal{Y}$ and $h : \mathcal{U} \to \mathcal{X}$ be morphisms of algebraic stacks. Assume that $\mathcal{Y}$ is locally Noetherian, that $f$ is locally of finite type and quasi-separated, that $h$ is of finite type, and that the image of $|h| : |\mathcal{U}| \to |\mathcal{X}|$ is dense in $|\mathcal{X}|$. If given any $2$-commutative diagram

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

where $A$ is a discrete valuation ring with field of fractions $K$ and $\gamma : y \circ j \to f \circ h \circ u$, the category of dotted arrows is either empty or a setoid with exactly one isomorphism class, then $f$ is separated.

Proof. We have to prove $\Delta$ is a proper morphism. Assume first that $\Delta$ is separated. Then we may apply Lemma 105.11.3 to the morphisms $\mathcal{U} \to \mathcal{X}$ and $\Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$. Observe that $\Delta$ is quasi-compact as $f$ is quasi-separated. Of course $\Delta$ is locally of finite type (true for any diagonal morphism, see Morphisms of Stacks, Lemma 100.3.3). Finally, suppose given a $2$-commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-u \ar[d]_ j & \mathcal{U} \ar[r]_ h & \mathcal{X} \ar[d]^\Delta \\ \mathop{\mathrm{Spec}}(A) \ar[rr]^-y \ar@{..>}[rru] & & \mathcal{X} \times _\mathcal {Y} \mathcal{X} }$

where $A$ is a discrete valuation ring with field of fractions $K$ and $\gamma : y \circ j \to \Delta \circ h \circ u$. By Morphisms of Stacks, Lemma 100.41.1 and the assumption in the lemma we find there exists a unique dotted arrow. This proves the last assumption of Lemma 105.11.3 holds and the result follows.

In the general case, it suffices to prove $\Delta$ is separated since then we'll be back in the previous case. In fact, we claim that the assumptions of the lemma hold for

$\mathcal{U} \to \mathcal{X} \quad \text{and}\quad \Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$

Namely, since $\Delta$ is representable by algebraic spaces, the category of dotted arrows for a diagram as in the previous paragraph is a setoid (see for example Morphisms of Stacks, Lemma 100.39.2). The argument in the preceding paragraph shows these categories are either empty or have one isomorphism class. Thus $\Delta$ is separated. $\square$

Lemma 105.11.5. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume that $\mathcal{Y}$ is locally Noetherian and that $f$ is of finite type. If given any $2$-commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-x \ar[d]_ j & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r]^-y & \mathcal{Y} }$

where $A$ is a discrete valuation ring with field of fractions $K$ and $\gamma : y \circ j \to f \circ x$ there exist an extension $K'/K$ of fields, a valuation ring $A' \subset K'$ dominating $A$ such that the category of dotted arrows for the induced diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K') \ar[r]_-{x'} \ar[d]_{j'} & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A') \ar[r]^-{y'} \ar@{..>}[ru] & \mathcal{Y} }$

with induced $2$-arrow $\gamma ' : y' \circ j' \to f \circ x'$ is nonempty (Morphisms of Stacks, Definition 100.39.1), then $f$ is universally closed.

Proof. Let $V \to \mathcal{Y}$ be a smooth morphism where $V$ is an affine scheme. The category of dotted arrows behaves well with respect to base change (Morphisms of Stacks, Lemma 100.39.4). Hence the assumption on existence of dotted arrows (after extension) is inherited by the morphism $\mathcal{X} \times _\mathcal {Y} V \to V$. Therefore the assumptions of the lemma are satisfied for the morphism $\mathcal{X} \times _\mathcal {Y} V \to V$. Hence we may assume $\mathcal{Y}$ is an affine scheme.

Assume $\mathcal{Y} = Y$ is a Noetherian affine scheme. (From now on we no longer have to worry about the $2$-arrows $\gamma$ and $\gamma '$, see Morphisms of Stacks, Lemma 100.39.3.) To prove that $f$ is universally closed it suffices to show that $|\mathcal{X} \times \mathbf{A}^ n| \to |Y \times \mathbf{A}^ n|$ is closed for all $n$ by Limits of Stacks, Lemma 101.7.2. Since the assumption in the lemma is inherited by the product morphism $\mathcal{X} \times \mathbf{A}^ n \to Y \times \mathbf{A}^ n$ (details omitted) we reduce to proving that $|\mathcal{X}| \to |Y|$ is closed.

Assume $Y$ is a Noetherian affine scheme. Let $T \subset |\mathcal{X}|$ be a closed subset. We have to show that the image of $T$ in $|Y|$ is closed. We may replace $\mathcal{X}$ by the reduced induced closed subspace structure on $T$; we omit the verification that property on the existence of dotted arrows is preserved by this replacement. Thus we reduce to proving that the image of $|\mathcal{X}| \to |Y|$ is closed.

Let $y \in |Y|$ be a point in the closure of the image of $|\mathcal{X}| \to |Y|$. By Lemma 105.11.1 we may choose a commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r] & Y }$

where $A$ is a discrete valuation ring and $K$ is its field of fractions mapping the closed point of $\mathop{\mathrm{Spec}}(A)$ to $y$. It follows immediately from the assumption in the lemma that $y$ is in the image of $|\mathcal{X}| \to |Y|$ and the proof is complete. $\square$

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