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68.19 Noetherian valuative criterion

We prove a version of the valuative criterion for properness using discrete valuation rings. More precise (and therefore more technical) versions can be found in Limits of Spaces, Section 69.21.

Lemma 68.19.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume

  1. $Y$ is locally Noetherian,

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

  3. for every commutative diagram

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

    where $A$ is a discrete valuation ring and $K$ its fraction field, there is at most one dotted arrow making the diagram commute.

Then $f$ is separated.

Proof. We have to show that the diagonal $\Delta : X \to X \times _ Y X$ is a closed immersion. We already know $\Delta $ is representable, separated, a monomorphism, and locally of finite type, see Morphisms of Spaces, Lemma 66.4.1. Choose an affine scheme $U$ and an étale morphism $U \to X \times _ Y X$. Set $V = X \times _{\Delta , X \times _ Y X} U$. It suffices to show that $V \to U$ is a closed immersion (Morphisms of Spaces, Lemma 66.12.1). Since $X \times _ Y X$ is locally of finite type over $Y$ we see that $U$ is Noetherian (use Morphisms of Spaces, Lemmas 66.23.2, 66.23.3, and 66.23.5). Note that $V$ is a scheme as $\Delta $ is representable. Also, $V$ is quasi-compact because $f$ is quasi-separated. Hence $V \to U$ is of finite type. Consider a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & V \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] \ar@{-->}[ru] & U } \]

of morphisms of schemes where $A$ is a discrete valuation ring with fraction field $K$. We can interpret the composition $\mathop{\mathrm{Spec}}(A) \to U \to X \times _ Y X$ as a pair of morphisms $a, b : \mathop{\mathrm{Spec}}(A) \to X$ agreeing as morphisms into $Y$ and equal when restricted to $\mathop{\mathrm{Spec}}(K)$. Hence our assumption (3) guarantees $a = b$ and we find the dotted arrow in the diagram. By Limits, Lemma 32.15.3 we conclude that $V \to U$ is proper. In other words, $\Delta $ is proper. Since $\Delta $ is a monomorphism, we find that $\Delta $ is a closed immersion (Étale Morphisms, Lemma 41.7.2) as desired. $\square$

Lemma 68.19.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume

  1. $Y$ is locally Noetherian,

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

  3. for every commutative diagram

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

    where $A$ is a discrete valuation ring and $K$ its fraction field, there is a unique dotted arrow making the diagram commute.

Then $f$ is proper.

Proof. It suffices to prove $f$ is universally closed because $f$ is separated by Lemma 68.19.1. To do this we may work étale locally on $Y$ (Morphisms of Spaces, Lemma 66.9.5). Hence we may assume $Y = \mathop{\mathrm{Spec}}(A)$ is a Noetherian affine scheme. Choose $X' \to X$ as in the weak form of Chow's lemma (Lemma 68.18.1). We claim that $X' \to \mathop{\mathrm{Spec}}(A)$ is universally closed. The claim implies the lemma by Morphisms of Spaces, Lemma 66.40.7. To prove this, according to Limits, Lemma 32.15.4 it suffices to prove that in every solid commutative diagram

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

where $A$ is a dvr with fraction field $K$ we can find the dotted arrow $a$. By assumption we can find the dotted arrow $b$. Then the morphism $X' \times _{X, b} \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(A)$ is a proper morphism of schemes and by the valuative criterion for morphisms of schemes we can lift $b$ to the desired morphism $a$. $\square$

Remark 68.19.3 (Variant for complete discrete valuation rings). In Lemmas 68.19.1 and 68.19.2 it suffices to consider complete discrete valuation rings. To be precise in Lemma 68.19.1 we can replace condition (3) by the following condition: Given any commutative diagram

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

where $A$ is a complete discrete valuation ring with fraction field $K$ there exists at most one dotted arrow making the diagram commute. Namely, given any diagram as in Lemma 68.19.1 (3) the completion $A^\wedge $ is a discrete valuation ring (More on Algebra, Lemma 15.43.5) and the uniqueness of the arrow $\mathop{\mathrm{Spec}}(A^\wedge ) \to X$ implies the uniqueness of the arrow $\mathop{\mathrm{Spec}}(A) \to X$ for example by Properties of Spaces, Proposition 65.17.1. Similarly in Lemma 68.19.2 we can replace condition (3) by the following condition: Given any commutative diagram

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

where $A$ is a complete discrete valuation ring with fraction field $K$ there exists an extension $A \subset A'$ of complete discrete valuation rings inducing a fraction field extension $K \subset K'$ such that there exists a unique arrow $\mathop{\mathrm{Spec}}(A') \to X$ making the diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K') \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(K) \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A') \ar[r] \ar[rru] & \mathop{\mathrm{Spec}}(A) \ar[r] & Y } \]

commute. Namely, given any diagram as in Lemma 68.19.2 part (3) the existence of any commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(L) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(K) \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(B) \ar[r] \ar[rru] & \mathop{\mathrm{Spec}}(A) \ar[r] & Y } \]

for any extension $A \subset B$ of discrete valuation rings will imply there exists an arrow $\mathop{\mathrm{Spec}}(A) \to X$ fitting into the diagram. This was shown in Morphisms of Spaces, Lemma 66.41.4. In fact, it follows from these considerations that it suffices to look for dotted arrows in diagrams for any class of discrete valuation rings such that, given any discrete valuation ring, there is an extension of it that is in the class. For example, we could take complete discrete valuation rings with algebraically closed residue field.


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