The Stacks project

Lemma 67.19.2. 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 67.19.1. To do this we may work ├ętale locally on $Y$ (Morphisms of Spaces, Lemma 65.9.5). 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 67.18.1). We claim that $X' \to \mathop{\mathrm{Spec}}(A)$ is universally closed. The claim implies the lemma by Morphisms of Spaces, Lemma 65.40.7. 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 $a$. $\square$


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