The Stacks project

66.44 Valuative criterion of properness

Here is a statement.

Lemma 66.44.1 (Valuative criterion for properness). Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is of finite type and quasi-separated. Then the following are equivalent

  1. $f$ is proper,

  2. the valuative criterion holds as in Definition 66.41.1,

  3. given any commutative solid 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 valuation ring with field of fractions $K$, there exists a unique dotted arrow, i.e., $f$ satisfies the valuative criterion as in Schemes, Definition 26.20.3.

Proof. Formal consequence of Lemma 66.43.3 and the definitions. $\square$


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