The Stacks project

[II Theorem 7.3.8, EGA]

Lemma 29.42.1 (Valuative criterion for properness). Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. Assume $f$ is of finite type and quasi-separated. Then the following are equivalent

  1. $f$ is proper,

  2. $f$ satisfies the valuative criterion (Schemes, Definition 26.20.3),

  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 making the diagram commute.

Proof. Part (3) is a reformulation of (2). Thus the lemma is a formal consequence of Schemes, Proposition 26.20.6 and Lemma 26.22.2 and the definitions. $\square$

Comments (3)

Comment #2709 by Ariyan Javanpeykar on

A reference for the valuative criterion of properness: EGA II, Theorem 7.3.8

Comment #3815 by Kestutis Cesnavicius on

Same comment as for

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