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

$f$ is proper,

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

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.

## Comments (3)

Comment #2709 by Ariyan Javanpeykar on

Comment #2746 by Takumi Murayama on

Comment #3815 by Kestutis Cesnavicius on