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.
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Comment #2709 by Ariyan Javanpeykar on
Comment #2746 by Takumi Murayama on
Comment #3815 by Kestutis Cesnavicius on