Lemma 32.16.3. Let f : X \to S and h : U \to X be morphisms of schemes. Assume that S is locally Noetherian, that f and h are of finite type, and that h(U) is dense in X. If given any commutative solid diagram
where A is a discrete valuation ring with field of fractions K, there exists a unique dotted arrow making the diagram commute, then f is proper.
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