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The Stacks project

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

\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & U \ar[r]^ h & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[rr] \ar@{-->}[rru] & & S }

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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