Lemma 32.16.2. Let f : X \to S and h : U \to X be morphisms of schemes. Assume that S is locally Noetherian, that f is locally of finite type, that h is 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 at most one dotted arrow making the diagram commute, then f is separated.
Proof.
We will apply Lemma 32.16.1 to the morphisms U \to X and \Delta : X \to X \times _ S X. We check the conditions. Observe that \Delta is quasi-compact by Properties, Lemma 28.5.4 (and Schemes, Lemma 26.21.13). Of course \Delta is locally of finite type and separated (true for any diagonal morphism). Finally, suppose given a commutative solid diagram
\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & U \ar[r]^ h & X \ar[d]^\Delta \\ \mathop{\mathrm{Spec}}(A) \ar[rr]^{(a, b)} \ar@{-->}[rru] & & X \times _ S X }
where A is a discrete valuation ring with field of fractions K. Then a and b give two dotted arrows in the diagram of the lemma and have to be equal. Hence as dotted arrow we can use a = b which gives existence. This finishes the proof.
\square
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