The Stacks project

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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