The Stacks project

Lemma 26.22.1. Let $f : X \to S$ be a morphism of schemes. If $f$ is separated, then $f$ satisfies the uniqueness part of the valuative criterion.

Proof. Let a diagram as in Definition 26.20.3 be given. Suppose there are two morphisms $a, b : \mathop{\mathrm{Spec}}(A) \to X$ fitting into the diagram. Let $Z \subset \mathop{\mathrm{Spec}}(A)$ be the equalizer of $a$ and $b$. By Lemma 26.21.5 this is a closed subscheme of $\mathop{\mathrm{Spec}}(A)$. By assumption it contains the generic point of $\mathop{\mathrm{Spec}}(A)$. Since $A$ is a domain this implies $Z = \mathop{\mathrm{Spec}}(A)$. Hence $a = b$ as desired. $\square$

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