Lemma 25.22.2 (Valuative criterion separatedness). Let $f : X \to S$ be a morphism. Assume

the morphism $f$ is quasi-separated, and

the morphism $f$ satisfies the uniqueness part of the valuative criterion.

Then $f$ is separated.

[II Proposition 7.2.3, EGA]

Lemma 25.22.2 (Valuative criterion separatedness). Let $f : X \to S$ be a morphism. Assume

the morphism $f$ is quasi-separated, and

the morphism $f$ satisfies the uniqueness part of the valuative criterion.

Then $f$ is separated.

**Proof.**
By assumption (1), Proposition 25.20.6, and Lemma 25.10.4 we see that it suffices to prove the morphism $\Delta _{X/S} : X \to X \times _ S X$ satisfies the existence part of the valuative criterion. Let a solid commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] \ar@{-->}[ru] & X \times _ S X } \]

be given. The lower right arrow corresponds to a pair of morphisms $a, b : \mathop{\mathrm{Spec}}(A) \to X$ over $S$. By (2) we see that $a = b$. Hence using $a$ as the dotted arrow works. $\square$

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