The Stacks project

Lemma 26.21.15. An affine scheme is separated. A morphism from an affine scheme to another scheme is separated.

Proof. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Then $U \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ has closed diagonal by Lemma 26.21.1. Thus $U$ is separated by Definition 26.21.3. If $U \to X$ is a morphism of schemes, then we can apply Lemma 26.21.13 to the morphisms $U \to X \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ to conclude that $U \to X$ is separated. $\square$

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