Definition 26.21.3. Let $f : X \to S$ be a morphism of schemes.

We say $f$ is

*separated*if the diagonal morphism $\Delta _{X/S}$ is a closed immersion.We say $f$ is

*quasi-separated*if the diagonal morphism $\Delta _{X/S}$ is a quasi-compact morphism.We say a scheme $Y$ is

*separated*if the morphism $Y \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is separated.We say a scheme $Y$ is

*quasi-separated*if the morphism $Y \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is quasi-separated.

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