The Stacks project

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

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

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

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

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