The Stacks project

Lemma 70.6.9. Notation and assumptions as in Situation 70.6.1. If $f$ is separated, then $f_ i$ is separated for some $i \geq 0$.

Proof. Apply Lemma 70.6.8 to the diagonal morphism $\Delta _{X_0/Y_0} : X_0 \to X_0 \times _{Y_0} X_0$. (Diagonal morphisms are locally of finite type and the fibre product $X_0 \times _{Y_0} X_0$ is quasi-compact and quasi-separated. Some details omitted.) $\square$

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