The Stacks project

Lemma 32.8.6. Notation and assumptions as in Situation 32.8.1. If $f$ is separated, then $f_ i$ is separated for some $i \geq 0$.

Proof. Apply Lemma 32.8.5 to the diagonal morphism $\Delta _{X_0/S_0} : X_0 \to X_0 \times _{S_0} X_0$. (This is permissible as diagonal morphisms are locally of finite type and the fibre product $X_0 \times _{S_0} X_0$ is quasi-compact and quasi-separated, see Schemes, Lemma 26.21.2, Morphisms, Lemma 29.15.5, and Schemes, Remark 26.21.18. $\square$

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