The Stacks project

Lemma 34.20.6. The property $\mathcal{P}(f) =$“$f$ is separated” is fpqc local on the base.

Proof. A base change of a separated morphism is separated, see Schemes, Lemma 25.21.12. Being separated is Zariski local on the base (from the definition or by Schemes, Lemma 25.21.7). Finally, let $S' \to S$ be a flat surjective morphism of affine schemes, and let $f : X \to S$ be a morphism. Assume that the base change $f' : X' \to S'$ is separated. This means that $\Delta ' : X' \to X'\times _{S'} X'$ is a closed immersion, hence universally closed. Note that $\Delta '$ is the base change of $\Delta : X \to X \times _ S X$ via $S' \to S$. By Lemma 34.20.3 this implies $\Delta $ is universally closed. Since it is an immersion (Schemes, Lemma 25.21.2) we conclude $\Delta $ is a closed immersion. Hence $f$ is separated. Therefore Lemma 34.19.4 applies and we win. $\square$


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