The Stacks project

Lemma 35.23.2. The property $\mathcal{P}(f) =$“$f$ is quasi-separated” is fpqc local on the base.

Proof. Any base change of a quasi-separated morphism is quasi-separated, see Schemes, Lemma 26.21.12. Being quasi-separated is Zariski local on the base (from the definition or by Schemes, Lemma 26.21.6). 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 quasi-separated. This means that $\Delta ' : X' \to X'\times _{S'} X'$ is quasi-compact. Note that $\Delta '$ is the base change of $\Delta : X \to X \times _ S X$ via $S' \to S$. By Lemma 35.23.1 this implies $\Delta $ is quasi-compact, and hence $f$ is quasi-separated. Therefore Lemma 35.22.4 applies and we win. $\square$

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