The Stacks project

Lemma 35.23.1. The property $\mathcal{P}(f) =$“$f$ is quasi-compact” is fpqc local on the base.

Proof. A base change of a quasi-compact morphism is quasi-compact, see Schemes, Lemma 26.19.3. Being quasi-compact is Zariski local on the base, see Schemes, Lemma 26.19.2. 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-compact. Then $X'$ is quasi-compact, and $X' \to X$ is surjective. Hence $X$ is quasi-compact. This implies that $f$ is quasi-compact. Therefore Lemma 35.22.4 applies and we win. $\square$

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