The Stacks project

Lemma 35.25.2. The property $\mathcal{P}(f)=$“$f$ is locally of finite type” is fppf local on the source.

Proof. Being locally of finite type is Zariski local on the source and the target, see Morphisms, Lemma 29.15.2. It is a property which is preserved under composition, see Morphisms, Lemma 29.15.3, and a flat morphism locally of finite presentation is locally of finite type, see Morphisms, Lemma 29.21.8. This proves (1), (2) and (3) of Lemma 35.23.4. The final condition (4) is Lemma 35.11.2. Hence we win. $\square$


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