Lemma 35.28.1 (036N)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 35: Descent
Section 35.28: Properties of morphisms local in the fppf topology on the source
Lemma 35.28.1 (cite)

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Lemma 35.28.1. The property $\mathcal{P}(f)=$ “ $f$ is locally of finite presentation” is fppf local on the source.

Proof. Being locally of finite presentation is Zariski local on the source and the target, see Morphisms, Lemma 29.21.2. It is a property which is preserved under composition, see Morphisms, Lemma 29.21.3. This proves (1), (2) and (3) of Lemma 35.26.4. The final condition (4) is Lemma 35.14.1. Hence we win. $\square$

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