The Stacks project

Lemma 35.23.27. The property $\mathcal{P}(f) =$“$f$ is smooth” is fpqc local on the base.

Proof. A morphism is smooth if and only if it is locally of finite presentation, flat, and has smooth fibres. We have seen already that being flat and locally of finite presentation are fpqc local on the base (Lemmas 35.23.15, and 35.23.11). Hence the result follows for smooth from Morphisms, Lemma 29.34.15. $\square$

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