The Stacks project

Lemma 34.20.29. The property $\mathcal{P}(f) =$“$f$ is étale” is fpqc local on the base.

Proof. A morphism is étale if and only if it flat and G-unramified. See Morphisms, Lemma 28.34.16. We have seen already that being flat and G-unramified are fpqc local on the base (Lemmas 34.20.15, and 34.20.28). Hence the result follows. $\square$

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