Lemma 35.23.29 (02VN)—The Stacks project

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Part 2: Schemes
Chapter 35: Descent
Section 35.23: Properties of morphisms local in the fpqc topology on the target
Lemma 35.23.29 (cite)

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Lemma 35.23.29. 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 29.36.16. We have seen already that being flat and G-unramified are fpqc local on the base (Lemmas 35.23.15, and 35.23.28). Hence the result follows. $\square$

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