Lemma 35.31.1 (036W)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 35: Descent
Section 35.31: Properties of morphisms local in the étale topology on the source
Lemma 35.31.1 (cite)

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Lemma 35.31.1. The property $\mathcal{P}(f)=$“$f$ is étale” is étale local on the source.

Proof. Combine Lemma 35.26.4 with Morphisms, Lemma 29.36.2 (local for Zariski on source and target), Morphisms, Lemma 29.36.3 (pre-composing), and Lemma 35.14.4 (part (4)). $\square$

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