The Stacks project

Lemma 29.36.3. The composition of two morphisms which are étale is étale.

Proof. In the proof of Lemma 29.36.2 we saw that being étale is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 29.14.5 combined with the fact that being étale is a property of ring maps that is stable under composition, see Algebra, Lemma 10.143.3. $\square$

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