Lemma 29.36.4. The base change of a morphism which is 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 lemma follows from Lemma 29.14.5 combined with the fact that being étale is a property of ring maps that is stable under base change, see Algebra, Lemma 10.143.3.
$\square$

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