Lemma 29.34.5. The base change of a morphism which is smooth is smooth.

**Proof.**
In the proof of Lemma 29.34.2 we saw that being smooth is a local property of ring maps. Hence the lemma follows from Lemma 29.14.5 combined with the fact that being smooth is a property of ring maps that is stable under base change, see Algebra, Lemma 10.137.4.
$\square$

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