Lemma 29.34.4 (01VA)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.34: Smooth morphisms
Lemma 29.34.4 (cite)

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Lemma 29.34.4. The composition of two morphisms which are 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 first statement of 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 composition, see Algebra, Lemma 10.137.14. $\square$

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