Proposition 10

Tag 0D9S, permanent. Cite it as [Bourbaki, Tag 0D9S].

Commutative Algebra, chapter X, § 7. Algèbres lisses

Let $A$ be a noetherian ring.

a) Let $A'$ be a noetherian $A$ -algebra and $B$ a smooth $A$ -algebra. Then the $A'$ -algebra $A' \otimes_A B$ is smooth.

b) Let $B$ be a smooth $A$ -algebra and $C$ a smooth $B$ -algebra. Then the $A$ -algebra $C$ is smooth.

c) Let $B$ and $C$ be two smooth $A$ -algebras. Then the $A$ -algebra $B \otimes_A C$ is smooth.

This follows from Prop. 4 of No. 2 and from the analogous statements for algebras essentially of finite type (§ 6, No. 1).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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