Corollary 2

Tag 06JO, permanent. Cite it as [Bourbaki, Tag 06JO].

— Let $K'$ be an extension of $K$ .

a) For a $K$ -algebra $A$ to be etale, it is necessary and sufficient that the $K'$ -algebra $A_{(K')}$ should be etale.

b) Let $A'$ be an algebra over $K'$ , not reduced to 0. For $A'$ to be etale over $K$ it is necessary and sufficient that $A'$ should be etale over $K'$ and $K'$ etale over $K$ .

We argue as for Cor. 1, applying this time (8) for a) and (9) for b).

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

Cited by

Theorem 4, § 6
Remark, § 7
Proposition 10, § 7
Proposition 9, § 7
Proposition 4, § 14
Proposition 10, § 16

Cites

Cor. 1