Corollaire

Tag 0D7U, permanent. Cite it as [Bourbaki, Tag 0D7U].

Commutative Algebra, chapter X, § 6. Extension des scalaires dans les algèbres régulières

The tensor product of two $k$ -algebras essentially of finite type is a $k$ -algebra essentially of finite type.

For let $A$ and $B$ be two $k$ -algebras essentially of finite type. Then $A \otimes_k B$ is essentially of finite type over $A$ (Prop. 2, d)), hence over $k$ (Prop. 3).

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

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