Corollary

Tag 0D11, permanent. Cite it as [Bourbaki, Tag 0D11].

Commutative Algebra, chapter X, § 1. Profondeur

Let $\rho : A \to B$ be a local homomorphism of noetherian local rings making B a flat A-module. One has

\operatorname{prof}(B) = \operatorname{prof}(A) + \operatorname{prof}(\kappa_A \otimes_A B),

\dim(B) = \dim(A) + \dim(\kappa_A \otimes_A B).

In fact the depth (resp. the dimension) of the B-module $\kappa_A \otimes_A B$ is equal to the depth (resp. the dimension) of the ring $\kappa_A \otimes_A B$ by the cor. of Prop. 4 (resp. by VIII, § 1, No. 4).

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

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