Remark 4

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

Commutative Algebra, chapter VIII, § 2. Dimension des algèbres

Let $\rho : A \to B$ be a local homomorphism of noetherian local rings satisfying condition (PM). We shall see later (§ 3, no 4, prop. 7) that in this case there is equality in (1). In the general case there may be strict inequality (p. 84, exercise 1).

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

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