Remark 4

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

Commutative Algebra, chapter VIII, § 1. Dimension de Krull d’un anneau

Let A be a ring of finite dimension, all maximal chains of whose prime ideals have the same length. Then A is catenary, one has ht(p) + dim(A/p) = dim(A) for every prime ideal p of A, and dim(A_p/qA_p) + dim(A/p) = dim(A/q) for every pair (p, q) of prime ideals of A such that q ⊂ p (No. 2, Proposition 5).

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

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