Corollary 2

Tag 0D17, permanent. Cite it as [Bourbaki, Tag 0D17].

Commutative Algebra, chapter X, § 1. Profondeur

Let A be a local noetherian ring and M a finitely generated A-module. One has the inequality

prof_A(M) \leq \inf_{p \in \operatorname{Ass}(M)} \dim(A/p) .

Let $p$ be a prime ideal associated with $M$ ; one has $prof_{A_p}(M_p) = 0$ (No. 1, remark 2). Prop. 13 applied to the ideals $p \subset m_A$ yields the inequality $prof_A(M) \leq \dim(A/p)$ , whence the corollary.

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

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