Proposition 12

Tag 0D12, permanent. Cite it as [Bourbaki, Tag 0D12].

Commutative Algebra, chapter X, § 1. Profondeur

Let A be a noetherian local ring, M a nonzero finitely generated A-module and J an ideal of A distinct from A. One has the sequence of inequalities

\operatorname{prof}_A(J; M) \leq \operatorname{codim}(\operatorname{Supp}(M) \cap V(J), \operatorname{Supp}(M)) \leq \dim(M) - \dim(M/JM)

\leq [J/m_A J : \kappa_A].

For every element $p$ of $\operatorname{Supp}(M) \cap V(J)$ , $\operatorname{prof}_A(J; M)$ is less than or equal to $\dim_{A_p}(M_p)$ (No. 5, Prop. 8 and No. 4, Cor. 2 of Th. 2), that is to say (VIII, § 1, No. 4, Prop. 9)

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

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