Remark 2

Tag 0D0S, permanent. Cite it as [Bourbaki, Tag 0D0S].

Commutative Algebra, chapter X, § 1. Profondeur

Suppose that the A-module M is finitely generated. By Remark 2 of No. 1 and Theorem 2 of No. 4, one has the following equivalences

\mathrm{prof}_F(M) = 0 \iff \mathrm{Ass}(M) \cap F \neq \varnothing \\ \mathrm{prof}_F(M) < +\infty \iff \mathrm{Supp}(M) \cap F \neq \varnothing .

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

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