Remark 6

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

Commutative Algebra, chapter VIII, § 4. Séries de Hilbert-Samuel

Under the hypotheses of Prop. 2, one therefore has $c_M = 0$ if and only if $M$ is a torsion $H$ -module, or again if and only if $\dim_H(M) < r$ (§ 1, No. 5, example 4).

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

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