Remark 1

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

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

One also has $d_q(M) = \sup_{m \in V(q)} \dim(M_m)$ , that is, $d_q(M) = \dim(\hat{M})$ , where $\hat{M}$ is the completion of $M$ for the q-adic topology ( $§ 3$ , No. 4, prop. 8).

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

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