Remark 1

Tag 0CS9, permanent. Cite it as [Bourbaki, Tag 0CS9].

Commutative Algebra, chapter VIII, § 7. Multiplicités

If q’ is an ideal of A contained in the radical of A and containing q, one has $e_{q'}(M) \leqslant e_q(M)$ and, if the q’-adic filtration of M is q-good, one has $e_{q'}(M) = e_q(M)$ (§ 4, No. 3, th. 2). 2) If M is of finite length, one has $e_q(M) = \operatorname{long}_A(M)$ (§ 4, No. 3, remark 3). 3) If $d > 0$ , one has

\operatorname{long}_{A/q}(q^nM/q^{n+1}M) = e_q(M) \frac{n^{d-1}}{(d-1)!} + \alpha_n n^{d-2}

where $\alpha_n$ tends to a limit as $n$ increases indefinitely (§ 4, No. 3, corollary to th. 2). 4) The calculation of multiplicities may be reduced to the case where A is local since, by § 4, No. 4, corollary to th. 3, one has

e_q(M) = \sum e_{q_m}(M_m)

the summation being extended to the maximal ideals m of A such that

m \in \operatorname{Supp}(M) \cap V(q) \quad \text{and} \quad \dim_{A_m}(M_m) = d .

It follows from remarks 2 and 3 that $e_q(M)$ depends only on the graded A/q-module $\operatorname{gr}_q(M)$ . Consequently:

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

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