Proposition 3

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

Commutative Algebra, chapter VIII, § 7. Multiplicités

Let M be a finitely generated A-module, of dimension d ≥ 0. Let Φ be the set of minimal elements p of Supp(M) such that dim(A/p) = d. Let q be an ideal of A, contained in the radical of A, and such that M/qM is of finite length. One has

e_q(M) = \sum_{p \in \Phi} \operatorname{long}_{A_p}(M_p).e_q(A/p).

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

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