Remark 2

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

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

To say that $M/qM$ is of finite length means that

\operatorname{Supp}(M/qM) = \operatorname{Supp}(M) \cap V(q)

consists of maximal ideals (IV, § 2, No. 5, Proposition 7). We shall see below (No. 4, corollary to Theorem 3) that

d_q(M)

is the least upper bound of the numbers

\dim_{A_m}(M_m)

, where

m

runs through the set

\operatorname{Supp}(M) \cap V(q)

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

Cited by

Nothing.

Nothing.