Definition 1

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

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

The series $H_{M,F}$ is called the Hilbert-Samuel series of the A-module M (relative to the q-good filtration F).

The mapping $n \mapsto \operatorname{long}_A(F_n/F_{n+1})$ is often called the Hilbert-Samuel function of M (relative to F).

This applies in particular to the case of the q-adic filtration ( $F_n = q^nM$ ); we then put $H_{M,F} = H_{M,q}$ . We therefore have

(9)

H_{M,q} = \sum_{n \in \mathbf{Z}} \operatorname{long}_{A/q}(q^nM/q^{n+1}M). T^n .

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

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