Proposition 3

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

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

a) If F is a q-good filtration on M, we have

(10)

H_{M,F}^{(1)} = \sum_{n \in \mathbf{Z}} \operatorname{long}_A(M/F_{n+1}). T^n .

b) If F and F’ are two q-good filtrations on M, there exists an integer m such that $H_{M,F'}^{(1)} \geq T^m H_{M,F}^{(1)}$ .

Part a) results immediately from the definition of $H_{M,F}^{(1)}$ ; part b) results from a) and Lemma 3.

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

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