Remark 4

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

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

One can prove (p. 88, exerc. 4) that under the hypotheses of th. 1, the $H_0$ -modules $\mathrm{Tor}_j^H(H_0, M)$ are of finite length, are zero for $j > r$ , and that one has

c_M = \sum_{j=0}^r (-1)^j \mathrm{long}_{H_0}(\mathrm{Tor}_j^H(H_0, M)).

More precisely, the

H

-modules

T_j = \mathrm{Tor}_j^H(H_0, M)

are naturally endowed with gradings and one has

Q_M = \sum_{j=0}^r (-1)^j P_{T_j}.

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

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