Remark 2

Tag 0CO7, permanent. Cite it as [Bourbaki, Tag 0CO7].

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

Let $0 \to M' \to M \to M'' \to 0$ be an exact sequence of graded H-modules and homomorphisms of degree 0 such that M is of finite type over H and $M_n$ of finite length over $H_0$ for each $n$ . Then, for each $n \in \mathbf{Z}$ , we have

\mathrm{long}_{H_0}(M_n) = \mathrm{long}_{H_0}(M'_n) + \mathrm{long}_{H_0}(M''_n),

hence

P_M = P_{M'} + P_{M''}

Q_M = Q_{M'} + Q_{M''}

and

c_M = c_{M'} + c_{M''}

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

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