Example 2

Tag 0CO9, permanent. Cite it as [Bourbaki, Tag 0CO9].

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

Suppose $H_0$ artinian, and let M be a free graded H-module generated by s homogeneous, linearly independent elements, of respective degrees $\delta_1, ..., \delta_s$ . Then M is isomorphic to $H(-\delta_1) \oplus \cdots \oplus H(-\delta_s)$ . By remarks 2 and 3 and example 1, we therefore have

P_M = \mathrm{long}(H_0)\left( \sum_{i=1}^s T^{\delta_i} \right)(1 - T)^{-r},

Q_M = \mathrm{long}(H_0)\left( \sum_{i=1}^s T^{\delta_i} \right),

c_M = s.\mathrm{long}(H_0).

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

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