Example 1

Tag 0CO4, permanent. Cite it as [Bourbaki, Tag 0CO4].

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

Suppose $H_0$ artinian and take $M = H$ . Then, with the preceding notations, we have $R = 0$ and $S = H'$ , hence by (5), $Q_H = Q_{H'}$ ; since we have $Q_{H_0} = \operatorname{long}(H_0)$ , we infer by induction that $Q_H = \operatorname{long}(H_0)$ , that is,

P_H = \operatorname{long}(H_0) \cdot \prod_{i \in I} (1 - T^{d_i})^{-1} .

Now suppose H endowed with the usual grading, for which $d_i = 1$ for all $i \in I$ , and put $r = \mathrm{Card}(I)$ ; we have $P_M = Q_M(T).(1 - T)^{-r}$ . Put $c_M = Q_M(1)$ . Then, by formula (2) of No. 1, we have:

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

Cited by

Nothing.

Cites

Nothing.