Remark 1

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

Commutative Algebra, chapter VIII, § 6. Dimension des anneaux gradués

With the notation of $c$ , let $d_i$ be the degree of $a_i$ ( $1 \leq i \leq n$ ). Then the Poincaré series $P_H = \sum_{n \in \mathbf{Z}} [H_n : H_0]. T^n$ is equal to $\prod_i (1 - T^{d_i})^{-1}$ (§ 4, No. 2, example 1). Consequently, if $H$ is a polynomial graded $H_0$ -algebra, one has

P_H = \prod_n (1 - T^n)^{-\delta(n)}, \quad \text{with} \quad \delta(n) = [(H_+/H_+^2)_n : 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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