Proposition 1

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

Lie Groups and Lie Algebras, chapter V, § 5. Invariants in the symmetric algebra

Let $S = \bigoplus_{n \geq 0} S_n$ be a commutative graded K-algebra with a system of generators $(x_1, x_2, \ldots, x_m)$ consisting of homogeneous and algebraically independent elements. Let $d_i$ be the degree of $x_i$ , and assume that $d_i > 0$ for all i. Then the $S_n$ are free and of finite rank over K, and

P_S(T) = \prod_{i=1}^m (1 - T^{d_i})^{-1}.

Indeed, S can be identified with the tensor product $K[x_1] \otimes \cdots \otimes K[x_m]$ , provided with the total grading. The Poincaré series of $K[x_i]$ is

\sum_{n \geq 0} T^{nd_i} = (1 - T^{d_i})^{-1},

and it suffices to apply (2).

Under the assumptions of Prop. 1, we shall say that S is a graded polynomial algebra over K.

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

Cited by

Proposition 3, § 5

Cites

Nothing.