Lemma 5

Let K be a commutative field, S a graded polynomial algebra over K, and R a graded subalgebra of S. Then S is a free graded R-module (Algebra, Chap. II, § 11, no. 2) if and only if the following two conditions are satisfied: a) R is a graded polynomial algebra over K; b) if $(\alpha_1, \ldots, \alpha_s)$ is a system of generators of the K-algebra R consisting of algebraically independent homogeneous elements, this system is an S-regular sequence.

4 In this number, we use results from chapters in preparation in the book on Commutative Algebra. We indicate them by the symbol "Commutative Algebra". 5 This means that, for all $i \in \{1, 2, \ldots, s\}$, the canonical image of $\alpha_i$ in the ring

is not a zero divisor in this ring.

When S is an R-module of finite type, b) is a consequence of a).

See Commutative Algebra for the proof.