Lemma 1

Let K be a commutative field, S a graded K-algebra of polynomials, and R a graded subalgebra of S of finite type such that the R-module S admits a basis $(z_\lambda)_{\lambda \in \Lambda}$ consisting of homogeneous elements. In the set of systems of generators of the ideal $R_+$ of R consisting of homogeneous elements, choose a minimal element $(\alpha_1, \ldots, \alpha_s)$. Assume that, for all i, the degree $k_i$ of $\alpha_i$ is coprime to the characteristic exponent p of K. Then the $\alpha_i$ generate the K-algebra R and are algebraically independent over K.

By Algebra, Chap. II, § 11, no. 4, Prop. 7, the assumption made about the $\alpha_i$ is equivalent to saying that they are homogeneous and that their images in the K-vector space $R_+/ (R_+)^2$ form a basis of this space. This condition is invariant under extension of the base field; we can thus reduce to the case where the latter is perfect.

The family $(\alpha_1, \ldots, \alpha_s)$ generates the algebra R by Commutative Algebra, Chap. III, § 1, no. 2, Prop. 1. We argue by contradiction and assume that this family is not algebraically independent over K.

1) We show first of all that there exist families

of homogeneous elements of S with the following properties:

Let $X_1, \ldots, X_s$ be indeterminates and give $K[X_1, \ldots, X_s]$ with the graded algebra structure for which $X_i$ has degree $k_i$. There exist non-zero homogeneous elements $H(X_1, \ldots, X_s)$ of $K[X_1, \ldots, X_s]$ such that $H(\alpha_1, \ldots, \alpha_s) = 0$; choose H to be of minimum degree. If $\partial H / \partial X_i \neq 0$, the polynomial $\frac{\partial H}{\partial X_i}(\alpha_1, \ldots, \alpha_s)$ is a non-zero homogeneous element of R; if $p \neq 1$, H is not of the form $H_1^p$ with $H_1 \in K[X_1, \ldots, X_s]$. Take

Since K is perfect, the polynomials $\partial H / \partial X_i \in K[X_1, \ldots, X_s]$ are not all zero (Algebra, Chap. V, § 1, no. 3, Prop. 4); in view of the assumption made about the $k_i$, neither are the $\beta_i$.

On the other hand, S can be identified with a graded algebra of polynomials $K[x_1, \ldots, x_r]$ for appropriate indeterminates $x_1, \ldots, x_r$ with suitable degrees $m_i > 0$. Let $D_k$ be the partial derivative with respect to $x_k$ on S. Take d_{ik} = k_i^{-1} D_k(\alpha_i). Then the equality (10) holds because its left-hand side is $D_k(H(\alpha_1, \ldots, \alpha_s))$. On the other hand, if we put $y_1 = m_1 x_1, \ldots, y_r = m_r x_r$, the equality (9) follows from the equality (5) of no. 1.

2) Let $b$ be the ideal of $R$ generated by the $\beta_i$; there exists a subset $J$ of

such that $(\beta_i)_{i \in J}$ is a minimal system of generators of $b$. Then $J \neq \varnothing$ since $b \neq 0$. We shall deduce from (9) and (10) that, if $i \in J$, $\alpha_i$ is an $R$-linear combination of the $\alpha_j$ for $j \neq i$, which will contradict the minimality of $(\alpha_1, \ldots, \alpha_s)$ and will complete the proof.

There exist homogeneous elements $\gamma_{ji}$ of $R$ ($i \in J, j \in I - J$) such that

Taking (11) into account, formula (10) gives Put Thus Write $u_{ik} = \sum_{\lambda \in \Lambda} \delta_{ik\lambda} z_\lambda$, where the $\delta_{ik\lambda}$ belong to $R$. Relation (14) implies that If one of the $\delta_{ik\lambda}$ had a non-zero homogeneous component of degree 0, the preceding equality would imply that one of the $\beta_i$ ($i \in J$) is an $R$-linear combination of the others, contradicting the minimality of $(\beta_i)_{i \in J}$. Thus $\delta_{ik\lambda} \in R_+$ and consequently $u_{ik} \in R_+ S$ for all $i$ and $k$. Thus, there exist $u_{ikh} \in S$ such that $u_{ik} = \sum_{h=1}^s u_{ikh} \alpha_h$, in other words, by (13), Multiply both sides of (15)$_{ik}$ by $y_k$ and add for $i$ in $J$ fixed and $k = 1, 2, \ldots, r$; in view of (9), we find that Take the homogeneous components of degree $k_i$ of both sides. Since $\deg y_k > 0$, $\alpha_i$ is an $S$-linear combination of the $\alpha_j$ for $j \neq i$. Since $S$ is free over $R$ and $\alpha_1, \ldots, \alpha_s \in R$, it follows that $\alpha_i$ is an $R$-linear combination of the $\alpha_j$ with $j \neq i$ (Commutative Algebra, Chap. I, § 3, no. 5, Prop. 9 d)).