Proposition 10

Let $A$ be a filtered commutative ring and $(A_n)$ its filtration; suppose that there exists a subring $C$ of $A_0$ such that $C \cap A_n = \{0\}$ and $C$ is identified with a subring of $\mathrm{gr}_0(A)$. Let $(x_i)_{1 \leq i \leq q}$ be a finite family of elements of $A$; suppose that $x_i \in A_n$, for $1 \leq i \leq q$ and let $\xi_i$ be the class of $x_i$ in $\mathrm{gr}_{n_i}(A)$ for $1 \leq i \leq q$.

(i) If the family $(\xi_i)$ of elements of $\mathrm{gr}(A)$ is algebraically free over $C$, the family $(x_i)$ is algebraically free over $C$.

(ii) If the filtration on $A$ is exhaustive and discrete and $(\xi_i)$ is a system of generators of the $C$-algebra $\mathrm{gr}(A)$, then $(x_i)$ is a system of generators of the $C$-algebra $A$.

Let $A'$ be the polynomial algebra $\mathbf{C}[X_1, \ldots, X_q]$ over $\mathbf{C}$; let $A'$ be given the graduation $(A'_n)$ of type $\mathbf{Z}$ where $A_n$ is the set of $\mathbf{C}$-linear combinations of the monomials $X_1^{s(1)} \ldots X_q^{s(q)}$ such that $\sum_{i=1}^q n_i s(i) = n$. Let $u$ be the homomorphism $f \mapsto f(x_1, \ldots, x_q)$ from the $\mathbf{C}$-algebra $A'$ to the $\mathbf{C}$-algebra $A$; by definition, $u(A'_n) \subset A_n$ for all $n \in \mathbf{Z}$ and hence $u$ is compatible with the filtrations $(A')$ being given its filtered ring structure associated with its graded ring structure, cf. no. 1, Example 1). This being so, the hypothesis of (i) means that

is injective; as the filtration on $A'$ is exhaustive and separated, Corollary 1 to Theorem 1 of no. 8 may be applied and $u$ is injective, which proves the conclusion of (i). Similarly, the hypothesis (ii) on the (\&) means that $\mathrm{gr}(u)$ is surjective; as $A$ is discrete and its filtration is exhaustive, Corollary 2 to Theorem 1 of no. 8 may be applied and $u$ is surjective, which proves the conclusion of (ii).