Proposition 11

Let $A$ be a complete Hausdorff filtered commutative ring, $C$ a subring of $A_0$ such that $C \cap A_n = 0$ and $(x_i)_{1 \leq i \leq q}$ a finite family of elements of $A$ such that $x_i \in A_{n_i}$ where $n_i > 0$ for $1 \leq i \leq q$; let $\xi_i$ be the class of $x_i$ in $\mathrm{gr}_{n_i}(A)$ for $1 \leq i \leq q$.

(i) There exists a unique $C$-homomorphism $v$ from the algebra of formal power series $A'' = \mathbf{C}[[X_1, \ldots, X_q]]$ to $A$ such that $v(X_i) = x_i$ for $1 \leq i \leq q$.

(ii) If the family $(\xi_i)$ is algebraically free over $C$, the homomorphism $v$ is injective.

(iii) If the filtration on $A$ is exhaustive and the family $(\xi_i)$ is a system of generators of the $C$-algebra $\mathrm{gr}(A)$, the homomorphism $v$ is surjective.

As $n_i \geq 1$ for all $i$, $\sum_{i=1}^q n_i s(i) \geq \sum_{i=1}^q s(i)$ for every monomial $X_1^{s(1)} \ldots X_q^{s(q)}$ and on the other hand $\sum_{i=1}^q n_i s(i) \leq r \cdot \sum_{i=1}^q s(i)$ if $r$ is the greatest of the $n_i$. If $A''_n$ denotes the set of formal power series whose non-zero terms $a_s X_1^{s(1)} \ldots X_q^{s(q)}$ satisfy $\sum_{i=1}^q n_i s(i) \geq n$, it follows from no. 6, Corollary to Proposition 6 that $A''$ is Hausdorff and complete with the exhaustive filtration $(A''_n)$ and that

is dense in $A''$; moreover the homomorphism $u$ defined in the proof of Proposition 10 is continuous on $A'$ and can be extended uniquely to a continuous homomorphism $v : A \to A$, since $A$ is Hausdorff and complete (General Topology, Chapter III, § 3, no. 3, Proposition 5), which proves (i); also, $\mathrm{gr}(A'') = \mathrm{gr}(A')$ and $\mathrm{gr}(v) = \mathrm{gr}(u)$; (ii) and (iii) then follow respectively from Corollaries 1 and 2 to Theorem 1 of no. 8 in view of the hypotheses on $A$.

The conclusion of (ii) (resp. (iii)) of the proposition is sometimes expressed by saying that the family $(x_i)$ is formally free over $C$ (resp. a formal system of generators of $A$).