Proposition 6

Let $A$ be a commutative ring, $B$ a ring satisfying Hensel’s conditions and $u : A \to B$ a homomorphism. For every family $x = (x_1, \ldots, x_n)$ of topologically nilpotent elements of $B$, there exists a unique homomorphism $\tilde{u}$ from $A[[X_1, \ldots, X_n]]$ to $B$ such that $\tilde{u}(a) = u(a)$ for all $a \in A$ and $\tilde{u}(X_i) = x_i$ for $1 \leq i \leq n$. Moreover, if $m$ denotes the ideal of series without constant term in $A[[X_1, \ldots, X_n]]$, $\tilde{u}$ is continuous for the m-adic topology.

Let $a$ be the finitely generated ideal generated in $B$ by the $x_i$ ($1 \leq i \leq n$); for every open ideal $\mathfrak{H}$ of $B$, the images of the $x_i$ in $B/\mathfrak{H}$ are nilpotent, hence the ideal $(a + \mathfrak{H})/\mathfrak{H}$ is nilpotent in $B/\mathfrak{H}$ and there exists an integer $k$ such that, for $\sum_{i=1}^n p_i \geq k$, $x_1^{p_1} \cdots x_n^{p_n} \in \mathfrak{H}$. As every element of $m^k$ is a finite sum of formal power series of the form $X_1^{p_1} \cdots X_n^{p_n} g(X_1, \ldots, X_n)$, where $\sum_{i=1}^n p_i \geq k$, it is seen that, if $\tilde{u}$ solves the problem, then $\tilde{u}(m^k) \subset \mathfrak{H}$, which proves the continuity of $\tilde{u}$. There obviously exists a unique homomorphism

such that $\nu(a) = u(a)$ for $a \in A$ and $\nu(X_i) = x_i$ for $1 \leq i \leq n$ and the above argument shows that $\nu$ is continuous with respect to the topology induced on $A[X_1, \ldots, X_n]$ by the m-adic topology. As $A[X_1, \ldots, X_r]$ is dense in $A[[X_1, \ldots, X_r]]$ with the m-adic topology and B is Hausdorff and complete, this completes the proof of the existence and uniqueness of $\tilde{u}$.

Note that this proposition gives us again as a special case (i) of Proposition 11 of § 2, no. 9.

If $A$ itself is linearly topologized, the restriction of $\tilde{u}$ to $A\{X_1, \ldots, X_r\}$ coincides with the homomorphism derived from $u$ in Proposition 4 of no. 2. This follows immediately from the fact that $A[X_1, \ldots, X_r]$ is dense in $A\{X_1, \ldots, X_r\}$ if this ring is given the topology with fundamental system of neighbourhoods of 0 the ideals $m^k \cap N_3$ (in the notation of no. 2, this topology is the least upper bound of the topology induced on $A\{X_1, \ldots, X_r\}$ by the m-adic topology on $A[[X_1, \ldots, X_n]]$ and the topology defined in no. 2).

If $B = A$ and $u$ is the identity mapping, we shall write $f(x_1, \ldots, x_r)$ or $f(x)$ for the element $\tilde{u}(f)$ for every formal power series $f \in A[[X_1, \ldots, X_r]]$; for every system $\mathbf{f} = (f_1, \ldots, f_r)$ of formal power series of $A[[X_1, \ldots, X_r]]$, let $\mathbf{f}(x)$ denote the element $(f_1(x), \ldots, f_r(x))$ of $A^r$, then it is said to be obtained by substituting the $x_i$ for the $X_i$ in $\mathbf{f}$. If $n \leq m$ and F is a formal power series of $A[[X_1, \ldots, X_m]]$, it is possible to consider F as a formal power series in $X_{n+1}, \ldots, X_m$ with coefficients in $A[[X_1, \ldots, X_n]]$; let

denote the formal power series in $A[[X_{n+1}, \ldots, X_m]]$ obtained by substituting the $x_i$ for the $X_i$ in the coefficients of F, for $1 \leq i \leq n$.

Let us take $B$ to be the ring of formal power series $A[[X_1, \ldots, X_r]]$ and let $n$ be the ideal of series in $B$ without constant term, so that $(B, n)$ satisfies Hensel's conditions ($§ 2$, no. 6, Corollary to Proposition 6). Proposition 6 may be applied by taking the $x_i \in B$ to be series without constant term; then, for every series $\mathbf{f} \in A[[X_1, \ldots, X_r]]$, $\tilde{u}(f)$ is just the formal power series $f(x_1, \ldots, x_n)$ defined in Algebra, Chapter IV, $§ 5$, no. 5. This is obvious if $f$ is a polynomial and it follows from the proposition in the general case by remarking that $f \mapsto f(x_1, \ldots, x_n)$ is continuous on $A[[X_1, \ldots, X_n]]$ with respect to the m-adic topology.