Exercise 8, § 4

Let $A$ be a linearly topologized commutative ring. The additive group $A[[X_1, \ldots, X_p]]$ is identified with the product group $A^{N^p}$ and is given the product topology $\mathscr{T}$.

(a) Show that $\mathscr{T}$ is compatible with the ring structure on $A[[X_1, \ldots, X_p]]$ and is a linear topology for which the mappings $f \mapsto \partial f / \partial X_i$ are continuous.

(b) For every element $P = \sum_e \alpha_{e,P} X^e$ of $A[[X_1, \ldots, X_p]]$ (notation of § 1, no. 6) and every complete Hausdorff topological A-algebra B ($§ 2$, Exercise 28), an element $x = (x_1, \ldots, x_r) \in B^p$ is said to be substitutable in $P$ if the family $(\alpha_{e,P} x^e)$ (where we have written $x^e = x_1^{e_1} \ldots x_p^{e_p}$ for $e = (e_1, \ldots, e_p)$) converges to 0 in B with respect to the filter of complements of finite subsets of $N^p$. This family is then summable and its sum is denoted by $P(x)$. Show that the set of formal power series $P \in A[[X_1, \ldots, X_p]]$ such that $x$ is substitutable in $P$ is a subring S, of $A[[X_1, \ldots, X_p]]$ and the mapping $P \mapsto P(x)$ is a homomorphism from S, to B.

(c) Show that, if $x$ is substitutable in $P$ and $y = (y_1, \ldots, y_p)$ is an element of $B^p$ such that the $y_i$ are topologically nilpotent for $1 \leq i \leq p$, then $x + y$ is substitutable in $P$.

In particular, if there exists in B a neighbourhood of 0 consisting of topologically nilpotent elements, then, for all $P \in A[[X_1, \ldots, X_p]]$, the set D of $x \in B^p$ which are substitutable in $P$ is open and the mapping $x \mapsto P(x)$ from D to B is continuous.

(d) Suppose from now on that A is Hausdorff and complete and let $A[[X_1, \ldots, X_p]]$ be given the topology $\mathscr{T}$. For a system of q formal power series $P_1, \ldots, P_q$ of $A[[X_1, \ldots, X_p]]$ to be substitutable in a formal power series

it is necessary and sufficient that the system $(P_1(0), \ldots, P_q(0))$ be substitutable in Q.

(e) Suppose that $x$ is substitutable in each of the $P_k$ ($1 \leq k \leq q$) and that the system $(P_1, \ldots, P_q)$ is substitutable in Q. Show that $x$ is substitutable in $Q(P_1, \ldots, P_q)$, that the system $(P_1(x), \ldots, P_q(x))$ is substitutable in Q and that $Q(P_1(x), \ldots, P_q(x)) = ((Q(P_1, \ldots, P_q))(x))$ iff further one of the following hypotheses assumed to be satisfied: (a) there is in B an ideal n such that the ordered pair $(B, n)$ satisfies Hensel's conditions and the $x_i$ ($1 \leq i \leq p$) belong to n; (β) the formal power series Q is restricted.

(f) Take $B = A = \mathbf{F}_2$ with the discrete topology, $P = X + X^2$, $Q = \sum_{n=0}^\infty X^{2^n}$; then $P$ is substitutable in $Q$, $x = 1$ is substitutable in $P$ and in $Q(P)$ and $P(x)$ is substitutable in $Q$, but $Q(P(x)) = 0$ and $(Q(P))(x) = 1$.