Exercise 26, § 1

Let M be the set of positive rational numbers whose denominator is a power of $p$, endowed with the monoid structure given by addition. Let $k$ be a perfect field of characteristic $p$. We endow $W(k)$ and its field of fractions K with the topology given by the valuation of $W(k)$. Let $n$ be a positive integer. Let C be the K-algebra of the product monoid $M^n$. For $1 \leqslant i \leqslant n$, we shall denote by $T_i$ the image in C of the element of M whose $i$-th component is 1 and whose other components are 0, and for $\alpha \in M^n$ we shall denote by $T^\alpha$ the image of $\alpha$. For $\alpha \in M^n$, we put

where $w_p$ denotes the $p$-adic valuation of $\mathbf{Q}$. We denote by A the ring $k[T_1, ..., T_n]$, by B the ring $W(k)[T_1, ..., T_n]$, by $\overline{B}$ the algebra of $M^n$ over $W(k)$, and by $\overline{A}$ the algebra of $M^n$ over $k$. We consider A as a subring of $\overline{A}$, B as a subring of $\overline{B}$, and $\overline{B}$ as a subring of C. Let E be the set of elements of C of the form $\sum_{\alpha \in M^n} \operatorname{den}(\alpha) a_\alpha T^\alpha$ where the family of finite support $(a_\alpha)$ consists of elements of $W(k)$. Let F be the automorphism of C coinciding with the Frobenius automorphism on $W(k)$ and satisfying $F(T_i^\alpha) = T_i^{p^\alpha}$ for every $i (1 \leqslant i \leqslant n)$ and every $\alpha \in M$, and let V be the automorphism of the additive group of C defined by $V = pF^{-1}$. a) E is a subring of C, containing B and stable under F and V. b) E is a $W(k)$-module, and is the sum of its submodules $V^iB$, for $i \in \mathbf{N}$. c) One has $\bigcap_{i \in \mathbf{N}} V^iE = 0$. d) For every $i \in \mathbf{N}$, one has $B \cap V^iE = p^iB$.

e) Let $\rho : \overline{B} \to \overline{A}$ denote the mapping obtained by reducing the coefficients of the elements of $M^n$ modulo $pW(k)$. Then $\rho(E) = A$, and $\rho$ induces an isomorphism $\rho'$ of $E/VE$ onto $A$. f) For every integer $r \geqslant 0$, $V^rE/V^{r+1}E$ is a module over $E/VE$. By means of ${\rho'}^{-1}$, it may be considered as an $A$-module. The mapping of $A$ into $V^rE/V^{r+1}E$, which to $a \in A$ associates the class of $V^r e$, for every element $e \in E$ satisfying $\rho(e) = a$, is an isomorphism of the $A$-module $\varphi'_*A$ (p. 44, Exercise 13) onto the $A$-module $V^rE/V^{r+1}E$. g) There exists a unique homomorphism of $W(k)$-algebras $\sigma : \overline{B} \to W(\overline{A})$ such that for $1 \leqslant i \leqslant n$ and $r \in M$, one has $\sigma(T_i^r) = \tau_{\overline{A}}(T_i^r)$. One has $\sigma \circ F = F_{\overline{A}}$ and $\sigma \circ V = V_{\overline{A}} \circ \sigma$ (taking Exercise 14, p. 44, as a model). h) The homomorphism $\sigma$ induces a homomorphism of $W(k)$-algebras $\tilde{\sigma} : E \to W(A)$, which is the only $W(k)$-homomorphism of $E$ into $W(A)$ satisfying $\tilde{\sigma}(T_i) = \tau_A(T_i)$ for $1 \leqslant i \leqslant n$ and $\tilde{\sigma} \circ V = V_A \circ \tilde{\sigma}$. i) For every integer $r \geqslant 1$, $\tilde{\sigma}$ induces an isomorphism of $E/V^rE$ onto $W_r(A)$ (use Exercise 13, p. 44); $\tilde{\sigma}$ is injective. j) Let $\hat{C}$ be the completion of $C$ for the $p$-adic topology. For every $x \in \hat{C}$, there exists a unique family $(a_\alpha)_{\alpha \in M^n}$ of elements of $K$, such that $a_\alpha$ tends to 0 when $\alpha$ tends to infinity along the filter of complements of finite subsets of $M^n$, and that $x$ is the sum of the family $a_\alpha T^\alpha$. k) Let $\hat{E}$ be the set of elements $x = \sum_{\alpha \in M^n} \mathrm{den}(\alpha) \alpha_\alpha T^\alpha$ of $\hat{C}$ such that one has $a_\alpha \in W(k)$ for every $\alpha \in M^n$. Prove that $\tilde{\sigma} : E \to W(A)$ extends to an isomorphism of $\hat{E}$ onto $W(A)$.