Exercise 3, § 2

Let n be a positive integer. Let C be a Cohen $p$-ring, with residue field k, and of length $n+1$. Let $(x_\lambda)_{\lambda \in \Lambda}$ be a family of elements of C lifting a $p$-basis $(\xi_\lambda)_{\lambda \in \Lambda}$ of k. We denote by $\rho_i : k \to C$ for every $i \in \mathbf{N}$ the mapping $\rho_{n,i}^C$ defined in Exercise 2, where one takes for $J_m$ the ideal $p^{m+1}C$. a) For $r \in \mathbf{N}$, let M be the set of multi-indices $m \in \mathbf{N}^{(\Lambda)}$ such that, for every $\lambda \in \Lambda$, one has $0 \leq m_\lambda < p^r$. Then, for every element $\alpha$ of C, there exists a unique family of finite support $(a_{i,m})$ $(0 \leq i \leq n,\ m \in M_{n-i})$ of elements of k, such that

where the notation $x^m$ denotes the product $\prod_{\lambda \in \Lambda} x_\lambda^{m_\lambda}$. b) Consider the ring U generated by generators $[x_\lambda]$, $\lambda$ running through $\Lambda$, and $[\rho_i(a)]$, i running through $\mathbf{N}$ and $a$ running through k, and subject only to the following relations (where the polynomials $R_i$ are those introduced in Exercise 3, p. 42). (i) $[\rho_i(a)] = 0$ for $a \in k,\ i > n$, (ii) $[\rho_0(1)] = 1$ and $[\rho_0(0)] = 0$, (iii) $[\rho_i(a)] [\rho_j(b)] = [\rho_{i+j}(a^{p^j} b^{p^i})]$ for $i, j$ in $\mathbf{N}$ and $a, b$ in $k$, (iv) $[\rho_i(a)] + [\rho_i(b)] = \sum_{m=0}^{n-i} [\rho_{i+m}(R_m(a, b))]$ for $i \in \mathbf{N}$ and $a, b$ in $k$, (v) $[x_\lambda]^{p^{n-i}} [\rho_i(a)] = [\rho_i(\xi_\lambda a)]$ for $0 \leq i \leq n,\ \lambda \in \Lambda,\ a \in k$. Prove that one has $[\rho_i(0)] = 0$ for every $i \in \mathbf{N}$. Prove, by means of Exercise 3, b), p. 42, that if p is odd, one has $[\rho_0(-1)] = -1$; if p is even, one has $\sum_{i=0}^n [\rho_i(1)] = -1$. c) There exists only one mapping of U into C which, to every family of finite support $(a_{i,m})_{0 \leq i \leq n,\ m \in M_{n-i}}$ of elements of k, assigns to the element of U the element $\sum_{i=0}^n \sum_{m \in M_{n-i}} \rho_i(a_{i,m}) x^m$ of C; it is an isomorphism of rings. d) Deduce from the above another proof of the results of § 2, No. 3.