Exercise 10, § 2

Let k be a field of characteristic p, $(\xi_\lambda)_{\lambda \in \Lambda}$ a p-basis of k, and for each integer n $\geqslant 1$, let C_n be the subring of W_n(k) generated by W_n(k^{p^{n-1}}) and the elements $\tau(\xi_\lambda) = [\xi_\lambda, 0, ..., 0]$, for $\lambda \in \Lambda$. For every integer n $\geqslant 1$, the projection of W_{n+1}(k) onto W_n(k) maps C_{n+1} into C_n. Let C denote the subring of W(k) which is the projective limit of the C_n. a) Let n be an integer $\geqslant 0$. Put A = W_{n+1}(k) and for every integer i $\geqslant 0$ let $\rho_i = \rho_{n,i}^\Lambda$ be the mapping of k into A defined in Exercise 2, p. 69. Prove that for $a \in k$, $\rho_i(a) = V^i \tau(a^{p^n})$. Deduce that the elements $\rho_i(a)$ for $a \in k$, generate the subring W_{n+1}(k^{p^n}) of A. b) Let $\tilde{C}$ be a p-ring with residue field k and of infinite length, and let $(x_\lambda)_{\lambda \in \Lambda}$ be a family of elements of $\tilde{C}$ lifting the p-basis $(\xi_\lambda)_{\lambda \in \Lambda}$. Using Exercise 4, p. 70, prove that there exists, for each integer n $\geqslant 1$, a unique ring homomorphism $\varphi_n : \tilde{C} \to W_n(k)$ inducing the identity on k by passage to quotients, and sending $x_\lambda$ to $\tau(\xi_\lambda)$ for every $\lambda \in \Lambda$. Deduce from Exercise 3, p. 69 that the image of $\varphi_n$ is C_n. c) The mappings $(\varphi_n)_{n \geqslant 1}$ form a projective system of mappings of $\tilde{C}$ into the C_n and $\varphi = \lim \varphi_n$ is an isomorphism of $\tilde{C}$ onto C. For each n $\geqslant 1$, the canonical projection $C \to C_n$ identifies C_n with C/p^nC.