Exercise 11, § 2

Let $k$ be a field of characteristic $p$, $(\xi_\lambda)_{\lambda \in \Lambda}$ a $p$-basis of $k$, $C$ the Cohen subring of $W(k)$ containing $\tau(\xi_\lambda)$ for all $\lambda \in \Lambda$. Let $(\varepsilon_j)_{j \in J}$ be a basis of $k^*/(k^{*p})$ as an $\mathbf{F}_p$-vector space. Let $n$ be an integer $\geqslant 0$ and $(x_j)_{j \in J}$ a family of elements of $k^*$ such that the image of $x_j$ in $k^*/k^{*p}$ is $\varepsilon_j$, for all $j \in J$. a) The group $k^*/k^{*p^n}$ is a free $(\mathbf{Z}/p^n\mathbf{Z})$-module with basis $(\overline{x}_j)_{j \in J}$, where $\overline{x}_j$ is the class of $x_j$ in $k^*/k^{*p^n}$. b) For each $j \in J$, let us write the decomposition of $x_j$ with respect to the basis $(\xi^m)_{m \in M_n}$ of $k$ over $k^{p^n}$ (with the notation of Exercise 3, p. 69):

Put

and

Then $\sigma_n$ extends, in a unique way, to a multiplicative mapping of $k^*$ into $C/p^{n+1}C$, again denoted by $\sigma_n$, and which, on passing to the quotient, determines the inclusion of $k^*$ into $k = C/pC$. c) Let $\sigma'_n$ be another multiplicative mapping of $k^*$ into $C/p^{n+1}C$ defining the inclusion of $k^*$ into $k$ on passing to the quotient. Then for all $x \in k^*$, we have $\sigma_n(x) = v(x) \sigma'_n(x)$, where $v$ is a homomorphism of $k^*$ into the multiplicative group $(1 + pC)/(1 + p^{n+1}C)$ trivial on $k^{*p^n}$. Any family $(\beta_j)_{j \in J}$ of elements of $C/p^nC$ defines such a homomorphism by the formula

d) For every integer $n \geqslant 0$ and every multiplicative mapping $s_n : k^* \to C/p^{n+1}C$ defining the inclusion of $k^*$ into $k$ on passing to the quotient, there exists a multiplicative mapping $s_{n+1} : k^* \to C/p^{n+2}C$ which determines $s_n$ on passing to the quotient. e) Prove that there exists a multiplicative section $s : k \to C$. One may require $s$ to satisfy $s(\xi_\lambda) = \tau(\xi_\lambda)$ for all $\lambda \in \Lambda$. f) Let $A$ be a local ring, separated and complete. Let $k$ be its residue field, $(\xi_\lambda)_{\lambda \in \Lambda}$ a $p$-basis of $k$, and for $\lambda \in \Lambda$, let $a_\lambda$ be a lifting of $\xi_\lambda$ in $A$. Now prove that there exists a multiplicative section $k^* \to A$ satisfying $s(\xi_\lambda) = a_\lambda$.