Exercise 12, § 2

Let $A$ be a complete local ring whose residue field is of characteristic p. a) Let $x \in m_A$. The mapping $n \mapsto (1 + x)^n$ from $\mathbf{Z}$ into $1 + m_A$ extends uniquely to a continuous mapping from $\mathbf{Z}_p$ into $1 + m_A$, denoted $\alpha \mapsto (1 + x)^\alpha$. One thus defines a $\mathbf{Z}_p$-module structure on the multiplicative group $1 + m_A$. b) Let $\varepsilon$ be the formal series $\exp(- \sum_{n=0}^\infty T^{p^n}/p^n)$ of Exercise 58, p. 68. The mapping $x \mapsto \varepsilon(x)$ from $m_A$ into $1 + m_A$ is bijective.

c) Let $k$ be a perfect field of characteristic $p$, and $\varphi$ a local homomorphism from $W(k)$ into $A$. For $\alpha \in W(k)$, consider the series $\varphi E_F(\alpha, T)$ obtained by applying $\varphi$ to the coefficients of the series $E_F(\alpha, T)$ of Exercise 58, f), p. 68. For $\alpha \in W(k)$, $x \in 1 + m_A$, set $x^\alpha = \varphi E_F(\alpha, \varepsilon^{-1}(x))$. Now prove that one thus defines an action of the additive group $W(k)$ on the set $1 + m_A$, which extends the action of $W(F_p)$ (identified with $\mathbf{Z}_p$) defined in a). Now prove that one has moreover, for $a \in \mathbf{Z}_p$ and $\alpha \in W(k)$, $x \in 1 + m_A$, the relation $(x^\alpha)^a = x^{a\alpha}$.

d) Using Exercise 15, p. 44, prove that one does not always have, for $a \in \mathbf{Z}_p$ and $\alpha \in W(k)$, $x \in 1 + m_A$, the relation $(x^a)^\alpha = x^{a\alpha}$. One does not always have either, for $\alpha \in W(k)$ and $x, y$ in $1 + m_A$, the relation $(xy)^\alpha = x^\alpha y^\alpha$.