Exercise 12, § 1

Let A be a ring of characteristic $p$. For every element $a = (a_n)_{n \in \mathbf{N}}$ of $W(A)$ and every integer $n \in \mathbf{N}$, put $a_+ = (a_{n+1})_{n \in \mathbf{N}}$. a) We have $a = \tau(a_0) + Va_+$. b) Let $\alpha : W(A) \to W(A)$ denote the mapping which associates to $a = (a_n)_{n \in \mathbf{N}}$

where (cf. previous exercise) we have put $\gamma_p(Va_+) = (p^{p-1}/p!) V(a_+^p)$. Prove that, in $W(A)$, one has the equality $Fa = a^p + p \alpha(a)$. c) Prove that, for every integer $n \geq 1$, $\alpha$ defines, by passing to quotients, a mapping $\alpha_n$ of $W_{n+1}(A)$ into $W_n(A)$.