Exercise 7, § 2

a) Let $R$ be a perfect ring of characteristic $p$, and $R'$ a ring of characteristic $p$. Let $f$ be a homomorphism of $R$ into $R'$. Then the homomorphism $W(f) : W(R) \to W(R')$ is the unique homomorphism making the following diagram commutative

(Use the preceding exercise.) In particular, taking for $f$ the raising to the $p$-th power in $R$, prove that $F_R$ is the unique endomorphism of $W(R)$ such that $\Phi_0 \circ F_R = f \circ \Phi_0$. It is characterized by the equalities

b) Let $A$ be a ring separated and complete for the $p$-adic topology. Suppose that multiplication by $p.1_A$ in $A$ is injective and that $A/pA$ is a perfect ring. Then there exists a unique endomorphism $\sigma$ of $A$ such that $\sigma(x) \equiv x^p \mod. pA$ for $x \in A$, and the inverse of the isomorphism $t_\sigma : A \to W(A/pA)$ described in Exercise 14, p. 44 is given by

where $\varphi : A/pA \to A$ is the unique mapping which yields the identity of $A/pA$ on passing to the quotient and satisfies $\varphi(x^p) = \varphi(x)^p$ for $x \in R$ (cf. Exercise 6, a)).