Exercise 14, § 1

Let A be a ring. Suppose that multiplication by $p \cdot 1_A$ is injective in A. Let $\sigma$ be an endomorphism of A such that $\sigma(x) \equiv x^p \mod pA$, for every $x \in A$. a) There exists a unique ring homomorphism $s_\sigma$ of A into $W(A)$ satisfying

and It is also the unique homomorphism $s_\sigma$ of A into $W(A)$ such that, for every positive integer $n$, $\Phi_n \circ s_\sigma = \sigma^n$. b) Let B be a ring. Suppose that multiplication by $p \cdot 1_B$ is injective in B. Let $\sigma'$ be an endomorphism of B such that $\sigma'(x) \equiv x^p \mod pB$ for every $x \in B$. Let $u$ be a homomorphism of A into B satisfying $u \circ \sigma = \sigma' \circ u$. Then one has $W(u) \circ s_\sigma = s_{\sigma'} \circ u$. c) If $t_\sigma$ denotes the composite of $s_\sigma$ and the canonical projection of $W(A)$ onto $W(A/pA)$, prove that $t_\sigma$ induces, for every integer $n \geq 1$, a homomorphism $t_{\sigma,n}$ of $A/p^nA$ into $W_n(A/pA)$. d) If $A/pA$ is perfect, prove that $t_{\sigma,n}$ is an isomorphism. (Endowing A with the filtration by the powers of the ideal $pA$ and denoting by $\mathrm{gr}_p(A)$ the associated graded ring, one will show that the homomorphism of $\mathrm{gr}_p(A)$ into $\mathrm{gr}_V(W(A/pA))$ (cf. Exercise 13) induced by $t_\sigma$ is an isomorphism.) e) If $A/pA$ is perfect, and if A is separated and complete for the $p$-adic topology, $t_\sigma$ is an isomorphism of A onto $W(A/pA)$.