Exercise 6, § 2

Let A be a ring and $(J_n)_{n \in \mathbf{N}}$ a decreasing sequence of ideals of A. Suppose that A is separated and complete for the topology defined by the filtration $(J_n)_{n \in \mathbf{N}}$ and that one has $pJ_n + J_n^p \subset J_{n+1}$ for every integer $n \geqslant 0$. Let $\pi_n$ denote the canonical mapping of A onto $A/J_n$. Let R be a perfect ring of characteristic $p$ and $\overline{\varphi}$ a ring homomorphism of R into $A/J_0$. a) There exists a mapping $\varphi$ of R into A, and only one, such that one has $\overline{\varphi} = \pi_0 \circ \varphi$ and $\varphi(x^p) = \varphi(x)^p$ for every $x \in R$. Moreover, one has $\varphi(1) = 1$ and $\varphi(xy) = \varphi(x) \varphi(y)$ for $x$ and $y$ in R. When A is a ring of characteristic $p$, $\varphi$ is a ring homomorphism. (One may use the preceding exercise.) b) For every $n \in \mathbf{N}$, there exists a unique ring homomorphism $v_n : W_{n+1}(A/J_0) \to A/J_n$ making the diagram

commutative. c) Put $u_n = v_n \circ W_{n+1}(\overline{\varphi}) \circ F^{-n}$. Then, for every $n \in \mathbf{N}$, the following diagram, in which the vertical arrows denote the canonical projections, is commutative

d) There exists a unique ring homomorphism $u$ making the following diagram commutative

The homomorphism $u$ is continuous when $W(R)$ is endowed with the $p$-adic topology, and for every $a = (a_n)_{n \in \mathbf{N}} \in W(R)$, one has $u(a) = \sum_{n=0}^{\infty} p^n \varphi(a_n^{p^{-n}})$.

(The existence of $u$ follows from the foregoing. To prove the uniqueness of $u$, its continuity and its explicit expression, use the equality $\varphi = u \circ \tau$ where $\varphi$ is the mapping of $R$ into $A$ constructed in $a$ and $\tau : R \to W(R)$ the mapping defined in § 1, No. 5.)

e) Give, for th. 2 of § 2, No. 4, a proof other than that of the text and that of exercise 4, c).