Exercise 39, § 1

Let $S$ be a nonempty subset of $\mathbf{J}$ such that for every $n$ in S, S contains $\mathbf{J}_n$. Let $\pi_S$ be the canonical projection of $A^\mathbf{J}$ onto $A^S$. a) The kernel of $\pi_S$ is an ideal of A. We shall denote by $U_S(A)$ the ring obtained by endowing $A^S$ with the structure of quotient ring. We have a commutative diagram of ring homomorphisms

where $\Phi_S((a_n)_{n \in S}) = (\Phi_n((a_d)_{d \in \mathbf{J}_n}))_{n \in S}$.

b) For every $n \in \mathbf{J}$, $\mathrm{Ker}(\pi_S)$ is stable by $V_n$, and $\mathrm{Ker}(\pi_S)$ contains $V_n(U(A))$ if $n \notin S$. By passing to the quotient $V_n$ defines an endomorphism, still denoted $V_n$, of the additive group of $U_S(A)$. c) If $n \in \mathbf{J}$ and if $nS \subset S$ then $\mathrm{Ker}(\pi_S)$ is stable by $F_n$, which defines, by passing to the quotient, an endomorphism, still denoted $F_n$, of the ring $U_S(A)$. d) Let $n \in \mathbf{J}$. The ring $U_S(A)$ will also be denoted by $U_n(A)$ if $S = J_n$, and by $U_{n^\infty}(A)$ if $S = \bigcup_{r \geq 1} J_{n^r}$.

Let $p$ be a prime number. Now prove that $U_{p^\infty}(A)$ is identified with the Witt ring $W(A)$, and that, for every $n \in \mathbf{N}$, $U_{p^n}(A)$ is identified with the ring $W_{n+1}(A)$ (the element $(u_1, u_p, u_{p^2}, ..., u_{p^{n-1}}, u_{p^n})$ of $U_{p^n}(A)$ will be identified with the Witt vector $[a_0, ..., a_n]$ with $a_i = u_{p^i}$ for $0 \leq i \leq n$). The group endomorphisms $V_p$ and $F_p$ of $U_{p^\infty}(A)$ correspond respectively, by this identification, to the shift and to the Frobenius endomorphism of $W(A)$.