Proposition 7

Let A be a perfect ring of characteristic p.

a) For every element $a = (a_n)_{n \in \mathbf{N}}$ of $W(A)$, the series with general term $p^n \tau(a_n^{p^{-n}})$ is convergent in $W(A)$, with sum $a$.

b) On $W(A)$, the $V_1(A)$-adic topology, the $p$-adic topology and the topology $\mathscr{G}$ coincide. More precisely, one has $V_n(A) = p^n.W(A) = (V_1(A))^n$ for every integer $n \geqslant 0$. In particular $\Phi_0$ defines an isomorphism of $W(A)/p.W(A)$ onto $A$.

By definition (A, V, p. 5), the mapping $a \mapsto a^p$ is an automorphism of the ring $A$. According to prop. 5, F is therefore an automorphism of the ring $W(A)$, and one has, for every $n \in \mathbf{N}$,

In particular, one has $(V_1(A))^n = (p.W(A))^n = p^n.W(A)$. The assertion b) follows from this.

According to prop. 5, one has

and the assertion a) follows from prop. 4 of no 6.