Exercise 15, § 1

a) Let A be a ring such that multiplication by $p \cdot 1_A$ in A is injective. Then there exists a unique ring homomorphism $s_A$ of $W(A)$ into $W(W(A))$ satisfying

and (where $\Phi_0$ is the projection of $W(W(A))$ onto $W(A)$). It is also the unique ring homomorphism satisfying $\Phi_n \circ s_A = F_A^n$ for every integer $n \in \mathbf{N}$ (where $\Phi_n : W(W(A)) \to W(A)$ is the $n$-th ghost component in $W(W(A))$). b) Consider the ring $\mathscr{A} = \mathbf{Z}[(X_n)_{n \in \mathbf{N}}]$ of polynomials in a family of indeterminates $(X_n)_{n \in \mathbf{N}}$. Let X be the element $(X_n)_{n \in \mathbf{N}}$ of $W(\mathscr{A})$. Put $s_{\mathscr{A}}(X) = (s_n(X))_{n \in \mathbf{N}}$, where $s_n(X) \in W(\mathscr{A})$. For

EXERCISES

every ring A, define the mapping $s_A$ of $W(A)$ into $W(W(A))$ by the formula $s_A(a) = (s_n(a))_{n \in \mathbf{N}}$. Prove that $s_A$ is a ring homomorphism satisfying

and

c) For every ring homomorphism $u : B \to A$, one has

d) The mappings $W(s_A) \circ s_A$ and $s_{W(A)} \circ s_A$ of $W(A)$ into $W(W(W(A)))$ are equal.

e) For every $x \in A$, one has $s_A(\tau_A(x)) = \tau_{W(A)}(\tau_A(x))$.

f) One has $s_A \circ V_A = V_{W(A)} \circ s_A$, and the mapping $s_A$ is continuous when $W(A)$ and $W(W(A))$ are endowed with the $\mathscr{G}$ topologies.