Exercise 16, § 1

a) Let A, B be two rings and $\varphi$ a homomorphism from A into B. Then the mapping $W(\varphi)$ makes it possible to endow the set $W(B)$ and, for every $m \in \mathbf{N}$, the set $W_m(B)$, with a structure of $W(A)$-module. If $m$ and $n$ are two integers $\geqslant 0$, such that $n \geqslant m$, the canonical mapping from $W_n(B)$ onto $W_m(B)$ is $W(A)$-linear. Moreover, $W(B)$ is the projective limit $W(A)$-module of the $W_m(B)$.

b) Let A, B be two rings and $\psi$ a homomorphism from $W(A)$ into B. Put $\tilde{\psi} = W(\psi) \circ s_A$ (cf. Exercise 15). The mapping $\tilde{\psi}$ from $W(A)$ into $W(B)$ makes it possible to endow the set $W(B)$ and, for every $m \in \mathbf{N}$, the set $W_m(B)$, with a structure of $W(A)$-module, and the last two assertions of a) are still true. If $\varphi$ is a homomorphism from A into B such that $\psi = \varphi \circ \Phi_0$, one has $\tilde{\psi} = W(\varphi)$.