Exercise 25, § 1

Let A be a perfect ring of characteristic $p$. a) Let B be an A-algebra and $\varphi : A \to B$ the structural homomorphism. For every integer $n \geqslant 1$, endow the group $W_n(B)$ with the $W(A)$-module structure defined by

Endowed with the action of F by the Frobenius homomorphism and of V by the shift homomorphism V, $W_n(B)$ is then a $\mathscr{D}_A$-module (Exercise 22). The mapping $V_n : W_n(B) \to W_{n+1}(B)$ induced by the shift is a homomorphism of $\mathscr{D}_A$-modules. By transport of structure, the group $CW^n(B)$ is endowed with a structure of $\mathscr{D}_A$-module. b) Let $\mathscr{A}$ be the polynomial ring $A[X]$ in a family $X = (X_i)_{i \in I}$ of indeterminates. For every integer $r \geqslant 0$, let $b_r$ be the ideal of $\mathscr{A}$ generated by the $X_{-n}$, for $n \geqslant r$. Let $r$ and $s$ be integers $\geqslant 1$. Then, for every element $a = (a_n)_{n \in \mathbf{N}}$ of $W(A)$, one has for every integer $m \geqslant r - 1$ if $s < p$, and for every integer $m \geqslant r - 1 + (s - p)/(p - 1)$ if $s \geqslant p$. c) Let B be an A-algebra and $\varphi : A \to B$ the structural homomorphism. Let $a = (a_n)_{n \in \mathbf{N}}$ be an element of $W(A)$, and $b = (b_i)_{i \in I}$ an element of $CW(B)$. For every integer $n \geqslant 0$, the sequence of elements $P_m(\varphi(a_0^{p^{-n-m}}), ..., \varphi(a_m^{p^{-n-m}}); b_{-m-n}, ..., b_{-n})$ is stationary. Denoting by $c_{-n}$ the limit of this sequence, the element $c = (c_i)_{i \in I}$ belongs to $CW(B)$. Put $c = a.b$. The mapping from $W(A) \times CW(B)$ into $CW(B)$ which associates $c$ with $(a, b)$ endows $CW(B)$ with a structure of $W(A)$-module, extending the structure of $W(A)$-module of $CW^n(A)$. For every $a \in A$ and every $b \in CW(B)$, one has and If, for every $b \in CW(B)$, one puts then F and V are continuous endomorphisms of $CW(B)$ and make it possible to endow $CW(B)$ with a structure of $\mathscr{D}_A$-module. The $\mathscr{D}_A$-module $CW^n(B)$ is a sub-$\mathscr{D}_A$-module of $CW(B)$.