Exercise 22, § 1

Let $A$ be a $\mathbf{Z}_{(p)}$-algebra. We denote by $\mathscr{D}_A$ the algebra over $W(A)$ generated by two indeterminates $F$ and $V$ subject to the relations

where $F$ and $V$ denote respectively the Frobenius and shift homomorphisms of $W(A)$. a) Let $B$ be an $A$-algebra. Making $F$ and $V$ act on $W(B)$ by the Frobenius and shift homomorphisms of $W(B)$, we endow $W(B)$ with a $\mathscr{D}_A$-module structure. Analogously, for each integer $n \geqslant 1$, $W_n(B)$ is a $\mathscr{D}_A$-module. b) For every element $x$ of $\mathscr{D}_A$, there exists a finite-support family $(a_i)_{i \in \mathbf{Z}}$ of elements of $W(A)$, characterized by the equality c) Suppose $W(A)$ is an integral domain. Let $x \in \mathscr{D}_A$. We denote by $\delta(x)$ the greatest integer $n$ such that $a_n$ is nonzero. If $x$ and $y$ are two nonzero elements of $\mathscr{D}_A$, one has $\delta(xy) = \delta(x) + \delta(y)$. Deduce that $\mathscr{D}_A$ is an integral domain. d) If $A$ is a perfect ring of characteristic $p$, one can replace the condition $VaF = Va$ for $a \in W(A)$ by the condition $VF = p$. The left ideal $\mathscr{D}_A V$ of $\mathscr{D}_A$ is two-sided. e) If $k$ is a perfect field of characteristic $p$, $\mathscr{D}_k$ is noetherian. (Consider $\mathscr{D}_k$ as quotient of the ring generated over $W(k)$ by two indeterminates $X$ and $Y$ subject to the relations $XY = YX$ and $Xa = (Fa)X, aY = Y(Fa)$ for $a \in W(k)$. Then apply III, § 2, No. 8, corollary 2 to Theorem 1 and Exercise 10.)