Exercise 23, § 3

Let $B$ be a Krull domain, $E$ its field of fractions, $A$ a derivation of $E$ such that $\Delta(B) \subset B$, $K$ the subfield of $E$ the kernel of $A$ and $A$ the Krull domain $B \cap K$; suppose that $K$ is of characteristic $p > 0$; then $E^p \subset K$ and $B^p \subset A$, so that $B$ is the integral closure of $A$ in $E$ and the canonical homomorphism $\tilde{i}: C(A) \to C(B)$ is defined ($§ 1$, no. 10). Let $U$ denote the group of invertible elements of $B$.

(a) If $b \in E$ is such that $\operatorname{div}_B(b)$ is the canonical image of a divisor in $D(A)$, show that $\Delta b/b \in B$ (note that for every prime ideal $\mathfrak{P}$ of $B$ of height 1 there exists $b' \in K$ such that $v_{\mathfrak{P}}(b) = v_{\mathfrak{P}}(b')$ and observe that $B_1$ is invariant under $A$). Let $L$ be the additive subgroup of $B$ consisting of the $\Delta b/b$ ("logarithmic derivatives") which belong to $B$ (for $b \in E$ or $b \in B$, which amounts to the same, and $b \neq 0$) and let $L'$ be the subgroup of $L$ consisting of the $\Delta u/u$, where $u \in U$. Show that, for every divisor $d \in D(A)$ such that the image of $d$ is a principal divisor in $D(B)$, the class mod. $L'$ of $\Delta b/b$ for all $b$ such that $i(d) = \operatorname{div}_B(b)$ depends only on the class of $d$ in $C(A)$ and derive a canonical injective homomorphism $\dagger$ of $\operatorname{Ker}(\tilde{i})$ to $L/L'$.

(b) Show that, if $\Delta(B)$ is contained in no prime ideal of $B$ of height 1 and $[E: K] = p$, $\phi$ is bijective. (Reduce it to showing that, if $b \in E$ is such that $\Delta b/b \in B$ and $\mathfrak{P}$ is a prime ideal of $B$ of height 1 such that $v_{\mathfrak{P}}(b)$ is not a multiple of $p$, then $e(\mathfrak{P}/p) = 1$, where $p = \mathfrak{P} \cap A$; for this, deduce from the hypothesis that, if $t$ is a uniformizer of $B_{\mathfrak{P}}$, then $\Delta t/t \in B_{\mathfrak{P}}$, so that $\mathfrak{P}B_{\mathfrak{P}}$ is invariant under $A$ and that $A$ therefore defines by taking quotients a derivation $\overline{\Delta}$ of the residue field $k = B_{\mathfrak{P}}/\mathfrak{P}B_{\mathfrak{P}}$; show that $\overline{\Delta} \neq 0$ and deduce that $f(\mathfrak{P}/p) = p.$)

(c) Let $k$ be a field of characteristic 2, $B$ the polynomial ring $k[X, Y, Z]$ and $A$ the derivation of $E = k(X, Y, Z)$ such that $\Delta(X) = Y^4, \Delta(Y) = X^2, \Delta(Z) = XYZ$; the field $K$ the kernel of $A$ is such that $[E : K] = 4$. Then $\Delta(Z)/Z \in B$, but show that $\operatorname{div}_B(Z)$ is not the canonical image of a divisor in $D(A)$. (Argue by reductio ad absurdum supposing that $e(\mathfrak{P}/p) = 1$ for $\mathfrak{P} = BZ, p = \mathfrak{P} \cap A$; there would then be in $A$ a uniformizer of $B$, necessarily of the form $a(X, Y, Z)Z$ where $b = a(X, Y, 0) \neq 0$; deduce that $\Delta b/b = -XY$ and obtain a contradiction by calculating $A(-XY)$.)