Exercise 23, § 3

Let $B$ be a Krull ring, $E$ its field of fractions, $\Delta$ a derivation of $E$ such that $\Delta(B) \subset B$, $K$ the subfield of $E$ which is the kernel of $\Delta$ and $A$ the Krull ring $B \cap K$; suppose that $K$ has characteristic $p > 0$; one has $E^p \subset K$ and $B^p \subset A$, so that $B$ is the integral closure of $A$ in $E$, and the canonical homomorphism $i : C(A) \to C(B)$ is defined (§ 1, No. 10). We denote by $U$ 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 of $D(A)$, now prove that one has $\Delta b/b \in B$ (note that for every prime ideal $\mathfrak{P}$ of height 1 of $B$, there exists $b' \in K$ such that $v_{\mathfrak{P}}(b) = v_{\mathfrak{P}}(b')$, and observe that $B_{\mathfrak{P}}$ is stable under $\Delta$). Let $L$ be the additive subgroup of $B$ formed by the $\Delta b/b$ (“logarithmic derivatives”) which belong to $B$ (for $b \in E$ or $b \in B$, which amounts to the same thing, and $b \neq 0$), and let $L'$ be the subgroup of $L$ formed by the $\Delta u/u$, where $u \in U$. Now prove 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 every $b$ such that $i(d) = \operatorname{div}_B(b)$ depends only on the class of $d$ in $C(A)$, and deduce a canonical injective homomorphism $\varphi$ of $\operatorname{Ker}(i)$ into $L/L'$.

(b) Prove that if $\Delta(B)$ is not contained in any prime ideal of height 1 of $B$ and if $[E : K] = p$, $\varphi$ is bijective. (Reduce to proving that if $b \in E$ is such that $\Delta b/b \in B$ and if $\mathfrak{P}$ is a prime ideal of height 1 in $B$ such that $v_{\mathfrak{P}}(b)$ is not multiple of $p$, then $e(\mathfrak{P}/p) = 1$, where $p = \mathfrak{P} \cap A$; for this, deduce from the assumption that if $t$ is a uniformizer of $B_{\mathfrak{P}}$, one has $\Delta t/t \in B_{\mathfrak{P}}$, so that $\mathfrak{P}B_{\mathfrak{P}}$ is stable under $\Delta$, and that $\Delta$ therefore defines by passing to quotients a derivation $\bar{\Delta}$ of the residue field $k = B_{\mathfrak{P}}/\mathfrak{P}B_{\mathfrak{P}}$; prove that $\bar{\Delta} \neq 0$, and deduce that $f(\mathfrak{P}/p) = p$.

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