Class groups and differential function fields

Abstract

Let V be an integral algebraic scheme over a fieldX and suppose we are given a finite set D of derivations on the function field L of V which commute and leave K globally invariant. Let V, be the set of all (not necessarily closed) points p E V such that fl& is a differential subring of L whose maximal ideal is differential (by differential we will always mean differential with respect to 0). The aim of this note is to give some information concerning the “size” of VD and location of the points of V, on V. Our approach will be via the divisor class group of V, the conclusion will be that if K is non-constant then there exist “plenty” of divisors on V whose intersection with VD is “small” (see Propositions 1 and 4 below). For instance, if K is algebraically closed and V is a smooth projective curve over K we show that Cl(V) is generated modulo torsion by VjV,. Notations and terminology will be borrowed from [2] and [6]. All fields appearing are assumed to have characteristic zero. We will use the notation V(K) for the set of K-rational points of V and we put VD(K) = VD f? V(K). For any differential ring B, Spec, B will denote the set of differential primes in-B. For any Krull domain A, we denote by A”’ the set all primes of A of height 1. For any S c Spec A we denote by A, the ring fipEs A,. Recall that if S c A”’ then A, is also a Krull domain and the morphism A --t A, induces an isomorphism (A,)“’ 7 S 13, p. 161. Furthermore there is a surjection Cl(A) + Cl@,) whose kernel is generated by the prime divisors in A”‘\S [3, p. 35 1. We begin with a non-differential lemma concerning Krull domains: