Residue Free Differentials and the Cartier Operator for Algebraic Function Fields of one Variable

Abstract

Let K be a field of characteristic p > 0 and let A be a separably generated algebraic function field of one variable with K as its exact constant field. Throughout this paper we shall use the following notations to classify differentials of A/K:D(A) : the K-module of all differentials,G(A) : the K-module of all differentials of the first kind,R(A) : the K-module of all residue free differentials in the sense of Chevalley [2, p. 48],E*(A) : the K-module of all pseudo-exact differentials in the sense of Lamprecht [7, p. 363], (compare the definition with our Lemma 8).