A note on traces of differential forms

Let A subset of B be integral domains with quotient fields F,L resp., L finite separable over F. Suppose that B is integral over A and that A is integrally closed. If PEL, we may consider the trace T(beta):= Tr-F(L)(beta) is an element of F. It is classical that, if beta is an element of beta, T(beta) is an element of A. Viewing elements of B as differential forms of degree 0, we consider here the analogous property when beta is a differential form of any degree. We prove that under above assumptions the analogous property indeed holds independently of the degree of the form. The result looks natural, but seems to have never been slated explicitly. We also give a simple geometric application. (C) 1999 Elsevier Science B.V. All rights reserved.