On the Steinitz module and capitulation of ideals

Abstract

Let L be a finite extension of a number field K with ring of integers O_L and O_K respectively. One can consider O_L as a projective module over O_K. The highest exterior power of O_L as an O_K module gives an element of the class group of O_K, called the Steinitz module. These considerations work also for algebraic curves where we prove that for a finite unramified cover Y of an algebraic curve X, the Steinitz module as an element of the Picard group of X is the sum of the line bundles on X which become trivial when pulled back to Y. We give some examples to show that this kind of result is not true for number fields. We also make some remarks on the capitulation problem for both number field and function fields. (An ideal in O_K is said to capitulate in L if its extension to O_L is a principal ideal.)