On the Steinitz module and capitulation of ideals

Abstract

Let $L$ be a finite extension of a number field $K$ with ring of integers ${\cal O}_L$ and ${\cal O}_K$ respectively. One can consider ${\cal O}_L$ as a projective module over ${\cal O}_K$. The highest exterior power of ${\cal O}_L$ as an ${\cal O}_K$ module gives an element of the class group of ${\cal O}_K$, 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 ${\cal O}_K$ is said to capitulate in $L$ if its extension to ${\cal O}_L$ is a principal ideal.)