First order Stickelberger modules over imaginary quadratic fields

Abstract

Let $K/k$ be a finite abelian extension of number fields of Galois group G with k imaginary quadratic. Let $n\ge 2$ be a rational integer, and for a certain finite set S of places of k, let $O_{K,S}$ be the ring of S-integers of K. We use generalized Stark elements to construct first order Stickelberger modules in odd higher algebraic K-groups of $O_{K,S}$. We show that the Fitting ideal (resp. index) of these modules inside the corresponding odd K-groups is exactly the Fitting ideal (resp. cardinality) of the even higher algebraic K-group $K_{2n-2}\left(O_{K,S}\right)$.