Relative K_0, annihilators, Fitting ideals and the Stickelberger phenomena

Abstract

When $G$ is abelian and $l$ is a prime we show how elements of the relative K-group $K_0({\mathbf{Z}}_l[G],{\mathbf{Q}}_l)$ give rise to annihilator/Fitting ideal relations of certain associated $\mathbf{Z}[G]$-modules. Examples of this phenomenon are ubiquitous. Particularly, we give examples in which $G$ is the Galois group of an extension of global fields and the resulting annihilator/Fitting ideal relation is closely connected to Stickelberger's Theorem and to the conjectures Coates-Sinnott and Brumer.