Relative $K_{0}$, annihilators, Fitting ideals and Stickelberger phenomena

Abstract

When $G$ is abelian and $l$ is a prime we show how elements of the relative K-group $K_{0}({\bf Z}_{l}[G], {\bf Q}_{l})$ give rise to annihilator/Fitting ideal relations of certain associated ${\bf Z}[G]$-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 of Coates and Sinnott, and Brumer. Higher Stickelberger ideals are defined in terms of special values of L-functions; when these vanish we show how to define fractional ideals, generalising the Stickelberger ideals, with similar annihilator properties. The fractional ideal is constructed from the Borel regulator and the leading term in the Taylor series for the L-function. En route, our methods yield new proofs, in the case of abelian number fields, of formulae predicted by Lichtenbaum for the orders of K-groups and étale cohomology groups of rings of algebraic integers.