Stickelberger elements, Fitting ideals of class groups of CM-fields, and dualisation

In this paper, we systematically construct abelian extensions of CM-fields over a totally real field whose Stickelberger elements are not in the Fitting ideals of the class groups. Our evidence indicates that Pontryagin duals of class groups behave better than the class groups themselves. We also explore the behaviour of Fitting ideals under projective limits and dualisation in a somewhat broader context.