Tate sequences and Fitting ideals of Iwasawa modules

We consider Abelian CM extensions $L/k$ of a totally real field k, and we essentially determine the Fitting ideal of the dualized Iwasawa module studied by the second author in the case where only places above p ramify. In doing so we recover and generalize the results mentioned above. Remarkably, our explicit description of the Fitting ideal, apart from the contribution of the usual Stickelberger element Θ at infinity, only depends on the group structure of the Galois group Gal(L/k) and not on the specific extension L. From our computation it is then easy to deduce that T Θ is not in the Fitting ideal as soon as the p-part of Gal(L/k) is not cyclic. We need a lot of technical preparations: resolutions of the trivial module Z over a group ring, discussion of the minors of certain big matrices that arise in this context, and auxiliary results about the behavior of Fitting ideals in short exact sequences.