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 $\dot\Theta$ at infinity, only depends on the group structure of the Galois group $\operatorname{Gal}(L/k)$ and not on the specific extension $L$. From our computation it is then easy to deduce that $\dot T\dot\Theta$ is not in the Fitting ideal as soon as the $p$-part of $\operatorname{Gal}(L/k)$ is not cyclic. We need a lot of technical preparations: resolutions of the trivial module $\mathbb Z$ 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.