Tate sequences and Fitting ideals of Iwasawa modules (Q2822784)

As a refinement of the main conjecture in the Iwasawa theory for ideal class groups, in certain cases we can describe the Fitting ideals of class groups as Galois modules (algebraic objects) by using the Stickelberger elements (analytic objects). But in general, the usual class group does not fit well with (étale) cohomology theory, and certain modified class groups are used in the theory of the leading terms of \(L\)-functions (for example, the \((S, T)\)-modified class group can be used in the theory of Stark's conjecture where \(S\) contains the ramified places, and \(T\) is used to get a torsion-free subgroup of the unit group). In order to treat the class group in the usual non-modified sense, we will need several devices. In this paper, the authors study these non-modified class groups and determine the Fitting ideals of certain Iwasawa modules related to them, in several new cases.NEWLINENEWLINEThe authors mainly consider abelian CM extensions \(L/k\) of a totally real field \(k\), and essentially determine the Fitting ideal of the dualized Iwasawa module studied by the second author in the case where only places above \(p\) ramify [Tokyo J. Math. 34, No. 2, 407--428 (2011; Zbl 1270.11117)]. They recover and generalize the results mentioned above. Remarkably, their 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 \(\text{Gal}(L/k)\) and not on the specific extension \(L\). From the detailed computation it is not difficult to deduce that \(\dot{T}\dot{\Theta}\) is not in the Fitting ideal as soon as the \(p\)-part of \(\text{Gal}(L/k)\) is not cyclic. A lot of technical preparations are needed and necessary: 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.