Geodesic
Tate sequences and Fitting ideals of Iwasawa modules
Algebra i analiz, Tome 27 (2015) no. 6, pp. 117-149.

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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.
Keywords: Tate sequences, class groups, cohomology, totally real fields, CM-fields. 
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     author = {C. Greither and M. Kurihara},
     title = {Tate sequences and {Fitting} ideals of {Iwasawa} modules},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_2015_27_6_a6/}
}
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C. Greither; M. Kurihara. Tate sequences and Fitting ideals of Iwasawa modules. Algebra i analiz, Tome 27 (2015) no. 6, pp. 117-149. http://geodesic.mathdoc.fr/item/AA_2015_27_6_a6/

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