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Tate sequences and Fitting ideals of Iwasawa modules


Authors: C. Greither and M. Kurihara
Original publication: Algebra i Analiz, tom 27 (2015), nomer 6.
Journal: St. Petersburg Math. J. 27 (2016), 941-965
MSC (2010): Primary 11R23, 11R29, 11R18
DOI: https://doi.org/10.1090/spmj/1428
Published electronically: September 30, 2016
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Abstract: 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 $ \mathrm {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 $ \mathrm {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.


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Additional Information

C. Greither
Affiliation: Institut für Theoretische Informatik und Mathematik, Universität der Bundeswehr, München, 85577 Neubiberg, Germany
Email: cornelius.greither@unibw.de

M. Kurihara
Affiliation: Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan
Email: kurihara@math.keio.ac.jp

DOI: https://doi.org/10.1090/spmj/1428
Keywords: Tate sequences, class groups, cohomology, totally real fields, CM-fields
Received by editor(s): June 15, 2015
Published electronically: September 30, 2016
Dedicated: To our colleague and friend Sergeĭ V. Vostokov on the occasion of his seventieth birthday
Article copyright: © Copyright 2016 American Mathematical Society