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 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 2015, Volume 27, Issue 6, Pages 117–149 (Mi aa1469)

Research Papers

Tate sequences and Fitting ideals of Iwasawa modules

C. Greithera, M. Kuriharab

a Institut für Theoretische Informatik und Mathematik, Universität der Bundeswehr, München, 85577 Neubiberg, Germany
b Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

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 $\operatorname{Gal}(L/k)$ 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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English version:
St. Petersburg Mathematical Journal, 2016, 27:6, 941–965

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Document Type: Article
Language: English

Citation: C. Greither, M. Kurihara, “Tate sequences and Fitting ideals of Iwasawa modules”, Algebra i Analiz, 27:6 (2015), 117–149; St. Petersburg Math. J., 27:6 (2016), 941–965

Citation in format AMSBIB
\Bibitem{GreKur15} \by C.~Greither, M.~Kurihara \paper Tate sequences and Fitting ideals of Iwasawa modules \jour Algebra i Analiz \yr 2015 \vol 27 \issue 6 \pages 117--149 \mathnet{http://mi.mathnet.ru/aa1469} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3589224} \elib{http://elibrary.ru/item.asp?id=26414161} \transl \jour St. Petersburg Math. J. \yr 2016 \vol 27 \issue 6 \pages 941--965 \crossref{https://doi.org/10.1090/spmj/1428} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000393181800007} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84999133838} 

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• http://mi.mathnet.ru/eng/aa/v27/i6/p117

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This publication is cited in the following articles:
1. C. Greither, M. Kurihara, “Fitting ideals of Iwasawa modules and of the dual of class groups”, Tokyo J. Math., 39:3 (2017), 619–642
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