Capitulation for even \(K\)-groups in the cyclotomic \(\mathbb Z_p\)-extension
type: Item
instance of: scholarly article
title: Capitulation for even \(K\)-groups in the cyclotomic \(\mathbb Z_p\)-extension
zbMATH Open document ID: 1254.11097
author: Romain Validire
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-10-26T00:00:00Z
full work available at URL: https://eudml.org/doc/10891
review text: Let \(F\) be a number field and let \(p\) be a prime (when \(p=2\) assume that \(F\) contains \(\sqrt{-1}\)). Let \(F_\infty/F\) be the cyclotomic \(\mathbb{Z}_p\)-extension of \(F\). For a finite layer \(F_n/F\) of \(F_\infty/F\), let \(A'_{F_n}\) denote the \(p\)-part of the class group of the ring of \(p\)-integers \(o'_{F_n}\) of \(F_n\). While studying the behaviour of \(A'_{F_n}\) for large \(n\) in the framework of classical Iwasawa theory, one is naturally led to consider the direct limit \(A'_{F_\infty}=\lim\limits_{\longrightarrow}A'_{F_n}\) (resp. the projective limit \(X'_{F_\infty}=\lim\limits_{\longleftarrow}A'_{F_n}\)) over the extension (or capitulation) maps \(A'_{F_n}\to A'_{F_m}\) (resp. over the norm maps \(A'_{F_m}\to A'_{F_n}\)) for \(m\geq n\). A well-known result of Iwasawa asserts that the kernels \[ \mathrm{Cap}(F_n)=\ker\left(A'_{F_n}\to A'_{F_\infty}\right) \] stabilize (i.e. become isomorphic) for large \(n\). Moreover, asymptotically, \(\mathrm{Cap}(F_n)\) is isomorphic to the maximal finite \(\Lambda\)-submodule \((X'_{F_\infty})^{0}\) of \(X'_{F_\infty}\) (where \(\Lambda=\mathbb{Z}_p[\![\mathrm{Gal}(F_\infty/F)]\!]\) denotes the Iwasawa algebra, as usual). Thanks to a result of \textit{M. Grandet} and \textit{J.-F. Jaulent} [J. Reine Angew. Math. 362, 213--217 (1985; Zbl 0564.12011)], one also knows that, if the torsion subgroup of \(X'_{F_\infty}\) is finite (i.e. the \(\mu\)-invariant of \(X'_{F_\infty}\) is trivial), then \(\mathrm{Cap}(F_n)\) is a direct summand of \(A'_{F_n}\) for large \(n\). The paper under review offers a cohomological generalization of the result of Grandet-Jaulent in the following context. The group \(A'_{F_n}\) has an interesting interpretation in terms of the étale cohomology of \(\mathrm{Spec}(o'_{F_n})\), namely \[ A'_{F_n}\cong t_{\mathbb{Z}_p}(H^2_{\acute{e}t}(o'_{F_n},\mathbb{Z}_p(1))) \] where \(\mathbb{Z}_p(1)\) is the (first) Tate twist of \(\mathbb{Z}_p\) and \(t_{\mathbb{Z}_p}(\--)\) denotes \(\…
reviewed by: Luca Caputo
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 19F99
zbMATH DE Number: 5620661
zbMATH Keywords: \(K\)-groups
zbMATH Keywords: \(\mathbb{Z}_p\)-extensions
zbMATH Keywords: étale cohomology
MaRDI profile type: Publication
OpenAlex ID: W2024038444
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DOI: 10.5802/JTNB.681
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description: scientific article; zbMATH DE number 5620661