A note on generators of even-numbered \(K\)-groups of rings of integers
type: Item
instance of: scholarly article
title: A note on generators of even-numbered \(K\)-groups of rings of integers
zbMATH Open document ID: 1194.11102
author: Wojciech Gajda
author: Cornelius Greither
published in: Mathematical Research Letters
publication date: 2010-05-14T00:00:00Z
review text: The aim of this paper is to give explicit generators for the étale \(K\)-groups \(K_{2n}^{\text{ét}}(\mathcal{O}_F[1/\ell])\) in the particular case where \(\mathcal{O}_F\) is the ring of integers of some cyclotomic field \(F\) and \(\ell\) is an odd prime number. The above groups were introduced by \textit{W. G. Dwyer} and \textit{E. M. Friedlander} [Trans. Am. Math. Soc. 292, 247--280 (1985; Zbl 0581.14012)]: they are related to both Quillen's \(K\)-theory and étale cohomology. In fact, throughout the paper the authors identify \(K_{2n}^{\text{ét}}(\mathcal{O}_F[1/\ell])\) with \(H^2(\mathcal{O}_F[1/\ell],\,\mathbb{Z}_\ell(n+1))\). Set now \(F=\mathbb{Q}(\mu_\ell)\) and \(E=\mathbb{Q}(\mu_{\ell^k})\) for some \(k\geq 1\). The main tool of the paper is a map \[ \alpha:\mathcal{O}_{E,\,S}^\times \otimes \mathbb{Z}/\ell^k(n)\longrightarrow K_{2n}^{\text{ét}}(\mathcal{O}_{F,\,S})[\ell^k] \] where \(S\) is a set of primes of \(E\) containing those above \(\ell\). For \(n=1\), this map is simply given by \(\alpha(x\otimes \zeta)=\text{Tr}_{E/F}\{x,\,\zeta\}\) where we identify étale \(K\)-theory with \(K\)-theory and the transfer \(\text{Tr}_{E/F}\) is applied to the symbol \(\{x,\,\zeta\}\). The general definition of \(\alpha\) is given by \[ \alpha:\mathcal{O}_{E,\,S}^\times\otimes \mathbb{Z}/\ell^k(n)\to H^1(\mathcal{O}_{E,\,S},\,\mathbb{Z}_\ell(1))\otimes H^0(\mathcal{O}_{E,\,S},\,\mathbb{Z}/\ell^k(n))\to H^2(\mathcal{O}_{E,\,S},\,\mathbb{Z}_\ell(n+1))[\ell^k] \] where the first map is the Kummer map in the first factor and trivial in the other, while the second map is the cup product followed by the canonical epimorphism \(H^1(\mathcal{O}_{E,\,S},\,\mathbb{Z}/\ell^k(n+1))\to H^2(\mathcal{O}_{E,\,S},\,\mathbb{Z}_\ell(n+1))[\ell^k]\). This map has also been studied by \textit{G. Banaszak} and \textit{W. Gajda} [Contemp. Math. 199, 7--18 (1996; Zbl 0866.19004)]. The first part of the paper is devoted to a special case, namely the case where the minus part of the class…
reviewed by: Luca Caputo
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 19F27
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 5707684
zbMATH Keywords: class group
zbMATH Keywords: cyclotomic fields
zbMATH Keywords: étale cohomology
zbMATH Keywords: K-groups
MaRDI profile type: Publication
full work available at URL: mrl.2009.v16.n5.a7
OpenAlex ID: W2063927122
DOI: 10.4310/MRL.2009.V16.N5.A7
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description: scientific article; zbMATH DE number 5707684