Algebraic K-groups as Galois modules
type: Item
instance of: scholarly article
title: Algebraic K-groups as Galois modules
zbMATH Open document ID: 1011.11074
published in: Progress in Mathematics
publication date: 2003-01-14T00:00:00Z
review text: The subject of this book (as aptly described by its title) is the study of the structure of K-groups K\(_i(R)\) as modules over the group ring \({\mathbb Z}[G]\), where \(R\) is a commutative ring on which the finite group \(G\) acts. Actually one assumes more: \(L= \text{Quot}(R)\) is a \(G\)-Galois extension of another field \(K\). Mostly, \(R\) is the ring of integers in a local or global field \(L\). It is usually difficult to find a priori results about any individual such K-group, and most results involve two ``adjacent'' K-groups K\(_{2n-1}(R)\) and K\(_{2n-2}(R)\). A prototypical example (\(n=1\)) is the analytic class number formula which expresses \(h_LR_L/w_L\) by arithmetic data; here \(h_L\), the class number, is the order of the torsion part of K\(_0(O_L)\), and \(R_L\) (the regulator) and \(w_L\) (the number of roots of unity in \(L\)) are quantities attached to \(O_L^*={}\)K\(_1(O_L)\). There are now quite a few invariants attached to a set of data \(L/K, G, n\). For \(n=1\) we have the now classical Chinburg invariants \(\Omega(L/K,i)\) with \(i=1,2,3\), which will be written \(\Omega_0(L/K,i)\) to leave room for notation of new invariants; for \(n\geq 2\) there are the constructions \(\Omega_{n-1}(L/K,3)\) of \textit{T. Chinburg, M. Kolster, G. Pappas}, and \textit{V. P. Snaith} [Fields Inst. Commun. 16, 1-29 (1997; Zbl 0886.11063)], along with an alternative construction of Snaith for \(n=2\). There are constructions of Burns and Flach as well; actually these have a much wider scope, and we only look at the case \(\Omega(L/K,{\mathbb Q}(-n))\). Then there are the ``lifted'' versions: on the one hand we have the construction \(T\Omega(L/K,{\mathbb Q}(-n))\) of Burns (again a special case of a more general construction based upon methods of Bloch and Kato, and of Fontaine and Perrin-Riou), and on the other hand the lifted root number invariants of Gruenberg, Ritter, and Weiss. By results of Burns, Flach, Chinburg, Kolster, and Snaith, these invarian…
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 19F27
zbMATH DE Number: 1853000
zbMATH Keywords: algebraic K-theory
zbMATH Keywords: Galois modules
author: Victor P. Snaith
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1853000