\(NK_{0}\) and \(NK_{1}\) of the groups \(C_{4}\) and \(D_{4}\)
type: Item
instance of: scholarly article
title: \(NK_{0}\) and \(NK_{1}\) of the groups \(C_{4}\) and \(D_{4}\)
zbMATH Open document ID: 1167.19003
DOI: 10.4171/CMH/164
published in: Commentarii Mathematici Helvetici
publication date: 2009-04-21T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_pdf.php
review text: Let \(R\) be a ring and \(K_i(R)\) denote the \(i\)-th \(K\)-theory group of \(R\), (\(i\in\mathbb{Z}\)). Recall that \(NK_i(R)\) is defined as the quotient \(K_{i}(R[X])/K_{i}(R)\). The author explicitly computes these groups in the following cases: \(R=\mathbb{Z}[G]\) (the integral group ring), \(G=D_{2}, C_{4}\) or \(D_4\) (dihedral of order \(2\), cyclic of order \(4\) and dihedral of order \(8\), respectively) and \(i=0,1\). The computations keep track of the module structure of these groups given by Verschiebung and Frobenius operators, as well as the continuous module structure over the ring of big Witt vectors \(W(\mathbb{Z})\). Let \(V\) denote the continuous \(W(\mathbb{F}_2)\)-module \(x\mathbb{F}_2[x]\): it is a countable direct sum of copies of \(\mathbb{F}_2\) on generator \(x^{i}\), \(i>0\) and the module structure is determined by \(V_{m}(x^{n})=x^{mn}\), \([a]x^{n}=a^{n}x^{n}; F_m(x^{n})=0\) if \( (m,n)=1\) \((m>1)\) and \( F_d(x^n)=dx^{n/d}\) wif \(d|n\). The results are as follows. Theorem 1.3: For \(D_2=C_2\times C_2\) we have \(NK_0(\mathbb{Z}[D_2])\cong V, NK_1(\mathbb{Z}[D_2])\cong \Omega_{\mathbb{F}_2[x]}\), Theorem 1.4: \(NK_1(\mathbb{Z}[C_4])\cong \Omega_{\mathbb{F}_2[x]}\) and \(NK_0(\mathbb{Z}[C_4])\cong V\), Theorem 2.5: \(NK_0(\mathbb{Z}[D_4])\) is isomorphic to the cyclic Cartier module \(NK_1(\mathbb{Z}[C_2],1-\tau)\) (described in the paper). As a group, this is the direct sum of a countable infinite free \(\mathbb{Z}/4\)-module and a countable infinite free \(\mathbb{Z}/2\)-module. The author also proves that \(NK_0(\mathbb{Z}[D_4])\) surjects onto \(NK_1(\mathbb{Z}[D_4])\).
Mathematics Subject Classification ID: 19D35
Mathematics Subject Classification ID: 19A31
Mathematics Subject Classification ID: 19B28
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 16E20
zbMATH DE Number: 5545773
zbMATH Keywords: Bass Nil-groups
zbMATH Keywords: lower algebraic K-theory
zbMATH Keywords: hyperbolic reflection groups
author: C. Weibel
reviewed by: Daniel Juan Pineda
MaRDI profile type: Publication
OpenAlex ID: W2054468337
description: scientific article; zbMATH DE number 5545773