Noncommutative localisation in algebraic \(K\)-theory. I
type: Item
instance of: scholarly article
title: Noncommutative localisation in algebraic \(K\)-theory. I
zbMATH Open document ID: 1083.18007
author: Andrew Ranicki
author: Amnon Neeman
published in: Geometry \& Topology
publication date: 2005-09-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0410620
full work available at URL: https://eudml.org/doc/125701
full work available at URL: paper38.abs.html
review text: The authors prove the existence of a long exact localization sequence in algebraic \(K\)-theory for a very general type of localization. Let \(A\) be an associative ring with unit, and let \(\sigma\) be any set of maps \(s_i: P_i \rightarrow Q_i\) of finitely generated projective (left) \(A\)-modules. A ring homomorphism \(A \rightarrow B\) is \(\sigma\)-inverting if for all \(i\) the map \(B\otimes_A P_i \rightarrow B \otimes_A Q_i\) is an isomorphism. The category of all \(\sigma\)-inverting homomorphisms \(A \rightarrow B\) has an initial object \(A \rightarrow \sigma^{-1}A\), which is the Cohn localization of \(A\). Under the assumption that \(\text{Tor}_n^A(\sigma^{-1}A,\sigma^{-1}A) = 0\) for all \(n>0\) the localization sequence is obtained from a description of the homotopy fiber of the map of spectra \(K(A) \rightarrow K(\sigma^{-1}A)\) as \(K(\mathcal R)\), where \(\mathcal R\) is a certain subcategory of the Waldhausen category of all perfect complexes of \(A\)-modules, i.e., bounded complexes of finitely generated projective \(A\)-modules.
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 19D10
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 18E35
zbMATH DE Number: 2206526
zbMATH Keywords: noncommuative localisation
zbMATH Keywords: algebraic \(K\)-theory
zbMATH Keywords: triangulated category
reviewed by: Manfred Kolster
MaRDI profile type: Publication
OpenAlex ID: W2611365983
arXiv ID: math/0410620
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DOI: 10.2140/GT.2004.8.1385
description: scientific article