Volume 8, issue 3 (2004)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Noncommutative localisation in algebraic $K$–theory I

Amnon Neeman and Andrew Ranicki

Geometry & Topology 8 (2004) 1385–1425

arXiv: math.RA/0410620

Abstract

This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic K–theory. The main result goes as follows. Let A be an associative ring and let AB be the localisation with respect to a set σ of maps between finitely generated projective A–modules. Suppose that TornA(B,B) vanishes for all n > 0. View each map in σ as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes Dperf(A). Denote by σ the thick subcategory generated by these complexes. Then the canonical functor Dperf(A)Dperf(B) induces (up to direct factors) an equivalence Dperf(A)σDperf(B). As a consequence, one obtains a homotopy fibre sequence

K(A,σ)K(A)K(B)

(up to surjectivity of K0(A)K0(B)) of Waldhausen K–theory spectra.

In subsequent articles [??] we will present the K– and L–theoretic consequences of the main theorem in a form more suitable for the applications to surgery. For example if, in addition to the vanishing of TornA(B,B), we also assume that every map in σ is a monomorphism, then there is a description of the homotopy fiber of the map K(A)K(B) as the Quillen K–theory of a suitable exact category of torsion modules.

Keywords
noncommutative localisation, $K$–theory, triangulated category
Mathematical Subject Classification 2000
Primary: 18F25
Secondary: 19D10, 55P60
References
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Publication
Received: 15 January 2004
Revised: 1 September 2004
Accepted: 11 October 2004
Published: 27 October 2004
Proposed: Bill Dwyer
Seconded: Thomas Goodwillie, Gunnar Carlsson
Authors
Amnon Neeman
Centre for Mathematics and its Applications
The Australian National University
Canberra
ACT 0200
Australia
Andrew Ranicki
School of Mathematics
University of Edinburgh
Edinburgh EH9 3JZ
Scotland
United Kingdom