Geometry & Topology, Vol. 8 (2004) Paper no. 38, pages 1385--1425.

Noncommutative localisation in algebraic K-theory I

Amnon Neeman, Andrew Ranicki

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 A-->B be the localisation with respect to a set sigma of maps between finitely generated projective A-modules. Suppose that Tor_n^A(B,B) vanishes for all n>0. View each map in sigma as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes D^perf(A). Denote by [sigma] the thick subcategory generated by these complexes. Then the canonical functor D^perf(A)-->D^perf(B) induces (up to direct factors) an equivalence D^perf(A)/[sigma]--> D^perf(B). As a consequence, one obtains a homotopy fibre sequence K(A,sigma)-->K(A)-->K(B) (up to surjectivity of K_0(A)-->K_0(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 Tor_n^A(B,B), we also assume that every map in sigma 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

AMS subject classification. Primary: 18F25. Secondary: 19D10, 55P60.

DOI: 10.2140/gt.2004.8.1385

E-print: arXiv:math.RA/0410620

Submitted to GT on 15 January 2004. (Revised 1 September 2004.) Paper accepted 11 October 2004. Paper published 27 October 2004.

Notes on file formats

Amnon Neeman, Andrew Ranicki

Centre for Mathematics and its Applications, The Australian National University
Canberra, ACT 0200, Australia
School of Mathematics, University of Edinburgh
Edinburgh EH9 3JZ, Scotland, UK


GT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to