#### 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

and

School of Mathematics, University of Edinburgh

Edinburgh EH9 3JZ, Scotland, UK

Email: Amnon.Neeman@anu.edu.au, a.ranicki@ed.ac.uk

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