#### Volume 8, issue 3 (2004)

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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 $A\to B$ be the localisation with respect to a set $\sigma$ of maps between finitely generated projective $A$–modules. Suppose that ${Tor}_{n}^{A}\left(B,B\right)$ 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 $Dperf\left(A\right)$. Denote by $〈\sigma 〉$ the thick subcategory generated by these complexes. Then the canonical functor $Dperf\left(A\right)\to Dperf\left(B\right)$ induces (up to direct factors) an equivalence $Dperf\left(A\right)∕〈\sigma 〉\to Dperf\left(B\right)$. As a consequence, one obtains a homotopy fibre sequence

$K\left(A,\sigma \right)\to K\left(A\right)\to K\left(B\right)$

(up to surjectivity of ${K}_{0}\left(A\right)\to {K}_{0}\left(B\right)$) 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 ${Tor}_{n}^{A}\left(B,B\right)$, we also assume that every map in $\sigma$ is a monomorphism, then there is a description of the homotopy fiber of the map $K\left(A\right)\to K\left(B\right)$ 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