Noncommutative localisation in algebraic K–theory I

Neeman, Amnon; Ranicki, Andrew

doi:10.2140/gt.2004.8.1385

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Noncommutative localisation in algebraic $K$–theory I

Amnon Neeman and Andrew Ranicki

Geometry & Topology 8 (2004) 1385–1425

DOI: 10.2140/gt.2004.8.1385

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

K (A, σ) \to K (A) \to K (B)

(up to surjectivity of $K_{0} (A) \to K_{0} (B)$ ) 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 $σ$ is a monomorphism, then there is a description of the homotopy fiber of the map $K (A) \to K (B)$ 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

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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

Volume 8, issue 3 (2004)