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→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 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)/⟨σ⟩→Dperf(B). As a consequence, one obtains a homotopy fibre sequence

(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 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)→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

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