Geometry & Topology
- Geom. Topol.
- Volume 8, Number 3 (2004), 1385-1425.
Noncommutative localisation in algebraic $K$–theory I
This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic –theory. The main result goes as follows. Let be an associative ring and let be the localisation with respect to a set of maps between finitely generated projective –modules. Suppose that vanishes for all . View each map in as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes . Denote by the thick subcategory generated by these complexes. Then the canonical functor induces (up to direct factors) an equivalence . As a consequence, one obtains a homotopy fibre sequence
(up to surjectivity of ) of Waldhausen –theory spectra.
In subsequent articles [?, ?] we will present the – and –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 , we also assume that every map in is a monomorphism, then there is a description of the homotopy fiber of the map as the Quillen –theory of a suitable exact category of torsion modules.
Geom. Topol., Volume 8, Number 3 (2004), 1385-1425.
Received: 15 January 2004
Revised: 1 September 2004
Accepted: 11 October 2004
First available in Project Euclid: 21 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 18F25: Algebraic $K$-theory and L-theory [See also 11Exx, 11R70, 11S70, 12- XX, 13D15, 14Cxx, 16E20, 19-XX, 46L80, 57R65, 57R67]
Secondary: 19D10: Algebraic $K$-theory of spaces 55P60: Localization and completion
Neeman, Amnon; Ranicki, Andrew. Noncommutative localisation in algebraic $K$–theory I. Geom. Topol. 8 (2004), no. 3, 1385--1425. doi:10.2140/gt.2004.8.1385. https://projecteuclid.org/euclid.gt/1513883471