Geometry & Topology

Noncommutative localisation in algebraic $K$–theory I

Amnon Neeman and Andrew Ranicki

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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 AB be the localisation with respect to a set σ of maps between finitely generated projective A–modules. Suppose that TornA(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)σDperf(B). As a consequence, one obtains a homotopy fibre sequence

K ( A , σ ) K ( A ) K ( B )

(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 TornA(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.

Article information

Source
Geom. Topol., Volume 8, Number 3 (2004), 1385-1425.

Dates
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
https://projecteuclid.org/euclid.gt/1513883471

Digital Object Identifier
doi:10.2140/gt.2004.8.1385

Mathematical Reviews number (MathSciNet)
MR2119300

Zentralblatt MATH identifier
1083.18007

Subjects
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

Keywords
noncommutative localisation $K$–theory triangulated category

Citation

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


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