Noncommutative localisation in algebraic K-theory I

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 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 D perf(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 D perf(A)/〈σ〉 → Dperf(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 [26, 27] 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.