Ranicki, Noncommutative localisation in algebraic K {theory

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 of maps between nitely generated projective A{modules. Suppose that TorAn (B; B) vanishes for all n> 0. 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 Dperf(A). Denote by hi the thick subcategory generated by these complexes. Then the canonical functor Dperf(A) ! Dperf(B) induces (up to direct factors) an equivalence Dperf(A)=hi ! Dperf(B). As a consequence, one obtains a homotopy bre sequence K(A; ) ! 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 TorAn (B; B), we also assume that every map in is a monomorphism, then there is a description of the homotopy ber of th