NONCOMMUTATIVE LOCALIZATION AND CHAIN COMPLEXES I. ALGEBRAIC K- AND L-THEORY

Abstract

The noncommutative (Cohn) localization (sigma)^−1 R of a ring R is defined for any collection (sigma) of morphisms of f.g. projective left R-modules. We exhibit (sigma)^−1 R as the endomorphism ring of R in an appropriate triangulated category. We use this expression to prove that if TorR i ((sigma)−1R, (sigma)−1R) = 0 for i GTE 1 then every bounded f.g. projective (sigma)−1R-module chain complex D with [D] 2 im(K0(R) -> K0((sigma)−1R)) is chain equivalent to (sigma)−1C for a bounded f.g. projective R-module chain complex C, and that there is a localization exact sequence in higher algebraic K-theory . . . -> Kn(R) -> Kn((sigma)−1R) -> Kn(R, (sigma)) -> Kn−1(R) −! . . . , extending to the left the sequence obtained for n LTE 1 by Schofield. For a noncommutative localization (sigma)−1R of a ring with involution R there are analogous results for algebraic L-theory, extending the results of Vogel from quadratic to symmetric L-theory.