Noncommutative localization and chain complexes I. Algebraic K- and L-theory

TL;DR

This paper explores noncommutative localization of rings, establishing connections with algebraic K- and L-theory, and providing new results on chain complexes and localization sequences under specific flatness conditions.

Contribution

It introduces a new perspective on noncommutative localization as endomorphism rings in triangulated categories and extends algebraic K- and L-theory results to this setting.

Findings

Characterization of S^{-1}R as endomorphism ring in a triangulated category

Proven chain complex equivalence under stable flatness condition

Extended localization exact sequences in algebraic K- and L-theory

Abstract

The noncommutative (Cohn) localization S^{-1}R of a ring R is defined for any collection S of morphisms of f.g. projective left R-modules. We exhibit S^{-1}R as the endomorphism ring of R in an appropriate triangulated category. We use this expression to prove that if S^{-1}R is "stably flat over R" (meaning that Tor^R_i(S^{-1}R,S^{-1}R)=0 for i>0) then every bounded f.g. projective S^{-1}R-module chain complex D with [D] \in im(K_0(R)-->K_0(S^{-1}R)) is chain equivalent to S^{-1}C for a bounded f.g. projective R-module chain complex C, and that there is a localization exact sequence in higher algebraic K-theory >... --> K_n(R) --> K_n(S^{-1}R) --> K_n(R,S) --> K_{n-1}(R) --> ..., extending to the left the sequence obtained for n<2 by Schofield. For a noncommutative localization S^{-1}R 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.