Noncommutative localization in algebraic $L$-theory

TL;DR

This paper develops a lifting theorem and localization exact sequences in algebraic L-theory for noncommutative localizations, extending known results in algebraic K-theory to a broader noncommutative setting.

Contribution

It introduces a new lifting theorem and localization exact sequences in algebraic L-theory for noncommutative localizations, generalizing previous K-theory results.

Findings

Established a lifting theorem for chain complexes over noncommutative localizations.

Derived localization exact sequences in algebraic L-theory.

Extended algebraic K-theory localization results to algebraic L-theory.

Abstract

Given a noncommutative (Cohn) localization $A \to \sigma^{-1}A$ which is injective and stably flat we obtain a lifting theorem for induced f.g. projective $\sigma^{-1}A$-module chain complexes and localization exact sequences in algebraic $L$-theory, matching the algebraic $K$-theory localization exact sequence of Neeman and Ranicki.