Noncommutative localization in algebraic L-theory

Given a noncommutative (Cohn) localization A→σ−1A which is injective and stably flat we obtain a lifting theorem for induced f.g. projective σ−1A-module chain complexes and localization exact sequences in algebraic L-theory, matching the algebraic K-theory localization exact sequence of Neeman–Ranicki [Amnon Neeman, Andrew Ranicki, Noncommutative localisation in algebraic K-theory I, Geom. Topol. 8 (2004) 1385–1425] and Neeman [Amnon Neeman, Noncommutative localisation in algebraic K-theory II, Adv. Math. 213 (2007) 785–819].