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

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The noncommutative (Cohn) localization S⁻¹R of a ring R is defined for any collection S of morphisms of f.g. projective left R-modules. We exhibit S⁻¹R as the endomorphism ring of R in an appropriate triangulated category. We use this expression to prove that if S⁻¹R is "stably flat over R" (meaning that Torʳᵢ(S⁻¹R,S⁻¹R)=0 for i>0) then every bounded f.g. projective S⁻¹R-module chain complex D with [D] ∈ im(K₀(R)-->K₀(S⁻¹R)) is chain equivalent to S⁻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(S⁻¹R) --> Kₙ(R,S) --> Kₙ₋₁(R) -->..., extending to the left the sequence obtained for n<2 by Schofield. For a noncommutative localization S⁻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.