Noncommutative localization and chain complexes I. Algebraic K- and L-theory
Amnon Neeman, Andrew Ranicki
Abstract
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.
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