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

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
\geq 1) 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\leq 1 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.

Amnon Neeman <Amnon.Neeman@anu.edu.au>

Andrew Ranicki <aar@maths.ed.ac.uk>