This article establishes, for an appropriate localisation of associative
rings, a long exact sequence in algebraic K–theory. The
main result goes as follows. Let A be an associative ring and let
A→B be the localisation with respect to a set σ of maps
between finitely generated projective A–modules. Suppose that
TornA(B,B) vanishes for all n>0. View each map
in σ as a complex (of length 1, meaning one non-zero map between two
non-zero objects) in the category of perfect complexes Dperf(A). Denote by
〈σ〉 the thick subcategory generated by these complexes.
Then the canonical functor Dperf(A)→Dperf(B) induces (up to direct
factors) an equivalence Dperf(A)/〈σ〉→Dperf(B).
As a consequence, one obtains a homotopy fibre sequence
(up to surjectivity of K0(A)→K0(B)) of
Waldhausen K–theory spectra.
In subsequent articles we will present the K– and L–theoretic
consequences of the main theorem in a form more suitable for the
applications to surgery. For example if, in addition to the vanishing
of TornA(B,B), we also assume that every map in
σ is a monomorphism, then there is a description of the homotopy
fiber of the map K(A)→K(B) as the Quillen K–theory of a
suitable exact category of torsion modules.