This article establishes, for an appropriate localisation
of associative rings, a long exact sequence in algebraic
–theory. The main result
goes as follows. Let be an
associative ring and let be the
localisation with respect to a set
of maps between finitely generated projective
–modules. Suppose
that vanishes
for all . View
each map in
as a complex (of length 1, meaning one non-zero map between
two non-zero objects) in the category of perfect complexes
. Denote
by
the thick subcategory generated by these complexes. Then the canonical functor
induces (up to direct
factors) an equivalence .
As a consequence, one obtains a homotopy fibre sequence
(up to surjectivity of )
of Waldhausen –theory
spectra.
In subsequent articles [?, ?] we will present the
– and
–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
, we also assume
that every map in
is a monomorphism, then there is a description of the homotopy fiber of the map
as the Quillen
–theory
of a suitable exact category of torsion modules.