Noncommutative localisation in algebraic $K$–theory I

Abstract

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 $A\to B$ be the localisation with respect to a set $\sigma$ of maps between finitely generated projective $A$–modules. Suppose that ${Tor}_n^A\left(B,B\right)$ vanishes for all $n>0$. View each map in $\sigma$ as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes $Dperf\left(A\right)$. Denote by $\langle \sigma \rangle$ the thick subcategory generated by these complexes. Then the canonical functor $Dperf\left(A\right)\to Dperf\left(B\right)$ induces (up to direct factors) an equivalence $Dperf\left(A\right)∕\langle \sigma \rangle \to Dperf\left(B\right)$. As a consequence, one obtains a homotopy fibre sequence

$$K\left(A,\sigma \right)\to K\left(A\right)\to K\left(B\right)$$

(up to surjectivity of $K_0\left(A\right)\to K_0\left(B\right)$) 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 ${Tor}_n^A\left(B,B\right)$, we also assume that every map in $\sigma$ is a monomorphism, then there is a description of the homotopy fiber of the map $K\left(A\right)\to K\left(B\right)$ as the Quillen $K$–theory of a suitable exact category of torsion modules.