Noncommutative localisation in algebraic \(K\)-theory. I
The authors prove the existence of a long exact localization sequence in algebraic \(K\)-theory for a very general type of localization. Let \(A\) be an associative ring with unit, and let \(\sigma\) be any set of maps \(s_i: P_i \rightarrow Q_i\) of finitely generated projective (left) \(A\)-modules. A ring homomorphism \(A \rightarrow B\) is \(\sigma\)-inverting if for all \(i\) the map \(B\otimes_A P_i \rightarrow B \otimes_A Q_i\) is an isomorphism. The category of all \(\sigma\)-inverting homomorphisms \(A \rightarrow B\) has an initial object \(A \rightarrow \sigma^{-1}A\), which is the Cohn localization of \(A\). Under the assumption that \(\text{Tor}_n^A(\sigma^{-1}A,\sigma^{-1}A) = 0\) for all \(n>0\) the localization sequence is obtained from a description of the homotopy fiber of the map of spectra \(K(A) \rightarrow K(\sigma^{-1}A)\) as \(K(\mathcal R)\), where \(\mathcal R\) is a certain subcategory of the Waldhausen category of all perfect complexes of \(A\)-modules, i.e., bounded complexes of finitely generated projective \(A\)-modules.
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