A generalization of Brauer's map of decomposition (Q1317447)

Let \(R\) be a Dedekind domain with field of fractions \(K\), let \(m\) be a maximal ideal of \(R\) and let \(k = R/m\). Further, let \(A\) be an \(R\)- algebra which is finitely generated projective as \(R\)-module. The author generalizes the classical Brauer decomposition homomorphism by producing homomorphisms \(d_ n\), \(n \geq 1\), which make the following diagrams commute: \[ \begin{matrix} G_ n (A) @>K \otimes_ ->> & G_ n (K \otimes A) \\ {_{k \otimes_ -} \searrow} & \downarrow_{d_ n} \\ & G_ n (k \otimes A) \end{matrix} \] If \(n \geq 2\) and \(k\) is finite, then \(d_ n\) is unique. The argument uses a reformulation of Quillen's localization sequence in terms of complicial bi-Waldhausen categories.