Ranks of \(K_n\) and \(G_n\) of orders and group rings of finite groups over integers in number fields
Let \(F\) be a number field with ring of integers \(R\), let \(\Lambda\) be an \(R\)-order which spans a semisimple \(F\)-algebra \(\Sigma\) and let \(\Gamma\) be a maximal \(R\)-order in \(\Sigma\) that contains \(\Lambda\). It is shown that, for all \(n\geq 2\), \[ \text{rank }K_n(\Lambda)= \text{rank }G_n(\Lambda)= \text{rank }K_n(\Gamma)= \text{rank }K_n(\Sigma). \] The proofs use the localization exact sequence in Quillen \(K\)-theory together with devissage arguments to establish that the modular terms are finite.
scientific article; zbMATH DE number 1308358
scientific article