On algebraic $K$-functors of crossed group rings and its applications

Abstract

Let $R[\pi, \sigma, \rho]$ be a crossed group ring. An induction theorem is proved for the functor $G_{0}^{R}(R[\pi, \sigma, \rho])$ and the Swan-Gersten higher algebraic $K$-functors $K_{i}(R[\pi,\sigma,\rho])$. Using this result, a theorem on reduction is proved for the discrete normalization ring $R$ with the field of quotients $K$: I f $P$ and $Q$ are finitely generated $R[\pi, \sigma, \rho]$-projective modules and $K\bigotimes_{R}P\simeq K\bigotimes_{R}Q$ as $K[\pi, \sigma, \rho]$-modules, then $P\simeq Q.$ Under some restrictions on $n=(\pi:1)$ it is shown that finitely generated $R[\pi,\sigma,\rho]$-projective modules are decomposed into the direct sum of left ideals of the ring $R[\pi,\sigma,\rho]$. More stronger results are proved when $\sigma=id$.