On algebraic $$K$$-functors of crossed group rings and its applications.(English)Zbl 1444.19003

Summary: 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 $$K$$: If $$P$$ and $$Q$$ are finitely generated $$R[\pi, \sigma, \rho]$$-projective modules and $$K\bigotimes_RP\simeq K\bigotimes_RQ$$ 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 $$\sigma=\mathrm{id}$$.

MSC:

 19D25 Karoubi-Villamayor-Gersten $$K$$-theory 13C10 Projective and free modules and ideals in commutative rings 19M05 Miscellaneous applications of $$K$$-theory
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