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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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References:

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