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June 2018 On algebraic $K$-functors of crossed group rings and its applications
Giorgi Rakviashvili
Tbilisi Math. J. 11(2): 1-15 (June 2018). DOI: 10.32513/tbilisi/1529460017

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 $\sigma=id$.

Citation

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Giorgi Rakviashvili. "On algebraic $K$-functors of crossed group rings and its applications." Tbilisi Math. J. 11 (2) 1 - 15, June 2018. https://doi.org/10.32513/tbilisi/1529460017

Information

Received: 28 December 2017; Accepted: 20 January 2018; Published: June 2018
First available in Project Euclid: 20 June 2018

zbMATH: 07172260
MathSciNet: MR3954179
Digital Object Identifier: 10.32513/tbilisi/1529460017

Subjects:
Primary: 13C10
Secondary: 19D25, 19M05

Rights: Copyright © 2018 Tbilisi Centre for Mathematical Sciences

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Vol.11 • No. 2 • June 2018
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