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Local cohomology associated to the radical of a group action on a noetherian algebra

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Abstract

An arbitrary group action on an algebra R results in an ideal r of R. This ideal r fits into the classical radical theory, and will be called the radical of the group action. If R is a noetherian algebra with finite GK-dimension and G is a finite group, then the difference between the GK-dimensions of R and that of R/r is called the pertinency of the group action. We provide some methods to find elements of the radical, which helps to calculate the pertinency of some special group actions. The r-adic local cohomology of R is related to the singularities of the invariant subalgebra RG. We establish an equivalence between the quotient category of the invariant subalgebra RG and that of the skew group ring R * G through the torsion theory associated to the radical r. With the help of the equivalence, we show that the invariant subalgebra RG will inherit certain a Cohen–Macaulay property from R.

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Acknowledgments

We would like to thank the referee for his/her valuble suggestions and comments. Thanks to James Zhang for many helpful conversations. J.-W. He is supported by NSFC (No. 11571239, 11671341) and ZJNSF (No. LY19A010011)., and Y. Zhang is supported by an FWO grant.

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Authors and Affiliations

  1. Department of Mathematics, Hangzhou Normal University, Hangzhou Zhejiang, 310036, China

    Ji-Wei He

  2. Department of Mathematics and Statistics, University of Hasselt, Universitaire Campus, 3590, Diepenbeek, Belgium

    Yinhuo Zhang

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  1. Ji-Wei He
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  2. Yinhuo Zhang
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Correspondence to Yinhuo Zhang.

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He, JW., Zhang, Y. Local cohomology associated to the radical of a group action on a noetherian algebra. Isr. J. Math. 231, 303–342 (2019). https://doi.org/10.1007/s11856-019-1855-9

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  • DOI: https://doi.org/10.1007/s11856-019-1855-9

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