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

- 10 Downloads

## 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 *R*^{G}. 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 *R*^{G} will inherit certain a Cohen–Macaulay property from *R*.

## Preview

Unable to display preview. Download preview PDF.

## References

- [AZ]M. Artin and J. J. Zhang,
*Noncommutative projective schemes*, Advances in Mathematics**109**(1994), 228–287.MathSciNetCrossRefzbMATHGoogle Scholar - [B]R.-O. Buchweitz,
*From platonic solids to preprojective algebras via the McKay correspondence*, in*Oberwolfach Jahresbericht Annual Report 2012*, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, 2014, pp. 18–28.Google Scholar - [BHZ1]Y.-H. Bao, J.-W. He and J. J. Zhang,
*Pertinency of Hopf actions and quotient categories of Cohen–Macaulay algebras*, Journal of Noncommutative Geometry, to appear, arXiv:1603.0234.Google Scholar - [BHZ2]Y.-H. Bao, J.-W. He and J. J. Zhang,
*Noncommutative Auslander Theorem*, Transactions of the American Mathematical Society**370**(2018), 8613–8638.MathSciNetCrossRefzbMATHGoogle Scholar - [BL]K. A. Brown and M. Lorenz,
*Grothendieck groups of invariant rings and of group rings*, Journal of Algebra**166**(1994), 423–454.MathSciNetCrossRefzbMATHGoogle Scholar - [CFM]M. Cohen, D. Fischman and S. Montgomery,
*Hopf Galois extensions*, Smash products, and Morita equivalence, Journal of Algebra**133**(1990), 351–372.zbMATHGoogle Scholar - [CH]O. Celikbas and H. Holm,
*Equivalences from tilting theory and commutative algebra from the adjoint functor point of view*, New York Journal of Mathematics**23**(2017), 1697–1721.MathSciNetzbMATHGoogle Scholar - [CKWZ1]K. Chan, E. Kirkman, C. Walton and J. J. Zhang,
*McKay Correspondence for semisimple Hopf actions on regular graded algebras*, I, Journal of Algebra**508**(2018), 512–538.MathSciNetCrossRefzbMATHGoogle Scholar - [CKWZ2]K. Chan, E. Kirkman, C. Walton and J. J. Zhang, McKay Correspondence for semisimple Hopf actions on regular graded algebras, part II, Journal of Noncommutative Geometry, to appear, arXiv:1610.01220.Google Scholar
- [GKMW]J. Gaddis, E. Kirkman, W. F. Moore and R. Won, Auslander’s Theorem for permutation actions on noncommutative algebras, Proceedings of the American Mathematical Society, to appear, https://doi.org/10.1090/proc/14363.Google Scholar
- [HVZ1]J.-W. He, F. Van Oystaeyen and Y. H. Zhang,
*Hopf dense Galois extensions with applications*, Journal of Algebra**476**(2017), 134–160.MathSciNetCrossRefzbMATHGoogle Scholar - [HVZ2]J.-W. He, F. Van Oystaeyen and Y. H. Zhang,
*Hopf algebra actions on differential graded algebras and applications*, Bulletin of the Belgian Mathematical Society. Simon Stevin**18**(2011), 99–111.MathSciNetzbMATHGoogle Scholar - [HZ]J.-W. He and Y. H. Zhang, Cohen–Macaulay invariant subalgebras of Hopf dense Galois extensions, Contemporary Mathematics, American Mathematical Society, Providence, RI, to appear, arXiv:1711.04197.Google Scholar
- [IW]O. Iyama and M. Wemyss,
*Maximal modifications and Auslander–Reiten duality for non-isolated singularities*, Inventiones Mathematicae**197**(2014), 521–586.MathSciNetCrossRefzbMATHGoogle Scholar - [JZ]P. Jørgensen and J. J. Zhang,
*Gourmet’s to guide to Gorensteinness*, Advances in Mathematics**151**(2000), 313–345.MathSciNetCrossRefzbMATHGoogle Scholar - [KKZ]E. Kirkman, J. Kuzmanovich and J. J. Zhang,
*Rigidity of graded regular algebras*, Transactions of the American Mathematical Society**360**(2008), 6331–6369.MathSciNetCrossRefzbMATHGoogle Scholar - [KL]G. R. Krause and T. H. Lenagan,
*Growth of Algebras and Gelfand–Kirillov Dimension*, Research Notes in Mathematics, Vol. 116, Pitman, Boston, MA, 1985.Google Scholar - [KMP]E. Kirkman, I. Musson and D. Passman,
*Noetherian down-up algebras*, Proceedings of the American Mathematical Society**127**(1999), 3161–3167.MathSciNetCrossRefzbMATHGoogle Scholar - [M]H. Matsumura,
*Commutative Ring Theory*, Cambridge Studies in Advanced Mathematics, Vol. 8, Cambridge University Press, Cambridge, 1989.Google Scholar - [MR]J. C. McConnell and J. C. Robson,
*Noncommutative Noetherian Rings*, Graduate Studies in Mathematics, Vol. 30, American Mathematical Society, Providence, RI, 1987.Google Scholar - [Mo]I. Mori,
*McKay type correspondence for AS-regular algebras*, Journal of the London Mathematical Society**88**(2013), 97–117.MathSciNetCrossRefzbMATHGoogle Scholar - [MU1]I. Mori and K. Ueyama,
*Ample group action on Artin–Schelter regular algebras and noncommutative graded isolated singularities*, Transactions of the American Mathematical Society**368**(2016), 7359–7383.MathSciNetCrossRefzbMATHGoogle Scholar - [MU2]I. Mori and K. Ueyama,
*Stable categories of graded Cohen–Macaulay modules over noncommutative quotient singularities*, Advances in Mathematics**297**(2016), 54–92.MathSciNetCrossRefzbMATHGoogle Scholar - [P]N. Popescu,
*Abelian Categories with Applications in Rings and Modules*, London Mathematical Society Monographs, Vol. 3, Academic Press, London–New York, 1973.Google Scholar - [S]F. A. Szasz,
*Radicals of Rings*, Akadémiai Kiadó, Budapest, 1981.zbMATHGoogle Scholar - [U]K. Ueyama,
*Graded maximal Cohen–Macaulay modules over noncommutative Gorenstein isolated singularities*, Journal of Algebra**383**(2013), 85–103.MathSciNetCrossRefzbMATHGoogle Scholar - [VdB]M. Van den Bergh,
*Existence theorems for dualizing complexes over noncommutative graded and filtered rings*, Journal of Algebra**195**(1997), 662–679.MathSciNetCrossRefzbMATHGoogle Scholar - [Z]M. R. Zargar,
*On the relative Cohen–Macaulay modules*, Journal of Algebra and its Applications**14**(2015), paper no. 1550042.Google Scholar