Noncommutative cyclic isolated singularities

Chan, Kenneth; Young, Alexander; Zhang, James

doi:10.1090/tran/8084

Noncommutative cyclic isolated singularities
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by Kenneth Chan, Alexander Young and James J. Zhang PDF

Trans. Amer. Math. Soc. 373 (2020), 4319-4358 Request permission

Abstract:

The question of whether a noncommutative graded quotient singularity $A^G$ is isolated depends on a subtle invariant of the $G$-action on $A$, called the pertinency. We prove a partial dichotomy theorem for isolatedness, which applies to a family of noncommutative quotient singularities arising from a graded cyclic action on the $(-1)$-skew polynomial ring. Our results generalize and extend some results of Bao, He, and the third-named author and results of Gaddis, Kirkman, Moore, and Won.

References

M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 228–287. MR 1304753, DOI 10.1006/aima.1994.1087
Maurice Auslander, On the purity of the branch locus, Amer. J. Math. 84 (1962), 116–125. MR 137733, DOI 10.2307/2372807
Yanhong Bao, Jiwei He, and James J. Zhang, Pertinency of Hopf actions and quotient categories of Cohen-Macaulay algebras, J. Noncommut. Geom. 13 (2019), no. 2, 667–710. MR 3988759, DOI 10.4171/JNCG/336
Y.-H. Bao, J.-W. He, and J. J. Zhang, Noncommutative Auslander theorem, Trans. Amer. Math. Soc. 370 (2018), no. 12, 8613–8638. MR 3864389, DOI 10.1090/tran/7332
Raf Bocklandt, Travis Schedler, and Michael Wemyss, Superpotentials and higher order derivations, J. Pure Appl. Algebra 214 (2010), no. 9, 1501–1522. MR 2593679, DOI 10.1016/j.jpaa.2009.07.013
Kenneth Chan, Ellen Kirkman, Chelsea Walton, and James J. Zhang, Quantum binary polyhedral groups and their actions on quantum planes, J. Reine Angew. Math. 719 (2016), 211–252. MR 3552496, DOI 10.1515/crelle-2014-0047
K. Chan, E. Kirkman, C. Walton, and J. J. Zhang, McKay correspondence for semisimple Hopf actions on regular graded algebras, I, J. Algebra 508 (2018), 512–538. MR 3810305, DOI 10.1016/j.jalgebra.2018.05.008
J. Chen, E. Kirkman, and J. J. Zhang, Auslander’s theorem for group coactions on noetherian graded down-up algebras, Transform. Groups (accepted). Preprint available at arXiv:1801.09020, 2018.
Michel Dubois-Violette, Multilinear forms and graded algebras, J. Algebra 317 (2007), no. 1, 198–225. MR 2360146, DOI 10.1016/j.jalgebra.2007.02.007
Akira Fujiki, On resolutions of cyclic quotient singularities, Publ. Res. Inst. Math. Sci. 10 (1974/75), no. 1, 293–328. MR 0385162, DOI 10.2977/prims/1195192183
Jason Gaddis, Ellen Kirkman, W. Frank Moore, and Robert Won, Auslander’s theorem for permutation actions on noncommutative algebras, Proc. Amer. Math. Soc. 147 (2019), no. 5, 1881–1896. MR 3937667, DOI 10.1090/proc/14363
Ji-Wei He and Yinhuo Zhang, Local cohomology associated to the radical of a group action on a noetherian algebra, Israel J. Math. 231 (2019), no. 1, 303–342. MR 3960009, DOI 10.1007/s11856-019-1855-9
J. Karmazyn, Superpotentials, Calabi-Yau algebras, and PBW deformations, J. Algebra 413 (2014), 100–134. MR 3216602, DOI 10.1016/j.jalgebra.2014.05.007
E. Kirkman, J. Kuzmanovich, and J. J. Zhang, Invariants of $(-1)$-skew polynomial rings under permutation representations, Recent advances in representation theory, quantum groups, algebraic geometry, and related topics, Contemp. Math., vol. 623, Amer. Math. Soc., Providence, RI, 2014, pp. 155–192. MR 3288627, DOI 10.1090/conm/623/12463
E. Kirkman, J. Kuzmanovich, and J. J. Zhang, Noncommutative complete intersections, J. Algebra 429 (2015), 253–286. MR 3320624, DOI 10.1016/j.jalgebra.2014.12.046
Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Revised edition, Graduate Studies in Mathematics, vol. 22, American Mathematical Society, Providence, RI, 2000. MR 1721834, DOI 10.1090/gsm/022
Thierry Levasseur, Some properties of noncommutative regular graded rings, Glasgow Math. J. 34 (1992), no. 3, 277–300. MR 1181768, DOI 10.1017/S0017089500008843
Ben Mathes, Matjaž Omladič, and Heydar Radjavi, Linear spaces of nilpotent matrices, Linear Algebra Appl. 149 (1991), 215–225. MR 1092879, DOI 10.1016/0024-3795(91)90335-T
J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. MR 934572
John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
Izuru Mori and S. Paul Smith, $m$-Koszul Artin-Schelter regular algebras, J. Algebra 446 (2016), 373–399. MR 3421098, DOI 10.1016/j.jalgebra.2015.09.016
Izuru Mori and Kenta Ueyama, Ample group action on AS-regular algebras and noncommutative graded isolated singularities, Trans. Amer. Math. Soc. 368 (2016), no. 10, 7359–7383. MR 3471094, DOI 10.1090/tran/6580
Izuru Mori and Kenta Ueyama, Stable categories of graded maximal Cohen-Macaulay modules over noncommutative quotient singularities, Adv. Math. 297 (2016), 54–92. MR 3498794, DOI 10.1016/j.aim.2016.04.009
David R. Morrison and Glenn Stevens, Terminal quotient singularities in dimensions three and four, Proc. Amer. Math. Soc. 90 (1984), no. 1, 15–20. MR 722406, DOI 10.1090/S0002-9939-1984-0722406-4
C. Năstăsescu and F. van Oystaeyen, Graded ring theory, North-Holland Mathematical Library, vol. 28, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 676974
Manuel Reyes, Daniel Rogalski, and James J. Zhang, Skew Calabi-Yau algebras and homological identities, Adv. Math. 264 (2014), 308–354. MR 3250287, DOI 10.1016/j.aim.2014.07.010
L. W. Small and R. B. Warfield Jr., Prime affine algebras of Gel′fand-Kirillov dimension one, J. Algebra 91 (1984), no. 2, 386–389. MR 769581, DOI 10.1016/0021-8693(84)90110-8
Darin R. Stephenson and James J. Zhang, Growth of graded Noetherian rings, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1593–1605. MR 1371143, DOI 10.1090/S0002-9939-97-03752-0
Kenta Ueyama, Graded maximal Cohen-Macaulay modules over noncommutative graded Gorenstein isolated singularities, J. Algebra 383 (2013), 85–103. MR 3037969, DOI 10.1016/j.jalgebra.2013.02.022
J. J. Zhang, Twisted graded algebras and equivalences of graded categories, Proc. London Math. Soc. (3) 72 (1996), no. 2, 281–311. MR 1367080, DOI 10.1112/plms/s3-72.2.281
James J. Zhang, Connected graded Gorenstein algebras with enough normal elements, J. Algebra 189 (1997), no. 2, 390–405. MR 1438182, DOI 10.1006/jabr.1996.6885