Noncommutative cyclic isolated singularities
HTML articles powered by AMS MathViewer
- 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
Additional Information
- Kenneth Chan
- Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
- MR Author ID: 1040750
- Email: kenhchan@math.washington.edu, ken.h.chan@gmail.com
- Alexander Young
- Affiliation: Department of Mathematics, DigiPen Institute of Technology, Redmond, Washington 98052
- MR Author ID: 947371
- Email: young.mathematics@gmail.com
- James J. Zhang
- Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
- MR Author ID: 314509
- Email: zhang@math.washington.edu
- Received by editor(s): March 10, 2019
- Received by editor(s) in revised form: October 21, 2019
- Published electronically: March 16, 2020
- Additional Notes: The third author was supported in part by the US National Science Foundation (No. DMS-1700825).
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 4319-4358
- MSC (2010): Primary 16E65, 16W22, 16S35, 16S38, 14J17
- DOI: https://doi.org/10.1090/tran/8084
- MathSciNet review: 4105525