Maximal Cohen-Macaulay modules over a noncommutative 2-dimensional singularity

doi:10.1007/s11464-019-0793-5

Front. Math. China

2019, Vol. 14

Issue (5) : 923-940 https://doi.org/10.1007/s11464-019-0793-5

RESEARCH ARTICLE

Maximal Cohen-Macaulay modules over a noncommutative 2-dimensional singularity

Xiaoshan QIN^1,², Yanhua WANG³, James ZHANG⁴(

)

¹. China Academy of Electronics and Information Technology, Beijing 100041, China
². School of Mathematical Sciences, Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China
³. School of Mathematics, Shanghai Key Laboratory of Financial Information Technology, Shanghai University of Finance and Economics, Shanghai 200433, China
⁴. Department of Mathematics, University of Washington, Seattle, WA 98195, USA

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Abstract

We study properties of graded maximal Cohen-Macaulay modules over an $ℕ$ -graded locally finite, Auslander Gorenstein, and Cohen-Macaulay algebra of dimension two. As a consequence, we extend a part of the McKay correspondence in dimension two to a more general setting.

Keywords Noncommutative quasi-resolution Artin-Schelter regular algebra Maximal Cohen-Macaulay module pretzeled quivers

Corresponding Author(s): James ZHANG

Issue Date: 22 November 2019

Cite this article:

Xiaoshan QIN,Yanhua WANG,James ZHANG. Maximal Cohen-Macaulay modules over a noncommutative 2-dimensional singularity[J]. Front. Math. China, 2019, 14(5): 923-940.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-019-0793-5
https://academic.hep.com.cn/fmc/EN/Y2019/V14/I5/923

1	K Ajitabh, S P Smith, J J Zhang. Auslander-Gorenstein rings. Comm Algebra, 1998, 26: 2159–2180 https://doi.org/10.1080/00927879808826267
2	K Brown, X-S Qin, Y-H Wang, J J Zhang. Pretzelization (in preparation)
3	D Chan, Q-S Wu, J J Zhang. Pre-balanced dualizing complexes. Israel J Math, 2002, 132: 285–314 https://doi.org/10.1007/BF02784518
4	K Chan, E Kirkman, C Walton, J J Zhang. Quantum binary polyhedral groups and their actions on quantum planes. J Reine Angew Math, 2016, 719: 211–252 https://doi.org/10.1515/crelle-2014-0047
5	K Chan, E Kirkman, C Walton, J J Zhang. McKay correspondence for semisimple Hopf actions on regular graded algebras, I. J Algebra, 2018, 508: 512–538 https://doi.org/10.1016/j.jalgebra.2018.05.008
6	K Chan, E Kirkman, C Walton, J J Zhang. McKay correspondence for semisimple Hopf actions on regular graded algebras, II. J Noncommut Geom, 2019, 13(1): 87–114 https://doi.org/10.4171/JNCG/305
7	M Dokuchaev, N M Gubareni, V M Futorny, M A Khibina, V V Kirichenko. Dynkin diagrams and spectra of graphs. São Paulo J Math Sci, 2013, 7(1): 83–104 https://doi.org/10.11606/issn.2316-9028.v7i1p83-104
8	D Happel, U Preiser, C M Ringel. Binary polyhedral groups and Euclidean diagrams. Manuscripta Math, 1980, 31(13): 317–329 https://doi.org/10.1007/BF01303280
9	J Herzog. Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen-Macaulay-Moduln. Math Ann, 1978, 233(1): 21–34 https://doi.org/10.1007/BF01351494
10	P Jørgensen. Finite Cohen-Macaulay type and smooth non-commutative schemes. Canad J Math, 2008, 60(2): 379–390 https://doi.org/10.4153/CJM-2008-018-0
11	G R Krause, T H Lenagan. Growth of Algebras and Gelfand-Kirillov Dimension. Res Notes in Math, Vol 116. Boston: Pitman Adv Publ Program, 1985
12	T Y Lam. A First Course in Noncommutative Rings. 2nd ed. Grad Texts in Math, Vol 131. New York: Springer-Verlag, 2001 https://doi.org/10.1007/978-1-4419-8616-0
13	T Levasseur. Some properties of non-commutative regular graded rings. Glasg J Math, 1992, 34: 277–300 https://doi.org/10.1017/S0017089500008843
14	R Martnez-Villa. Introduction to Koszul algebras. Rev Un Mat Argentina, 2007, 48(2): 67–95
15	X-S Qin, Y-H Wang, J J Zhang. Noncommutative quasi-resolutions. J Algebra, 2019, 536: 102–148 https://doi.org/10.1016/j.jalgebra.2019.07.015
16	M Reyes, D Rogalski. A twisted Calabi-Yau toolkit. arXiv: 1807.10249
17	M Reyes, D Rogalski. Growth of graded twisted Calabi-Yau algebras. arXiv: 1808.10538
18	M Reyes, D Rogalski, J J Zhang. Skew Calabi-Yau algebras and homological identities. Adv Math, 2014, 264: 308–354 https://doi.org/10.1016/j.aim.2014.07.010
19	J H Smith. Some properties of the spectrum of a graph. In: Guy R, Hanani H, Sauer N, Schönheim J, eds. Combinatorial Structures and their Applications. New York: Gordon and Breach, 1970, 403–406
20	K Ueyama. Graded maximal Cohen-Macaulay modules over noncommutative graded Gorenstein isolated singularities. J Algebra, 2013, 383: 85–103 https://doi.org/10.1016/j.jalgebra.2013.02.022
21	M Van den Bergh. Existence theorems for dualizing complexes over non-commutative graded and filtered rings. J Algebra, 1997, 195(2): 662–679 https://doi.org/10.1006/jabr.1997.7052
22	M Van den Bergh. Three-dimensional fiops and noncommutative rings. Duke Math J, 2004, 122(3): 423–455 https://doi.org/10.1215/S0012-7094-04-12231-6
23	M Van den Bergh. Non-commutative crepant resolutions. In: Laudal O A, Piene R, eds. The Legacy of Niels Henrik Abel. Berlin: Springer, 2004, 749–770 https://doi.org/10.1007/978-3-642-18908-1_26
24	S Weispfenning. Properties of the fixed ring of a preprojective algebra. J Algebra, 2019, 517: 276–319 https://doi.org/10.1016/j.jalgebra.2018.10.005
25	A Yekutieli. Dualizing complexes over noncommutative graded algebras. J Algebra, 1992, 153(1): 41{84 https://doi.org/10.1016/0021-8693(92)90148-F
26	J J Zhang. Connected graded Gorenstein algebras with enough normal elements. J Algebra, 1997, 189(3): 390–405 https://doi.org/10.1006/jabr.1996.6885

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