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

Abstract

We study properties of graded maximal Cohen-Macaulay modules over an \mathbb{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

References

[1] Ajitabh K, Smith S P, Zhang J J. Auslander-Gorenstein rings. Comm Algebra, 1998, 26: 2159–2180

[2] Brown K, Qin X-S, Wang Y-H, Zhang J J. Pretzelization (in preparation)

[3] Chan D, Wu Q-S, Zhang J J. Pre-balanced dualizing complexes. Israel J Math, 2002, 132: 285–314

[4] Chan K, Kirkman E, Walton C, Zhang J J. Quantum binary polyhedral groups and their actions on quantum planes. J Reine Angew Math, 2016, 719: 211–252

[5] Chan K, Kirkman E, Walton C, Zhang J J. McKay correspondence for semisimple Hopf actions on regular graded algebras, I. J Algebra, 2018, 508: 512–538

[6] Chan K, Kirkman E, Walton C, Zhang J J. McKay correspondence for semisimple Hopf actions on regular graded algebras, II. J Noncommut Geom, 2019, 13(1): 87–114

[7] Dokuchaev M, Gubareni N M, Futorny V M, Khibina M A, Kirichenko V V. Dynkin diagrams and spectra of graphs. São Paulo J Math Sci, 2013, 7(1): 83–104

[8] Happel D, Preiser U, Ringel C M. Binary polyhedral groups and Euclidean diagrams. Manuscripta Math, 1980, 31(13): 317–329

[9] Herzog J. Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen-Macaulay-Moduln. Math Ann, 1978, 233(1): 21–34

[10] Jørgensen P. Finite Cohen-Macaulay type and smooth non-commutative schemes. Canad J Math, 2008, 60(2): 379–390

[11] Krause G R, Lenagan T H. Growth of Algebras and Gelfand-Kirillov Dimension. Res Notes in Math, Vol 116. Boston: Pitman Adv Publ Program, 1985

[12] Lam T Y. A First Course in Noncommutative Rings. 2nd ed. Grad Texts in Math, Vol 131. New York: Springer-Verlag, 2001

[13] Levasseur T. Some properties of non-commutative regular graded rings. Glasg J Math, 1992, 34: 277–300

[14] Martnez-Villa R. Introduction to Koszul algebras. Rev Un Mat Argentina, 2007, 48(2): 67–95

[15] Qin X-S, Wang Y-H, Zhang J J. Noncommutative quasi-resolutions. J Algebra, 2019, 536: 102–148

[16] Reyes M, Rogalski D. A twisted Calabi-Yau toolkit. arXiv: 1807.10249

[17] Reyes M, Rogalski D. Growth of graded twisted Calabi-Yau algebras. arXiv: 1808.10538

[18] Reyes M, Rogalski D, Zhang J J. Skew Calabi-Yau algebras and homological identities. Adv Math, 2014, 264: 308–354

[19] Smith J H. 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] Ueyama K. Graded maximal Cohen-Macaulay modules over noncommutative graded Gorenstein isolated singularities. J Algebra, 2013, 383: 85–103

[21] Van den Bergh M. Existence theorems for dualizing complexes over non-commutative graded and filtered rings. J Algebra, 1997, 195(2): 662–679

[22] Van den Bergh M. Three-dimensional fiops and noncommutative rings. Duke Math J, 2004, 122(3): 423–455

[23] Van den Bergh M. Non-commutative crepant resolutions. In: Laudal O A, Piene R, eds. The Legacy of Niels Henrik Abel. Berlin: Springer, 2004, 749–770

[24] Weispfenning S. Properties of the fixed ring of a preprojective algebra. J Algebra, 2019, 517: 276–319

[25] Yekutieli A. Dualizing complexes over noncommutative graded algebras. J Algebra, 1992, 153(1): 41{84

[26] Zhang J J. Connected graded Gorenstein algebras with enough normal elements. J Algebra, 1997, 189(3): 390–405

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