Ample group action on AS-regular algebras and noncommutative graded isolated singularities
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- by Izuru Mori and Kenta Ueyama PDF
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Abstract:
In this paper, we introduce a notion of ampleness of a group action $G$ on a right noetherian graded algebra $A$, and show that it is strongly related to the notion of $A^G$ to be a graded isolated singularity introduced by the second author of this paper. Moreover, if $S$ is a noetherian AS-regular algebra and $G$ is a finite ample group acting on $S$, then we will show that $\mathcal {D}^b(\operatorname {tails} S^G)\cong \mathcal {D}^b(\operatorname {mod} \nabla S*G)$ where $\nabla S$ is the Beilinson algebra of $S$. We will also explicitly calculate a quiver $Q_{S, G}$ such that ${\mathcal D}^b(\operatorname {tails} S^G)\cong {\mathcal D}^b(\operatorname {mod} kQ_{S, G})$ when $S$ is of dimension 2.References
- Claire Amiot, Osamu Iyama, and Idun Reiten, Stable categories of Cohen-Macaulay modules and cluster categories, Amer. J. Math. 137 (2015), no. 3, 813–857. MR 3357123, DOI 10.1353/ajm.2015.0019
- 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
- K. Chan, E. Kirkman, C. Walton, and J. J. Zhang, Quantum binary polyhedral groups and their actions on quantum planes, preprint.
- Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR 935124, DOI 10.1017/CBO9780511629228
- Osamu Iyama and Ryo Takahashi, Tilting and cluster tilting for quotient singularities, Math. Ann. 356 (2013), no. 3, 1065–1105. MR 3063907, DOI 10.1007/s00208-012-0842-9
- Osamu Iyama and Yuji Yoshino, Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (2008), no. 1, 117–168. MR 2385669, DOI 10.1007/s00222-007-0096-4
- N. Jing and J. J. Zhang, Gorensteinness of invariant subrings of quantum algebras, J. Algebra 221 (1999), no. 2, 669–691. MR 1728404, DOI 10.1006/jabr.1999.8023
- Peter Jørgensen and James J. Zhang, Gourmet’s guide to Gorensteinness, Adv. Math. 151 (2000), no. 2, 313–345. MR 1758250, DOI 10.1006/aima.1999.1897
- Helmut Lenzing, Hereditary Noetherian categories with a tilting complex, Proc. Amer. Math. Soc. 125 (1997), no. 7, 1893–1901. MR 1423314, DOI 10.1090/S0002-9939-97-04122-1
- 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
- Hiroyuki Minamoto and Izuru Mori, The structure of AS-Gorenstein algebras, Adv. Math. 226 (2011), no. 5, 4061–4095. MR 2770441, DOI 10.1016/j.aim.2010.11.004
- Susan Montgomery, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Mathematics, vol. 818, Springer, Berlin, 1980. MR 590245
- Izuru Mori, McKay-type correspondence for AS-regular algebras, J. Lond. Math. Soc. (2) 88 (2013), no. 1, 97–117. MR 3092260, DOI 10.1112/jlms/jdt005
- Idun Reiten and Christine Riedtmann, Skew group algebras in the representation theory of Artin algebras, J. Algebra 92 (1985), no. 1, 224–282. MR 772481, DOI 10.1016/0021-8693(85)90156-5
- 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
- Idun Reiten and Michel Van den Bergh, Grothendieck groups and tilting objects, Algebr. Represent. Theory 4 (2001), no. 1, 1–23. Special issue dedicated to Klaus Roggenkamp on the occasion of his 60th birthday. MR 1825805, DOI 10.1023/A:1009902810813
- 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
- Kazushi Ueda, Triangulated categories of Gorenstein cyclic quotient singularities, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2745–2747. MR 2399037, DOI 10.1090/S0002-9939-08-09470-7
- 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
- Yuji Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, vol. 146, Cambridge University Press, Cambridge, 1990. MR 1079937, DOI 10.1017/CBO9780511600685
Additional Information
- Izuru Mori
- Affiliation: Department of Mathematics, Faculty of Science, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan
- Email: simouri@ipc.shizuoka.ac.jp
- Kenta Ueyama
- Affiliation: Department of Mathematics, Graduate School of Science, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan
- Address at time of publication: Department of Mathematics, Faculty of Education, Hirosaki University, 1 Bunkyocho, Hirosaki, Aomori 036-8560, Japan
- Email: skueyam@ipc.shizuoka.ac.jp, k-ueyama@hirosaki-u.ac.jp
- Received by editor(s): October 27, 2013
- Received by editor(s) in revised form: June 18, 2014, June 20, 2014, and October 3, 2014
- Published electronically: December 9, 2015
- Additional Notes: The first author was supported by Grant-in-Aid for Scientific Research (C) 25400037. The second author was supported by JSPS Fellowships for Young Scientists No. 23-2233.
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 7359-7383
- MSC (2010): Primary 14A22, 16W22, 16S35, 18E30
- DOI: https://doi.org/10.1090/tran/6580
- MathSciNet review: 3471094