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Ample group action on AS-regular algebras and noncommutative graded isolated singularities


Authors: Izuru Mori and Kenta Ueyama
Journal: Trans. Amer. Math. Soc. 368 (2016), 7359-7383
MSC (2010): Primary 14A22, 16W22, 16S35, 18E30
DOI: https://doi.org/10.1090/tran/6580
Published electronically: December 9, 2015
MathSciNet review: 3471094
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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.


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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

DOI: https://doi.org/10.1090/tran/6580
Keywords: AS-regular algebra, group action, graded isolated singularity, skew group algebra, derived equivalence
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.
Article copyright: © Copyright 2015 American Mathematical Society