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Tilting and cluster tilting for quotient singularities

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Abstract

We shall show that the stable categories of graded Cohen–Macaulay modules over quotient singularities have tilting objects. In particular, these categories are triangle equivalent to derived categories of finite dimensional algebras. Our method is based on higher dimensional Auslander–Reiten theory, which gives cluster tilting objects in the stable categories of (ungraded) Cohen–Macaulay modules.

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Acknowledgments

The authors express their gratitude to R.-O. Buchweitz, who suggested constructing tilting objects by using syzygies. This led them to our main Theorem 2.7. They are also grateful to Y. Yoshino and H. Krause for valuable suggestions on skew group algebras and the tilting theorem for algebraic triangulated categories. Results in this paper were presented in Nagoya (June 2008), Banff (September 2008), Sherbrooke (October 2008), Lincoln (November 2008), Kyoto (November 2008) and Osaka (November 2009). The authors thank the organizers of these meetings and seminars. Finally, the authors thank the referee for his/her careful reading.

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Author notes

  1. Ryo Takahashi

    Present address: Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan

Authors and Affiliations

  1. Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan

    Osamu Iyama

  2. Department of Mathematical Sciences, Faculty of Science, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano, 390-8621, Japan

    Ryo Takahashi

  3. Department of Mathematics, University of Nebraska, Lincoln, NE, 68588-0130, USA

    Ryo Takahashi

Authors
  1. Osamu Iyama
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  2. Ryo Takahashi
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Correspondence to Ryo Takahashi.

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Dedicated to Professor Shiro Goto on the occasion of his 65th birthday.

O. Iyama was partially supported by JSPS Grant-in-Aid for Scientific Research 21740010 and 21340003. R. Takahashi was partially supported by JSPS Grant-in-Aid for Young Scientists (B) 22740008 and by JSPS Postdoctoral Fellowships for Research Abroad.

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Iyama, O., Takahashi, R. Tilting and cluster tilting for quotient singularities. Math. Ann. 356, 1065–1105 (2013). https://doi.org/10.1007/s00208-012-0842-9

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  • DOI: https://doi.org/10.1007/s00208-012-0842-9

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