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Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions

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Abstract

In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our description of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (NCCR) of the Segre product of polynomial rings. Furthermore, applying the operation called mutation, we give other modules giving NCCRs of it.

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Acknowledgements

The authors thank Špela Špenko for valuable discussions concerning NCCRs and for explaining results in [28]. The authors also thank the anonymous referee for valuable comments. The first author is partially supported by JSPS Grant-in-Aid for Young Scientists (B) 17K14177. The second author is supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, and JSPS Grant-in-Aid for Young Scientists (B) 17K14159.

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Authors and Affiliations

  1. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, 565-0871, Japan

    Akihiro Higashitani

  2. Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba, 277-8583, Japan

    Yusuke Nakajima

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  1. Akihiro Higashitani
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  2. Yusuke Nakajima
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Correspondence to Yusuke Nakajima.

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Higashitani, A., Nakajima, Y. Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions. Sel. Math. New Ser. 25, 78 (2019). https://doi.org/10.1007/s00029-019-0523-6

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