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Generation in singularity categories of hypersurfaces of countable representation type

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Abstract

The Orlov spectrum and Rouquier dimension are invariants of a triangulated category to measure how big the category is, and they have been studied actively. In this paper, we investigate the singularity category \(\textsf {D} _{\textsf {sg} }(R)\) of a hypersurface R of countable representation type. For a thick subcategory \({\mathcal {T}}\) of \(\textsf {D} _{\textsf {sg} }(R)\) and a full subcategory \(\mathcal {X}\) of \({\mathcal {T}}\), we calculate the Rouquier dimension of \({\mathcal {T}}\) with respect to \(\mathcal {X}\). Furthermore, we prove that the level in \(\textsf {D} _{\textsf {sg} }(R)\) of the residue field of R with respect to each nonzero object is at most one.

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Acknowledgements

The authors thank the referee for reading the paper carefully and giving useful suggestions.

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Authors and Affiliations

  1. Department of Applied Science, Faculty of Science, Okayama University of Science, Ridaicho, Kitaku, Okayama, 700-0005, Japan

    Tokuji Araya

  2. Department of Liberal Studies, National Institute of Technology, Nara College, 22 Yata-cho, Yamatokoriyama, Nara, 639-1080, Japan

    Kei-ichiro Iima

  3. Department of Mathematics, Okayama University, Okayama, 700-8530, Japan

    Maiko Ono

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

    Ryo Takahashi

  5. Department of Mathematics, University of Kansas, Lawrence, KS, 66045-7523, USA

    Ryo Takahashi

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  1. Tokuji Araya
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  2. Kei-ichiro Iima
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  3. Maiko Ono
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  4. Ryo Takahashi
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Correspondence to Ryo Takahashi.

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RT was partly supported by JSPS Grants-in-Aid for Scientific Research 16K05098 and 16KK0099. MO was partly supported by Foundation of Research Fellows, The Mathematical Society of Japan.

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Araya, T., Iima, Ki., Ono, M. et al. Generation in singularity categories of hypersurfaces of countable representation type. Arch. Math. 113, 603–615 (2019). https://doi.org/10.1007/s00013-019-01374-x

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  • DOI: https://doi.org/10.1007/s00013-019-01374-x

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