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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The authors thank the referee for reading the paper carefully and giving useful suggestions.
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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
Keywords
- Hypersurface
- Countable representation type
- Singularity category
- Cohen–Macaulay module
- Level
- Rouquier dimension