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Relative singularity categories and singular equivalences

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Abstract

Let R be a right noetherian ring. We introduce the concept of relative singularity category \(\Delta _{\mathcal {X} }(R)\) of R with respect to a contravariantly finite subcategory \(\mathcal {X} \) of \({\text {{mod{-}}}}R.\) Along with some finiteness conditions on \(\mathcal {X} \), we prove that \(\Delta _{\mathcal {X} }(R)\) is triangle equivalent to a subcategory of the homotopy category \(\mathbb {K} _\mathrm{{ac}}(\mathcal {X} )\) of exact complexes over \(\mathcal {X} \). As an application, a new description of the classical singularity category \(\mathbb {D} _\mathrm{{sg}}(R)\) is given. The relative singularity categories are applied to lift a stable equivalence between two suitable subcategories of the module categories of two given right noetherian rings to get a singular equivalence between the rings. In different types of rings, including path rings, triangular matrix rings, trivial extension rings and tensor rings, we provide some consequences for their singularity categories.

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Acknowledgements

This research is supported by a grant from University of Isfahan. The author would like to thank his wife for her support and patience during this work.

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Authors and Affiliations

  1. Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, 81746-73441, Iran

    Rasool Hafezi

  2. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Rasool Hafezi

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  1. Rasool Hafezi
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Correspondence to Rasool Hafezi.

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Communicated by Emily Riehl.

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Hafezi, R. Relative singularity categories and singular equivalences. J. Homotopy Relat. Struct. 16, 487–516 (2021). https://doi.org/10.1007/s40062-021-00289-1

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  • DOI: https://doi.org/10.1007/s40062-021-00289-1

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