Abstract
A triangulated category \(\mathcal {T}\) whose suspension functor \(\Sigma \) satisfies \(\Sigma ^m \cong {{\,\mathrm{\textsf{Id}}\,}}_{\mathcal {T}}\) as additive functors is called an m-periodic triangulated category. Such a category does not have a tilting object by the periodicity. In this paper, we introduce the notion of an m-periodic tilting object in an m-periodic triangulated category, which is a periodic analogue of a tilting object in a triangulated category, and prove that an m-periodic triangulated category having an m-periodic tilting object is triangle equivalent to the m-periodic derived category of an algebra under some homological assumptions. As an application, we construct a triangle equivalence between the stable category of finitely generated modules over a self-injective algebra and the m-periodic derived category of a hereditary algebra.
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This question is posed by Haruhisa Enomoto. See [1, Remark 3.20] for the application.
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Acknowledgements
The author would like to thank his supervisor Shintarou Yanagida for valuable suggestions and discussion. He is also very grateful to Norihiro Hanihara. The author learned tilting theory for triangulated categories and DG categories from him, and discussions with him played a crucial role in the early stages of this project. He would like to thank Osamu Iyama, Ryo Takahashi and Haruhisa Enomoto for insightful comments and suggestions. He would also like to thank the anonymous referees for their careful reading and various helpful suggestions. This work is supported by JSPS KAKENHI Grant Number JP21J21767.
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Saito, S. Tilting theory for periodic triangulated categories. Math. Z. 304, 47 (2023). https://doi.org/10.1007/s00209-023-03304-8
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DOI: https://doi.org/10.1007/s00209-023-03304-8