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Relative Oppermann–Thomas Cluster Tilting Objects in \((n+2)\)-Angulated Categories

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Abstract

Let \(\mathcal {T}\) be a k-linear Hom-finite \((n+2)\)-angulated category with n-suspension functor \(\Sigma ^n\), split idempotents, and Serre functor \(\mathbb {S}\). Let T be an Oppermann–Thomas cluster tilting object in \(\mathcal {T}\) with endomorphism algebra \(\Gamma = \mathrm {End}_\mathcal {T}(T)\). We introduce the notions of relative Oppermann–Thomas cluster tilting objects and support \(\tau _n\)-tilting pairs, and show that there is an bijection between the set of isomorphism classes of basic relative Oppermann–Thomas cluster tilting objects in \(\mathcal {T}\) and the set of isomorphism classes of basic support \(\tau _n\)-tilting pairs in an n-cluster tilting subcategory of \(\mathrm {mod}~\Gamma \). As applications, we recover the Yang–Zhu bijection (Trans Am Math Soc 371:387–412, 2019) and Adachi–Iyama–Reiten bijection (Compos Math 150:415–452, 2014), and we give a natural partial order for relative Oppermann–Thomas cluster tilting objects.

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Acknowledgements

The authors wish to express their sincere thanks to the anonymous referee for her/his carefully reading and helpful comments. This research was partially supported by National Natural Science Foundation of China (11971388, 11901463).

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Authors and Affiliations

  1. College of Computer Science and Engineering, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    Zongyang Xie

  2. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    Zhongkui Liu & Xiaoyan Yang

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  1. Zongyang Xie
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  2. Zhongkui Liu
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  3. Xiaoyan Yang
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Correspondence to Zongyang Xie.

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Communicated by Wendy Lowen.

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Xie, Z., Liu, Z. & Yang, X. Relative Oppermann–Thomas Cluster Tilting Objects in \((n+2)\)-Angulated Categories. Appl Categor Struct 30, 805–823 (2022). https://doi.org/10.1007/s10485-022-09673-1

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