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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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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DOI: https://doi.org/10.1007/s10485-022-09673-1
Keywords
- \((n+2)\)-Angulated categories
- n-Abelian categories
- Cluster tilting objects
- n-Rigid objects
- Support \(\tau _n\)-tilting pairs