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Relative Rigid Subcategories and τ-Tilting Theory

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Abstract

Let be an extriangulated category with enough projectives \(\mathcal P\) and enough injectives \(\mathcal I\), and let be a contravariantly finite rigid subcategory of which contains \(\mathcal P\). We have an abelian quotient category \(\\mathcal{H} / \mathcal{R} \subseteq / \mathcal{B} / \mathcal{R} \) which is equivalent to \(\mod (\mathcal{R} / \mathcal{P})\). In this article, we find a one-to-one correspondence between support τ-tilting (resp. τ-rigid) subcategories of / and maximal relative rigid (resp. relative rigid) subcategories of , and show that support tilting subcategories in / is a special kind of support τ-tilting subcategories. We also study the relation between tilting subcategories of / and cluster tilting subcategories of when is cluster tilting.

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Acknowledgements

The authors would like to thank the referee for reading the paper carefully and for many suggestions on mathematics and English expressions.

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Authors and Affiliations

  1. School of Mathematics, Southwest Jiaotong University, 610031, Chengdu, Sichuan, People’s Republic of China

    Yu Liu

  2. College of Mathematics, Hunan Institute of Science and Technology, 414006, Yueyang, Hunan, People’s Republic of China

    Panyue Zhou

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  1. Yu Liu
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  2. Panyue Zhou
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Correspondence to Panyue Zhou.

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Presented by: Henning Krause

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The authors would like to thank Professor Dong Yang and Professor Bin Zhu for helpful discussions.

Yu Liu was supported by the National Natural Science Foundation of China (Grant No. 11901479). Panyue Zhou was supported by the National Natural Science Foundation of China (Grant No. 11901190) and by the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 19B239).

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Liu, Y., Zhou, P. Relative Rigid Subcategories and τ-Tilting Theory. Algebr Represent Theor 25, 1699–1722 (2022). https://doi.org/10.1007/s10468-021-10082-6

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