#### Vol. 301, No. 2, 2019

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Triangulated categories with cluster tilting subcategories

### Wuzhong Yang, Panyue Zhou and Bin Zhu

Vol. 301 (2019), No. 2, 703–740
DOI: 10.2140/pjm.2019.301.703
##### Abstract

For a triangulated category $\mathsc{C}$ with a cluster tilting subcategory $\mathsc{T}$ which contains infinitely many indecomposable objects, the notion of weak $\mathsc{T}\left[1\right]$-cluster tilting subcategories of $\mathsc{C}$ is introduced. We use them to study the $\tau$-tilting theory in the module category over $\mathsc{T}$. Inspired by the work of Iyama, Jørgensen and Yang (2014), we introduce the notion of $\tau$-tilting subcategories of mod $\mathsc{T}$, and show that there exists a bijection between weak $\mathsc{T}\left[1\right]$-cluster tilting subcategories of $\mathsc{C}$ and support $\tau$-tilting subcategories of mod $\mathsc{T}$. Moreover, we describe the subcategories of mod $\mathsc{T}$ which correspond to cluster tilting subcategories of $\mathsc{C}$. This generalizes and improves results by Adachi, Iyama and Reiten (2014), Beligiannis (2013), and Yang and Zhu (2019).

 Dedicated to Professor Idun Reiten on the occasion of her 76th birthday
##### Keywords
ghost cluster tilting subcategories, (support) $\tau$-tilting subcategories, tilting subcategories, cluster tilting subcategories
##### Mathematical Subject Classification 2010
Primary: 16G20, 18E30
Secondary: 16G99, 16G70
##### Milestones
Received: 25 February 2018
Revised: 16 January 2019
Accepted: 18 January 2019
Published: 24 October 2019
##### Authors
 Wuzhong Yang School of Mathematics Northwest University 710127, Xi’an Shaanxi China Panyue Zhou College of Mathematics Hunan Institute of Science and Technology 414006, Yueyang Hunan China Bin Zhu Department of Mathematical Sciences Tsinghua University 100084, Beijing China