#### 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