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