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

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

ghost cluster tilting subcategories, (support) $\tau$-tilting subcategories, tilting subcategories, cluster tilting subcategories
Mathematical Subject Classification 2010
Primary: 16G20, 18E30
Secondary: 16G99, 16G70
Received: 25 February 2018
Revised: 16 January 2019
Accepted: 18 January 2019
Published: 24 October 2019
Wuzhong Yang
School of Mathematics
Northwest University
710127, Xi’an
Panyue Zhou
College of Mathematics
Hunan Institute of Science and Technology
414006, Yueyang
Bin Zhu
Department of Mathematical Sciences
Tsinghua University
100084, Beijing