2017/11/12 by Wuzhong Yang, Panyue Zhou, Bin Zhu · 6 citations
Mathematics · #Algebraic structures and combinatorial models #Bijection #Cluster (spacecraft) #Finite Group Theory Research #Generalization #Homotopy and Cohomology in Algebraic Topology #Subcategory #Triangulated category #math.RT #msc:16G20 #msc:16G70 #msc:18E30
paper · pdf · doi:10.2140/pjm.2019.301.703
published in Pacific Journal of Mathematics 301(2), 703-740 (Mathematical Sciences Publishers) · All comments welcome
arxiv created 2017/11/12 · openalex created_date 2017/12/04 · openalex publication_date 2019/10/24 · arxiv updated 2019/10/30 · openalex updated_date 2026/08/05
Let \C be a triangulated category with a cluster tilting subcategory \T. We introduce the notion of \T[1]-cluster tilting subcategories (also called ghost cluster tilting subcategories) of \C, which are a generalization of cluster tilting subcategories. We first develop a basic theory on ghost cluster tilting subcategories. Secondly, we study links between ghost cluster tilting theory and τ-tilting theory: Inspired by the work of Iyama, J\orgensen and Yang \citeijy, we introduce the notion of τ-tilting subcategories and tilting subcategories of \mod\T. We 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 figure out the subcategories of \mod\T which correspond to cluster tilting subcategories of \C. This generalizes and improves several results by Adachi-Iyama-Reiten \citeAIR, Beligiannis \citeBe2, and Yang-Zhu \citeYZ. Finally, we prove that the definition of ghost cluster tilting objects is equivalent to the definition of relative cluster tilting objects introduced by the first and the third author in \citeYZ.