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Triangulated categories with cluster tilting subcategories. (English) Zbl 1454.16020

Summary: For a triangulated category \(\mathscr{C}\) with a cluster tilting subcategory \(\mathcal{T}\) which contains infinitely many indecomposable objects, the notion of weak \(\mathcal{T}[1]\)-cluster tilting subcategories of \(\mathscr{C}\) is introduced. We use them to study the \(\tau \)-tilting theory in the module category over \(\mathcal{T}\). Inspired by the work of O. Iyama et al. [Algebra Number Theory 8, No. 10, 2413–2431 (2014; Zbl 1305.18048)], we introduce the notion of \(\tau \)-tilting subcategories of mod \(\mathcal{T}\), and show that there exists a bijection between weak \(\mathcal{T}[1]\)-cluster tilting subcategories of \(\mathscr{C}\) and support \(\tau \)-tilting subcategories of mod \(\mathcal{T}\). Moreover, we describe the subcategories of \(\bmod \mathcal{T}\) which correspond to cluster tilting subcategories of \(\mathscr{C}\). This generalizes and improves results by T. Adachi et al. [Compos. Math. 150, No. 3, 415–452 (2014; Zbl 1330.16004)], A. Beligiannis [Math. Z. 274, No. 3–4, 841–883 (2013; Zbl 1279.18010)], and W. Yang and B. Zhu [Trans. Am. Math. Soc. 371, No. 1, 387–412 (2019; Zbl 1456.16011)].

MSC:

16G20 Representations of quivers and partially ordered sets
18M05 Monoidal categories, symmetric monoidal categories
16G99 Representation theory of associative rings and algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
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