Cluster-tilting subcategories in extriangulated categories. (English) Zbl 1408.18029
Summary: Let \((\mathcal{C},\mathbb{E},\mathfrak{s})\) be an extriangulated category. We show that certain quotient categories of extriangulated categories are equivalent to module categories by some restriction of functor \(\mathbb{E}\), and in some cases, they are abelian. This result can be regarded as a simultaneous generalization of S. Koenig and B. Zhu [Math. Z. 258, No. 1, 143–160 (2008; Zbl 1133.18005)] and L. Demonet and Y. Liu [J. Pure Appl. Algebra 217, No. 12, 2282–2297 (2013; Zbl 1408.18021)]. In addition, we introduce the notion of maximal rigid subcategories in extriangulated categories. Cluster tilting subcategories are obviously strongly functorially finite maximal rigid subcategories, we prove that the converse is true if the 2-Calabi-Yau extriangulated categories admit a cluster tilting subcategories, which generalizes a result of A. B. Buan et al. [Compos. Math. 145, No. 4, 1035–1079 (2009; Zbl 1181.18006)] and Y. Zhou and B. Zhu [J. Algebra 348, No. 1, 49–60 (2011; Zbl 1248.16013)].
MSC:
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 18E10 | Abelian categories, Grothendieck categories |
Keywords:
extriangulated categories; cluster-tilting subcategories; rigid subcategories; maximal rigid subcategories; quotient categoriesReferences:
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