Lifting to two-term relative maximal rigid subcategories in triangulated categories
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Lifting to two-term relative maximal rigid subcategories in triangulated categories PANYUE ZHOU College of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, Hunan, People’s Republic of China E-mail: [email protected]
MS received 5 August 2019; revised 25 November 2019; accepted 24 February 2020 Abstract. Let C be a triangulated category with shift functor [1] and R a contravariantly rigid subcategory of C . We show that a tilting subcategory of mod R lifts to a two-term maximal R[1]-rigid subcategory of C . As an application, our result generalizes a result by Xie and Liu (Proc. Amer. Math. Soc. 141(10) (2013) 3361–3367) for maximal rigid objects and a result by Fu and Liu (Comm. Algebra 37(7) (2009) 2410–2418) for cluster tilting objects. Keywords.
Tilting subcategories; maximal rigid objects; cluster tilting objects.
Mathematics Subject Classification.
18E30, 16D90.
1. Introduction Let k be an algebraically closed field and be a finite-dimensional algebra. Let mod be the category of finite-dimensional right -modules. For an -module T , let add T denote the full subcategory of mod with objects all direct summands of direct sums of copies of T . Then T is called a tilting module in mod if • pd T ≤ 1; • Ext1 (T, T ) = 0; • there exists an exact sequence 0 → → T 0 → T 1 → 0, with T 0 , T 1 in add T . Fu and Liu proved the following result. Theorem 1.1 [8, Theorem 3.3]. Let C be a 2-Calabi–Yau triangulated category with a cluster tilting object T and let be the endomorphism algebra of T . If M is a tilting module over , then M lifts to a cluster tilting object in C . Note that each cluster tilting object is a maximal rigid in a 2-Calabi–Yau triangulated category. But the converse is not true, in general. The counter-examples can be found in tube categories [7] or in categories of Cohen–Macaulay modules over an isolated hypersurface singularity [4]. Xie and Liu [14] showed that tilting modules over such algebras lift to maximal rigid objects in the corresponding 2-Calabi–Yau triangulated category. Namely, they proved the following. © Indian Academy of Sciences 0123456789().: V,-vol
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Theorem 1.2 [14, Theorem 2.2]. Let C be a 2-Calabi–Yau triangulated category with a maximal rigid object T and let be the endomorphism algebra of T . If M is a tilting module over , then M lifts to a maximal rigid object in C . The notion of tilting module was generalized to abelian categories by Beligiannis [3]. We recall this definition here: Let A be an abelian category with enough projective objects. A contravariantly finite subcategory M of A is called a tilting subcategory if • Ext 1A (M, M) = 0. • pd A M ≤ 1, for any M ∈ M. • for any projective object P in A, there exists a short exact sequence 0 −→ P −→ M0 −→ M1 −→ 0 where M0 , M1 ∈ M. In this note, we give a similar result of Fu and Liu [8] and Xie and Liu [14]. Namely, we prove the following. Theorem 1.3 (see Theorem 3.3 for more details). Let C be a triangulated
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