We study the properties of torsion pairs in triangulated category by introducing the notions of d-Ext-projectivity and d-Ext-injectivity. In terms of -mutation of torsion pairs, we investigate the properties of torsio...We study the properties of torsion pairs in triangulated category by introducing the notions of d-Ext-projectivity and d-Ext-injectivity. In terms of -mutation of torsion pairs, we investigate the properties of torsion pairs in triangulated category under some conditions on subcategories and in .展开更多
In this paper, we consider two kinds of 2-Calabi-Yau triangulated categories with finitely many indecomposable objects up to isomorphisms, called An,t =D^b（KA（2t＋1）（n＋1）-3）/τ^t（n＋1）-1[1], where n,t ≥ 1, a...In this paper, we consider two kinds of 2-Calabi-Yau triangulated categories with finitely many indecomposable objects up to isomorphisms, called An,t =D^b（KA（2t＋1）（n＋1）-3）/τ^t（n＋1）-1[1], where n,t ≥ 1, and Dn,t = Db（KD2t（n＋1））/τ^（n＋1）φ^n, where n,t ≥ 1, and φ is induced by an automorphism of D2t（n＋1） of order 2. Except the categories An,1, they all contain non-zero maximal rigid objects which are not cluster tilting. An,1 contain cluster tilting objects. We define the cluster complex of An,t （resp. Dn,t） by using the geometric description of cluster categories of type A （resp. type D）. We show that there is an isomorphism from the cluster complex of An,t （resp. Dn,t） to the cluster complex of root system of type Bn. In particular, the maximal rigid objects are isomorphic to clusters. This yields a result proved recently by Buan-Palu-Reiten： Let RAn,t, resp. RDn,t, be the full subcategory of An,t, resp. Dn,t, generated by the rigid objects. Then RAn,t≈RAn,1 and TDn,t≈TAn,1 as additive categories, for all t 〉 1.展开更多
Let C be a triangulated category.We first introduce the notion of balanced pairs in C,and then establish the bijective correspondence between balanced pairs and proper classesξwith enoughξ-projectives andξ-injectiv...Let C be a triangulated category.We first introduce the notion of balanced pairs in C,and then establish the bijective correspondence between balanced pairs and proper classesξwith enoughξ-projectives andξ-injectives.Assume thatξ:=ξX=ξ^(Y) is the proper class induced by a balanced pair(X,Y).We prove that(C,Eξ,sξ)is an extriangulated category.Moreover,it is proved that(C,Eξ,sξ)is a triangulated category if and only if X=Y=0,and that(C,Eξ,sξ)is an exact category if and only if X=Y=C.As an application,we produce a large variety of examples of extriangulated categories which are neither exact nor triangulated.展开更多
In this paper,we prove that if a triangulated category D admits a recollement relative to triangulated categories D' and D″,then the abelian category D/T admits a recollement relative to abelian categories D'...In this paper,we prove that if a triangulated category D admits a recollement relative to triangulated categories D' and D″,then the abelian category D/T admits a recollement relative to abelian categories D'/i(T) and D″/j(T) where T is a cluster tilting subcategory of D and satisfies i i (T) T,j j (T) T.展开更多
We introduce and study （pre）resolving subcategories of a triangulated category and the homological dimension relative to these subcategories. We apply the obtained properties to relative Gorenstein categories.
Let C be a triangulated category with a proper class g of triangles. We prove that there exists an Avramov-Martsinkovsky type exact sequence in g, which connects ε-cohomology, ε-Tate cohomology and ε-Corenstein coh...Let C be a triangulated category with a proper class g of triangles. We prove that there exists an Avramov-Martsinkovsky type exact sequence in g, which connects ε-cohomology, ε-Tate cohomology and ε-Corenstein cohomology.展开更多
Let A be a finite dimensional algebra over a field k. We consider a subfunc- tor F of Ext1A（-, -）, which has enough projectives and injectives such that P（F） is of finite type, where P（F） denotes the set of F-pr...Let A be a finite dimensional algebra over a field k. We consider a subfunc- tor F of Ext1A（-, -）, which has enough projectives and injectives such that P（F） is of finite type, where P（F） denotes the set of F-projectives. One can get the relative derived category Db（A） of A-rood. For an F-self-orthogonal module TF, we discuss the relation between the relative quotient triangulated category Db（A）/Kb（addTF） and the relative stable category of the Frobenius category of TF-Cohen-Macaulay modules. In particular, for an F-Gorenstein algebra A and an F-tilting A-module TF, we get a triangle equiva- lence between DbF（A）/Kb（add TF） and the relative stable category of TF-Cohen-Macaulay modules. This gives the relative version of a result of Chen and Zhang.展开更多
Given an odd-periodic algebraic triangulated category, we compare Bridgeland's Hall algebra in the sense of Bridgeland(2013) and Gorsky(2014), and the derived Hall algebra in the sense of Ten(2006), Xiao and Xu(20...Given an odd-periodic algebraic triangulated category, we compare Bridgeland's Hall algebra in the sense of Bridgeland(2013) and Gorsky(2014), and the derived Hall algebra in the sense of Ten(2006), Xiao and Xu(2008) and Xu and Chen(2013), and show that the former one is the twisted form of the tensor product of the latter one and a suitable group algebra.展开更多
文摘We study the properties of torsion pairs in triangulated category by introducing the notions of d-Ext-projectivity and d-Ext-injectivity. In terms of -mutation of torsion pairs, we investigate the properties of torsion pairs in triangulated category under some conditions on subcategories and in .
基金Supported by the NSF of China(Grant No.11671221)
文摘In this paper, we consider two kinds of 2-Calabi-Yau triangulated categories with finitely many indecomposable objects up to isomorphisms, called An,t =D^b（KA（2t＋1）（n＋1）-3）/τ^t（n＋1）-1[1], where n,t ≥ 1, and Dn,t = Db（KD2t（n＋1））/τ^（n＋1）φ^n, where n,t ≥ 1, and φ is induced by an automorphism of D2t（n＋1） of order 2. Except the categories An,1, they all contain non-zero maximal rigid objects which are not cluster tilting. An,1 contain cluster tilting objects. We define the cluster complex of An,t （resp. Dn,t） by using the geometric description of cluster categories of type A （resp. type D）. We show that there is an isomorphism from the cluster complex of An,t （resp. Dn,t） to the cluster complex of root system of type Bn. In particular, the maximal rigid objects are isomorphic to clusters. This yields a result proved recently by Buan-Palu-Reiten： Let RAn,t, resp. RDn,t, be the full subcategory of An,t, resp. Dn,t, generated by the rigid objects. Then RAn,t≈RAn,1 and TDn,t≈TAn,1 as additive categories, for all t 〉 1.
基金Xianhui Fu was supported by YDZJ202101ZYTS168 and the NSF of China(12071064)Jiangsheng Hu was supported by the NSF of China(12171206)+2 种基金the Natural Science Foundation of Jiangsu Province(BK20211358)Haiyan Zhu was supported by Zhejiang Provincial Natural Science Foundation of China(LY18A010032)the NSF of China(12271481).
文摘Let C be a triangulated category.We first introduce the notion of balanced pairs in C,and then establish the bijective correspondence between balanced pairs and proper classesξwith enoughξ-projectives andξ-injectives.Assume thatξ:=ξX=ξ^(Y) is the proper class induced by a balanced pair(X,Y).We prove that(C,Eξ,sξ)is an extriangulated category.Moreover,it is proved that(C,Eξ,sξ)is a triangulated category if and only if X=Y=0,and that(C,Eξ,sξ)is an exact category if and only if X=Y=C.As an application,we produce a large variety of examples of extriangulated categories which are neither exact nor triangulated.
基金supported by National Natural Science Foundation of China (Grant No.10931006)the PhD Programs Foundation of Ministry of Education of China (Grant No.20060384002)the Scientific Research Foundation of Huaqiao University (Grant No.08BS506)
文摘In this paper,we prove that if a triangulated category D admits a recollement relative to triangulated categories D' and D″,then the abelian category D/T admits a recollement relative to abelian categories D'/i(T) and D″/j(T) where T is a cluster tilting subcategory of D and satisfies i i (T) T,j j (T) T.
基金Supported by the National Natural Science Foundation of China(Grant No.11571164)a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions,Postgraduate Research and Practice Innovation Program of Jiangsu Province(Grant No.KYZZ16 0034)Nanjing University Innovation and Creative Program for PhD candidate(Grant No.2016011)
文摘We introduce and study （pre）resolving subcategories of a triangulated category and the homological dimension relative to these subcategories. We apply the obtained properties to relative Gorenstein categories.
基金Supported by National Natural Science Foundation of China(Grant Nos.11401476,11361052,11261050)
文摘Let C be a triangulated category with a proper class g of triangles. We prove that there exists an Avramov-Martsinkovsky type exact sequence in g, which connects ε-cohomology, ε-Tate cohomology and ε-Corenstein cohomology.
文摘Let A be a finite dimensional algebra over a field k. We consider a subfunc- tor F of Ext1A（-, -）, which has enough projectives and injectives such that P（F） is of finite type, where P（F） denotes the set of F-projectives. One can get the relative derived category Db（A） of A-rood. For an F-self-orthogonal module TF, we discuss the relation between the relative quotient triangulated category Db（A）/Kb（addTF） and the relative stable category of the Frobenius category of TF-Cohen-Macaulay modules. In particular, for an F-Gorenstein algebra A and an F-tilting A-module TF, we get a triangle equiva- lence between DbF（A）/Kb（add TF） and the relative stable category of TF-Cohen-Macaulay modules. This gives the relative version of a result of Chen and Zhang.
基金supported by National Natural Science Foundation of China(Grant Nos.11301533 and 11471177)
文摘Given an odd-periodic algebraic triangulated category, we compare Bridgeland's Hall algebra in the sense of Bridgeland(2013) and Gorsky(2014), and the derived Hall algebra in the sense of Ten(2006), Xiao and Xu(2008) and Xu and Chen(2013), and show that the former one is the twisted form of the tensor product of the latter one and a suitable group algebra.
