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
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
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 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
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
We revisit Auslander-Buchweitz approximation theory and find some relations with cotorsion pairs and model category structures.From the notion of weak-cogenerators,we introduce the concept of Frobenius pair(X,ω)in a ...We revisit Auslander-Buchweitz approximation theory and find some relations with cotorsion pairs and model category structures.From the notion of weak-cogenerators,we introduce the concept of Frobenius pair(X,ω)in a triangulated category T.We show how to construct from a Frobenius pair(X,ω)a triangulated model structure on X^(∧).展开更多
We consider the existence of cluster-tilting objects in a d-cluster category such that its endomorphism algebra is self-injective,and also the properties for cluster-tilting objects in d-cluster categories.We get the ...We consider the existence of cluster-tilting objects in a d-cluster category such that its endomorphism algebra is self-injective,and also the properties for cluster-tilting objects in d-cluster categories.We get the following results:(1)When d>1,any almost complete cluster-tilting object in d-cluster category has only one complement.(2)Cluster-tilting objects in d-cluster categories are induced by tilting modules over some hereditary algebras.We also give a condition for a tilting module to induce a cluster-tilting object in a d-cluster category.(3)A 3-cluster category of finite type admits a cluster-tilting object if and only if its type is A1,A3,D2n-1(n>2).(4)The(2m+1)-cluster category of type D2n-1 admits a cluster-tilting object such that its endomorphism algebra is self-injective,and its stable category is equivalent to the(4m+2)-cluster category of type A4mn-4m+2n-1.展开更多
Let C be a triangulated category which has Auslander-Reiten triangles, and Ra functorially finite rigid subcategory of C. It is well known that there exist Auslander-Reiten sequences in rood R. In this paper, we give ...Let C be a triangulated category which has Auslander-Reiten triangles, and Ra functorially finite rigid subcategory of C. It is well known that there exist Auslander-Reiten sequences in rood R. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod R and the Auslander-Reiten functors, triangles in C, respectively. Furthermore, if T is a cluster-tilting subcategory of C and mod T- is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod T- corresponding to the ones in C. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Probenius, then its stable category is (2d-1)-Calabi-Yau. This result was first proved by Keller and Reiten in the case d= 2, and then by Dugas in the general case, using different methods. 2010 Mathematics Subject Classification: 16G20, 16G70展开更多
Let R be any ring. We motivate the study of a class of chain complexes of injective R-modules that we call A C-injective complexes, showing that K(AC-Inj), the chain homotopy category of all AC-injective complexes, ...Let R be any ring. We motivate the study of a class of chain complexes of injective R-modules that we call A C-injective complexes, showing that K(AC-Inj), the chain homotopy category of all AC-injective complexes, is always a compactly generated triangulated category. In general, all DG- injective complexes are AC-injective and in fact there is a recollement linking K(AC-Inj) to the usual derived category D(R). This is based on the author's recent work inspired by work of Krause and Stovicek. Our focus here is on giving straightforward proofs that our categories are compactly generated.展开更多
Let T be a triangulated category and ζ a proper class of triangles. Some basics properties and diagram lemmas are proved directly from the definition of ζ.
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 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.
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.
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 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.
文摘We revisit Auslander-Buchweitz approximation theory and find some relations with cotorsion pairs and model category structures.From the notion of weak-cogenerators,we introduce the concept of Frobenius pair(X,ω)in a triangulated category T.We show how to construct from a Frobenius pair(X,ω)a triangulated model structure on X^(∧).
文摘We consider the existence of cluster-tilting objects in a d-cluster category such that its endomorphism algebra is self-injective,and also the properties for cluster-tilting objects in d-cluster categories.We get the following results:(1)When d>1,any almost complete cluster-tilting object in d-cluster category has only one complement.(2)Cluster-tilting objects in d-cluster categories are induced by tilting modules over some hereditary algebras.We also give a condition for a tilting module to induce a cluster-tilting object in a d-cluster category.(3)A 3-cluster category of finite type admits a cluster-tilting object if and only if its type is A1,A3,D2n-1(n>2).(4)The(2m+1)-cluster category of type D2n-1 admits a cluster-tilting object such that its endomorphism algebra is self-injective,and its stable category is equivalent to the(4m+2)-cluster category of type A4mn-4m+2n-1.
文摘Let C be a triangulated category which has Auslander-Reiten triangles, and Ra functorially finite rigid subcategory of C. It is well known that there exist Auslander-Reiten sequences in rood R. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod R and the Auslander-Reiten functors, triangles in C, respectively. Furthermore, if T is a cluster-tilting subcategory of C and mod T- is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod T- corresponding to the ones in C. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Probenius, then its stable category is (2d-1)-Calabi-Yau. This result was first proved by Keller and Reiten in the case d= 2, and then by Dugas in the general case, using different methods. 2010 Mathematics Subject Classification: 16G20, 16G70
文摘Let R be any ring. We motivate the study of a class of chain complexes of injective R-modules that we call A C-injective complexes, showing that K(AC-Inj), the chain homotopy category of all AC-injective complexes, is always a compactly generated triangulated category. In general, all DG- injective complexes are AC-injective and in fact there is a recollement linking K(AC-Inj) to the usual derived category D(R). This is based on the author's recent work inspired by work of Krause and Stovicek. Our focus here is on giving straightforward proofs that our categories are compactly generated.
基金Supported by National Natural Science Foundation of China(Grant No.11001222)
文摘Let T be a triangulated category and ζ a proper class of triangles. Some basics properties and diagram lemmas are proved directly from the definition of ζ.