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 consider stable representations of non-Dynkin quivers with respect to a central charge.These attract a lot of interest in mathematics and physics since they can be identified with so-called BPS states.Another motiv...We consider stable representations of non-Dynkin quivers with respect to a central charge.These attract a lot of interest in mathematics and physics since they can be identified with so-called BPS states.Another motivation is the work of Dimitrov et al.on the phases of stable representations of the generalized Kronecker quiver.One aim is to explain for general Euclidean and wild quivers the behavior of phases of stable representations well known in some examples.In addition,we study especially the behavior of preinjective,postprojective and regular indecomposable modules.We show that the existence of a stable representation with self-extensions implies the existence of infinitely many stables without self-extensions for rigid central charges.In this case the phases of the stable representations approach one or two limit points.In particular,the phases are not dense in two arcs.The category of representations of acyclic quivers is a special case of rigid Abelian categories which show this behavior for rigid central charges.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.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
We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D^b(A)and the subcategory K^b(P) of perfect complexes in D^b(A), by giving two classes of abelian categories A with enough p...We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D^b(A)and the subcategory K^b(P) of perfect complexes in D^b(A), by giving two classes of abelian categories A with enough projective objects such that D_(hf)~b(A) = K^b(P), and finding an example such that D_(hf)~b(A)≠K^b(P). We realize the bounded derived category D^b(A) as a Verdier quotient of the relative derived category D_C^b(A), where C is an arbitrary resolving contravariantly finite subcategory of A. Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension.We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dimT <∞ such that ~⊥T is finite, then D^b(modA) admits a categorical resolution; and that for a CM(Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant.展开更多
Motivated by the concept of a torsion pair in a pre-triangulated category induced by Beligiannis and Reiten, the notion of a left (right) torsion pair in the left (right) triangulated category is introduced and invest...Motivated by the concept of a torsion pair in a pre-triangulated category induced by Beligiannis and Reiten, the notion of a left (right) torsion pair in the left (right) triangulated category is introduced and investigated. We provide new connections between different aspects of torsion pairs in one-sided triangulated categories, pre-triangulated categories, stable categories and derived categories.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.
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
Given a triangle functor F : A → B, the authors introduce the half image hIm F,which is an additive category closely related to F. If F is full or faithful, then hIm F admits a natural triangulated structure. However...Given a triangle functor F : A → B, the authors introduce the half image hIm F,which is an additive category closely related to F. If F is full or faithful, then hIm F admits a natural triangulated structure. However, in general, one can not expect that hIm F has a natural triangulated structure. The aim of this paper is to prove that hIm F admits a natural triangulated structure if and only if F satisfies the condition(SM). If this is the case, hIm F is triangle-equivalent to the Verdier quotient A/Ker F.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
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
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.
文摘We consider stable representations of non-Dynkin quivers with respect to a central charge.These attract a lot of interest in mathematics and physics since they can be identified with so-called BPS states.Another motivation is the work of Dimitrov et al.on the phases of stable representations of the generalized Kronecker quiver.One aim is to explain for general Euclidean and wild quivers the behavior of phases of stable representations well known in some examples.In addition,we study especially the behavior of preinjective,postprojective and regular indecomposable modules.We show that the existence of a stable representation with self-extensions implies the existence of infinitely many stables without self-extensions for rigid central charges.In this case the phases of the stable representations approach one or two limit points.In particular,the phases are not dense in two arcs.The category of representations of acyclic quivers is a special case of rigid Abelian categories which show this behavior for rigid central charges.
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.
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.11271251 and 11431010)
文摘We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D^b(A)and the subcategory K^b(P) of perfect complexes in D^b(A), by giving two classes of abelian categories A with enough projective objects such that D_(hf)~b(A) = K^b(P), and finding an example such that D_(hf)~b(A)≠K^b(P). We realize the bounded derived category D^b(A) as a Verdier quotient of the relative derived category D_C^b(A), where C is an arbitrary resolving contravariantly finite subcategory of A. Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension.We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dimT <∞ such that ~⊥T is finite, then D^b(modA) admits a categorical resolution; and that for a CM(Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10671161) the Cultivation Fund of the Key Scientific and Technical Innovation Project of Ministry of Education of China (Grant NO. 704004)the Natural Science Foundation of Fujian Province (Grant No.Z0511021)
文摘Motivated by the concept of a torsion pair in a pre-triangulated category induced by Beligiannis and Reiten, the notion of a left (right) torsion pair in the left (right) triangulated category is introduced and investigated. We provide new connections between different aspects of torsion pairs in one-sided triangulated categories, pre-triangulated categories, stable categories and derived categories.
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.
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 the National Natural Science Foundation of China(Nos.11401001,11571329)the Project of Introducing Academic Leader of Anhui University(No.01001770)the Research Project of Anhui Province(No.KJ2015A101)
文摘Given a triangle functor F : A → B, the authors introduce the half image hIm F,which is an additive category closely related to F. If F is full or faithful, then hIm F admits a natural triangulated structure. However, in general, one can not expect that hIm F has a natural triangulated structure. The aim of this paper is to prove that hIm F admits a natural triangulated structure if and only if F satisfies the condition(SM). If this is the case, hIm F is triangle-equivalent to the Verdier quotient A/Ker F.
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.
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 .
