Balanced Pairs on Triangulated Categories

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.

Resolving Subcategories of Triangulated Categories and Relative Homological Dimension

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.

Cohomology Theories in Triangulated 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 cohomology.

Relative Quotient Triangulated 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-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.

On two Hall algebra approaches to odd periodic triangulated categories

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.

Cluster Structures in 2-Calabi-Yau Triangulated Categories of Dynkin Type with Maximal Rigid Objects

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.

A Hovey Triple Arising from a Frobenius Pair

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^(∧).

Cluster-tilting objects in higher cluster categories

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.

Phases of Stable Representations of Euclidean and Wild Quivers

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.

Categorical resolutions of a class of derived categories

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.

Auslander-Reiten Sequences or Triangles Related to Rigid Subcategories

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

A note on the equivalence of m-periodic derived categories

Let A and B be finite-dimensional algebras over a field k of finite global dimension. Using some results of Gorsky in "Semi-derived Hall algebras and tilting invariance of Bridgeland-Hall algebras",we prove that if A and B are derived equivalent,then the corresponding m-periodic derived categories are triangulated equivalent.

On the homotopy category of 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.

Notes on Proper Class of Triangles

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.

On one-sided torsion pair

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.

Objective triangle functors

An additive functor F:A→B between additive categories is said to be objective,provided any morphism f in A with F(f)=0 factors through an object K with F(K)=0.We concentrate on triangle functors between triangulated categories.The first aim of this paper is to characterize objective triangle functors F in several ways.Second,we are interested in the corresponding Verdier quotient functors VF:A→A/Ker F,in particular we want to know under what conditions VF is full.The third question to be considered concerns the possibility to factorize a given triangle functor F=F2F1with F1a full and dense triangle functor and F2a faithful triangle functor.It turns out that the behavior of splitting monomorphisms and splitting epimorphisms plays a decisive role.