Auslander-Reiten Theory And Auslander Bijection In Extriangulated Categories

The notion of extriangulated categories was introduced by Nakaoka and Palu,which unifies exact categories and extension-closed subcategories of triangulated categories.Assume that(C,E,s)is an Ext-finite Krull-Schmidt R-linear extriangulated category,where R is a commutative Artin ring.Recently,Iyama,Nakaoka and Palu introduced the notions of almost split extensions(or almost split 5-triangles)and Auslander-Reiten-Serre duality in extriangulated categories.They proved that the existence of almost split extensions is equivalent to that of Auslander-Reiten-Serre duality.In this paper,we study the existence of almost split s-triangles from different opinions,and investigate the restricted Auslander bijection in extriangulated categories.First of all,we give the judging theorem for being almost split s-triangles in terms of the injective property of functors and the almost vanishing property of stable endomorphism algebras.Meanwhile,we give the existence theorem of almost split s-triangles in terms of the injective property of functors and the socle’s property.If the extriangulated category C admits almost split s-triangles,then the Ext-finite property of C is equivalent to the Hom-finite property of projectively stable category C(or injectively stable category C).We now fix an extension-closed subcategory ε of the extriangulated category C.In order to study the existence of almost split sε-triangles,we introduce the notion of right injectively(or left projectively)stable ε-approximations.Under the condition that C admits right(or left)almost split s-triangles,we give a necessary and sufficient conditions such that ε admits right(or left)almost split sε-triangles.Secondly,we introduce two subcategories Cr={X ∈ C| the functor DE(X,-):C→ mod R is representable},Cl:={X∈C | the functor DIE(-,X):C→mod R is representable}in an extriangulated category,where mod R is the category of finitely generated R-modules,and D is the Matlis duality.We construct an equivalent functor τ:Cr→Cl and its quasiinverse τ-:Cl→Cr,and give the notion of generalized Auslander-Reiten-Serre duality.We prove that there are only two kinds of indecomposable objects in Cr:s-projective objects,or the ending terms of almost split s-triangles.We also investigate the morphisms determined by objects in extriangulated categories,and give the judging theorem for the existence of right determiners of s-defiations:an s-deflation is right determined by some object if and only if its intrinsic weak kernel belongs to Cl.Meanwhile,we also give another classification of indecomposable objects in Cr:s-projective objects,or intrinsic weak cokernels of s-inflations being left determined by some objects.Furthermore,we introduce the notion of Serre duality in extriangulated categories.To study the existence of Serre duality,we introduce the notions of right deflation-classified objects and left inflation-classified objects.Following this,we get the existence theorem of Serre duality:the extriangulated category C admits Serre duality if and only if C has right determined defaltions and left determined inflations.Meanwhile,we give the relation of Serre duality and Auslander-Reiten-Serre duality.On the other hand,we give the notion of restricted Auslander bijection in extriangulated categories.We prove that the restricted Auslander bijection holds if the extriangulated category admits Auslander-Reiten-Serre duality.In particular,the Auslander bijection holds in extriangulated categories admitting Serre duality.Finally,we give some applications.