The Generalized Auslander-reiten Duality And Morphisms Determined By Objects On An Exact Category

We introduce the notion of generalized Auslander-Reiten duality on a Hom-finite Krull-Schmidt exact category.We study morphisms determined by objects in an exact category,and then give some characterizations for the generalized Auslander-Reiten duality.We study the generalized Auslander-Reiten duality on the category of finitely presented representations of a strongly locally finite quiver.We characterize projective objects in the category of locally finite dimensional representations of an interval finite quiver.For a Hom-finite Krull-Schmidt exact category C over a commutative artin ring k,we denote by C.and C the projectively and injectively stable category,respectively.we introduce a pair of full subcategoriesCr = {X ∈ C|the functor D ExtC1(X,-):C→ mod k is representable}andCl = {X ∈C|the functor D ExtC1(-,X):C→ mod k is representable}.Here,mod k is the category of finitely generated κ-modules and D is the Matlis duality.We obtain the generalized Auslander-Reiten translation functorsτ:Cr→Cl and τ:Cl→Cr,and the generaliezd Auslander-Reiten duality{Cr,Cl,φ,ψ,τ,τ-}.Here,τ-and τ are a pair of mutually quasi-inverse equivalences,which form an adjoint pair.We show that a non-projective indecomposable object lies in Cr if and only if it appears as the third term of an almost split conflation;a non-injective indecomposable object lies in Cl if and only if it appears as the first term of an almost split conflation.We study morphisms determined by objects in C.We generalize two main theorems of Auslander about morphisms determined by objects from categories of modules to exact categories.More precisely,we prove an existence theorem of deflations,and that a deflation is right determined by some object if and only if its intrinsic kernel lies in Cl.We give some characterizations for objects in Cτ via morphisms determined by objects.We introduce the notions of right stably determined deflations and left stably de-termined inflations.We prove that the following statements are equivalent.(1)C has Auslander-Reiten duality.(2)C has right stably determined deflations.(3)C has left stably determined inflations.As an application,we describe the generalized Auslander-Reiten duality on the category of finitely presented representations of a strongly locally finite quiver.For further investigation of the generalized Auslander-Reiten duality on the cat-egory of representations of infinite quivers,we study the projective objects.For each tail-equivalence class[p]of right infinite path,we define an explicit representation X[p].We introduce the notion of uniformly interval finite quiver.We show that X[p]is an in-decomposable projective object in rep(Q),when the convex hull of each right infinite path in[p]is uniformly interval finite.We then classify the indecomposable projective objects in rep(Q).