Serre Duality And Auslander-Reiten Triangles

In this thesis, we always consider the triangulated category to be k-linear Hom-finite Krull-Schmidt. In this condition, it satisfies the Uniqueness Decomposition The-orem, all idempotents split, and all indecomposable objects have local endomorphism rings. It has been known for some time that there is a connection between classical Serre duality and existence of almost split sequences in finite dimensional algebras. There is a strong analogy between the Serre duality formula for curves and the formula DExtΛ(N, M)≌Hom(M, DTrN) for artin algebras on which the existence of almost split sequences is based, where D=Homk(-, k). Actually, existence of almost split se-quences in some sheaf categories for curves can be proved either by using an analogous formula for graded maximal Cohen-Macaulay modules or by using Serre duality. The notion of almost split sequences was extended to the notion of Auslander-Reiten trian-gles in triangulated categories, and existence of such was proved for Kb(mod A) when Λ is a k-algebra of finite global dimension. In this case the corresponding translate is given by an equivalence of categories. This thesis is organized as follows:Chapter0, we introduce the background of the work which is done in this thesis.Chapter1, Auslander-Reiten triangles can be described by left minimal almost split morphisms or right minimal almost split morphisms.Chapter2, Serre duality can be characterized by non-degeneracy, which is fre-quently used in linear algebra. In particular, the Serre duality is always an isomor-phism.Chapter3, we can perfectly construct an equivalence between Serre duality and Auslander-Reiten triangles in k-linear Hom-finite Krull-Schmidt triangulated cate-gories.Chapter4, we try to find a triangulated version of Wakamatsu’s Lemma, and the idea is derived from relative homological algebra. Finally, We generalize Auslander-Reiten theory to subcategories of triangulated categories, and combine it with covers and envelopes.