| This thesis for Ph.D degree is on triangulated categories and their applications. It mainly consists of the following three parts.1. For any abelian category A and a self-orthogonal full additive subcategory ω, we introduce in Chapter 2 the notion of relative singularity category D_ω(A). This generalizes Orlov's notion of singularity category for Noetherian rings [93]. We introduce a Frobenius exact category α(ω): it is an additive full subcategory of A, whose subcategory of projective-injective objects are exactly the additive closure of ω. The main result of Chapter 2 asserts that the natural functor from the stable category α(ω) to D_ω(A) is a fully-faithful exact functor of triangulated categories; moreover, under some reasonable conditions, it is even an equivalence. Using this equivalence, we describe the relative singularity category D_ω(A) via the homotopy category of unbounded acyclic complexes over ω.Applying the results above to some concrete singularity categories, we obtain:1). For a noncommutative Gorenstein ring, its singularity category is triangle-equivalent to the stable category of the full subcategory consisting of maximal Cohen-Macaulay modules. This is an unpublished result of Buchweitz [28];2). For a Gorenstein category with finite Gorenstein dimension, its singularity category is triangle-equivalent to the stable category α(ω), where ω is any functorially-finite tilting subcategory. This generalizes a corresponding result of Happel [51].2. For any Hom-finite additive category C, we introduce the notion of generalized Serre structure (duality) on C. It is a six-tuple (S, C_r, C_l, φ-,-, (-,-), Tr_), where C_r and C_l are certain full subcategories of C, S : C_r→C_r is an equivalence, which is called the generalized Serre functor, and for the definitions of φ-,-, (-,-)and Tr_see Chapter 3. Note that any one of the last three terms in the six-tuple is uniquely determined by any other. For a Krull-Schmidt pre-triangulated category C, the main result of Chapter 3 asserts that:1). Both C_r and C_l are thick triangulated subcategories of C;2). The generalized Serre functor S is an exact functor of (pre-)triangulated categories; 3). Let X ∈ C be indecomposable. Then X ∈ C_r (resp. X ∈ C_l) if and only if there exists an Auslander-Reiten triangle with X being its right term (resp. left term).Note that by the well-known result of Reiten and Van den Bergh [98], the existence of a Serre functor is equivalent to the existence of Auslander-Reiten triangles in C. By the result above, the generalized Serre functor always exists. Thus, in particular, our result generalizes Reiten-Van den Bergh's theorem.For the bounded derived categories of finite-dimensional algebras and some noncom-mutative projective schemes, we explicitly compute their generalized Serre structures. As two applications, we give a new characterization for finite-dimensional Gorenstein algebras; and we strengthen a remarkable theorem of Rickard [101] via a short proof.3. Note that for Hom-finite additive categories, the idempotent-split property is equivalent to the property of being Krull-Schmidt. In Chapter 4, we study the idempotent-split property of triangulated categories. We prove the following two fundamental results, which seem to be known but have no exact references:1). For any idempotent-split category C, the bounded homotopy category K~b(C) is also idempotent-split;2). For any abelian category A, its bounded derived category D~b (A) is always idempotent-split.
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