Torsion theories and Auslander-Reiten sequences

Abstract

Chapter 0 gives a gentle background to the thesis. It begins with some general notions and concepts from homological algebra. For example, not only are the notions of universal property and of duality central to the flavour of the subject, they are also suggestive in understanding mathematics at another depth. In category theory, objects and morphisms are the two main elements in a category, and notions such as kernels and cokernels are defined in terms of objects together with morphisms. In accordance with it, the morphisms are given a very subtle signifcance within a category. The chapter then introduces the notion of a triangulated category, where due to the lack of uniqueness of certain morphisms described in the axioms, is allowed to be far from an abelian category. A few examples of triangulated categories are given, the homotopy category, the derived category and certain stable cat- egories. The chapter ends with a little description of an Auslander-Reiten quiver de ned on a Krull-Schmidt category, as well as the notions of Serre functor and of Auslander-Reiten triangles in subcategories. The introduction chapter selects lemmas and theorems not only to be referenced in later chapters, but also those which can induce good intuition on the reader, for example, in their capacity of being analogues to each other, in the interplay between them and in their different suggestiveness in approximating or generalizing concepts in different ways and directions. Chapter 1 studies torsion pairs in abelian categories and torsion theories with torsion theory triangles in triangulated categories. It then gives a necessary and su cient condition for the existence of certain adjoint functors in triangulated categories. Intuitively, they are all different expressions of subcategories approximating their ambient categories. The chapter goes on to introduce two special cases of torsion theories, namely t-structures and split torsion theories, and finishes with a characterization of a split torsion theory and a classiffcation of split torsion theories in a chosen derived category. There is a very close and subtle relationship between the existence of torsion theory triangles and the existence of Auslander-Reiten triangles. Chapter 2 studies the existence of Auslander-Reiten sequences in subcategories of mod( ), where is a nite-dimensional k-algebra over the eld k, based on the theory of the existence of Auslander-Reiten triangles in subcategories developed by J rgensen. The existence theorems strengthen the results by Auslander and Smal and by Kleiner. Chapter 3 sees that quotients of certain triangulated categories are triangulated and are in addition derived categories, appealing to a theorem which is a slight variation of the results by Rickard and Keller. In this chapter, the Auslander-Reiten triangles play a predominant role in re ecting the tri angulation structure of a triangulated category, and the Auslander-Reiten triangles can be read o from the Auslander-Reiten quiver. The cluster category D of Dynkin type A1 was introduced by J rgensen. One of its several de nitions, which is completely analogous to the de nition of the cluster category of type An, motivates us to say that D is a cluster category of type A1. In the result by Holm and J rgensen, the cluster tilting subcategories of D were shown to be in bijection with certain maximal sets of non-crossing arcs connecting non-neighbouring integers. Chapter 4 generalizes the result by giving a bijection between torsion theories in D and certain configurations of arcs connecting non-neighbouring integers. Finally, a few examples, characterizing all t-structures and co-t-structures in D, are given.