Representation Theory Of Pre-additive Categories And Its Applications

Enlighten by B. Mitchell's long paper—Rings with Several Objects, which published in Advances in Mathematics for a whole issue, this dissertation, which is divided into three parts, is to develop the representation theory of pre-additive categories.The first chapter presents the background as well as the structure of this dis-sertation.Chapter two is devoted to the study of the existence of recollement situations of representation categories of small pre-additive categories and the Morita equiv-alences. A criterion for the existence of recollement situations of representation categories of small pre-additive categories is put forward, which entitled as the Konig theorem for representation categories of small pre-additive categories, and a Morita theorem of the representation categories of small pre-additive categories's version is also given in this part. After that, we will apply these two main theo-rems to (graded) module categories and obtain related recollements and Morita equivalences'results. As to the graded module categories, we give a new method to construct the Morita theorem, and the Konig theorem for graded module categories has generalized the related results of many papers, which were aimed at dealing with Morita equivalences.Chapter three studies the Morita equivalence by investigate the functors be-tween pre-additive categories. For the sake of studying Morita equivalence in a more general way, the first section of this chapter focuses on the construction of a recollement situation of functor categories. Using the techniques of Godement products, we construct a left (or right) pre-recollement situation of functor cate-gories from two given left (or right) pre-recollements of pre-additive categories. As an application of this result, we will show that it is natural to get the recollement situations of the representation categories of the related small pre-additive categories from the given recollement situations of small pre-additive categories. Combine the results of Representable functors in section two, we put forward four new criterions for Morita equivalence of two pre-additive categories, which induced by two given additive functors between them in the last section.