Classification of Subcategories in Abelian Categories and Triangulated Categories

Date
2016-09
Authors
Liu, Yong
Journal Title
Journal ISSN
Volume Title
Publisher
Faculty of Graduate Studies and Research, University of Regina
Abstract

Two approaches for classifying subcategories of a category are given. We examine the class of Serre subcategories in an abelian category as our first target, using the concepts of monoform objects and the associated atom spectrum [13]. Then we generalize this idea to give a classification of nullity classes in an abelian category, using premonoform objects instead to form a new spectrum so that there is a bijection between the collection of nullity classes and that of closed and extension closed subsets of the spectrum. Additionally, we impose a natural sheaf structure induced by the center of a category on the atom spectrum, over which the sheaves of modules over the structure sheaf are also discussed. The second approach is enlightened by the lattice structure implicitly shown in the statements of classification of the subcategories in an abelian category. We introduce a new concept of classifying space of subcategories, those subcategories satifying finitely many closure operations, in an either abelian or triangulated category. We show that a class of subcategories is classified by a topological space if these subcategories are primely generated. Many well-known results fit into our framework, such as Neeman’s classification [19] of localizing subcategories of the derived category D(R) of a commutative Noetherian ring R, etc.

Description
A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics, University of Regina. v, 118 p.
Keywords
Citation