Quotient Categories Of Abelian Categories

This thesis mainly studies the quotient categories of an Abelian category and its related problems.The main results hold for exact categories,so we expand on them in exact categories.Firstly,we reviewed the methods for localization categories;secondly,this thesis gives a direct proof of the fact that the localization of an exact category by biresolving subcategories is a triangulated category;finally,starting from the known result that the quotient of an exact category by cluster tilting subcategory is an Abelian category,the concept of a pseudocluster tilting subcategory is introduced.It is proved that the quotient of an exact category by pseudo-cluster tilting subcategories is a semi-Abelian category and it is an Abelian category if and only if it satisfies some self-orthogonal condition.At last,we proved that the quotient category of the category of short exact sequences by the subcategory of splitting short exact sequences is an Abelian category,and there exists a unique exact structure making it a cluster quotient.