Presilting Categories In Triangulated Categories

Presilting categories are newly introduced in triangulated categories,lots of its properties are to be discussed. In this thesis, we are mainly focused on presilting categories and its mutations. This master’s degree thesis consists of four chapters.In first chapter, we recall some basic definitions,backgrounds needed in the sequel and list the main results of this thesis.In second chapter, firstly, we present notions and properties of presilting, pretilting categories in triangulated categories,then we give definitions of mu-tations of presilting,pretilting categories.Furthermore, we get the equivalent conditions of the mutation of pretilting categories is pretilting.In third chapter, we introduce the definitions of presilting,partial silt-ing, almost complete silting objects firstly. After that,we get some results which can be used in triangulated category, and by these facts,we explicitly describe the relationship among the three objects.Then,we give properties of mutation-s of partial silting, almost complete silting objects, and base on these,we reach a conclusion that complements of an almost complete silting object are not unique. Finally,we conclude that two quotient rings made of arbitrary two complements to an almost complete silting object are isomorphic.In forth chapter,we present the dual definition of presilting categories-prelilting categories,and on the basis of prelilting categories,we introduce a new categories-milting categories and give corresponding properties about these.