Studies On ?-tilting Theory

Tilting theory has been central in the representation theory of algebras since 1970s,in which the notion of tilting modules is fundamental.Moreover,in representation theory of algebras,the notion of mutation often plays important roles.Mutation is an operation for a certain class of objects in a fixed category to construct a new object from a given one by replacing a direct summand.It is quite useful and effective in studying the structure of categories.In[21],Happel and Unger proved an important fact that mutation of tilting modules is often impossible depending on the choice of indecomposable direct summands.Thanks to the work[2]of Adachi,Iyama and Reiten,we know that mutation of support ?-tilting modules is always possible and this is a big advantage of support ?-tilting modules.Now,many algebraists paid much attention to it.On the other hand,the notion of skew group algebras was introduced in[31].For a fixed algebra and a finite group acting on A,the corresponding skew group algebra is obtained.Note that an algebra and its skew group algebra have a lot of properties in common.In this thesis,we mainly study ?-tilting theory and skew group algebras,and combine their properties efficiently.In Chapter 1,some background and main results are given.In Chapter 2,we combine ?-tilting theory with skew group algebras.For a finite-dimensional algebra ? with action by G a finite group,we introduce the notion of G-stable support ?-tilting modules.Then we connect it with torsion classes,silting complexes and cluster-tilting objects,and establish bijections among them.Moreover,we establish the relation between stable support ?-tilting modules over ? and its skew group algebra.In Chapter 3,we study mutation of ?-tilting modules.Mutation of ?-tilting mod-ules is a basic operation to construct a new support-r-tilting module from a given one by replacing a direct summand.We give a positive answer to an open question posed by Adachi,Iyama and Reiten in[2]about mutation of ?-tilting modules.