?-Tilting Modules And Algebra Extensions

Tilting theory is one of the theoretical foundations of the representation theory of algebras.It is important to construct a new tilting module from a given one,and mutation of tilting modules is a very effective way to do it.However,mutation of tilting modules may not be realized.As a generalization of tilting modules,the concept of support ?-tilting modules over finite dimensional algebras was introduced by Adachi,Iyama and Reiten.It was shown that support ?-tilting modules are in bijection with several other important classes including functorially finite torsion classes,2-term silting complexes,cluste?-tilting objects in the cluster category and so on.What is more,mutation of any support ?-tilting module always exists.In this thesis,our aims are to construct support ?-tilting modules by using algebra extensions(such as one-point extensions,triangular matrix algebras,split-by-nilpotent extensions)and calculate the numbers of ?-tilting modules and support ?-tilting modules over some algebras.In Chapter 1,we introduce the backgroud and main results.In Chapter 2,we construct support ?-tilting modules over split-by-nilpotent extensions in two different ways.Firstly,we obtain a necessary and sufficient condition for constructing support ?-tilting modules over split-by-nilpotent extensions by using the tensor functor.Morover,we show that the tensor functor preserves left mutation.Secondly,we construct semibricks over split-by-nilpotent extensions and support ?-tilting modules over split-by-nilpotent extensions which are ?-tilting finite.In Chapter 3,we construct silting modules over triangular matrix rings.Then for finite dimensional algebras,we give a method for constructing support ?-tilting modules over triangular matrix algebras.In Chapter 4,we calculate the number of ?-tilting modules over Nakayama algebras.Adachi gave a recursion relation about the number of ?-tilting modules over linear Nakayama algebras Anr.We show that there is the same recursion relation for selfinjective Nakayama algebras(?).As an application,we give a new proof of a result by Asai about the recursion relations on the number of support ?-tilting modules over?nr and(?).In Chapter 5,we classify the ?-tilting modules over the one-point extension by a simple module at a source point.We also get two formulas about the numbers of?-tilting modules and support ?-tilting modules.As a consequence,we calculate the numbers of ?-tilting modules and support ?-tilting modules over linear Dynkin type algebras whose square radicals are zero.