τ-Rigid Modules, Local Algebras And Tilted Algebras

Tilting theory is one of the main objects in the representation theory of algebras.It originated with the study of reflection functors. The first set of axioms for a tilting module is due to Brenner and Butler, the one generally accepted now is due to Happel and Ringel. The main ideal of tilting theory is that when the representation theory of an algebra A is difficult to study directly, it can be convenient to replace A with another simpler algebra B and to reduce the problem on A to a problem on B. Some important results can be achieved by constructing tilting modules M.Recently, T. Adachi, O. Iyama and I. Reiten introduced τ-tilting theory to generalize the classical tilting theory. Note that a τ-tilting module is a direct sum of indecomposable τ-rigid modules. Therefore, as long as we find the indecomposableτ-rigid modules we can get the τ-tilting modules. The main parts of this thesis are as follows.(1) τ-rigid modules and projective modules. For a special class of algebras, a method on constructing indecomposable τ-rigid modules from simple modules is given. As a result, it is proved that a basic and connected algebra A with radical square zero is local if all τ-rigid A -modules are projective. Moreover, by mutation we can conclude that for a basic indecomposable finite dimensional algebra A , if allτ-rigid A -modules are projective, then A is local.(2) τ-rigid modules and cotilting modules. For any basic indecomposable finite dimensional algebra B , then the injective module DB is τ-rigid if and only if the injective dimension of B is at most one. Then we give an example to show that there exists a class of algebras B satisfying that all indecomposable injective-B-modules are τ-rigid modules but DB is not a τ-rigid module. Then, we show the relationship between injective modules, cotilting modules and τ-rigid modules.(3) τ-rigid modules over tilted algebras. By using tilting theorem,we can give a characterizaion of indecomposable τ-rigid modules with projective dimension less than or equal to 1 over tilted algebras.