Some Studies On τ-tilting And Tilting Modules

Tilting theory is one of the main tools in the representation theory of finite di-mensional algebras,which describes the method of using tilting modules and related module categories of tiling functors for two related 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 idea of tilting theory is that when the representation theory of an algebra A is difficult to study directly,it may be convenient to replace A with another simpler algebra B and to reduce the problem on A to a problem on B.We can construct an A-module T,called a tilting module,which is close to the Morita progenerators such that B=End T_Aand the categories mod A and mod B are reasonably close to each other.Many important results are achieved by constructing tilting modules T.In 2014,T.Adachi,O.Iyama and I.Reiten introducedτ-tilting theory to generalize the classical tilting theory from the viewpoint of mutation.It is well known that an almost complete tilting module for any finite-dimensional algebra over a field is a direct summand of exactly one or two tilting modules.Adachi,Iyama and Reiten generalized tilting modules to what we callτ-tilting modules,and showed that an almost complete supportτ-tilting module has exactly two complements for any finite dimensional algebra.Note that aτ-tilting module is a direct sum of indecomposableτ-rigid modules.Therefore,as long as we find the indecomposableτ-rigid modules,then we can get theτ-tilting modules.In this thesis,we study the related properties ofτ-tilting and tilting modules.The main results are as follows.（1）We give some equivalent characterizations for Iwanaga-Gorenstein algebras with self-injective dimension at most one in terms ofτ-rigid modules.We show that every indecomposable module over iterated tilted algebras of Dynkin type isτ-rigid.Moreover,we give aτ-tilting theorem on homological dimension which is an analog to that of classical tilting modules.（2）We give a recurrence relation about the number of supportτ-tilting modules over Nakayama algebras with radical square zero.Then we give a criterion for com-puting the number of tilting modules over Auslander algebras of Nakayama algebras with radical cube zero.（3）We introduce the notions of Gorenstein projective supportτ-tilting modules and Gorenstein projective supportτ-tilting pairs.Then we show that there is a bi-jection between Gorenstein projective supportτ-tilting pairs and Gorenstein injective supportτ^-1-tilting pairs.Furthermore,we introduce the notion of CM-τ-tilting finite algebras and show that algebras of radical square zero are CM-τ-tilting finite.