Tilting Modules And Functorially Finite Subcategories

Tilting theory is the core content in the representation theory of algebras,which appearance promoted the development and prosperity of the representation theory of algebras Tilting theory can be back to the study of reflection functors in the representation theory of algebras.The first to put forward the concept of the tilting module are English mathematicians,Brenner and Butler.Afterwards,German mathematicians Happel and Ringel put forward the concept of tilting modules and partial tilting modules on the base of the study of the hereditary algebra.Then,Miyashita and Happel generalized the classical tilting modules to tilting modules of finite projective dimension independently.It is well-known that cotilting modules were obtained by applying the classical duality functor to tilting modules Nowadays,most scholars focus on the properties and characteristics of one-sided tilting modules.But little is known for the relation between left and right tilting modules.Therefore,it’s far from meaningless to study the number of left and right tilting modulesIn this thesis,we focus on the connection between left tilting modules and right tilting modules over an algebra and prove the following resultTheorem 1.For a Gorenstein algebra A,the number of tilting right A-modules is equal to the number of tilting left A-modulesDuring the development of the tilting theory,Auslander and Smal(?)put forward the concepts of left,right minimal morphisms.Then they put forward the concepts of approximations and minimal approximations on the basis of minimal morphisms and introduced the concepts of covariantly finite subcategories and contravariantly finite subcategories.Moreover,they obtained the relation between minimal approximations and functorially finite subcategories.And they established the relation between tilting modules and contravariantly finite subcategories.It is noted that the classical tilting modules have good properties,for example,the existence of tilting theorem and derived equivalence Therefore,it is meaningful to determine when a rigid module is a classical tilting module.By using the properties of functorially finite subcategories,Auslander and Reiten gave a necessary and sufficient condition for a rigid module to be a partial cotilting module.Dual to their conclusion,we give the following result:Theorem 2.Let A be an algebra,T ∈ mod A and ExtA1(T,T)=0.Denote by X(T)={X ∈ mod A | ExtA1(T,X)=0},Y(T)={Y∈ mod A| ExtA1(Y,X(T))=0}.The following conditions are equivalent:(1)pdAT ≤1;(2)X(T)(?)mod A is covariantly finite and Y(T)only consists injective modules.