The Form Of The Category Of The Maximal Cohen-macaulay Modules And Torsion Theory

This thesis is devoted to two topics on the representation theory of Artin algberas.Fist, we study the characteristic of the category of Gorenstein projective modules(i.e., maximal Cohen-Macaulay modules) over an Artin algebra. Especially, we areinterested in fnding the concrete information of this subcategory from some abstractinformation. And we are also interested in the role of this subcategory in the modulecategory. The main results are as follows.(1) We fnd sufcient and necessary conditions for the category of Gorensteinprojective modules of an artin algebra being an abelian category, and give anotherproof for Auslander-Solberg correspondence which demonstrates the concrete formof the category of Gorenstein projective modules. Then we fnd a characterizationfor this category of Gorenstein projective modules. And we give an example of thiscorrespondence.(2) We prove that the relative Auslander algebra of a CM-fnite algebra is CM-free. Thus, the category of the Gorenstein-projective modules over a CM-fnite algebraturns out to be the category of the projective modules over a CM-free algebra.(3) WeprovethattheGorensteindefectcategoryofaCM-fnitealgebraistriangle-equivalent to the singularity category of its relative Auslander algebra. Thus, theGorenstein defect category of a CM-fnite algebra is uniquely determined by the cate-gory of its Gorenstein-projective modules.Second, we study the structure of torsion pairs on the category of the fnitely gen-erated modules over an Artin algebra. We generalize the concept of torsion pair andstudy its structure. As a trial of obtaining all torsion pairs, we decompose torsion pairs by projective modules and injective modules. Then we calculate torsion pairs on thealgebra KAnand tube categories. At last we study the structure of torsion pairs on themodule categories of fnite-dimensional hereditary algebras.