Some Research On Relative Homological Theory And Relative Singularity Categories

Since the 1940s,homological algebra has gradually formed a complete theoretic framework and research methods,in which the abelian category is the core concept of homological algebra.Both the module category and the category of complexes are abelian categories,and the module category is a full subcategory of the category of complexes.Some characterizations of rings or algebras can be obtained by studying the homological properties of modules and complexes over rings or algebras.Since the late 1950s,the relative projective modules,especially Gorenstein projective modules and pure projective modules,have been used to replace projective modules,resulting in relative homological algebras,namely the Gorenstein homological algebra and the pure homological algebra.Moreover,the research on relative homological theory of the category of complexes has been developed to an advanced level.Some categories consisting of natural research objects are not abelian categories,such as homotopy category.It can be seen that the homotopy category,derived category and singularity category are triangular categories.The introduction of triangular categories brings new research ideas and methods to homological algebra and representation theory.Its relative homological theory and recollements theory are also the research topics of many scholars.For a given ring or algebra,it is not trivial to construct all the relative projective modules over it.In this thesis,we characterize some relative projective modules over Morita context rings by using the theories and methods in homological algebra.We discuss the relative homological theory of the category of complexes and generalize some conclusions on the module category.We also investigate some questions about relative derived categories and relative singularity categories.The details are as follows.In chapter 2,we assume that(?)is a Morita context ring.We study the functors Hom,? and derived functors Ext,Tor over a Morita context ring Λ(0,0),and on this basis,we describe the structures of Gorenstein projective modules and Gorenstein flat modules over A(0,0),respectively.We also determine all the finitely presented modules,pure projective modules,coherent modules,and FP-injective modules.We study the quasi-heredity of the Morita context ring Λ(0,0).Finally,we give some research of the Gorenstein hereditary rings.In chapter 3,based on the homological theory of the category of complexes,we generalize some conclusions on module category.We determine when a finitely generated Gorenstein projective complex is finitely presented.Applying this result,we introduce the notion of Gorenstein transpose of complexes,and we establish a relation between the transpose and the Gorenstein transpose.In addition,we give several cases when a complex is pure injective.In chapter 4,under some suitable conditions,we give a relation between Gorenstein projective modules and pure projective modules.Applying this relation,we establish a relation between the Gorenstein derived category and the pure derived category.Moreover,we also establish a relation between the Gorenstein singularity category and the pure singularity category.Finally,we investigate the behavior of pure singularity categories along a morphism of rings.In particular,we discuss pure singularity categories over triangular matrix rings.