Research On The Homological Dimensions And Related Problems In Abelian Category

In the study of algebra,homological dimension is widely used.Compared with other categories,the abelian category is a more important category,recollements of abelian categories are discussed and studied on the basis of abelian category.The application of abelian category and homological algebra are very extensive.In this paper,we study the necessary conditions for abelian categories and its object's finite presentation dimension,presentation dimension and the homological dimension of ho-mological heart to be finite.At the same time,the properties of FP-projective objects,FP-projective dimensions,FP-injective objects and FP-injective dimensions in abelian categories are further studied.The details are as follows:In the first part,we mainly discuss the abelian categories and the finite presentation dimension and the presentation dimension of objects.Firstly,we give the relationship between the dimensions of categories;Then we further discuss the dimension relation-ship between objects for the exact sequences and the long exact sequences;At last,the paper discusses the necessary conditions of the global dimensions,the finite pre-sentation dimensions and the presentation dimensions of a category with the help of homological heart.In the second part,we give the definition of FP-injective object and FP-injective dimension in an abelian category and study their related properties.Firstly,we give the necessary and sufficient conditions that an object is a finite presentation object;Then we discuss some related properties of FP-injective objects and give some equivalent propositions about FP-injective dimensions;At last,the dimensions relation between objects in an exact sequence are given in abelian category.In the third part,we introduce the definition of FP-projective objects and FP-projective dimensions in abelian category and study their related properties.Firstly,we give the definition and properties of FP-projective objects;then we give the re-lations between FP-projective dimensions,global dimensions and finite presentation dimensions,and some equivalent propositions about FP-projective dimensions;sec-ondly we give the relations between FP-projectivity and FP-injectivity in equivalent categories;Finally,the cotorsion theory of FP-projective objects and FP-injective ob-jects is given.