Relative Gorenstein Homological Modules And Dimensions

The theory of homological algebra is closely related to algebraic topology,and topo-logical data analysis can capture the effective information of high-dimensional data space,so the study of homological algebra is helpful for data analysis and statistical judgment in the current era of big data.In the tools for studying algebra,homological dimensions are widely used.Projective,injective and flat modules are important objects in homological algebra.The abelian category is one of the most important categories in homological algebra,and many of the conclusions in the abelian category can be applied to the mod-ule category.In addition,formal triangular matrix rings are typical non-commutative rings.This paper explore the finiteness of relative Gorenstein dimension on the abelian category,as well as the properties and dimensions of the relative Gorenstein flat modules over formal triangular matrix rings.First of all,we study the finiteness of relative Gorenstein dimension in abelian cate-gory.Let（X,Y）a pair of classes of objects in abelian category.Inspired by Bouchiba’s work on generalized Gorenstein projective modules,we give a new way of measuring（X,Y）-Gorenstein projective dimension by defining a complete n-（X,Y）resolution.This leads to the exploration of relation of the relative global Gorenstein homological dimen-sion to the invariants the supremum of the injective dimension of projective modules and the supremum of the projective dimension of injective modules under some conditions.Furthermore,this paper proves that,in the setting of a left and right coherent ring R,the supremum of Ding projective dimensions of all finitely presented（left or right）R-modules and the（left or right）Gorenstein weak global dimension are identical,generalizing a theorem of Ding,Li and Mao.Secondly,we study relative Gorenstein flat modules and dimensions over formal tri-angular matrix rings.Let T be a formal triangular matrix ring,we construct a w⁺-tilting module over T by using the corresponding ones over rings A and B.And,for left T-module C,left A-module C₁and left B-module C₂,this paper proves that when U is weakly C⁺-compatible,G_CF（T）is closed under extensions,C₁is w⁺-tilting and G_C1F（A）is closed under extensions,a left T-module M is G_C-flat if and only if M₁is a G_C1-flat left A-module,Cokerφ^Mis a G_C2-flat left B-module and the morphismφ^M:U（?）_AM₁→M₂is a monomorphism.Furthermore,this paper gives an estimate of Relative Gorenstein flat dimension of a left T-module by the relative dimensions A and B under some conditions.Finally,we study relative Ding projective modules and dimensions over formal trian-gular matrix rings.Suppose that C₁and C₂are semidualizing.Let C=p（C₁,C₂）and U Ding C-compatible,we prove that a left T-module M is D_C-projective if and only if M₁is a D_C1-projective left A-module,Cokerφ^Mis a D_C2-projective left B-module and the morphismφ^M:U（?）_AM₁→M₂is a monomorphism.Furthermore,we give an estimate of relative Ding projective dimension of a left T-module by the relative dimensions of A and B.