Properties Of Homology And Relative Homology On Triangular Matrix Algebras

In this parper,A =(?)is the lower t,riangular matrix ring,?L is left A mod in module category of ring ?,We know that ?L can be represented by the triple(?).We mainly discuss the influence of the homological dimension of A-Mod AX and B-Mod BY on the homological dimension of A-Mod ?L and the influence of A-Mod AX,B-Mod BY and B module morphism/on properties of ?-Mod ?L.Wa mainly discuss under what circumstances ?-Mod?L is(n,m)-strong Gorenst.ein projection module.In the first chapter,we introduce the research background and properties of triangular matrix ring and the preparatory knowledge of homology and Gorenstein relative homology.In the second chapter,firstly,through The equivalent conditions of pro-jective modules and injective modules[7],we first obtain that when MA is a flat module,the projective dimension of ?L equals the projective dimension of AX,and when BM is a projective module,the injective dimension of AL equals the injective dimension of BY.Then we discuss the relationship between the range of the projective dimension of X? and the projective dimension of X and Y when BM is a projective module.Further,we discuss the relationship between the range of the projective dimension(injective dimension)of ?L and the projective dimension(injective dimension)of AX and BY when projective dimension(flat dimension)of BM(MA)is finite.Next,the relationship be-tween the global dimension of lower triangular matrix ring ? and the global dimension of ring A,B and the projective dimension of BM is studied.Fi-nally,the projective dimension of modules of n order lower triangular matrix algebras is studied by matrix partitioning method.In Chapter 3,we make the ring B in triangular matrix ring A Gorenstein regular algebra.Firstly,we discuss when the module ?L of ring A is n-Strongly Gorenstein projective module,and give the necessary and sufficient condition that XA is n-Strongly Gorenstein projective module under certain conditions.Then,we introduce(n,m)-Strongly Gorenstein projective module,and dis-cuss the sufficient conditions for A module to be(n,m)-Strongly Gorenstein projective module under certain conditions.In Chapter 4,we give two constructions of tilting modules in lower trian-gular matrix Artin algebra.It is proved that A-Mod(?)is a tilting module,if Ext1(T,M)= 0,and B-Mod T is a tilting module;It is also proved that if A-Mod F is a tilting module and left A-module M is projective module,A-Mod(?)is a tilting module.