Research On The Properties Of Generalized Local Cohomology Modules With Respect To A Pair Of Ldeals

The generalized local cohomology modules with respect to a pair of ideals are extension of the local cohomology modules and generalized local cohomology modules.This paper is mainly concerned with the properties of generalized local cohomology modules with respect to a pair of ideals when R is a commutative Noetherian ring with nonzero identity and I.J ideals of R.M a finitely generalized non-zero R-module,N an arbitrary non-zero R-module.Firstly,we use the direct limit(?)Hαi(M,N)to describe the i-th generalized local cohomology module HI,JiJ(M,N)with respect to a pair of ideals(I,J).And based on it,we study the properties on support set,associated prime ideal set.Krull dimension and get some equivalent conditions about dim HI,Ji(M,N)≤k,for ?i<t.Secondly,we study the propeties on splitting theorem.Suppose that there exist an integer t with t>1 such that HI,Ji(M,N)is a finitely generated for i=t,t+1.Then there exist α∈W(I,J)and a positive integer m such that,for every α-filter regular element x on N,HI,Jt(N,N/χmN)(?)HI,Jt(M,N)⊕HI,Jt+1(M,N/ΓI,J(N)).Finally,in the Serre subcategory of the R-module category,we study the relative properties of generalized local cohomology modules with respect to a pair of ideals(I,J).Some relationships between HI,Ji(N)and HI,Ji(M,N)are considered with the help of Grothendieck spectral sequences theorem.