Cofiniteness Of Generalized Local Cohomology Modules

Let R be commutative Noether ring with non-zero identity,I,J be two ideals of R,M,N be finitely generated R-modules.We study the cofiniteness of generalized local cohomology modules HIi(M,N)and generalized local cohomology modules H7I,Ji(M,N)with respect to a pair of ideals by using a spectral sequence argument.Firstly,we study the cofiniteness of generalized local cohomology modules HIi(M,N)under the conditions of small dimensions,it is show that if either dimM ≤2 and HIi(N)are I-cofinite R-module for all i≥0,or HIi(N)is an I-cofinite module of dimension ≤1 for each i≥ 1,or q(I,R)≤1,then the R-modules HIt(M,N)are cofinite.As an application,we prove that if ExtRp(M,HIq(N))is finitely generated for all p,q≥ 0,then R-module HIt(M,N)is finitely generated.Secondly,we investigate the cofiniteness of HI,Ji(M,N)when pdRM<∞,prove that if N is(I,J)-torsion R-module,then HI,Ji(M,N)is cofinite module for all i;if HI,Jt(M,R/p)is minimax cofinite R-module for all p ∈SuppRN,then HI,Jt(M,N)is minimax cofinite.Finally,we discuss that the minimaxness and(I,J)-cofiniteness of HI,Ji(M,N)when cd(I,J,N)≤1.