Cofiniteness of Local Cohomlogy Based on a Non–Closed Support Defined by a Pair of Ideals

Let I, J be ideals of a commutative Noetherian ring R and let t be a non-negative integer. Let M be an R-module such that Ext tR(R/I,M) is a finite R-module. If t is the first integer such that the local cohomology module with respect to (I, J) is non- (I, J)-cofinite, then we show that Hom R(R/I, HTI,J(M)) is finite. Also, we study the finiteness of Ext iR(R/I, HtI,J(M)), for i = 1, 2. In addition, for a finite R-module M, we show that the associated primes of HtI, J(M) have an equal grade, when t = inf{i|HiI, J(M) = 0}