Filter Depth and Cofiniteness of Local Cohomology Modules Defined by a Pair of Ideals

Let I, J be ideals of a commutative Noetherian local ring (R, m) and let M be a finite R-module. The f-depth of M with respect to I is the least integer r such that H-I(r)(M) is not Artinian. In this paper we show that inff-depth(a, M) a (W) over tilde (I, J)} is the least integer such that the local cohomology module with respect to a pair of ideals I, J is not Artinian. As a consequence, it follows that H-I, J(i)(M) is (I, J)-cofinite for all i < inff-depth(a, M) a <(W)over tilde> I, J)}. In addition, we show that for a Serre subcategory S, if H-I, J(i)(M) belongs to S for all i > n and if b is an ideal of R such that H-I, J(n)(M/bM) belongs to S, then the module H-I, J(n)(M)/bH(I, J)(n)(M) belongs to S.