On the attached prime ideals of local cohomology modules defined by a pair of ideals

Let I and J be two ideals of a commutative Noetherian ring R and M be an R-module of dimension d. For each i is an element of N-0 let H-I(i),(J)(-) denote the i-th right derived functor of Gamma(I,J)(-), where Gamma(I,J)(M) := {x is an element of M : I-n x subset of Jx for n >> 1}. If R is a complete local ring and M is finite, then attached prime ideals of H-I,J(d-1)(M) are computed by means of the concept of co-localization. Moreover, we illustrate the attached prime ideals of WI, j(M) on a nonlocal ring R, for t - dim M and t - cd (I, J, M), where cd (I, J, M) is the last nonvanishing level of H-I(,J)i(M).