Local Cohomology With Respect To A Pair Of Sets Of Prime Ideals

Ia [12], the authors introduced an idea for generalization of a local cohomology module, a local cohomology module with respect to a pair of ideals I, J.In this paper, we try to generalize this idea to a pair of sets of prime ideals.The main results are the following:(1)With X,Y (?) Spec(R), for integer i≥0, we define the i-th local cohomology module of M with respect to (X, Y), W_X,Yⁱ(M).And we get in f{i|H_X,Yⁱ(M)≠0} = in f{depthM_p|p∈W(X,Y)} for Noether ring R and finite R module M;(2)For an ideal J in R, and R module M, we introduce the concept of (J, M) regular element and (J, M) weak regular sequence, and we get theorem: if there exists a (J, M) weak regular sequence of length r in 7, then H_I,Jⁱ(M) = 0, for i < r.