On the associated prime ideals and support of local cohomology modules on systems of ideals

Let R be a commutative Noetherian ring and M an Rmodule. We begin by some results about the connection between the general local cohomology modules with respect to a system of ideals of R and an arbitrary Serre subcategory of R-modules. Let n = dim R and a be an ideal of R. We show that SuppR(H j a (M)) ⊆ A ∗ (a) ∪ ( ∪n−j i=1 Suppj+i R (H j a (M))) for all j ≥ 0. As a consequence, if R is a semi-local ring and M is a minimax R-module of dimension at most three, then the R-modules H j a (M) have finite sets of associated prime ideals for all j ≥ 0. Moreover, we give some results on the finiteness of the Bass numbers and the Betti numbers and cofiniteness of the ordinary local cohomology modules.