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) s ( sn−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.