Ideal topologies, local cohomology and connectedness

Let a be an ideal of a local ring (R, m) and let N be a finitely generated R-module of dimension d. It is shown that H-a(d)(N) similar or equal to H-m(d)(N)/Sigma (n is an element ofN)<m > (0: (Hmd(N))a(n)), where for an Artinian R-module X we put <m >X = boolean AND (n is an element ofN) m(n)X. As a consequence several vanishing and connectedness results are deduced.