Let [afr ] be an ideal of a local ring (R, [mfr ]) and let N be a finitely generated R-module of dimension d: It is shown that Hd[afr ](N) ≃ Hd[mfr ](N)/[sum ]n∈ℕ〈[mfr ]〉 (0: Hd[mfr ](N)[afr ]n); where for an Artinian R-module X we put 〈[mfr ]〉X = ∩n∈ℕ[mfr ]nX. As a consequence several vanishing and connectedness results are deduced.
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