Attached primes of the top local cohomology modules with respect to an ideal (II)

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For a finitely generated module M, over a commutative Noetherian local ring (R, m), it is shown that there exist only a finite number of non--isomorphic top local cohomology modules Hₐdim (M)(M), for all ideals a of R. We present a reduced secondary representation for the top local cohomology modules with respect to an ideal. It is also shown that for a given integer r≥ 0, if Hₐʳ(R/p) is zero for all p in Supp(M), then Hₐⁱ(M)= 0 for all i≥ r.