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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 localeohomology modules Hdim(M) (M) for all ideals a of RIt is also shown that for a giveninteger r ≥ 0, if Hra(R/p) is zero for all p in Supp(M), then Hia(M)=0 for all I ≥ r.