Let R be a commutative Noetherian ring, a an ideal of R and
M a finitely generated Rmodule. Let t be a nonnegative
integer. It is known that if the local cohomology module
H^{i}_{\fraka}(M) is finitely generated for all i < t, then
Hom_{R}(R/\fraka, H^{i}_{\fraka}(M)) is finitely generated. In
this paper it is shown that if H^{i}_{\fraka}(M) is Artinian
for all i < t, then Hom_{R}(R/\fraka, H^{i}_{\fraka}(M))
need not be Artinian, but it has a finitely generated submodule
N such that Hom_{R}(R/\fraka, H^{i}_{\fraka}(M)) is
Artinian.
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