We show that if R = Ln2N0 Rn is a Noetherian homogeneous ring with local base ring (R0,m0), irrelevant ideal R+, and M a finitely generated graded R-module, then Hj m0R(HtR + (M)) is Artinian for j = 0, 1 where t = inf{i 2 N0 : HiR + (M) is not finitely generated}. Also, we prove that if cd(R+,M) = 2, then for each i 2 N0, Him 0R(H2R + (M)) is Artinian if and only if Hi+2 m0R(H1R + (M)) is Artinian, where cd(R+,M) is the cohomological dimension of M with respect to R+. This improves some results of R. Sazeedeh.
