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Paper IPM / M / 8718  


Abstract:  
Let R = ⊕_{n ∈ \mathbbN0}R_{n} be a Noetherian
homogeneous ring with local base ring (R_{0}, \frakm_{0}) and
irre1evant ideal R+, let M be a finitely generated graded
Rmodule. In this paper we show that
H^{1}_{\frakm0R}(H^{1}_{R+}(M)) is Artinian and
H^{i}_{\frakm0R}(H^{1}_{R+}(M)) is Artinian for each i in
the case where R+ is principal. Moreover, for the case where
ara(R+) = 2, we prove that, for each i ∈ \mathbbN_{0},
H^{i}_{\frakm0R}(H^{1}_{R+}(M)) is Artinian if and only if
H^{i+2}_{\frakm0R}(H^{1}_{R+}(M)) is Artinian. We also prove
that H^{d}_{\frakm0}R(H^{c}_{R+}(M)) is Artinian, where d = dim(R_{0}) and c is the cohomological dimension of M with
respect to R+. Finally we present some examples which show that
H^{2}_{\frakm0R}(H^{1}_{R+}(M)) and H^{3}_{\frakm0R}(H^{1}_{R+}(M)) need not be Artinian.
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