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Let $R = \oplus_{n \in \mathbb{N}_{0}}R_{n}$ be a Noetherian
homogeneous ring with local base ring $(R_{0}, \frak{m}_{0})$ and
irre1evant ideal $R+$, let $M$ be a finitely generated graded
$R$-module. In this paper we show that
$H^{1}_{\frak{m}{0}R}(H^{1}_{R+}(M))$ is Artinian and
$H^{i}_{\frak{m}{0}R}(H^{1}_{R+}(M))$ 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 $i \in \mathbb{N}_{0}$,
$H^{i}_{\frak{m}{0}R}(H^{1}_{R+}(M))$ is Artinian if and only if
$H^{i+2}_{\frak{m}{0}R}(H^{1}_{R+}(M))$ is Artinian. We also prove
that $H^{d}_{\frak{m}{0}}R(H^{c}_{R+}(M))$ 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
$H^{2}_{\frak{m}{0}R}(H^{1}_{R+}(M))$ and $H^{3}_{\frak{m}{0}R}(H^{1}_{R+}(M))$ need not be Artinian.
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