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Assume that $R =\oplus_{i\in \mathbb{N}_{0}}$ $R_{i}$ is a
homogeneous graded Noetherian ring, and that $M$ is a
$\mathbb{Z}$-graded R-module, where $\mathbb{N}_0$ (resp.
$\mathbb{Z}$) denote the set all non-negative integers (resp.
integers). The set of all homogeneous attached prime ideals of the
top non-vanishing local cohomology module of a finitely generated
module $M$, $H^{c}_{R_{+}}(M)$, with respect to the irrelevant
ideal $R_{+}: \oplus_{i\geq 1} R_{i}$ and the set of associated
primes of $H^{i}_{R_{+}}(M)$ is studied. The asymptotic behavior
of Hom$_{R}(R/R_{+}, H^{s}_{R_+}(M))$ for $s\geq f(M)$ is
discussed, where $f(M)$ is the finiteness dimension of $M$. It is
shown that $H^{h}_{R_{+}}(M)$ is tame if $H^{i}_{R_+}$ is Artinian
for all $i> h$.
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