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Let $A$ be a commutative Noetherian ring, let $M$ be a finitely
generated $A$-module, and let ${\frak a},{\frak b}$ be ideals of
$A$ with $\frak{b}\subseteq \frak{a}$. In this paper, firstly, we
determine precisely the set of associated primes of the first
non-finitely generated local cohomology module $H^n_{\frak{a}}
(M)$. Then we give an affirmative answer, in certain cases, to the
following question: If, for each prime ideal $\frak{p}$ of $A$,
there exists an integer $k(\frak{p})$ such that
$\frak{b}^{k(\frak{p})} H^i_{\frak{a} A_{\frak{p}}}
(M_{\frak{p}})=0$ for every $i$ less than a fixed integer $n$,
then does there exist an integer $k$ such that $\frak{b}^k
H^i_{\frak{a}} (M)=0$ for all $i< n$. A formulation of this
question is referred as the generalized local-global principle for
the finiteness of local cohomology modules.
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