Let $R$ be a commutative Noetherian ring with non-zero identity,
$fa$ an ideal of $R$, and $X$ an $R$--module. Here, for fixed
integers $s, t$ and a finite $fa$--torsion $R$--module $N$, we
first study the membership of $Ext^{s+t}_{R}(N, X)$ and
$Ext^{s}_{R}(N, H^{t}_{fa}(X))$ in the Serre subcategories of
the category of $R$--modules. Then, we present some conditions
which ensure the existence of an isomorphism between them.
Finally, we introduce the concept of the Serre cofiniteness as a
generalization of cofiniteness and study this property for
certain local cohomology modules.
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