  Let $R$ be a commutative Noetherian ring with non-zero identity, $\fa$ an
ideal of $R$, and $X$ an $R$--module. In this paper, 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 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 Serre cofiniteness as a generalization of cofiniteness
and study this property for certain local cohomology modules.
