Let $A$ be a commutative noetherian ring, $\mathfrak{a}$ an ideal of $A$, and $m,n$ non-negative integers. $M$ $A$-module such that $\operatorname{Ext}^i\_A(A/\mathfrak{a},M)$ is finitely generated for all $i\leq m+n$. We define class $\mathcal{S}n(\mathfrak{a})$ modules we assume $H{\mathfrak{a}}^s(M)\in\mathcal{S}{n}(\mathfrak{a})$ $s\leq m$. show $H{\mathfrak{a}}^s(M)$ $\mathfrak{a}$-cofinit...

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 exi...