On the finiteness properties of extension and torsion functors of local cohomology modules

Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ and $\mathfrak{b}$ ideals of $R$ with $\mathfrak{a} \subseteq \mathfrak{b}$, and $M$ a finitely generated $R$-module. In this paper, for fixed integers $j$ and $n$, we study the finiteness of $\operatorname{Ext}^j_R(R/\mathfrak{b},H^n_{\mathfrak{a}}(M))$ and $Tor_j^R(R/\mathfrak{b},H^n_{\mathfrak{a}}(M))$ in several cases.