Some Finiteness Results for Local Cohomology Modules with Respect to a Pair of Ideals

Suppose that $R$ is a commutative Noetherian ring with identity, $I$, $J$ are ideals of $R$, and let $M$ be a finitely generated $R$-module. Let $H^i_{I,J}(-)$ be the $i$th local cohomology functor with respect to $(I, J)$. In this paper, we show that the $R$-module
$$\mathrm{Hom}_R(R/I,H^1_{I,J}(M)/JH^1_{I,J}(M))$$
is always finitely generated. Moreover, we provide sufficient conditions such that the modules
$$ \mathrm{Hom}_R(R/I,H^i_{I,J}(M)/JH^i_{I,J}(M)) \qquad \mathrm{or} \qquad \mathrm{Tor}^R_j(R/I,H^i_{I,J}(M)/JH^i_{I,J}(M)) $$
is finitely generated.