On the vanishing of local homology modules

Let $R$ be a commutative Noetherian ring, $\text{fa}$ an ideal of $R$ and $M$ an $R$-module. We intend to establish the dual of Grothendieck’s Vanishing Theorem for local homology modules. We conjecture that $H_i^{\text{fa}}(M)=0$ for all $i>{\text{Mag}}_{R}M$. We prove this in several cases.