  Let $R$ be a commutative Noetherian ring, $\fa$ 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 $H^{\fa}_i(M)=0$ for all
$i>\Mag_RM$. We prove this in several cases.
