  For a horizontally linked module, over a commutative semiperfect Noetherian
ring $R$, the connections of its invariants reduced grade, Gorenstein dimension
and depth are studied. It is shown that under certain conditions the depth of a
horizontally linked module is equal to the reduced grade of its linked module.
The connection of the Serre condition $(S_n)$ on an $R$--module of finite
Gorenstein dimension with the vanishing of the local cohomology groups of its
linked module is discussed.
