  Let $R$ be semiperfect commutative Noetherian ring and $C$ be a semidualizing
$R$--module. The connection of the Serre condition $(S_n)$ on a horizontally
linked $R$-module of finite $\gc$-dimension with the vanishing of certain
cohomology modules of its linked module is discussed. As a consequence, it is
shown that under some conditions Cohen-Macaulayness is preserved under
horizontally linkage.
