  The notion of linkage with respect to a semidualizing module is introduced.
It is shown that over a Cohen-Macaulay local ring with canonical module, every
Cohen-Macaulay module of finite Gorenstein injective dimension is linked with
respect to the canonical module. For a linked module $M$ with respect to a
semidualizing module, the connection between the Serre condition $(S_n)$ on $M$
with the vanishing of certain local cohomology modules of its linked module is
discussed.
