  Let $(R,\fm)$ be a Cohen-Macaulay local ring. If $R$ has a canonical module,
then there are some interesting results about duality for this situation. In
this paper, we show that one can indeed obtain similar these results in the
case $R$ has not a canonical module. Also, we give some characterizations of
complete big Cohen-Macaulay modules of finite injective dimension and by using
it some characterizations of Gorenstein modules over the $\fm$-adic completion
of $R$ are obtained.
