  Let $R$ and $S$ be rings and $_R\omega_S$ a semidualizing bimodule. We prove
that there exists a Morita equivalence between the class of
$\infty$-$\omega$-cotorsionfree modules and a subclass of the class of
$\omega$-adstatic modules. Also we establish the relation between the relative
homological dimensions of a module $M$ and the corresponding standard
homological dimensions of ${\rm Hom}(\omega,M)$. By investigating the
properties of the Bass injective dimension of modules (resp. complexes), we get
some equivalent characterizations of semi-tilting modules (resp. Gorenstein
artin algebras). Finally we obtain a dual version of the Auslander-Bridger's
approximation theorem. As a consequence, we get some equivalent
characterizations of Auslander $n$-Gorenstein artin algebras.
