  We study homological behavior of modules satisfying the Auslander condition.
Assume that $\mathcal{AC}$ is the class of left $R$-modules satisfying the
Auslander condition. It is proved that each cycle of an exact complex with each
term in $\mathcal{AC}$ belongs to $\mathcal{AC}$ for any ring $R$. As a
consequence, we show that for any left Noetherian ring $R$, $\mathcal{AC}$ is a
resolving subcategory of the category of left $R$-modules if and only if $_RR$
satisfies the Auslander condition if and only if each Gorenstein projective
left $R$-module belongs to $\mathcal{AC}$. As an application, we prove that,
for an Artinian algebra $R$ satisfying the Auslander condition, $R$ is
Gorenstein if and only if $\mathcal{AC}$ coincides with the class of Gorenstein
projective left $R$-modules if and only if $({\mathcal{AC}^{<
\infty}},(\mathcal{AC}^{<\infty})^\bot)$ is a tilting-like cotorsion pair if
and only if (${\mathcal{AC}^{< \infty}},\mathcal{I}$) is a tilting-like
cotorsion pair, where $\mathcal{AC}^{<\infty}$ is the class of left $R$-modules
with finite $\mathcal{AC}$-dimension and $\mathcal{I}$ is the class of
injective left $R$-modules. This leads to some criteria for the validity of the
Auslander and Reiten conjecture which says that an Artinian algebra satisfying
the Auslander condition is Gorenstein.
