Duality Pairs Induced by Auslander and Bass Classes

Let $R$ and $S$ be any rings and $_RC_S$ a semidualizing bimodule, and let $\mathcal{A}_C(R^{op})$ and $\mathcal{B}_C(R)$ be the Auslander and Bass classes respectively. Then both the pairs $$(\mathcal{A}_C(R^{op}),\mathcal{B}_C(R))\ {\rm and}\ (\mathcal{B}_C(R),\mathcal{A}_C(R^{op}))$$ are coproduct-closed and product-closed duality pairs and both $\mathcal{A}_C(R^{op})$ and $\mathcal{B}_C(R)$ are covering and preenveloping; in particular, the former duality pair is perfect. Moreover, if $\mathcal{B}_C(R)$ is enveloping in $\Mod R$, then $\mathcal{A}_C(S)$ is enveloping in $\Mod S$. Then some applications to the Auslander projective dimension of modules are given.