Homological dimensions under Foxby equivalence

Let $R$ and $S$ be rings and $_{R}C_{S}$ a semidualizing bimodule, and let $\mathscr{T}$ be a subcategory of the Auslander class $\mathcal{A}_C(S)$ and $\mathscr{H}=\{C\otimes_ST \mid T \in \mathscr{T}\}$. Then for any left $R$-module $M$, the $\mathscr{T}$-projective dimension of $\mathrm{Hom}_R(C,M)$ is at most the $\mathscr{H}$-projective dimension of $M$, and they are identical when $M$ is in the Bass class $\mathcal{B}_C(R)$. If $_RC_S$ is faithful and $\mathscr{T}$ is resolving, then in a short exact sequence of left $R$-modules, the $\mathscr{H}$-projective dimensions of any two terms can determine an upper bound of that of the third term. Furthermore, we apply these results to the cases of $\mathscr{T}$ being the subcategories of (weak) flat modules, projective modules and $\mathcal{A}_C(S)$ respectively. Some known results are obtained as corollaries.