and by Vasconcelos [@vasconcelos:dtmc §4.4] for arbitrary $C$, with different notation.

\[notation08d\] Let $C$ be a semidualizing $R$-module. The *Auslander class* of $C$ is the class ${{\mathcal{A}}_C}(R)$ of $R$-modules $M$ such that ${\operatorname{Tor}}^R_{{\geqslant}1}(C,M)=0={\operatorname{Ext}}_R^{{\geqslant}1}(C,C\otimes_R M)$, and the natural map $M\to{\operatorname{Hom}_{R}(C,C\otimes_R M)}$ is an isomorphism. The *Bass class* of $C$ is the class ${{\mathcal{B}}_C}(R)$ of $R$-modules $N$ such that ${\operatorname{Ext}}_R^{{\geqslant}1}(C,M)=0={\operatorname{Tor}}^R_{{\geqslant}1}(C,{\operatorname{Hom}_{R}(C,M)})$, and the natural evaluation map $C\otimes_R{\operatorname{Hom}_{R}(C,N)}\to N$ is an isomorphism.

\[disc110530a\] It is straightforward to show that the Auslander and Bass classes for $C=R$ are trivial: ${{\mathcal{A}}}_{R}(R)$ and ${{\mathcal{B}}}_R(R)$ both contain all $R$-modules.

The following notion was introduced and studied by Holm and Jørgensen [@holm:smarghd] and White [@white:gpdrsm]. Note that in the special case $C=R$, we recover the class of projective $R$-modules and the classical projective dimension.

\[notation08bxy\] Let $C$ be a semidualizing $R$-module. Let ${{\mathcal{P}}_C}(R)$ denote the class of modules $M\cong P\otimes_R C$ with $P$ projective. Modules in ${{\mathcal{P}}_C}(R)$ are called *$C$-projective*. We let ${{\mathcal{P}}_C}\text{-}{\operatorname{pd}}_R(-)$ denote the homological dimension obtained from resolutions in ${{\mathcal{P}}_C}(R)$, with the convention that ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R(0)=-\infty$.[^6]

\[disc01xyz\] Let $C$ be a semidualizing $R$-module. The Bass class ${{\mathcal{B}}_C}(R)$ contains all $R$-modules of finite ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}$; see [@holm:fear Lemmas 4.1 and 5.1, and Corollary 6.3]. Given an $R$-module $M$, one has 