generated $R$-module $G$ is *totally $C$-reflexive* if the natural map $G\to{\operatorname{Hom}_{R}({\operatorname{Hom}_{R}(G,C)},C)}$ is an isomorphism, and ${\operatorname{Ext}_{R}^{{\geqslant}1}(G,C)}=0={\operatorname{Ext}_{R}^{{\geqslant}1}({\operatorname{Hom}_{R}(G,C)},C)}$. Let ${{\mathcal{G}}_C}(R)$ denote the class of totally $C$-reflexive $R$-modules, and set ${{\mathcal{G}}}(C)={{\mathcal{G}}_C}(R)\cap{{\mathcal{B}}_C}(R)$. In the case $C=R$ we use the more common terminology “totally reflexive” and the notation ${{\mathcal{G}}}(R)={{\mathcal{G}}}_R(R)={{\mathcal{G}}}(R)\cap{{\mathcal{B}}}_R(R)$. We abbreviate as follows: $$\begin{aligned}
{{\mathrm{G}_{C}\text{-}\!\dim}}_R(-)
&=\text{the homological dimension obtained from resolutions  in ${{\mathcal{G}}_C}(R)$}
\\
{{\mathcal{G}}}(C)\text{-}{\operatorname{pd}}_R(-)
&=\text{the homological dimension obtained from resolutions  in ${{\mathcal{G}}}(C)$}
\\
{\mathrm{G}\text{-}\!\dim}_R(-)
&=\text{the homological dimension obtained from resolutions  in ${{\mathcal{G}}}(R)$.}\end{aligned}$$

The following facts are included for perspective.

Let $C$ be a semidualizing $R$-module, and let $M$ be a finitely generated $R$-module. Because of the containments ${{\mathcal{P}}_C}(R)\subseteq{{\mathcal{G}}}(C)\subseteq{{\mathcal{G}}_C}(R)$, one has ${{\mathrm{G}_{C}\text{-}\!\dim}}_R(M){\leqslant}{{\mathcal{G}}}(C)\text{-}{\operatorname{pd}}_R(M){\leqslant}{{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R(M)$ with equality to the left of any finite quantity. In particular, the case $C=R$ says that ${\mathrm{G}\text{-}\!\dim}_R(M)={{\mathcal{G}}}(R)\text{-}{\operatorname{pd}}_R(M){\leqslant}{\operatorname{pd}}_R(M)$ since ${{\mathcal{G}}}(R)={{\mathcal{G}}}_R(R)$, with equality holding when ${\operatorname{pd}}_R(M)<\infty$.

The next lemma explains how these homological dimensions are connected.

\[lem0701x\] Let $C$ be a semidualizing $R$-module. For a finitely generated $R$-module $M$, the following conditions are equivalent:

1.  \[lem0701xi\] ${{\mathcal{G}}}(C)\text{-}{\operatorname{pd}}_R(M)<\infty$;

2.  \[lem0701xii\] ${{\mathrm{G}_{C}\text{-}\!\dim}}_R(M)<\infty$ and $M\in{{\mathcal{B}}_C}(R)$; and

3.  \[lem0701xiii\] ${\mathrm{G}\text{-}\!\dim}_R({{\operatorname{Hom}_{R}(},C)}M)<\infty$ and $M\in{{\mathcal{B}}_C}(R)$.

When these conditions are satisfied, we have $$\begin{aligned}
{{\mathcal{G}}}(C)\text{-}{\operatorname{pd}}_R(M)
&={{\mathrm{G}_{C}\text{-}\!\dim}}_R(M)
={\mathrm{G}\text{-}\!\dim}_R({{\operatorname{Hom}_{R}(},C)}M).\end{aligned}$$

Now we are in position to complete the proof of Theorem \[thm110527b\].

\[proof110530b\] Assume that $M$ 