since $N'$ has locally finite injective dimension, it is in ${{\mathcal{B}}_C}(R)$, so we have $N'\cong {C\otimes_{R}N}$. This implies that ${\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}N})})={\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,N')})
{\leqslant}r$, as desired.

In the next definition, the special case $C=R$ recovers the complete injective resolutions and Gorenstein injective modules.

\[notation08b’\] Let $C$ be a semidualizing $R$-module. A *complete ${{\mathcal{I}}_C}{{\mathcal{I}}}$-coresolu-tion* is a complex $Y$ of $R$-modules such that $Y$ is exact and ${\operatorname{Hom}_{R}(U,Y)}$ is exact for each $U\in{{\mathcal{I}}_C}(R)$, $Y_i$ is injective for $i{\leqslant}0$, and $Y_i$ is $C$-injective for $i>0$. An $R$-module $N$ is *$\text{G}_C$-injective* if there exists a complete ${{\mathcal{I}}_C}{{\mathcal{I}}}$-coresolution $Y$ such that $N\cong{\operatorname{Ker}}(\partial^Y_0)$, and $Y$ is a *complete ${{\mathcal{I}}_C}{{\mathcal{I}}}$-coresolution of $N$*. Let ${{\mathcal{GI}}_C}(R)$ denote the class of $\text{G}_C$-injective $R$-modules, and set ${{\mathcal{G}}}({{\mathcal{I}}_C})={{\mathcal{GI}}_C}(R)\cap{{\mathcal{A}}_C}(R)$. We let ${{\mathcal{GI}}_C}\text{-}{\operatorname{id}}_R(-)$ and ${{\mathcal{G}}}({{\mathcal{I}}_C})\text{-}{\operatorname{id}}_R(-)$ denote the homological dimensions obtained from coresolutions in ${{\mathcal{GI}}_C}(R)$ and ${{\mathcal{G}}}({{\mathcal{I}}_C})$, respectively.

An $R$-module $M$ has *locally finite ${{\mathcal{G}}}({{\mathcal{I}}_C})\text{-}{\operatorname{id}}$* provided that ${{\mathcal{G}}}({{\mathcal{I}}}_{C_{{{\mathfrak{m}}}}})\text{-}{\operatorname{id}}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}})<\infty$ for each maximal ideal ${{\mathfrak{m}}}\subset R$. An $R$-module $M$ has *locally finite ${{\mathcal{G}}}{{\mathcal{I}}_C}\text{-}{\operatorname{id}}$* provided that ${{\mathcal{G}}}{{\mathcal{I}}}_{C_{{{\mathfrak{m}}}}}\text{-}{\operatorname{id}}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}})<\infty$ for each maximal ideal ${{\mathfrak{m}}}\subset R$.

The next lemma explains the relation between the different Gorenstein injective dimensions.

\[lem0701’\] Let $C$ be a semidualizing $R$-module. For an $R$-module $M$, the following conditions are equivalent:

1.  \[lem0701’i\] ${{\mathcal{G}}}({{\mathcal{I}}_C})\text{-}{\operatorname{id}}_R(M)<\infty$;

2.  \[lem0701’ii\] ${{\mathcal{G}}}{{\mathcal{I}}_C}\text{-}{\operatorname{id}}_R(M)<\infty$ and $M\in{{\mathcal{A}}_C}(R)$; and

3.  