= z_1, \ldots , z_r$, non-negative integers $n_0, n_1, \ldots , n_r$ with $n_r{\geqslant}1$, and an exact sequence $$0 \to M \to Z \to N \to 0$$ such that $N$ has locally finite ${{\mathcal{GI}}_C}\text{-}{\operatorname{id}}$, and $Z = \oplus_{i=0}^r({C^{\dagger}}/(\underline z_{[i]}){C^{\dagger}})^{n_i}$, $N \in {{\mathcal{A}}_C}(R)$ and ${\mathrm{G}\text{-}\!\dim}({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}N})}){\leqslant}r$.

2.  \[thm110530b2\] If $M$ has locally finite ${{\mathcal{I}}_C}\text{-}{\operatorname{id}}$, then there exist an integer $$r = {\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}M})})$$ an $R$-regular sequence $\underline z = z_1, \ldots , z_r$, non-negative integers $n_0, n_1, \ldots , n_r$ with $n_r{\geqslant}1$, and an exact sequence $$0 \to M \to Z \to N \to 0$$ such that $N$ has locally finite ${{\mathcal{I}}_C}\text{-}{\operatorname{id}}$, and $Z = \oplus_{i=0}^r({C^{\dagger}}/(\underline z_{[i]}){C^{\dagger}})^{n_i}$ and ${\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}N})}){\leqslant}r$.

We summarize the organization of this paper. Section \[sec110530a\] contains the proof of Theorem \[thm110527b\], along with the necessary background material. Section \[sec110608a\] similarly treats Theorems \[thm110608a\] and \[thm110530b\]; this section also contains several lemmas about modules of locally finite G-injective dimension, locally finite ${\text{G}_C}$-injective dimension, and locally finite ${{\mathcal{I}}_C}$-injective dimension where $C$ is a semidualizing module, as these notions have not been developed in the literature as best we know.[^4] Note that we do not develop the background material in the most optimal manner, focusing instead on accessibility. To be specific, we present 