M$. Thus, an application of ${\operatorname{Hom}_{R}(C,-)}$ to the sequence  yields the next exact sequence: $$\label{proof110609c2}
0 \to M \to {\operatorname{Hom}_{R}(C,Z')} \to {\operatorname{Hom}_{R}(C,N')} \to 0.$$ Note that $Z'$ has locally finite injective dimension, so we have $Z'\in{{\mathcal{B}}_C}(R)$, which implies that ${\operatorname{Hom}_{R}(C,Z')}$ and ${\operatorname{Hom}_{R}(C,N')}$ are in ${{\mathcal{A}}_C}(R)$ by Fact \[disc01xy\](b)–(c). With $Z:={\operatorname{Hom}_{R}(C,Z')}$, we have ${\mathrm{G}\text{-}\!\dim}_R({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}N})}){\leqslant}r$ and $Z \cong \oplus_{i=0}^r({C^{\dagger}}/(\underline z_{[i]}){C^{\dagger}})^{n_i}$, as in Proof \[proof110609b\].

Acknowledgments {#acknowledgments .unnumbered}
===============

We are grateful to the referee for her/his thoughtful comments.

[10]{}

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