generated $R$-module with locally finite G-injective dimension. Lemma \[lem110610a\] implies that $M \in \mathcal B_{\omega}(R)$. Using Foxby equivalence (from Fact \[disc01xy\](c)) we conclude that ${\operatorname{Hom}_{R}(\omega,M)}\in{{\mathcal{A}}}_{\omega}(R)$, so Lemma \[lem110610b\] implies that $r:={\mathrm{G}\text{-}\!\dim}_R({\operatorname{Hom}_{R}(\omega,M)})<\infty$. The fact that $R$ is noetherian implies that ${\operatorname{Hom}_{R}(\omega,M)}$ is finitely generated. (The proof concludes as in Proof \[proof110530b\]. We include the details for the sake of thoroughness.)

Since $M\neq 0$, we have ${\operatorname{Hom}_{R}(\omega,M)}\neq 0$, by Fact \[fact110527a\](b), so Theorem \[thm110527a\] implies that there are an $R$-regular sequence $\underline z = z_1, \ldots, z_r$, integers $n_0, n_1, \ldots , n_r  {\geqslant}0$ with $n_r{\geqslant}1$, and an exact sequence $$\label{proof110609a1}
0 \to {\operatorname{Hom}_{R}(\omega,M)} \to Z' \to N' \to 0$$ with $Z' = \oplus_{i=0}^r(R/(\underline z_{[i]}))^{n_i}$ and ${\mathrm{G}\text{-}\!\dim}_R(N') {\leqslant}r$. Fact \[disc01xy\](a) implies that $Z'\in{{\mathcal{A}}}_\omega(R)$. From Fact \[disc01xy\](b), we conclude that $N'\in{{\mathcal{A}}}_\omega(R)$, which implies that ${\operatorname{Tor}^{R}_{1}(\omega,N')}=0$. Thus, an application of ${\omega\otimes_{R}-}$ to the sequence  yields the next exact sequence: $$\label{proof110609a2}
0 \to {\omega\otimes_{R}{\operatorname{Hom}_{R}(\omega,M)}} \to {\omega\otimes_{R}Z'} \to {\omega\otimes_{R}N'} \to 0.$$ The assumption $M\in{{\mathcal{B}}}_\omega(R)$ gives an isomorphism ${\omega\otimes_{R}{\operatorname{Hom}_{R}(\omega,M)}}\cong M$. The equality $Z' = \oplus_{i=0}^r(R/(\underline z_{[i]}))^{n_i}$ implies that $Z:={\omega\otimes_{R}Z'} = \oplus_{i=0}^r(\omega/(\underline z_{[i]})\omega)^{n_i}$.

The condition $N'\in{{\mathcal{A}}}_\omega(R)$ implies that $N:={\omega\otimes_{R}N'}\in{{\mathcal{B}}}_\omega(R)$ by Fact \[disc01xy\](c), and $N'\cong{\operatorname{Hom}_{R}(\omega,N)}$ by the definition of what it means for $N'$ to be in ${{\mathcal{A}}}_\omega(R)$. Thus, 