sequence $$0 \to M \to Z \to N \to 0$$ with $Z = \oplus_{i=0}^r(\omega/(\underline z_{[i]})\omega)^{n_i}$, such that $N$ has locally finite injective dimension and ${\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,N)}){\leqslant}r$.

We extend this in two directions in Theorems \[thm110608a\] and \[thm110530b\]. First, we have the version for G-injective dimension. Second, we give versions relative to a semidualizing module. These are proved in Proofs \[proof110609a\], \[proof110609b\], and \[proof110609c\].

\[thm110608a\] Let $R$ be a Cohen-Macaulay ring with a pointwise dualizing module $\omega$. For any non-zero finitely generated $R$-module $M$ with locally finite G-injective dimension, there exist an integer $r = {\mathrm{G}\text{-}\!\dim}_R({\operatorname{Hom}_{R}(\omega,M)})$, 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 $$0 \to M \to Z \to N \to 0$$ with $Z = \oplus_{i=0}^r(\omega/(\underline z_{[i]})\omega)^{n_i}$ such that $N$ has locally finite G-injective dimension and ${\mathrm{G}\text{-}\!\dim}_R({\operatorname{Hom}_{R}(\omega,N)}){\leqslant}r$.

\[thm110530b\] Let $R$ be a Cohen-Macaulay ring with a pointwise dualizing module $\omega$. Let $M$ a non-zero finitely generated $R$-module, and let $C$ be a semidualizing $R$-module. Set ${C^{\dagger}}={\operatorname{Hom}_{R}(C,\omega)}$. Then

1.  \[thm110530b1\] If $M$ has locally finite ${{\mathcal{GI}}_C}\text{-}{\operatorname{id}}$ and $M$ is in the Auslander class ${{\mathcal{A}}_C}(R)$, then there exist an integer $r = {\mathrm{G}\text{-}\!\dim}({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}M})})$, an $R$-regular sequence $\underline z 