which are similar to the proof of [@hochster:etmf Theorem 4.2].

\[thm110527b\] Let $M$ be a non-zero finitely generated $R$-module, and let $C$ be a semidualizing $R$-module. Then

1.  \[thm110527b1\] If ${{\mathrm{G}_{C}\text{-}\!\dim}}_R(M) = r < \infty$ and $M$ is in the Bass class ${{\mathcal{B}}_C}(R)$, then there exist 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 $$0 \to M \to Z \to N \to 0$$ with $Z = \oplus_{i=0}^r(C/(\underline z_{[i]})C)^{n_i}$ and ${{\mathrm{G}_{C}\text{-}\!\dim}}_R(N) {\leqslant}r$ and $N\in{{\mathcal{B}}_C}(R)$.

2.  \[thm110527b2\] If ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R(M) = r < \infty$, then there exist 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 $$0 \to M \to Z \to N \to 0$$ with $Z = \oplus_{i=0}^r(C/(\underline z_{[i]})C)^{n_i}$ and ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R(N) {\leqslant}r$.

The next result we extend is the following:

\[thm110530a\] Let $R$ be a Cohen-Macaulay ring with a pointwise dualizing module $\omega$. Then, for any non-zero finitely generated $R$-module $M$ with locally finite injective dimension, there exist an integer $r = {\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,M)})$, 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 