n_r  {\geqslant}0$ with $n_r{\geqslant}1$, and an exact sequence $$0 \to M \to Z \to N \to 0$$ with $Z = \oplus_{i=0}^r(R/(\underline z_{[i]}))^{n_i}$ and ${\mathrm{G}\text{-}\!\dim}_R(N) {\leqslant}r$.

2.  \[thm110527a2\] If ${\operatorname{pd}}_R(M) = r < \infty$, 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 $Z = \oplus_{i=0}^r(R/(\underline z_{[i]}))^{n_i}$ and ${\operatorname{pd}}_R(N) {\leqslant}r$.

The point of Theorem \[thm110527a\], as Hochster and Yao indicate, is that the given module of finite G-dimension or projective dimension embeds in a module that *obviously* has finite projective dimension, with some extra control on the G-dimension or projective dimension of the cokernel.

The first of our results, stated next, is for modules of finite ${\text{G}_C}$-dimension and modules of finite ${{\mathcal{P}}_C}$-projective dimension, where $C$ is a semidualizing $R$-module, after Golod [@golod:gdagpi], Holm and Jørgensen [@holm:smarghd] and White [@white:gpdrsm].[^3] (See Section \[sec110530a\] for definitions and background material.) On the one hand, this result is a corollary to Theorem \[thm110527a\]. On the other hand, Theorem \[thm110527a\] is the special case $C=R$, so our result generalizes this one. This result is proved in Proofs \[proof110530a\] and \[proof110530b\], 