and Theorems 1.6 and 2.5]. By definition, the condition “$M$ has locally finite G-injective dimension” is a local condition. Fact \[disc01xy\](b) shows that the condition “$M\in{{\mathcal{B}}}_{\omega}(R)$” is also a local condition. Note that Fact \[fact110527a\](a) implies that for each maximal ideal ${{\mathfrak{m}}}\subset R$, the localization $\omega_{{{\mathfrak{m}}}}$ is a dualizing module for the Cohen-Macaulay local ring $R_{{{\mathfrak{m}}}}$. Thus, the general result follows from the local case.

The next result shows that the G-dimension of a finitely generated module behaves like its projective dimension.

\[lem110610b\] Let $M$ be a finitely generated $R$-module. The following conditions are equivalent:

1.  \[lem110610b1\] ${\mathrm{G}\text{-}\!\dim}_{R}(M)<\infty$;

2.  \[lem110610b2\] ${\mathrm{G}\text{-}\!\dim}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}})<\infty$ for each maximal ideal ${{\mathfrak{m}}}\subset R$.

When $R$ is Cohen-Macaulay with a pointwise dualizing module $\omega$, these conditions are equivalent to the following:

1.  \[lem110610b3\] $M\in{{\mathcal{A}}}_{\omega}(R)$.

The implication $\eqref{lem110610b1}\implies\eqref{lem110610b2}$ follows from the inequality ${\mathrm{G}\text{-}\!\dim}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}}){\leqslant}{\mathrm{G}\text{-}\!\dim}_R(M)$, which is straightforward to verify. The implication $\eqref{lem110610b2}\implies\eqref{lem110610b1}$ is from [@avramov:rrc1 Theorem 3.3]. When $R$ is Cohen-Macaulay with a pointwise dualizing module $\omega$, the equivalence $\eqref{lem110610b2}\iff$(iii) follows from the local results [@enochs:fdgipm Proposition 1.3 and Theorems 1.6 and 2.1] as in the proof of Lemma \[lem110610a\].[^9]

\[proof110609a\] Assume that $R$ is Cohen-Macaulay with a pointwise dualizing module $\omega$ and that $M$ is a non-zero finitely 