definition due to Enochs and Jenda [@enochs:gipm].

\[notation110609a\] A *complete injective resolution* is an exact complex $Y$ of injective $R$-modules such that ${\operatorname{Hom}_{R}(J,Y)}$ is exact for each injective $R$-module $J$. An $R$-module $N$ is *G-injective* if there exists a complete injective resolution $Y$ such that $N\cong{\operatorname{Ker}}(\partial^Y_0)$, and $Y$ is a *complete injective resolution of $N$*. Let ${{\mathcal{GI}}}(R)$ denote the class of G-injective $R$-modules, and let ${\operatorname{Gid}}_R(-)$ denote the homological dimension obtained from coresolutions in ${{\mathcal{GI}}}(R)$. We say that an $R$-module $M$ has *locally finite G-injective dimension* provided that ${\operatorname{Gid}}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}})<\infty$ for each maximal ideal ${{\mathfrak{m}}}\subset R$.

\[disc110726a\] Using existing technology, the modules of finite G-injective dimension behave best when $R$ is Cohen-Macaulay with a dualizing module.[^7] For instance, in general it is not known whether the G-injective dimension localizes; but it is known to localize when $R$ has a dualizing module. This is due to the following connection with Bass classes, the local case of which is from [@enochs:fdgipm].[^8]

\[lem110610a\] Assume that $R$ is Cohen-Macaulay with a pointwise dualizing module. Let $M$ be an $R$-module. Then an $R$-module $M$ has locally finite G-injective dimension if and only if $M\in{{\mathcal{B}}}_{\omega}(R)$.

If $R$ is local, then the result follows from [@enochs:fdgipm Proposition 1.4 