dimension” after the work of Enochs, Jenda, and others. However, this is a generalization of Auslander and Bridger’s original notion for non-finitely-generated modules. Since we are focused on finitely generated modules here, we continue with Auslander and Bridger’s terminology.

[^3]: It is worth noting that these notions appeared (more or less) implicitly in several places. However, to the best of our knowledge these are the first places where ${\text{G}_C}$-dimension and ${{\mathcal{P}}_C}$-projective dimension were studied explicitly.

[^4]: Ths can be partially explained by the fact that, at the time of the writing of this article, the localization question for G-injective dimension is still not completely answered. Remark \[disc110726a\].

[^5]: In [@hochster:etmf] the term “global canonical module” is used in place of “pointwise dualizing module”. Our choice follows the terminology of Grothendieck and Hartshorne [@hartshorne:rad].

[^6]: We observe the same convention for any homological dimension of the zero module.

[^7]: Most good behavior is also known when $R$ has a dualizing complex, but we restrict our attention to the Cohen-Macaulay case. For instance, the conclusion of Lemma \[lem110610a\] holds when $R$ is only assumed to have a pointwise dualizing complex; the proof is the same, using results from [@christensen:ogpifd] in place of 