background material as it is needed in Sections \[sec110530a\] and \[sec110608a\] instead of putting it all in its own section. Also, we feel that the definition of locally finite ${{\mathcal{GI}}_C}\text{-}{\operatorname{id}}$ is a bit much to swallow on the first bite, so we first discuss the special case of ${{{\mathcal{G}}\text{-}{\operatorname{id}}}}$ and work our way up.

Proof of Theorem \[thm110527b\] {#sec110530a}
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\[notn1xx\] We let ${\operatorname{pd}}_R(-)$ and ${\operatorname{id}}_R(-)$ denote the classical projective dimension and the classical injective dimension, respectively.

The study of semidualizing modules was initiated independently (with different names) by Foxby [@foxby:gmarm], Golod [@golod:gdagpi], and Vasconcelos [@vasconcelos:dtmc]. For example, a finitely generated projective $R$-module of rank 1 is semidualizing.

\[notation08a\] A finitely generated $R$-module $C$ is *semidualizing* if the natural map $R\to{\operatorname{Hom}_{R}(C,C)}$ is an isomorphism, and ${\operatorname{Ext}}_R^{{\geqslant}1}(C,C)=0$. An $R$-module $C$ is *pointwise dualizing* if it is semidualizing and ${\operatorname{id}}_{R_{{{\mathfrak{m}}}}}(C_{{{\mathfrak{m}}}})<\infty$ for all maximal ideals ${{\mathfrak{m}}}\subset R$.[^5] An $R$-module $C$ is *dualizing* if it is semidualizing and ${\operatorname{id}}_R(C)<\infty$.

\[fact110527a\] (a) It is straightforward to show that the semidualizing property is local. That is, if $C$ is a semidualizing $R$-module, then for each multiplicatively closed subset $U\subseteq R$ the localization $U^{-1}C$ is a semidualizing $U^{-1}R$-module; and conversely, if $M$ is a finitely generated $R$-module 