Takahashi and White that is documented in [@sather:tate1].

\[lem110612a\] Assume that $R$ is Cohen-Macaulay with a pointwise dualizing module $\omega$. For each finitely generated $R$-module $M$, one has ${{\mathcal{P}}}_{\omega}\text{-}{\operatorname{pd}}_R(M)<\infty$ if and only if $M$ has locally finite injective dimension.

When $\omega$ is a dualizing module for $R$, i.e., when $R$ has finite Krull dimension, this result is from [@sather:tate1 Lemma 2.7]. In particular, this takes care of the local case. The general case follows from the local case, by Lemma \[lem110611d\].

Here are some more notions from [@holm:smarghd].

\[notation08b\] Let $C$ be a semidualizing $R$-module, and let ${{\mathcal{I}}_C}(R)$ denote the class of modules $N\cong {\operatorname{Hom}_{R}(C,I)}$ with $I$ injective. Modules in ${{\mathcal{I}}_C}(R)$ are called *$C$-injective*. We let ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}_R(-)$ denote the homological dimension obtained from coresolutions in ${{\mathcal{I}}_C}(R)$. We say that an $R$-module $M$ has *locally finite ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}$* provided that $\text{I}_{C_{{{\mathfrak{m}}}}}\text{-}{\operatorname{id}}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}})<\infty$ for each maximal ideal ${{\mathfrak{m}}}\subset R$.

\[fact110612a\] Let $C$ be a semidualizing $R$-module. The Auslander class ${{\mathcal{A}}_C}(R)$ contains every $R$-module of finite ${{\mathcal{I}}}_C$-injective dimension; see [@holm:fear Lemmas 4.1 and 5.1]. Given an $R$-module $M$, one has ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}_R(M)={\operatorname{id}}_R({C\otimes_{R}M})$ and ${\operatorname{id}}_R(M)={{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}_R({\operatorname{Hom}_{R}(C,M)})$ by [@takahashi:hasm Theorem 2.11]. In particular, one has ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}_R(M)<\infty$ if and only if ${\operatorname{id}}_R({C\otimes_{R}M})<\infty$, and one has ${\operatorname{id}}_R(M)<\infty$ if and only if 