we have ${\mathrm{G}\text{-}\!\dim}({\operatorname{Hom}_{R}(\omega,N)})={\mathrm{G}\text{-}\!\dim}_R(N') {\leqslant}r$. So, the sequence  satisfies the conclusion of Theorem \[thm110608a\].

For our next results, we need a better understanding of the relationship between semidualizing modules and a pointwise dualizing module.

\[lem110611a\] Assume that $R$ is Cohen-Macaulay with a pointwise dualizing module $\omega$. If $C$ is a semidualizing $R$-module, then so is ${\operatorname{Hom}_{R}(C,\omega)}$.

In the local case, this result is standard; see, e.g., [@sather:bnsc Facts 1.18–1.19]. Since the semidualizing and (pointwise) dualizing conditions are local by Fact \[fact110527a\](a), the general case of the result follows.

The next two lemmas elaborate on the local-global behavior for our invariants.

\[lem110611d\] Let $M$ be a finitely generated $R$-module, and let $C$ be a semidualizing $R$-module. Then ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_{R}(M)<\infty$ if and only if $\text{P}_{C_{{{\mathfrak{m}}}}}\text{-}{\operatorname{pd}}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}})<\infty$ for each maximal ideal ${{\mathfrak{m}}}\subset R$.

The forward implication follows from the inequality $\text{P}_{C_{{{\mathfrak{m}}}}}\text{-}\dim_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}}){\leqslant}{{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R(M)$, which is straightforward to verify. For the converse, assume that $\text{P}_{C_{{{\mathfrak{m}}}}}\text{-}{\operatorname{pd}}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}})<\infty$ for each maximal ideal ${{\mathfrak{m}}}\subset R$. It follows that the module ${\operatorname{Hom}_{R}(C,M)}_{{{\mathfrak{m}}}}\cong{\operatorname{Hom}_{R_{{{\mathfrak{m}}}}}(C_{{{\mathfrak{m}}}},M_{{{\mathfrak{m}}}})}$ has finite projective dimension over $R_{{{\mathfrak{m}}}}$ for all ${{\mathfrak{m}}}$. Hence, the finitely generated $R$-module ${\operatorname{Hom}_{R}(C,M)}$ has finite projective dimension, so Fact \[disc01xyz\] implies that ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_{R}(M)$ is finite.

The next lemma is a souped-up version of a special case of a result of 