such that $M_{{{\mathfrak{m}}}}$ is a semidualizing $R_{{{\mathfrak{m}}}}$-module for each maximal ideal ${{\mathfrak{m}}}\subset R$, then $M$ is a semidualizing $R$-module. Similarly, if $C$ is a (pointwise) dualizing module for $R$, then for each multiplicatively closed subset $U\subseteq R$ the localization $U^{-1}C$ is a (pointwise) dualizing module for $U^{-1}R$; and conversely, if $M$ is a finitely generated $R$-module such that $M_{{{\mathfrak{m}}}}$ is a dualizing module for $R_{{{\mathfrak{m}}}}$ for each maximal ideal ${{\mathfrak{m}}}\subset R$, then $M$ is a pointwise dualizing module for $R$. Moreover, an $R$-module $M$ is a dualizing module for $R$ if and only if it is a pointwise dualizing module for $R$ and the Krull dimension of $R$ is finite.

\(b) Let $C$ be a semidualizing $R$-module. The isomorphism $R\cong{\operatorname{Hom}_{R}(C,C)}$ implies that ${\operatorname{Supp}}_R(C)={\operatorname{Spec}}(R)$ and ${\operatorname{Ass}}_R(C)={\operatorname{Ass}}(R)$. Thus, an element $x\in R$ is $C$-regular if and only if it is $R$-regular. When $x$ is $R$-regular, the quotient $C/xC$ is a semidualizing $R/xR$-module. Thus, by induction, a sequence $\underline z=z_1,\ldots,z_r\in R$ is $R$-regular if and only if it is $C$-regular; see [@frankild:rrhffd Theorem 4.5]. Also, from [@holm:fear Proposition 3.1] we know that, for all $R$-modules $M\neq 0$, we have ${C\otimes_{R}M}\neq 0\neq{\operatorname{Hom}_{R}(C,M)}$.

The next categories were introduced by Foxby [@foxby:gdcmr] when $C$ is dualizing, 