Dual of Bass numbers and dualizing modules

Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, by using relative homological dimensions with respect to $C$, we impose various conditions on $C$ to be dualizing. First, we show that $C$ is dualizing if and only if there exists a Cohen-Macaulay $R$-module of type 1 and of finite G$_C$-dimension. This result extends Takahashi \cite[Theorem 2.3]{T} as well as Christensen \cite[Proposition 8.4]{C}. Next, as a generalization of Xu \cite[Theorem 3.2]{X2}, we show that $C$ is dualizing if and only if for an $R$-module $M$, the necessary and sufficient condition for $M$ to be $C$-injective is that $\pi_i(\fp , M) = 0$ for all $\fp \in \Spec(R)$ and all $i \neq \h(\fp)$, where $\pi_i$ is the invariant dual to the Bass numbers defined by E.Enochs and J.Xu \cite{EX}. We use the later result to give an explicit structure of the minimal flat resolution of \H_{\fm}^d(R) , where $(R, \fm)$ is a $d$-dimensional Cohen-Macaulay local ring possessing a canonical module. As an application, we compute the torsion product of these local cohomology modules.