Vanishing Properties of Dual Bass numbers

Let $R$ be a Noetherian ring, $M$ an Artinian $R$-module, $\text{p}\in {\text{Cos}}_{R}M$. Then ${\text{cograde}}_{R_{\text{p}}}{\text{Hom}}_R(R_{\text{p}},M)=\inf \{i\mathrm{\mid }{\pi }_i(\text{p},M)>0\}$ and $${\pi }_i(\text{p},M)>0\Rightarrow {\text{cograde}}_{R_{\text{p}}}{\text{Hom}}_R(R_{\text{p}},M)\le i\le {\text{fd}}_{R_{\text{p}}}{\text{Hom}}_R(R_{\text{p}},M),$$ where ${\pi }_i(\text{p},M)$ is the $i$-th dual Bass number of $M$ with respect to $\text{p}$, the integer ${\text{cograde}}_{R_{\text{p}}}{\text{Hom}}_R(R_{\text{p}},M)$ is the common length of any maximal ${\text{Hom}}_R(R_{\text{p}},M)$-quasi co-regular sequence contained in $\text{p}{R}_{\text{p}}$, and ${\text{fd}}_{R_{\text{p}}}{\text{Hom}}_R(R_{\text{p}},M)$ is the flat dimension of $R_{\text{p}}$-module ${\text{Hom}}_R(R_{\text{p}},M)$ (Theorem \ref{Thm:Main}). Besides, we also study the relations among cograde, co-dimension and flat dimension of co-localization module ${\text{Hom}}_R(R_{\text{p}},M)$.