Let $R$ be a Noetherian ring, $M$ an Artinian $R$-module, $\p\in\Cos_RM$. Then $\cograde_{ R_{ \p}}\Hom_{ R}(R_{ \p},M)=\inf\{ i | \pi_{ i}(\p,M)>0\}$ and $$\pi_{ i}(\p,M)>0\Rightarrow\cograde_{ R_{ \p}}\Hom_{ R}(R_{ \p},M)\leq i\leq\fd_{ R_{ \p}}\Hom_{ R}(R_{ \p},M),$$ where $\pi_{ i}(\p,M)$ is the $i$-th dual Bass number of $M$ with respect to $\p$, the integer $\cograde_{ R_{ \p}}\Hom_{ R}(R_{ \p},M)$ is the common length of any maximal $\Hom_{ R}(R_{ \p},M)$-quasi co-regular sequence contained in $\p R_{ \p}$, and $\fd_{ R_{ \p}}\Hom_{ R}(R_{ \p},M)$ is the flat dimension of $R_{ \p}$-module $\Hom_{ R}(R_{ \p},M)$ (Theorem \ref{ Thm:Main}). Besides, we also study the relations among cograde, co-dimension and flat dimension of co-localization module $\Hom_{ R}(R_{ \p},M)$.
Categories: math.AC