Vanishing Properties of Dual Bass numbers
arXiv:1005.1754
Abstract
Let be a Noetherian ring, an Artinian -module, $\p\in\Cos_RM$. Then $\cograde_{R_{\p}}\Hom_{R}(R_{\p},M)=\inf\{i | π_{i}(\p,M)>0\}$ and $$π_{i}(\p,M)>0\Rightarrow\cograde_{R_{\p}}\Hom_{R}(R_{\p},M)\leq i\leq\fd_{R_{\p}}\Hom_{R}(R_{\p},M),$$ where $π_{i}(\p,M)$ is the -th dual Bass number of 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 $\Hom_{R}(R_{\p},M)$.
arXiv comment14 pages