Vanishing Properties of Dual Bass numbers

TL;DR

This paper investigates the vanishing and properties of dual Bass numbers of Artinian modules over Noetherian rings, establishing relationships with cograde, co-dimension, and flat dimension, and providing new insights into their interplay.

Contribution

It introduces new vanishing properties of dual Bass numbers and explores their connections with cograde, co-dimension, and flat dimension in the context of Artinian modules.

Findings

Established the equality between cograde and the infimum of indices with positive dual Bass numbers.

Derived bounds relating dual Bass numbers to cograde and flat dimension.

Analyzed the relationships among cograde, co-dimension, and flat dimension of co-localization modules.

Abstract

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 $\Hom_{R}(R_{\p},M)$.