Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Vanishing Properties of Dual Bass numbers   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 $\Hom_{R}(R_{\p},M)$.

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The categories of this paper are: supr-con
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: cs
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The categories of this paper are: quant-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
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The categories of this paper are: q-alg
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The categories of this paper are: adap-org