Vanishing Theorem of Dual Bass Numbers

Details Vanishing Theorem of Dual Bass Numbers Vanishing Theorem of Dual Bass Numbers

2010-05-11

arxiv.org/abs/1005.1754v4

Mathematics

Commutative Algebra

16 pages

Scientific paper

In this paper, we prove a vanishing theorem of Dual Bass numbers (Theorem 5.10). Precisely, let $R$ be a $U$ ring, $M$ an Artinian $R$-module, $\frak{p}\in\textmd{Cos}_{R}~M$, if $\pi_{i}(\frak{p},M)>0$, then $$\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)\leq i\leq\textmd{fd}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M),$$ where $\pi_{i}(\frak{p},M)=\textmd{dim}_{k(\frak{p})}\textmd{Tor}^{R_{\frak{p}}}_{i}(k(\frak{p}),\textmd{Hom}_{R}(R_{\frak{p}},M))$ is the $i$-th Dual Bass number of $M$ with respect to $\frak{p}$, and the flat dimension $\textmd{fd}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)$ is possibly infinite. Moveover, if $\textmd{Cograde}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)=s, ~\textmd{fd}_{R_{\frak{p}}}\textmd{Hom}_{R}(R_{\frak{p}},M)=t<\infty$, then $\pi_{s}(\frak{p},M)>0$ and $\pi_{t}(\frak{p},M)>0$.

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