The local-global principle for the artinianness dimensions

Abstract

Let $R$ be a commutative noetherian ring and $\mathfrak{a}$ an ideal of $R$. The goal of this paper is to establish the local-global principle for the artinianness dimension $r_{\mathfrak{a}}(M)$, where $r_{\mathfrak{a}}(M)$ is the smallest integer such that the local homology module of $M$ is not artinian. For an artinian $R$-module $M$ with the set ${\mathrm{Coass}}_R{\mathrm{H}}_{r_{\mathfrak{a}}(M)}^{\mathfrak{a}}(M)$ finite, we show that $r_{\mathfrak{a}}(M)=\inf \{r_{\mathfrak{a}{R}_{\mathfrak{p}}}({\mathrm{Hom}}_R(R_{\mathfrak{p}},M))|\mathfrak{p}\in \mathrm{Spec}R\}$. And the class of all modules $N$ such that ${\mathrm{Coass}}_{R}N$ is finite is studied.

9 pages