Coartinianess of local homology modules for ideals of small dimension

Abstract

Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$ and $M$ an $R$-module with Cosupport in $\mathrm{V}(\mathfrak{a})$. We show that $M$ is $\mathfrak{a}$-coartinian if and only if ${\mathrm{Ext}}_R^i(R/\mathfrak{a},M)$ is artinian for all $0\le i\le \mathrm{cd}(\mathfrak{a},M)$, which provides a computable finitely many steps to examine $\mathfrak{a}$-coartinianness. We also consider the duality of Hartshorne's questions: for which rings $R$ and ideals $\mathfrak{a}$ are the modules ${\mathrm{H}}_i^{\mathfrak{a}}(M)$ $\mathfrak{a}$-coartinian for all $i\ge 0$; whether the category $\mathcal{C}(R,\mathfrak{a}{)}_{coa}$ of $\mathfrak{a}$-coartinian modules is an Abelian subcategory of the category of all $R$-modules, and establish affirmative answers to these questions in the case $\mathrm{cd}(\mathfrak{a},R)\le 1$ and $\dim R/\mathfrak{a}\le 1$.

14 pages. Comments welcome