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 $0\leq i\leq \mathrm{cd}(\mathfrak{a},M)$, 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 $\mathrm{H}^{\mathfrak{a}}_{i}(M)$ $\mathfrak{a}$-coartinian for all $i\geq 0$; 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 $\mathrm{cd}(\mathfrak{a},R)\leq 1$ and $\mathrm{dim}R/\mathfrak{a}\leq 1$.