Coartinianess of local homology modules for ideals of small dimension

TL;DR

This paper characterizes $rak{a}$-coartinian modules over noetherian rings using Ext modules, and explores duality questions related to local homology modules, providing affirmative results for rings with small dimension or cohomological dimension.

Contribution

It provides a new criterion for $rak{a}$-coartinianness via Ext modules and addresses duality questions for local homology modules in low-dimensional cases.

Findings

$rak{a}$-coartinian modules characterized by Ext conditions.

Established duality results for modules with small cohomological dimension.

Proved the category of $rak{a}$-coartinian modules is Abelian in certain cases.

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 $\mathrm{dim}R/\mathfrak{a}\leq 1$.