Coartinianess of local homology modules for ideals of small dimension

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