Coartinianess of local homology modules for ideals of small dimension.

Let be an ideal of a commutative noetherian ring R and M an R -module with Cosupport in V (). We show that M is -coartinian if and only if Ext R i (R / , M) is artinian for all 0 ≤ i ≤ cd (, M) , which provides finite steps to examine -coartinianess. We also consider the duality of Hartshorne's questions: for which rings R and ideals are the modules H i (M) -coartinian for every artinian R -module M and all i ≥ 0 ; whether the category (R ,) coa of -coartinian modules is an abelian subcategory of the category of R -modules, and establish affirmative answers to these questions in the case cd (, R) ≤ 1 and dim R / ≤ 1. [ABSTRACT FROM AUTHOR]