Coartinianess of local homology modules for ideals of small dimension

Abstract

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