Modules cofinite and weakly cofinite with respect to an ideal

作者：Bahmanpour Kamal; Naghipour Reza^*; Sedghi Monireh

来源：Journal of Algebra and Its Applications, 2018, 17(3): 1850056.

DOI：10.1142/S0219498818500561

• Abelian category
• cofinite module
• local cohomology
• minimax module
• Serre category
• weakly cofinite module
• weakly Laskerian module

The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal a of a Noetherian ring R. It is shown that an R-module M is cofinite with respect to a, if and only if, Ext(R)(i) (R/a, M) is finitely generated for all i <= cd(a, M) + 1, whenever dim R/a = 1. In addition, we show that if M is finitely generated and H-a(i) (M) are weakly Laskerian for all i <= t - 1, then H-a(i) (M) are a-cofinite for all i <= t - 1 and for any minimax submodule K of H-a(t) (M), the R-modules Hom (R)(R/a, H-a(t) (M)/K) and Ext(R)(1) (R/a, H-a(t) (M)/K) are finitely generated, where t is a non-negative integer. Finally, we explore a criterion for weakly cofiniteness of modules with respect to an ideal of dimension one. Namely, for such ideals it suffices that the two first Ext-modules in the definition for weakly cofiniteness are weakly Laskerian. As an application of this result, we deduce that the category of all a-weakly cofinite modules over R forms a full Abelian subcategory of the category of modules.

• 出版日期2018-3