2017/03/02 by Kamal Bahmanpour, Bahmanpour, Kamal, Reza Naghipour +3
Computer Science · Mathematics · #13D45 #13E05 #14B15 #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1703.00766
openalex publication_date 2017/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal \frak a of a Noetherian ring R. It is shown that an R-module M is cofinite with respect to \frak a, if and only if, \ExtiR(R/\frak a,M) is finitely generated for all i≤ \rm cd(\frak a,M)+1, whenever dim R/\frak a=1. In addition, we show that if M is finitely generated and Hi\frak a(M) are weakly Laskerian for all i≤ t-1, then Hi\frak a(M) are \frak a-cofinite for all i≤ t-1 and for any minimax submodule K of Ht\frak a(M), the R-modules \HomR(R/\frak a, Ht\frak a(M)/K) and \Ext1R(R/\frak a, Ht\frak a(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 \frak a-weakly cofinite modules over R forms a full Abelian subcategory of the category of modules.