Cominimaxness With Respect to Ideals of Dimension One

Abstract

Abstract. Let R denote a commutative Noetherian (not necessarily local) ring and let I be an ideal of R of dimension one. The main purpose of this note is to show that the category M (R, I)com of I-cominimax Rmodules forms an Abelian subcategory of the category of all R-modules. This assertion is a generalization of the main result of Melkersson in [15]. As an immediate consequence of this result we get some conditions for cominimaxness of local cohomology modules for ideals of dimension one. Finally, it is shown that the category C 1 B (R) of all R-modules of dimension at most one with finite Bass numbers forms an Abelian subcategory of the category of all R-modules.

Cominimaxness of certain general local cohomology modules

“…The general local cohomology module H j Φ (M ) is defined to be Φ-cominimax if there exists an ideal I ∈ Φ such that Ext i R (R/I, H j Φ (M )) is minimax, for all i, j ≥ 0. Recently many authors studied the minimaxness and cominimaxness of local cohomology modules and answered the Conjecture 1.1 and Question 1.2 in the class of minimax modules in some cases (see [1,3,9,14,24,27,29]). The purpose of this note is to make a suitable generalization of Conjecture 1.1 and Question 1.2 in terms of minimax modules instead of finitely generated modules for general local cohomology modules.…”

“…it is shown that Hartshorne's question is true for the category of all I-cofinite R-modules M with dim M ≤ 1 (resp. the class of I-cofinite FD ≤1 modules), for all ideals I in a commutative Noetherian ring R. Also in[27] (resp [3]…”

“…Proposition 2.6] and[3,27, Proposition 2.4]. In fact, we remove I-torsion condition from these propositions.…”

Cominimaxness of local cohomology modules

“…Recently many authors have studied the minimaxness and cominimaxness of local cohomology modules and answered Conjecture 1.1 and Question 1.2 in the class of minimax modules in some cases (see [1], [7], [20], [22] [24], [26]). The purpose of this note is to make a suitable generalization of Conjecture 1.1 and Question 1.2 in terms of minimax modules instead of finitely generated modules.…”

“…On the other hand, in [13], Delfino and Marley extended this result to arbitrary complete local rings. Recently, Kawasaki in [23] generalized the Delfino and Marley's result to an arbitrary ideal I of dimension one in a local ring R. Finally, Melkersson in [31] completely removed the local assumption on R. More recently, in [2] and [10] it is shown that Hartshorne's question is true for the category of all I-cofinite R-modules M with dim M 1 and the class of I-cofinite FD 1 modules, respectively, for all ideals I in a commutative Noetherian ring R. Irani in [22], Theorem 2.5, proved that the category of all I-cominimax R-modules M with dim M 1 is Abelian. One of the main results of this section is to prove that the class of I-cominimax weakly Laskerian (WL(R, I) comin ) and I-cominimax FD 1 (F D 1 (R, I) comin ) modules are Abelian categories.…”