Weakly cofiniteness of local cohomology modules

Abstract

Let I be an ideal of a Noetherian ring R and M be a finitely generated R-module. We introduce the class of extension modules of finitely generated modules by the class of all modules T with dim T ≤ n and we show it by FD ≤n where n ≥ −1 is an integer. We prove that for any FD ≤0 (or minimax) submodule N of H t I (M ) the R-modules Hom R (R/I, H t I (M )/N ) and Ext 1 R (R/I, H t I (M )/N ) are finitely generated, whenever the modules H 0 I (M ), H 1 I (M ), ..., H t−1 I (M ) are FD ≤1 ( or weakly Laskerian). As a consequence, it follows that the associated primes of H t I (M )/N are finite. This generalizes the main results of Bahmanpour and Naghipour [4] and [5], Brodmann and Lashgari [7], Khashyarmanesh and Salarian [21] and Hong Quy [18]. We also show that the category F D 1 (R, I) cof of I-cofinite FD ≤1 R-modules forms an Abelian subcategory of the category of all R-modules.

Cominimaxness of local cohomology modules

“…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. In this direction in Section 2, we generalize [2], Theorem 3.4 and Corollaries 3.5 and 3.6. More precisely, we will show: Theorem 1.3 (See Theorem 2.7 and Corollary 2.10).…”

“…Although the conjecture is not true in general as Hartshorne showed in [19], some authors proved that for some numbers t, the module Hom R (R/I, H t I (M )) is finite under some conditions. See [2], Theorem 3.4, [3], Theorem 3.3, [7], Theorem 2.3, [8], Theorem 2.6, [14], Theorem 2.1, and [15], Theorem 6.3.9. Hartshorne also defined a module M to be I-cofinite if Supp R (M ) ⊆ V (I) and Ext i R (R/I, M ) is finitely generated for all i 0, and posed the following question: Question 1.2.…”

“…This question was studied by several authors in [2], [6], [8], [13], [19], [21], [34], [27], [29], and [30].…”

“…As a special case of [35], Definition 2.1, and a generalization of FSF modules (see [32], Definition 2.1), in [2], Definition 2.1, the author of the present paper and Bahmanpour introduced the class of FD n modules. A module M is said to be an FD n module, if there exists a finitely generated submodule N of M such that dim M/N n. For more details about properties of this class see [2], Lemma 2.3. Note that the class of FD −1 is the same as that of finitely generated R-modules.…”

Lower bounds of certain general local cohomology modules

“…It is easy to see that the above definition of system of ideals and general local cohomology modules is equivalent to the [12, Definition 2.1.10 and Notation 2.2.2]. General local cohomology modules was studied by several authors in [10,11,15,4,1,2,12].…”