Cofinite Modules Research Articles

On the cofinite modules with respect to an ideal of cohomological dimension not exceeding one

On the cofinite modules with respect to an ideal of cohomological dimension not exceeding one

Locally cohomologically complete intersection ideals and cofiniteness

Let [Formula: see text] be a unitary commutative local Noetherian ring and [Formula: see text] be an ideal. The aim of this paper is twofold: In the first part of this paper, we consider locally cohomologically complete intersection ideals of pure height [Formula: see text]. Next, we deal with cofinite modules. In particular, we investigate conditions for cofiniteness of [Formula: see text] and conditions under which, the modules [Formula: see text] are of finite length for all [Formula: see text] and all finitely generated [Formula: see text]-module [Formula: see text].

Correction: Injective dimension of cofinite modules and local cohomology

Correction: Injective dimension of cofinite modules and local cohomology

On the Abelian categories of cofinite modules

Let [Formula: see text] be a commutative Noetherian ring and [Formula: see text] be an ideal of [Formula: see text] such that the [Formula: see text]-modules [Formula: see text] are [Formula: see text]-cofinite, for all finitely generated [Formula: see text]-modules [Formula: see text] and all [Formula: see text]. In this paper, we investigate a question of Hartshorne concerning the Abelianness of the category of all [Formula: see text]-cofinite modules.

Cofiniteness of torsion functors of a pair of cofinite modules (II)

Let I be an ideal of a commutative Noetherian ring R such that the R-modules are I-cofinite, for all finitely generated R-modules U and all . Also, assume that the category of all I-cofinite modules is an Abelian subcategory of the category of R-modules and R-homomorphisms. It is shown that for each pair of I-cofinite modules M and N the R-modules are I-cofinite for all . Also, some applications of this result will be included.

Cofiniteness of torsion functors of a pair of cofinite modules

Let I be an ideal of a commutative Noetherian ring R. It is shown that the R-modules are I-cofinite for all if M and N are I-cofinite R-modules and N is of dimension at most 1. Also, it is shown that R-modules are I-weakly cofinite for all if M and N are I-cofinite R-modules, N is of dimension at most 2, and R is a local ring.

Abelian Category of Weakly Cofinite Modules and Local Cohomology

Let R be a commutative Noetherian ring, $${\mathfrak {a}}$$ an ideal of R, and M an R-module. We prove that the category of $${\mathfrak {a}}$$ -weakly cofinite modules is a Melkersson subcategory of R-modules whenever $$\dim R\le 1$$ and is an Abelian subcategory whenever $$\dim R\le 2$$ . We also prove that if $$(R,{\mathfrak {m}})$$ is a local ring with $$\dim R/{\mathfrak {a}}\le 2$$ and $${{\,\mathrm{Supp}\,}}_R(M)\subseteq {{\,\mathrm{V}\,}}({\mathfrak {a}})$$ , then M is $${\mathfrak {a}}$$ -weakly cofinite if (and only if) $${{\,\mathrm{Hom}\,}}_R(R/{\mathfrak {a}}, M)$$ , $${{\,\mathrm{Ext}\,}}_{R}^{1}(R/{\mathfrak {a}}, M)$$ and $${{\,\mathrm{Ext}\,}}_{R}^{2}(R/{\mathfrak {a}}, M)$$ are weakly Laskerian. In addition, we prove that if $$(R,{\mathfrak {m}})$$ is a local ring with $$\dim R/{\mathfrak {a}}\le 2$$ and $$n\in \mathbb {N}_0$$ , such that $${{{\,\mathrm{Ext}\,}}}^{i}_R(R/{\mathfrak {a}},M)$$ is weakly Laskerian for all i, then $${{\,\mathrm{H}\,}}_{{\mathfrak {a}}}^{i}(M)$$ is $${\mathfrak {a}}$$ -weakly cofinite for all i if (and only if) $${{\,\mathrm{Hom}\,}}_R(R/{\mathfrak {a}}, {{\,\mathrm{H}\,}}_{{\mathfrak {a}}}^{i}(M))$$ is weakly Laskerian for all i.

A characterization of cominimax and weakly cofinite modules

Let R be a commutative Noetherian ring and I be an ideal of R. Let M be an arbitrary R-module. In this paper we establish some results concerning the cofiniteness properties of modules. It is shown that, M is I-cominimax if and only if there is an ideal with such that M is -cofinite and the R-module is finitely generated, for each Moreover, it is shown that M is I-weakly cofinite if and only if there is an ideal such that V(J) is a finite set and M is -cofinite.

A study of cofiniteness through minimal associated primes

In this article, we shall investigate the concepts of cofiniteness of local cohomology modules and Abelian categories of cofinite modules over arbitrary Noetherian rings. Then we shall improve some of the well known results in the area.

Modules cofinite and weakly cofinite with respect to an ideal

The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal [Formula: see text] of a Noetherian ring [Formula: see text]. It is shown that an [Formula: see text]-module [Formula: see text] is cofinite with respect to [Formula: see text], if and only if, [Formula: see text] is finitely generated for all [Formula: see text], whenever [Formula: see text]. In addition, we show that if [Formula: see text] is finitely generated and [Formula: see text] are weakly Laskerian for all [Formula: see text], then [Formula: see text] are [Formula: see text]-cofinite for all [Formula: see text] and for any minimax submodule [Formula: see text] of [Formula: see text], the [Formula: see text]-modules [Formula: see text] and [Formula: see text] are finitely generated, where [Formula: see text] 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 [Formula: see text]-modules in the definition for weakly cofiniteness are weakly Laskerian. As an application of this result, we deduce that the category of all [Formula: see text]-weakly cofinite modules over [Formula: see text] forms a full Abelian subcategory of the category of modules.

Some properties of Artinian modules and applications

Let [Formula: see text] be a Noetherian local ring and [Formula: see text] be an Artinian [Formula: see text]-module. Consider the following property for [Formula: see text] : [Formula: see text] In this paper, we study the property (∗) of [Formula: see text] in order to investigate the relation of system of parameters between [Formula: see text] and the ring [Formula: see text]. We also show that the property (∗) of [Formula: see text] has strong connection with the structure of base ring. Some applications to cofinite Artinian module are given. These are generalizations of [N. Abazari and K. Bahmanpour, A note on the Artinian cofinite modules, Comm. Algebra. 42 (2014) 1270–1275; G. Ghasemi, K. Bahmanpour and J. Azami, On the cofiniteness of Artinian local cohomology modules, J. Algebra Appl. 15(4) (2016), Article ID: 1650070, 8 pp.] A generalization of Lichtenbaum–Hartshorne Vanishing Theorem is also given in this paper.

Cohomological dimension, cofiniteness and Abelian categories of cofinite modules

Let R be a commutative Noetherian ring, I⊆J be ideals of R and M be a finitely generated R-module. In this paper it is shown that q(J,M)≤q(I,M)+cd(J,M/IM). Furthermore, it is shown that, for any ideal I of R and any finitely generated R-module M with q(I,M)≤1, the local cohomology modules HIi(M) are I-cofinite for all integers i≥0. As a consequence of this result it is shown that, if q(I,R)≤1, then for any finitely generated R-module M, the local cohomology modules HIi(M) are I-cofinite for all integers i≥0. Finally, it is shown that the category of all I-cofinite R-modules C(R,I)cof is an Abelian subcategory of the category of all R-modules, whenever (R,m) is a complete Noetherian local ring and I is an ideal of R with q(I,R)≤1. These assertions answer affirmatively two questions raised by R. Hartshorne in [16], in the some special cases.

Serre subcategories and the cofiniteness of generalized local cohomology modules with respect to a pair of ideals

We study some properties of the generalized local cohomology modules [Formula: see text] of [Formula: see text] with respect to a pair of ideals [Formula: see text] in Serre subcategories. Some results concerning to [Formula: see text]-cofinite modules are also given in this paper.

On the category of cofinite modules which is Abelian

Let R R denote a commutative Noetherian (not necessarily local) ring and I I an ideal of R R of dimension one. The main purpose of this paper is to generalize, and to provide a short proof of, K. I. Kawasaki’s theorem that the category M ( R , I ) c o f \mathscr {M}(R, I)_{cof} of I I -cofinite modules over a commutative Noetherian local ring R R forms an Abelian subcategory of the category of all R R -modules. Consequently, this assertion answers affirmatively the question raised by R. Hartshorne in his article Affine duality and cofiniteness [Invent. Math. 9 (1970), 145-164] for an ideal of dimension one in a commutative Noetherian ring R R .

Cofiniteness of torsion functors of cofinite modules

Cofiniteness of torsion functors of cofinite modules

The category of cofinite modules for ideals of dimension one and codimension one

The category of cofinite modules for ideals of dimension one and codimension one

Cofiniteness of extension functors of cofinite modules

Let R be a commutative Noetherian ring, I an ideal of R and let M and N be non-zero R-modules. It is shown that the R-modules Ext R i ( N , M ) are I-cofinite, for all i ⩾ 0 , whenever M is I-cofinite and N is finitely generated of dimension d ⩽ 2 . Also, we prove that the R-modules Ext R i ( N , M ) are I-cofinite, for all i ⩾ 0 , whenever N is finitely generated and M is I-cofinite of dimension d ⩽ 1 . This immediately implies that if I has dimension one (i.e., dim R / I = 1 ) then Ext R i ( N , H I i ( M ) ) is I-cofinite for all i ⩾ 0 , and all finitely generated R-modules M and N. Also, we prove that if R is local then the R-modules Ext R i ( N , M ) are I-weakly cofinite, for all i ⩾ 0 , whenever M is I-cofinite and N is finitely generated of dimension d ⩽ 3 . Finally, it is shown that the R-modules Ext R i ( N , M ) are I-weakly cofinite, for all i ⩾ 0 , whenever R is local, N is finitely generated and M is I-cofinite of dimension d ⩽ 2 .

On a category of cofinite modules which is Abelian

Let A be a noetherian local ring, and I an ideal of A of dimension one. Our purpose of this paper is to prove that the category \({\mathcal{M} (A, I)_{cof}}\) of cofinite modules is Abelian. Consequently, this assertion answers affirmatively the questions given by Hartshorne (Invent Math 9:147, 1970; cf. Sect. 2. Four Questions and One theorem) for an ideal of dimension one over a local ring.

Cofiniteness of local cohomology modules for ideals of small dimension

Let R be a commutative Noetherian ring and let M be a non-zero finitely generated R-module. Let I be an ideal of R and t a non-negative integer such that dim Supp H I i ( M ) ⩽ 1 for all i < t . It is shown that the R-modules H I 0 ( M ) , H I 1 ( M ) , … , H I t − 1 ( M ) are I-cofinite and the R-module Hom R ( R / I , H I t ( M ) ) is finitely generated. This immediately implies that if I has dimension one (i.e., dim R / I = 1 ), then H I i ( M ) is I-cofinite for all i ⩾ 0 . This is a generalization of the main results of Delfino and Marley [D. Delfino, T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra 121 (1997) 45–52] and Yoshida [K.I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147 (1997) 179–191] for an arbitrary Noetherian ring R. Also, we prove that if R is local and dim Supp H I i ( M ) ⩽ 2 for all i < t , then the R-modules Ext R j ( R / I , H I i ( M ) ) and Hom R ( R / I , H I t ( M ) ) are weakly Laskerian for all i < t and all j ⩾ 0 . As a consequence, it follows that the set of associated primes of H I i ( M ) is finite for all i ⩾ 0 , whenever dim R / I ⩽ 2 .

Co-support and Coartinian Modules

We show that for a linearly compact module over a local ring (R,𝔪), the concept of co-support of modules provided by Melkersson is equivalent with Yassemi's concept. We also introduce the definition of coartinian modules which is in some sense dual to Hartshorne's definition of cofinite modules and show that [Formula: see text] is I-coartinian in case M is a semi-discrete linearly compact R-module with Ndim M=d&gt; 0.