On Weakly Laskerian and Weakly Cofinite Modules

2012 ◽  
Vol 19 (04) ◽  
pp. 693-698
Author(s):  
Kazem Khashyarmanesh ◽  
M. Tamer Koşan ◽  
Serap Şahinkaya
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Negative Integer ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Cofinite Modules ◽  
Gorenstein Local Ring

Let R be a commutative Noetherian ring with non-zero identity, 𝔞 an ideal of R and M a finitely generated R-module. We assume that N is a weakly Laskerian R-module and r is a non-negative integer such that the generalized local cohomology module [Formula: see text] is weakly Laskerian for all i < r. Then we prove that [Formula: see text] is also weakly Laskerian and so [Formula: see text] is finite. Moreover, we show that if s is a non-negative integer such that [Formula: see text] is weakly Laskerian for all i, j ≥ 0 with i ≤ s, then [Formula: see text] is weakly Laskerian for all i ≤ s and j ≥ 0. Also, over a Gorenstein local ring R of finite Krull dimension, we study the question when the socle of [Formula: see text] is weakly Laskerian?

Cominimaxness with respect to ideals of dimension two and local cohomology

2020 ◽  
pp. 2150081
Author(s):  
Hamidreza Karimirad ◽  
Moharram Aghapournahr
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Equivalent Conditions

Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] an [Formula: see text]-module with [Formula: see text]. We get equivalent conditions for top local cohomology module [Formula: see text] to be Artinian and [Formula: see text]-cofinite Artinian separately. In addition, we prove that if [Formula: see text] is a local ring such that [Formula: see text] is minimax, for each [Formula: see text], then [Formula: see text] is minimax [Formula: see text]-module for each [Formula: see text] and for each finitely generated [Formula: see text]-module [Formula: see text] with [Formula: see text] and [Formula: see text]. As a consequence we prove that if [Formula: see text] and [Formula: see text], then [Formula: see text] is [Formula: see text]-cominimax if (and only if) [Formula: see text], [Formula: see text] and [Formula: see text] are minimax. We also prove that if [Formula: see text] and [Formula: see text] such that [Formula: see text] is minimax for all [Formula: see text], then [Formula: see text] is [Formula: see text]-cominimax for all [Formula: see text] if (and only if) [Formula: see text] is minimax for all [Formula: see text].

Artinian Local Cohomology Modules

2007 ◽  
Vol 50 (4) ◽  
pp. 598-602 ◽  
Author(s):  
Keivan Borna Lorestani ◽  
Parviz Sahandi ◽  
Siamak Yassemi
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Negative Integer ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

AbstractLet R be a commutative Noetherian ring, α an ideal of R and M a finitely generated R-module. Let t be a non-negative integer. It is known that if the local cohomology module is finitely generated for all i < t, then is finitely generated. In this paper it is shown that if is Artinian for all i < t, then need not be Artinian, but it has a finitely generated submodule N such that /N is Artinian.

Some Rigidity Results for Highest Order Local Cohomology Modules

2007 ◽  
Vol 14 (03) ◽  
pp. 497-504 ◽  
Author(s):  
Mohammad T. Dibaei ◽  
Siamak Yassemi
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Rigidity Results ◽  
Local Cohomology Modules ◽  
Formal Behaviour ◽  
Cohomology Modules

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M a finitely generated R-module of finite Krull dimension n. We describe the (finite) sets [Formula: see text] and [Formula: see text] of primes associated and attached to the highest local cohomology module [Formula: see text] in terms of the local formal behaviour of 𝔞.

On the Finiteness Dimension of Local Cohomology Modules

2014 ◽  
Vol 21 (03) ◽  
pp. 517-520 ◽  
Author(s):  
Hero Saremi ◽  
Amir Mafi
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Negative Integer ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M a non-zero finitely generated R-module. Let t be a non-negative integer. In this paper, it is shown that [Formula: see text] for all i < t if and only if there exists an ideal 𝔟 of R such that dim R/𝔟 ≤ 1 and [Formula: see text] for all i < t. Moreover, we prove that [Formula: see text] for all i.

scholarly journals Stable under specialization sets and cofiniteness

2019 ◽  
Vol 18 (01) ◽  
pp. 1950015 ◽  
Author(s):  
K. Divaani-Aazar ◽  
H. Faridian ◽  
M. Tousi
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring

Let [Formula: see text] be a commutative noetherian ring, and [Formula: see text] a stable under specialization subset of [Formula: see text]. We introduce a notion of [Formula: see text]-cofiniteness and study its main properties. In the case [Formula: see text], or [Formula: see text], or [Formula: see text] is semilocal with [Formula: see text], we show that the category of [Formula: see text]-cofinite [Formula: see text]-modules is abelian. Also, in each of these cases, we prove that the local cohomology module [Formula: see text] is [Formula: see text]-cofinite for every homologically left-bounded [Formula: see text]-complex [Formula: see text] whose homology modules are finitely generated and every [Formula: see text].

Cofiniteness of local cohomology modules for a pair of ideals for small dimensions

2018 ◽  
Vol 17 (02) ◽  
pp. 1850020 ◽  
Author(s):  
Moharram Aghapournahr
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] and [Formula: see text] be two ideals of [Formula: see text] and [Formula: see text] be an [Formula: see text]-module (not necessary [Formula: see text]-torsion). In this paper among other things, it is shown that if dim [Formula: see text], then the [Formula: see text]-module [Formula: see text] is finitely generated, for all [Formula: see text], if and only if the [Formula: see text]-module [Formula: see text] is finitely generated, for [Formula: see text]. As a consequence, we prove that if [Formula: see text] is finitely generated and [Formula: see text] such that the [Formula: see text]-module [Formula: see text] is [Formula: see text] (or weakly Laskerian) for all [Formula: see text], then [Formula: see text] is [Formula: see text]-cofinite for all [Formula: see text] and for any [Formula: see text] (or minimax) submodule [Formula: see text] of [Formula: see text], the [Formula: see text]-modules [Formula: see text] and [Formula: see text] are finitely generated. Also it is shown that if dim [Formula: see text] (e.g. dim [Formula: see text]) for all [Formula: see text], then the local cohomology module [Formula: see text] is [Formula: see text]-cofinite for all [Formula: see text].

Non-Artinian Local Cohomology with Respect to a Pair of Ideals

2013 ◽  
Vol 20 (04) ◽  
pp. 637-642 ◽  
Author(s):  
Abolfazl Tehranian ◽  
Atiyeh Pour Eshmanan Talemi
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Negative Integer ◽  
Nilpotent Ideal ◽  
Commutative Noetherian Ring

Let R be a commutative Noetherian ring, I, J two ideals of R, and M an R-module. For a non-negative integer t, we show: (a) If [Formula: see text] for all j < t and [Formula: see text] are finite (resp., Artinian), then [Formula: see text] is finite (resp., Artinian). (b) If [Formula: see text] for all j < t and [Formula: see text] are finite (resp., Artinian), then [Formula: see text] is finite (resp., Artinian). In addition, if (R,𝔪) is a local ring, J a non-nilpotent ideal, and M a finite R-module, then we show that [Formula: see text] is not Artinian for some i ∈ ℕ0.

scholarly journals A generalization of the cofiniteness problem in local cohomology modules

2003 ◽  
Vol 75 (3) ◽  
pp. 313-324 ◽  
Author(s):  
J. Asadollahi ◽  
K. Khashyarmanesh ◽  
SH. Salarian
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules ◽  
Nonzero Identity

AbstractLetRbe a commutative Noetherian ring with nonzero identity and letMbe a finitely generated R-module. In this paper, we prove that if an idealIofRis generated by a u.s.d-sequence onMthen the local cohomology module(M) isI-cofinite. Furthermore, for any system of ideals Φ of R, we study the cofiniteness problem in the context of general local cohomology modules.

scholarly journals On the asymptotic behaviour of associated primes of generalized local cohomology modules

2007 ◽  
Vol 83 (2) ◽  
pp. 217-226 ◽  
Author(s):  
Kazem Khashyarmaneshs ◽  
Ahmad Abbasi
Keyword(s):  
Asymptotic Behavior ◽  
Local Cohomology ◽  
Associated Primes ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Graded Modules ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

AbstractLetMandNbe finitely generated and graded modules over a standard positive graded commutative Noetherian ringR, with irrelevant idealR+. Letbe thenth component of the graded generalized local cohomology module. In this paper we study the asymptotic behavior of AssfR+() as n → –∞ wheneverkis the least integerjfor which the ordinary local cohomology moduleis not finitely generated.

On ideals preserving generalized local cohomology modules

2015 ◽  
Vol 15 (01) ◽  
pp. 1650019 ◽  
Author(s):  
Tsutomu Nakamura
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules ◽  
The Ideal

Let R be a commutative Noetherian ring, 𝔞 an ideal of R and M, N two finitely generated R-modules. Let t be a positive integer or ∞. We denote by Ωt the set of ideals 𝔠 such that [Formula: see text] for all i < t. First, we show that there exists the ideal 𝔟t which is the largest in Ωt and [Formula: see text]. Next, we prove that if 𝔡 is an ideal such that 𝔞 ⊆ 𝔡 ⊆ 𝔟t, then [Formula: see text] for all i < t.

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