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.

Cohomological Dimension of Generalized Local Cohomology Modules

2008 ◽  
Vol 15 (02) ◽  
pp. 303-308 ◽  
Author(s):  
Jafar Amjadi ◽  
Reza Naghipour
Keyword(s):  
Exact Sequence ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Cohomological Dimension ◽  
Algebraic Varieties ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

The study of the cohomological dimension of algebraic varieties has produced some interesting results and problems in local algebra. Let 𝔞 be an ideal of a commutative Noetherian ring R. For finitely generated R-modules M and N, the concept of cohomological dimension cd 𝔞(M, N) of M and N with respect to 𝔞 is introduced. If 0 → N' → N → N'' → 0 is an exact sequence of finitely generated R-modules, then it is shown that cd 𝔞(M, N) = max { cd 𝔞(M, N'), cd 𝔞(M, N'')} whenever proj dim M < ∞. Also, if L is a finitely generated R-module with Supp (N/Γ𝔞(N)) ⊆ Supp (L/Γ𝔞(L)), then it is proved that cd 𝔞(M, N) ≤ max { cd 𝔞(M,L), proj dim M}. Finally, as a consequence, a result of Brodmann is improved.

Cofiniteness of Generalized Local Cohomology Modules for One-Dimensional Ideals

2012 ◽  
Vol 55 (1) ◽  
pp. 81-87 ◽  
Author(s):  
Kamran Divaani-Aazar ◽  
Alireza Hajikarimi
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
One Dimensional ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

AbstractLet a be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Our main result asserts that if dim R/a ≤ 1, then all generalized local cohomology modules (M,N) are a-cofinite.

FINITENESS DIMENSION AND BASS NUMBERS OF GENERALIZED LOCAL COHOMOLOGY MODULES

2013 ◽  
Vol 12 (07) ◽  
pp. 1350036 ◽  
Author(s):  
HERO SAREMI ◽  
AMIR MAFI
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Bass Numbers ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M, N two nonzero finitely generated R-modules. Let t be a non-negative integer. It is shown that dim Supp [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. As a consequence all Bass numbers and all Betti numbers of generalized local cohomology modules [Formula: see text] are finite for all i < t, provided that the projective dimension pd (M) is finite.

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 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.

Remarks on the local-global principle for a subcategory consisting of extension modules

2019 ◽  
Vol 18 (12) ◽  
pp. 1950236
Author(s):  
Takeshi Yoshizawa
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Global Principles ◽  
Finitely Generated Modules ◽  
Cohomology Modules ◽  
Global Principle

Faltings presented the local-global principle for the finiteness dimension of local cohomology modules. This paper deals with the local-global principle for an extension subcategory over a commutative Noetherian ring. We prove that finitely generated modules satisfy the local-global principles for certain extension subcategories. Additionally, we provide a generalization of Faltings’ local-global principle, which also includes the local-global principles for the Artinianness and Minimaxness of local cohomology modules.

scholarly journals Bass numbers of generalized local cohomology modules

2015 ◽  
Vol 97 (111) ◽  
pp. 233-238 ◽  
Author(s):  
Sh. Payrovi ◽  
S. Babaei ◽  
I. Khalili-Gorji
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Modules ◽  
Bass Numbers ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

Let R be a Noetherian ring, M a finitely generated R-module and N an arbitrary R-module. We consider the generalized local cohomology modules, with respect to an arbitrary ideal I of R, and prove that, for all nonnegative integers r, t and all p ? Spec(R) the Bass number ?r(p,HtI (M,N)) is bounded above by ?tj=0?r(p, t?jExtR (M,HjI (N))). A corollary is that Ass (HtI (M,N)? Utj=0 Ass (t?jExtR (M,HjI(N))). In a slightly different direction, we also present some well known results about generalized local cohomology modules.

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].

Family of quotients of some special rings

2018 ◽  
Vol 17 (12) ◽  
pp. 1850233 ◽  
Author(s):  
Maryam Salimi
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Proper Ideal ◽  
Rees Algebra ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
Local Cohomology Modules ◽  
Cohomology Modules ◽  
The Ideal

Let [Formula: see text] be a commutative Noetherian ring and let [Formula: see text] be a proper ideal of [Formula: see text]. We study some properties of a family of rings [Formula: see text] that are obtained as quotients of the Rees algebra associated with the ring [Formula: see text] and the ideal [Formula: see text]. We deal with the strongly cotorsion property of local cohomology modules of [Formula: see text], when [Formula: see text] is a local ring. Also, we investigate generically Cohen–Macaulay, generically Gorenstein, and generically quasi-Gorenstein properties of [Formula: see text]. Finally, we show that [Formula: see text] is approximately Cohen–Macaulay if and only if [Formula: see text] is approximately Cohen–Macaulay, provided some special conditions.

The first non-isomorphic local cohomology modules with respect to their ideals

2018 ◽  
Vol 17 (12) ◽  
pp. 1850230
Author(s):  
Ali Fathi
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Regular Sequences ◽  
Maximal Ideals ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let [Formula: see text] be ideals of a commutative Noetherian ring [Formula: see text] and [Formula: see text] be a finitely generated [Formula: see text]-module. By using filter regular sequences, we show that the infimum of integers [Formula: see text] such that the local cohomology modules [Formula: see text] and [Formula: see text] are not isomorphic is equal to the infimum of the depths of [Formula: see text]-modules [Formula: see text], where [Formula: see text] runs over all prime ideals of [Formula: see text] containing only one of the ideals [Formula: see text]. In particular, these local cohomology modules are isomorphic for all integers [Formula: see text] if and only if [Formula: see text]. As an application of this result, we prove that for a positive integer [Formula: see text], [Formula: see text] is Artinian for all [Formula: see text] if and only if, it can be represented as a finite direct sum of [Formula: see text] local cohomology modules of [Formula: see text] with respect to some maximal ideals in [Formula: see text] for any [Formula: see text]. These representations are unique when they are minimal with respect to inclusion.

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