scholarly journals On the primefulness of local cohomology modules

2021 ◽  
Vol 3 (1) ◽  
pp. 108-112
Author(s):  
Payman Mahmood Hamaali ◽  
Adil Kadir Jabbar
Keyword(s):  
Noetherian Ring ◽  
Projective Module ◽  
Integral Domain ◽  
Local Cohomology ◽  
Projective Resolution ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let be a commutative Noetherian ring with identity For a non-zero module . We prove that a multiplication primeful module and are I-cofinite and primeful, for each where is an ideal of with . As a consequence, we deduce that, if and are multiplication primeful R- modules, then is primeful. Another result is, for a projective module over an integral domain, admits projective resolution such that each is primeful (faithfully flat).

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.

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.

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.

On the Endomorphism Rings of Local Cohomology Modules

2010 ◽  
Vol 53 (4) ◽  
pp. 667-673 ◽  
Author(s):  
Kazem Khashyarmanesh
Keyword(s):  
Nonnegative Integer ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Ring Homomorphism ◽  
Endomorphism Rings ◽  
Commutative Noetherian Ring ◽  
Ring Of Fractions ◽  
Local Cohomology Modules ◽  
Canonical Ring ◽  
Cohomology Modules

AbstractLet R be a commutative Noetherian ring and a a proper ideal of R. We show that if n := gradeRa, then . We also prove that, for a nonnegative integer n such that = 0 for every i ≠ n, if for all i > 0 and z ∈ a, then is a homomorphic image of R, where Rz is the ring of fractions of R with respect to a multiplicatively closed subset ﹛z j | j ⩾ 0﹜ of R. Moreover, if HomR(Rz , R) = 0 for all z ∈ a, then is an isomorphism, where is the canonical ring homomorphism R → .

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].

scholarly journals Artinian and Non-Artinian Local Cohomology Modules

2011 ◽  
Vol 54 (4) ◽  
pp. 619-629 ◽  
Author(s):  
Mohammad T. Dibaei ◽  
Alireza Vahidi
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Maximal Element ◽  
Local Cohomology ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Finite Module ◽  
Bass Numbers ◽  
Cohomology Modules

AbstractLet M be a finite module over a commutative noetherian ring R. For ideals a and b of R, the relations between cohomological dimensions of M with respect to a, b, a ⋂ b and a + b are studied. When R is local, it is shown that M is generalized Cohen–Macaulay if there exists an ideal a such that all local cohomology modules of M with respect to a have finite lengths. Also, when r is an integer such that 0 ≤ r < dimR(M), any maximal element q of the non-empty set of ideals ﹛a : (M) is not artinian for some i, i ≥ r} is a prime ideal, and all Bass numbers of (M) are finite for all i ≥ r.

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.

Noetherianness and Local Cohomology Modules

2012 ◽  
Vol 19 (spec01) ◽  
pp. 807-812
Author(s):  
Kazem Khashyarmanesh
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Vanishing Theorem ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M an R-module. We show that, whenever [Formula: see text], M is Noetherian if and only if there exists a submodule N of M such that the R-modules M/𝔞 N and [Formula: see text] are Noetherian. By using this result, we establish Noetherian properties for local cohomology modules [Formula: see text] in several cases. For instance, we obtain a new version of the Lichtenbaum-Hartshorne Vanishing Theorem.

Betti Numbers and Flat Dimensions of Local Cohomology Modules

2015 ◽  
Vol 58 (3) ◽  
pp. 664-672 ◽  
Author(s):  
Alireza Vahidi
Keyword(s):  
Upper Bound ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Betti Numbers ◽  
Upper Bounds ◽  
Commutative Noetherian Ring ◽  
Flat Dimension ◽  
Local Cohomology Modules ◽  
Cohomology Modules

AbstractAssume that R is a commutative Noetherian ring with non-zero identity, 𝔞 is an ideal of R, and X is an R-module. In this paper, we first study the finiteness of Betti numbers of local cohomology modules . Then we give some inequalities between the Betti numbers of X and those of its local cohomology modules. Finally, we present many upper bounds for the flat dimension of X in terms of the flat dimensions of its local cohomology modules and an upper bound for the flat dimension of in terms of the flat dimensions of the modules , and that of X.

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