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.

Finiteness Results on Generalized Local Cohomology Modules

2009 ◽  
Vol 16 (01) ◽  
pp. 65-70
Author(s):  
Naser Zamani
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Associated Primes ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

Let (R, 𝔪) be a local ring, 𝔞 an ideal of R, and M, N be two finitely generated R-modules. We show that r = gdepth (M/𝔞M, N) is the least integer such that [Formula: see text] has infinite support. Also, we prove that the first non-Artinian generalized local cohomology module has finitely many associated primes.

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.

scholarly journals On the cominimaxness of generalized local cohomology modules

2020 ◽  
Vol 23 (1) ◽  
pp. 479-483
Author(s):  
Cam Thi Hong Bui ◽  
Tri Minh Nguyen
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Local Cohomology ◽  
Cohomology Theory ◽  
Negative Integer ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

The local cohomology theory plays an important role in commutative algebra and algebraic geometry. The I-cofiniteness of local cohomology modules is one of interesting properties which has been studied by many mathematicians. The I-cominimax modules is an extension of I-cofinite modules which was introduced by Hartshorne. An R-module M is I-cominimax if Supp_R(M)\subseteq V(I) and Ext^i_R(R/I,M) is minimax for all  i\ge 0. In this paper, we show some conditions such that the generalized local cohomology module H^i_I(M,N) is I-cominimax for all i\ge 0. We show that if H^i_I(M,K) is I-cofinite for all i<t and all finitely generated R-module K, then  H^i_I(M,N) is I-cominimax for all i<t  and all minimax R-module N.  If M is a finitely generated R-module, N is a minimax R-module and t is a non-negative integer such that  dim Supp_R(H^i_I(M,N))\le 1 for all i<t then H^i_I(M,N) is I-cominimax for all  i<t. When  dim R/I\le 1 and H^i_I(N) is I-cominimax for all  i\ge 0 then H^i_I(M,N) is I-cominimax for all i\ge 0.

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].

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.

On the Finiteness Property of Generalized Local Cohomology Modules

2005 ◽  
Vol 12 (02) ◽  
pp. 293-300 ◽  
Author(s):  
K. Khashyarmanesh ◽  
M. Yassi
Keyword(s):  
Local Cohomology ◽  
Prime Ideals ◽  
Local Cohomology Module ◽  
Finiteness Property ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Associated Prime Ideals ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules ◽  
Associated Prime

Let [Formula: see text] be an ideal of a commutative Noetherian ring R, and let M and N be finitely generated R-modules. Let [Formula: see text] be the [Formula: see text]-finiteness dimension of N. In this paper, among other things, we show that for each [Formula: see text], (i) the set of associated prime ideals of generalized local cohomology module [Formula: see text] is finite, and (ii) [Formula: see text] is [Formula: see text]-cofinite if and only if [Formula: see text] is so. Moreover, we show that whenever [Formula: see text] is a principal ideal, then [Formula: see text] is [Formula: see text]-cofinite for all n.

scholarly journals Tameness and Artinianness of Graded Generalized Local Cohomology Modules

2015 ◽  
Vol 22 (01) ◽  
pp. 131-146 ◽  
Author(s):  
M. Jahangiri ◽  
N. Shirmohammadi ◽  
Sh. Tahamtan
Keyword(s):  
Local Cohomology ◽  
Sufficient Conditions ◽  
Cohomological Dimension ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Graded Ring ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Quotient Modules ◽  
Cohomology Modules

Let R=⊕n ≥ 0 Rn be a standard graded ring, 𝔞 ⊇ ⊕n > 0 Rn an ideal of R, and M, N two finitely generated graded R-modules. This paper studies the homogeneous components of graded generalized local cohomology modules. We show that for any i ≥ 0, the n-th graded component [Formula: see text] of the i-th generalized local cohomology module of M and N with respect to 𝔞 vanishes for all n ≫ 0. Some sufficient conditions are proposed to satisfy the equality [Formula: see text]. Also, some sufficient conditions are proposed for the tameness of [Formula: see text] such that [Formula: see text] or i= cd 𝔞(M,N), where [Formula: see text] and cd 𝔞(M,N) denote the R+-finiteness dimension and the cohomological dimension of M and N with respect to 𝔞, respectively. Finally, we consider the Artinian property of some submodules and quotient modules of [Formula: see text], where j is the first or last non-minimax level of [Formula: see text].

scholarly journals ATTACHED PRIMES OF THE TOP GENERALIZED LOCAL COHOMOLOGY MODULES

2009 ◽  
Vol 79 (1) ◽  
pp. 59-67 ◽  
Author(s):  
YAN GU ◽  
LIZHONG CHU
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

AbstractLet (R,𝔪) be a commutative Noetherian local ring, letIbe an ideal ofRand letMandNbe finitely generatedR-modules. Assume that$\mathrm {pd} (M)=d\lt \infty $,$\dim N=n\lt \infty $. First, we give the formula for the attached primes of the top generalized local cohomology moduleHId+n(M,N); later, we prove that if Att(HId+n(M,N))=Att(HJd+n(M,N)), thenHId+n(M,N)=HJd+n(M,N).

FINITENESS PROPERTIES OF LOCAL COHOMOLOGY MODULES AND GENERALIZED REGULAR SEQUENCES

2010 ◽  
Vol 09 (02) ◽  
pp. 315-325
Author(s):  
KAMAL BAHMANPOUR ◽  
SEADAT OLLAH FARAMARZI ◽  
REZA NAGHIPOUR
Keyword(s):  
Nonnegative Integer ◽  
Local Cohomology ◽  
Regular Sequence ◽  
Associated Primes ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Regular Sequences ◽  
Local Cohomology Modules ◽  
Finiteness Properties ◽  
Cohomology Modules

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M an R-module. The purpose of this paper is to show that if M is finitely generated and dim M/𝔞M > 1, then the R-module ∪{N|N is a submodule of [Formula: see text] and dim N ≤ 1} is 𝔞-cominimax and for some x ∈ R is Rx + 𝔞-cofinite, where t ≔ gdepth (𝔞, M). For any nonnegative integer l, it is also shown that if R is semi-local and M is weakly Laskerian, then for any submodule N of [Formula: see text] with dim N ≤ 1 the associated primes of [Formula: see text] are finite, whenever [Formula: see text] for all i < l. Finally, we show that if (R, 𝔪) is local, M is finitely generated, [Formula: see text] for all i < l, and [Formula: see text] then there exists a generalized regular sequence x1, …, xl ∈ 𝔞 on M such that [Formula: see text].

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