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.

On the Finiteness Results of the Generalized Local Cohomology Modules

2009 ◽  
Vol 16 (02) ◽  
pp. 325-332 ◽  
Author(s):  
Amir Mafi
Keyword(s):  
Positive Integer ◽  
Local Cohomology ◽  
Prime Ideals ◽  
Local Cohomology Module ◽  
Local Cohomology Modules ◽  
Associated Prime Ideals ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules ◽  
Infinite Set ◽  
Associated Prime

Let 𝔞 be an ideal of a commutative Noetherian local ring R, and let M and N be two finitely generated R-modules. Let t be a positive integer. It is shown that if the support of the generalized local cohomology module [Formula: see text] is finite for all i < t, then the set of associated prime ideals of the generalized local cohomology module [Formula: see text] is finite. Also, if the support of the local cohomology module [Formula: see text] is finite for all i < t, then the set [Formula: see text] is finite. Moreover, we prove that gdepth (𝔞+ Ann (M),N) is the least integer t such that the support of the generalized local cohomology module [Formula: see text] is an infinite set.

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.

scholarly journals Localization at countably infinitely many prime ideals and applications

2016 ◽  
Vol 15 (03) ◽  
pp. 1650045 ◽  
Author(s):  
Kamal Bahmanpour ◽  
Pham Hung Quy
Keyword(s):  
Local Cohomology ◽  
Prime Ideals ◽  
Local Cohomology Modules ◽  
Technical Lemma ◽  
Associated Prime Ideals ◽  
Cohomology Modules ◽  
Associated Prime

In this paper we present a technical lemma about localization at countably infinitely many prime ideals. We apply this lemma to get many results about the finiteness of associated prime ideals of local cohomology modules.

ON THE BEHAVIOR OF APPLYING DERIVED FUNCTORS OF TORSION FUNCTORS ON LOCAL COHOMOLOGY MODULES

2012 ◽  
Vol 11 (06) ◽  
pp. 1250113
Author(s):  
K. KHASHYARMANESH ◽  
F. KHOSH-AHANG
Keyword(s):  
Local Cohomology ◽  
Cohomological Dimension ◽  
Prime Ideals ◽  
Local Cohomology Modules ◽  
Associated Prime Ideals ◽  
Special Cases ◽  
Derived Functors ◽  
Cohomology Modules ◽  
Associated Prime

In this note, by using some properties of the local cohomology functors of weakly Laskerian modules, we study the behavior of right and left derived functors of torsion functors. In fact, firstly we gain some isomorphisms in the context of these functors, grade and cohomological dimension. Then we study their supports and their sets of associated prime ideals in special cases.

A finite result for the set of associated prime ideals of formal local cohomology modules

2018 ◽  
Vol 13 (02) ◽  
pp. 2050046
Author(s):  
Pham Huu Khanh
Keyword(s):  
Maximal Ideal ◽  
Local Cohomology ◽  
Prime Ideals ◽  
Local Cohomology Modules ◽  
Associated Prime Ideals ◽  
Finite Set ◽  
Formal Local Cohomology ◽  
Cohomology Modules ◽  
Associated Prime ◽  
Finite Result

Let [Formula: see text] be a Noetherian local ring, [Formula: see text] two ideals of [Formula: see text], and [Formula: see text] two finitely generated [Formula: see text]-modules. It is first shown that [Formula: see text] is a finite set. We also prove that except the maximal ideal [Formula: see text], the set [Formula: see text] is stable for large [Formula: see text], where we use [Formula: see text] to denote [Formula: see text]-module [Formula: see text] or [Formula: see text] and [Formula: see text] is the eventual value of [Formula: see text].

A Note on the Artinianness and Vanishing of Local Cohomology and Generalized Local Cohomology Modules

2009 ◽  
Vol 16 (03) ◽  
pp. 517-524 ◽  
Author(s):  
K. Khashyarmanesh ◽  
F. Khosh-Ahang
Keyword(s):  
Local Cohomology ◽  
Cohomological Dimension ◽  
Prime Ideals ◽  
Lower And Upper Bounds ◽  
Commutative Noetherian Ring ◽  
Reflexive Module ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Attached Prime ◽  
Cohomology Modules

The first part of this paper is concerned with the Artinianness of certain local cohomology modules [Formula: see text] when M is a Matlis reflexive module over a commutative Noetherian complete local ring R and 𝔞 is an ideal of R. Also, we characterize the set of attached prime ideals of [Formula: see text], where n is the dimension of M. The second part is concerned with the vanishing of local cohomology and generalized local cohomology modules. In fact, when R is an arbitrary commutative Noetherian ring, M is an R-module and 𝔞 is an ideal of R, we obtain some lower and upper bounds for the cohomological dimension of M with respect to 𝔞.

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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