On the Attached Prime Ideals of Local Cohomology Modules

2013 ◽  
Vol 41 (10) ◽  
pp. 3648-3651
Author(s):  
Jafar A'zami
Keyword(s):  
Local Cohomology ◽  
Prime Ideals ◽  
Local Cohomology Modules ◽  
Attached Prime ◽  
Cohomology Modules

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.

scholarly journals On the attached prime ideals of certain Artinian local cohomology modules

1981 ◽  
Vol 24 (1) ◽  
pp. 9-14 ◽  
Author(s):  
R. Y. Sharp
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Local Problem ◽  
Algebraic Varieties ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Attached Prime ◽  
Cohomology Modules

The study of the cohomological dimensions of algebraic varieties has produced some interesting results and problems in local algebra: the general local problem is that posed by Hartshorne and Speiser in (4, p. 57). We consider a (commutative, Noetherian) local ring A (with identity), a proper ideal a of A, and ask the following question.

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.

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.

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.

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.

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