scholarly journals ON THE ASSOCIATED PRIMES OF LOCAL COHOMOLOGY

2018 ◽  
Vol 237 ◽  
pp. 1-9 ◽  
Author(s):  
HAILONG DAO ◽  
PHAM HUNG QUY
Keyword(s):  
Local Cohomology ◽  
Positive Characteristic ◽  
Short Proof ◽  
Singular Locus ◽  
Prime Ideals ◽  
Associated Primes ◽  
Commutative Noetherian Ring ◽  
Regular Sequences ◽  
Associated Prime Ideals ◽  
Associated Prime

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. In this paper, we give a short proof using filter regular sequences that the set of associated prime ideals of $H_{I}^{t}(R)$ is finite for any ideal $I$ and for any $t\geqslant 0$ when $R$ has finite $F$-representation type or finite singular locus. This extends a previous result by Takagi–Takahashi and gives affirmative answers for a problem of Huneke in many new classes of rings in positive characteristic. We also give a criterion about the singularities of $R$ (in any characteristic) to guarantee that the set $\operatorname{Ass}H_{I}^{2}(R)$ is always finite.

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.

ASSOCIATED PRIMES OVER ORE EXTENSION RINGS

2004 ◽  
Vol 03 (02) ◽  
pp. 193-205 ◽  
Author(s):  
SCOTT ANNIN
Keyword(s):  
Recent Work ◽  
Polynomial Rings ◽  
Prime Ideals ◽  
Associated Primes ◽  
Ore Extension ◽  
Skew Polynomial Rings ◽  
Associated Prime Ideals ◽  
Polynomial Module ◽  
Skew Polynomial ◽  
Associated Prime

The study of the prime ideals in Ore extension rings R[x,σ,δ] has attracted a lot of attention in recent years and has proven to be a challenging undertaking ([5], [7], [12], et al.). The present article makes a contribution to this study for the associated prime ideals. More precisely, we aim to describe how the associated primes of an R-module MR behave under passage to the polynomial module M[x] over an Ore extension R[x,σ,δ]. If we impose natural σ-compatibility and δ-compatibility assumptions on the module MR (see Sec. 2 below), we can describe all associated primes of the R[x,σ,δ]-module M[x] in terms of the associated primes of MR in a very straightforward way. This result generalizes the author's recent work [1] on skew polynomial rings.

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.

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].

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.

Associated Prime Ideals in Non-Noetherian Rings

1984 ◽  
Vol 36 (2) ◽  
pp. 344-360 ◽  
Author(s):  
Juana Iroz ◽  
David E. Rush
Keyword(s):  
Prime Ideal ◽  
Finite Subset ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Associated Primes ◽  
The Past ◽  
Associated Prime Ideals ◽  
Associated Prime

The theory of associated prime ideals is one of the most basic notions in the study of modules over commutative Noetherian rings. For modules over non-Noetherian rings however, the classical associated primes are not so useful and in fact do not exist for some modules M. In [4] [22] a prime ideal P of a ring R is said to be attached to an R-module M if for each finite subset I of P there exists m ∊ M such that I ⊆ annR(m) ⊆ P. In [4] the attached primes were compared to the associated primes and the results of [4], [22], [23], [24] show that the attached primes are a useful alternative in non-Noetherian rings to associated primes. Several other methods of associating a set of prime ideals to a module M over a non-Noetherian ring have proven very useful in the past. The most common of these is the set Assf(M) of weak Bourbaki primes of M [2, pp. 289-290].

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