scholarly journals General formal local cohomology modules

2020 ◽  
Vol 30 (2) ◽  
pp. 254-266
Author(s):  
Sh. Rezaei ◽  
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Local Cohomology Modules ◽  
Formal Local Cohomology ◽  
Cohomology Modules

Let (R,m) be a local ring, Φ a system of ideals of R and M a finitely generated R-module. In this paper, we define and study general formal local cohomology modules. We denote the ith general formal local cohomology module M with respect to Φ by FiΦ(M) and we investigate some finiteness and Artinianness properties of general formal local cohomology modules.

Artinianness and Attached Primes of Formal Local Cohomology Modules

2014 ◽  
Vol 21 (02) ◽  
pp. 307-316 ◽  
Author(s):  
Mohammad Hasan Bijan-Zadeh ◽  
Shahram Rezaei
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Upper Bounds ◽  
Finitely Generated ◽  
Lower And Upper Bounds ◽  
Local Cohomology Modules ◽  
Formal Local Cohomology ◽  
Cohomology Modules

Let 𝔞 be an ideal of a local ring (R, 𝔪) and M a finitely generated R-module. In this paper we study the Artinianness properties of formal local cohomology modules and we obtain the lower and upper bounds for Artinianness of formal local cohomology modules. Additionally, we determine the set [Formula: see text] and we show that the set of all non-isomorphic formal local cohomology modules [Formula: see text] is finite.

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.

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).

On the Attached Primes and Shifted Localization Principle for Local Cohomology Modules

2013 ◽  
Vol 20 (04) ◽  
pp. 671-680 ◽  
Author(s):  
Tran Nguyen An
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Localization Principle ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let (R,𝔪) be a Noetherian local ring and M a finitely generated R-module. For an integer i ≥ 0, the Artinian i-th local cohomology module [Formula: see text] is said to satisfy the shifted localization principle if [Formula: see text] for all 𝔭 ∈ Spec (R). In this paper we study the attached primes of [Formula: see text] and give some conditions for [Formula: see text] to satisfy the shifted localization principle.

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.

Some homological properties of generalized local cohomology modules

2019 ◽  
Vol 18 (12) ◽  
pp. 1950238
Author(s):  
Yavar Irani ◽  
Kamal Bahmanpour ◽  
Ghader Ghasemi
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

Let [Formula: see text] be a Noetherian local ring and [Formula: see text], [Formula: see text] be two finitely generated [Formula: see text]-modules. In this paper, it is shown that [Formula: see text] and [Formula: see text] for each [Formula: see text] and each integer [Formula: see text]. In particular, if [Formula: see text] then [Formula: see text]. Moreover, some applications of these results will be included.

On the cofiniteness of Artinian local cohomology modules

2016 ◽  
Vol 15 (04) ◽  
pp. 1650070 ◽  
Author(s):  
Ghader Ghasemi ◽  
Kamal Bahmanpour ◽  
Jafar A’zami
Keyword(s):  
Local Ring ◽  
Homomorphic Image ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Gorenstein Local Ring ◽  
Cohomology Modules

Let [Formula: see text] be a commutative Noetherian local ring, which is a homomorphic image of a Gorenstein local ring and [Formula: see text] an ideal of [Formula: see text]. Let [Formula: see text] be a nonzero finitely generated [Formula: see text]-module and [Formula: see text] be an integer. In this paper we show that, the [Formula: see text]-module [Formula: see text] is nonzero and [Formula: see text]-cofinite if and only if [Formula: see text]. Also, several applications of this result will be included.

Gorenstein Injective Resolutions, Cousin Complexes and Top Local Cohomology Modules

2009 ◽  
Vol 16 (01) ◽  
pp. 95-101
Author(s):  
Kazem Khashyarmanesh
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Local Cohomology Module ◽  
Injective Resolution ◽  
Local Cohomology Modules ◽  
Gorenstein Local Ring ◽  
Gorenstein Injective ◽  
Cousin Complexes ◽  
Cohomology Modules ◽  
Cousin Complex

Let R be a Gorenstein local ring. We show that for a balanced big Cohen–Macaulay module M over R, the Cousin complex [Formula: see text] provides a Gorenstein injective resolution of M. Also, over a d-dimensional Gorenstein local ring R with maximal ideal 𝔪, we show that [Formula: see text], the dth local cohomology module of M with respect to 𝔪, is Gorenstein injective if (a) M is a balanced big Cohen–Macaulay R-module, or (b) M ∈ G(R), where G(R) is the Auslander's G-class of R.

scholarly journals On Injective and Gorenstein Injective Dimensions of Local Cohomology Modules

2015 ◽  
Vol 22 (spec01) ◽  
pp. 935-946 ◽  
Author(s):  
Majid Rahro Zargar ◽  
Hossein Zakeri
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Proper Ideal ◽  
Finitely Generated ◽  
Gorenstein Rings ◽  
Noetherian Local Ring ◽  
Dualizing Complex ◽  
Local Cohomology Modules ◽  
Gorenstein Injective ◽  
Cohomology Modules

Let (R, 𝔪) be a commutative Noetherian local ring and M an R-module which is relative Cohen-Macaulay with respect to a proper ideal 𝔞 of R, and set n := ht M𝔞. We prove that injdim M < ∞ if and only if [Formula: see text] and that [Formula: see text]. We also prove that if R has a dualizing complex and Gid RM < ∞, then [Formula: see text]. Moreover if R and M are Cohen-Macaulay, then Gid RM < ∞ whenever [Formula: see text]. Next, for a finitely generated R-module M of dimension d, it is proved that if [Formula: see text] is Cohen-Macaulay and [Formula: see text], then [Formula: see text]. The above results have consequences which improve some known results and provide characterizations of Gorenstein rings.

Generalized Regular Sequence and Finiteness of Local Cohomology Modules

2008 ◽  
Vol 15 (03) ◽  
pp. 457-462 ◽  
Author(s):  
A. Mafi ◽  
H. Saremi
Keyword(s):  
Positive Integer ◽  
Local Ring ◽  
Local Cohomology ◽  
Regular Sequence ◽  
Natural Homomorphism ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Finite Set ◽  
Cohomology Modules

Let R be a commutative Noetherian local ring, 𝔞 an ideal of R, and M a finitely generated generalized f-module. Let t be a positive integer such that [Formula: see text] and t > dim M - dim M/𝔞M. In this paper, we prove that there exists an ideal 𝔟 ⊇ 𝔞 such that (1) dim M - dim M/𝔟M = t; and (2) the natural homomorphism [Formula: see text] is an isomorphism for all i > t and it is surjective for i = t. Also, we show that if [Formula: see text] is a finite set for all i < t, then there exists an ideal 𝔟 of R such that dim R/𝔟 ≤ 1 and [Formula: see text] for all i < t.

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