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 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 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).

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.

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.

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 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.

scholarly journals On Vanishing of Generalized Local Cohomology Modules

2005 ◽  
Vol 12 (02) ◽  
pp. 213-218 ◽  
Author(s):  
K. Divaani-Aazar ◽  
R. Sazeedeh ◽  
M. Tousi
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Gorenstein Local Ring ◽  
Cohomology Modules

Let [Formula: see text] denote an ideal of a d-dimensional Gorenstein local ring R, and M and N two finitely generated R-modules with pd M < ∞. It is shown that [Formula: see text] if and only if [Formula: see text] for all [Formula: see text].

scholarly journals COFINITENESS AND FINITENESS OF GENERALIZED LOCAL COHOMOLOGY MODULES

2009 ◽  
Vol 80 (2) ◽  
pp. 244-250 ◽  
Author(s):  
LIZHONG CHU
Keyword(s):  
Positive Integer ◽  
Local Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Finitely Generated Modules ◽  
Cohomology Modules

AbstractLet I be an ideal of a commutative Noetherian local ring R, and M and N two finitely generated modules. Let t be a positive integer. We mainly prove that (i) if HIi(M,N) is Artinian for all i<t, then HIi(M,N) is I-cofinite for all i<t and Hom(R/I,HIt(M,N)) is finitely generated; (ii) if d=pd(M)<∞ and dim N=n<∞, then HId+n(M,N) is I-cofinite. We also prove that if M is a nonzero cyclic R-module, then HIi(N) is finitely generated for all i<t if and only if HIi(M,N) is finitely generated for all i<t.

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.

Bass numbers of generalized local cohomology modules with respect to a pair of ideals

2018 ◽  
Vol 11 (02) ◽  
pp. 1850019
Author(s):  
M. Lotfi Parsa
Keyword(s):  
Local Ring ◽  
Principal Ideal ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Bass Numbers ◽  
Generalized Local Cohomology Modules ◽  
The Relationship ◽  
Cohomology Modules

Let [Formula: see text] be a Noetherian local ring, [Formula: see text] and [Formula: see text] are ideals of [Formula: see text], and [Formula: see text] and [Formula: see text] are [Formula: see text]-modules. We study the relationship between the Bass numbers of [Formula: see text] and [Formula: see text]. As a consequence, it follows that if one of the following holds: (a) [Formula: see text] is a principal ideal of [Formula: see text], (b) [Formula: see text], (c) [Formula: see text] (when [Formula: see text] is local and [Formula: see text] is finitely generated), (d) [Formula: see text] (when [Formula: see text] is local), (e) [Formula: see text] (when [Formula: see text] is local), then [Formula: see text] is finite for all [Formula: see text] and [Formula: see text], whenever [Formula: see text] is finitely generated and flat, [Formula: see text] is minimax, and [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document