scholarly journals On cominimaxness of generalized local cohomology modules

2019 ◽  
Vol 16 (3) ◽  
pp. 50
Author(s):  
Nguyen Thanh Nam ◽  
Tran Tuan Nam ◽  
Nguyen Minh Tri
Keyword(s):  
Nonnegative Integer ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

This research introduces and focuses on (I, M)-cominimax modules. The paper shows that if t is an nonnegative integer, M is a finitely generated projective R-module and N is an R-module such that  is minimax and  is (I, M)-cominimax for all  then  is minimax and  is finite.

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.

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.

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.

Cohomological Dimension of Generalized Local Cohomology Modules

2008 ◽  
Vol 15 (02) ◽  
pp. 303-308 ◽  
Author(s):  
Jafar Amjadi ◽  
Reza Naghipour
Keyword(s):  
Exact Sequence ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Cohomological Dimension ◽  
Algebraic Varieties ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

The study of the cohomological dimension of algebraic varieties has produced some interesting results and problems in local algebra. Let 𝔞 be an ideal of a commutative Noetherian ring R. For finitely generated R-modules M and N, the concept of cohomological dimension cd 𝔞(M, N) of M and N with respect to 𝔞 is introduced. If 0 → N' → N → N'' → 0 is an exact sequence of finitely generated R-modules, then it is shown that cd 𝔞(M, N) = max { cd 𝔞(M, N'), cd 𝔞(M, N'')} whenever proj dim M < ∞. Also, if L is a finitely generated R-module with Supp (N/Γ𝔞(N)) ⊆ Supp (L/Γ𝔞(L)), then it is proved that cd 𝔞(M, N) ≤ max { cd 𝔞(M,L), proj dim M}. Finally, as a consequence, a result of Brodmann is improved.

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 Bass numbers of generalized local cohomology modules

2015 ◽  
Vol 97 (111) ◽  
pp. 233-238 ◽  
Author(s):  
Sh. Payrovi ◽  
S. Babaei ◽  
I. Khalili-Gorji
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Modules ◽  
Bass Numbers ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

Let R be a Noetherian ring, M a finitely generated R-module and N an arbitrary R-module. We consider the generalized local cohomology modules, with respect to an arbitrary ideal I of R, and prove that, for all nonnegative integers r, t and all p ? Spec(R) the Bass number ?r(p,HtI (M,N)) is bounded above by ?tj=0?r(p, t?jExtR (M,HjI (N))). A corollary is that Ass (HtI (M,N)? Utj=0 Ass (t?jExtR (M,HjI(N))). In a slightly different direction, we also present some well known results about generalized local cohomology modules.

Cofiniteness of Generalized Local Cohomology Modules for One-Dimensional Ideals

2012 ◽  
Vol 55 (1) ◽  
pp. 81-87 ◽  
Author(s):  
Kamran Divaani-Aazar ◽  
Alireza Hajikarimi
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
One Dimensional ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

AbstractLet a be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Our main result asserts that if dim R/a ≤ 1, then all generalized local cohomology modules (M,N) are a-cofinite.

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].

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 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].

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