Artinian local cohomology modules of cofinie modules

2020 ◽  
pp. 2250029
Author(s):  
Kamal Bahmanpour
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Proper Ideal ◽  
Dimension Formula ◽  
Complete Local Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let [Formula: see text] be a commutative Noetherian complete local ring and [Formula: see text] be a proper ideal of [Formula: see text]. Suppose that [Formula: see text] is a nonzero [Formula: see text]-cofinite [Formula: see text]-module of Krull dimension [Formula: see text]. In this paper, it shown that [Formula: see text] As an application of this result, it is shown that [Formula: see text], for each [Formula: see text] Also it shown that for each [Formula: see text] the submodule [Formula: see text] and [Formula: see text] of [Formula: see text] is [Formula: see text]-cofinite, [Formula: see text] and [Formula: see text] whenever the category of all [Formula: see text]-cofinite [Formula: see text]-modules is an Abelian subcategory of the category of all [Formula: see text]-modules. Also some applications of these results will be included.

Cofiniteness of Local Cohomology Modules

2014 ◽  
Vol 21 (04) ◽  
pp. 605-614 ◽  
Author(s):  
Kamal Bahmanpour ◽  
Reza Naghipour ◽  
Monireh Sedghi
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Partial Answer ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Noetherian Local Ring ◽  
Dimension Formula ◽  
Complete Local Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let M be a non-zero finitely generated module over a commutative Noetherian local ring (R, 𝔪). In this paper we consider when the local cohomology modules are finitely generated. It is shown that if t ≥ 0 is an integer and [Formula: see text], then [Formula: see text] is not 𝔭-cofinite. Then we obtain a partial answer to a question raised by Huneke. Namely, if R is a complete local ring, then [Formula: see text] is finitely generated if and only if 0 ≤ n ∉ W, where [Formula: see text]. Also, we show that if J ⊆ I are 1-dimensional ideals of R, then [Formula: see text] is J-cominimax, and [Formula: see text] is finitely generated (resp., minimax) if and only if [Formula: see text] is finitely generated for all [Formula: see text] (resp., [Formula: see text]). Moreover, the concept of the J-cofiniteness dimension [Formula: see text] of M relative to I is introduced, and we explore an interrelation between [Formula: see text] and the filter depth of M in I. Finally, we show that if R is complete and dim M/IM ≠ 0, then [Formula: see text].

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.

Family of quotients of some special rings

2018 ◽  
Vol 17 (12) ◽  
pp. 1850233 ◽  
Author(s):  
Maryam Salimi
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Proper Ideal ◽  
Rees Algebra ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
Local Cohomology Modules ◽  
Cohomology Modules ◽  
The Ideal

Let [Formula: see text] be a commutative Noetherian ring and let [Formula: see text] be a proper ideal of [Formula: see text]. We study some properties of a family of rings [Formula: see text] that are obtained as quotients of the Rees algebra associated with the ring [Formula: see text] and the ideal [Formula: see text]. We deal with the strongly cotorsion property of local cohomology modules of [Formula: see text], when [Formula: see text] is a local ring. Also, we investigate generically Cohen–Macaulay, generically Gorenstein, and generically quasi-Gorenstein properties of [Formula: see text]. Finally, we show that [Formula: see text] is approximately Cohen–Macaulay if and only if [Formula: see text] is approximately Cohen–Macaulay, provided some special conditions.

scholarly journals Top Local Cohomology Modules with Specified Attached Primes

2008 ◽  
Vol 15 (02) ◽  
pp. 341-344 ◽  
Author(s):  
Mohammad T. Dibaei ◽  
Raheleh Jafari
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Proper Subset ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let (R, 𝔪) be a complete Noetherian local ring and let M be a finite R-module of positive Krull dimension n. It is shown that any subset T of Assh R(M) can be expressed as the set of attached primes of the top local cohomology module [Formula: see text] for some ideal 𝔞 of R. Moreover, if 𝔞 is an ideal of R such that the set of attached primes of [Formula: see text] is a non-empty proper subset of Assh R(M), then [Formula: see text] for some ideal 𝔟 of R with dim R(R/𝔟) = 1.

scholarly journals Local cohomology modules and their properties

2021 ◽  
Vol 73 (2) ◽  
pp. 268-274
Author(s):  
J. Azami ◽  
M. Hasanzad
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Proper Ideal ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Bass Numbers ◽  
Cohomology Modules ◽  
The Ideal

UDC 512.5 Let be a complete Noetherian local ring and let be a generalized Cohen-Macaulay -module of dimension We show thatwhere and is the ideal transform functor. Also, assuming that is a proper ideal of a local ring , we obtain some results on the finiteness of Bass numbers, cofinitness, and cominimaxness of local cohomology modules with respect to

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.

Some properties of Artinian modules and applications

2018 ◽  
Vol 17 (02) ◽  
pp. 1850019
Author(s):  
Tran Nguyen An
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Vanishing Theorem ◽  
Strong Connection ◽  
Artinian Modules ◽  
Local Cohomology Modules ◽  
Artinian Module ◽  
Base Ring ◽  
System Of Parameters ◽  
Cohomology Modules

Let [Formula: see text] be a Noetherian local ring and [Formula: see text] be an Artinian [Formula: see text]-module. Consider the following property for [Formula: see text] : [Formula: see text] In this paper, we study the property (∗) of [Formula: see text] in order to investigate the relation of system of parameters between [Formula: see text] and the ring [Formula: see text]. We also show that the property (∗) of [Formula: see text] has strong connection with the structure of base ring. Some applications to cofinite Artinian module are given. These are generalizations of [N. Abazari and K. Bahmanpour, A note on the Artinian cofinite modules, Comm. Algebra. 42 (2014) 1270–1275; G. Ghasemi, K. Bahmanpour and J. Azami, On the cofiniteness of Artinian local cohomology modules, J. Algebra Appl. 15(4) (2016), Article ID: 1650070, 8 pp.] A generalization of Lichtenbaum–Hartshorne Vanishing Theorem is also given in this paper.

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.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document