Regular sequences and vanishing of the homological functors

2018 ◽  
Vol 17 (11) ◽  
pp. 1850220
Author(s):  
Kamal Bahmanpour
Keyword(s):  
Positive Integer ◽  
Local Ring ◽  
Regular Sequence ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Dimension Formula ◽  
Regular Sequences ◽  
Equivalent Conditions

Let [Formula: see text] be a Noetherian local ring of dimension [Formula: see text] and let [Formula: see text] be a nonzero finitely generated [Formula: see text]-module. Let [Formula: see text] be a positive integer and let [Formula: see text] be an [Formula: see text]-regular sequence. In this paper we shall present some equivalent conditions for the vanishing of the [Formula: see text]-modules [Formula: see text] and [Formula: see text].

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.

scholarly journals A Characterization of Gorenstein Rings and Grothendieck's Conjecture

2009 ◽  
Vol 16 (04) ◽  
pp. 653-660
Author(s):  
Kazem Khashyarmanesh
Keyword(s):  
Local Ring ◽  
Regular Sequence ◽  
Negative Integer ◽  
Finitely Generated ◽  
Gorenstein Rings ◽  
Noetherian Local Ring ◽  
Filter Regular Sequence ◽  
System Of Parameters

Given a commutative Noetherian local ring (R, 𝔪), it is shown that R is Gorenstein if and only if there exists a system of parameters x1,…,xd of R which generates an irreducible ideal and [Formula: see text] for all t > 0. Let n be an arbitrary non-negative integer. It is also shown that for an arbitrary ideal 𝔞 of a commutative Noetherian (not necessarily local) ring R and a finitely generated R-module M, [Formula: see text] is finitely generated if and only if there exists an 𝔞-filter regular sequence x1,…,xn∈ 𝔞 such that [Formula: see text] for all t > 0.

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 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.

scholarly journals Maximal depth property of finitely generated modules

2018 ◽  
Vol 17 (11) ◽  
pp. 1850202 ◽  
Author(s):  
Ahad Rahimi
Keyword(s):  
Local Ring ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Maximal Depth ◽  
Finitely Generated Modules ◽  
Associated Prime

Let [Formula: see text] be a Noetherian local ring and [Formula: see text] a finitely generated [Formula: see text]-module. We say [Formula: see text] has maximal depth if there is an associated prime [Formula: see text] of [Formula: see text] such that depth [Formula: see text]. In this paper, we study finitely generated modules with maximal depth. It is shown that the maximal depth property is preserved under some important module operations. Generalized Cohen–Macaulay modules with maximal depth are classified. Finally, the attached primes of [Formula: see text] are considered for [Formula: see text].

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

2021 ◽  
Vol 28 (01) ◽  
pp. 13-32
Author(s):  
Nguyen Tien Manh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Graded Modules ◽  
Fiber Cone

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

scholarly journals Toward a theory of generalized Cohen-Macaulay modules

1986 ◽  
Vol 102 ◽  
pp. 1-49 ◽  
Author(s):  
Ngô Viêt Trung
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Finitely Generated ◽  
Noetherian Local Ring

Throughout this paper, A denotes a noetherian local ring with maximal ideal m and M a finitely generated A-module with d: = dim M≥1.

scholarly journals Linearity defect of the residue field of short local rings

2016 ◽  
Vol 16 (09) ◽  
pp. 1750163
Author(s):  
Rasoul Ahangari Maleki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Residue Field ◽  
Positive Answer ◽  
Local Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Linear Resolution

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text] and residue field [Formula: see text]. The linearity defect of a finitely generated [Formula: see text]-module [Formula: see text], which is denoted [Formula: see text], is a numerical measure of how far [Formula: see text] is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether [Formula: see text] implies [Formula: see text], in the case when [Formula: see text].

Annihilator of local cohomology of homogeneous parts of a graded module

2020 ◽  
pp. 2150092
Author(s):  
Tran Do Minh Chau ◽  
Nguyen Thi Kieu Nga ◽  
Le Thanh Nhan
Keyword(s):  
Asymptotic Stability ◽  
Local Ring ◽  
Local Cohomology ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Graded Module

Let [Formula: see text] be a homogeneous graded ring, where [Formula: see text] is a Noetherian local ring. Let [Formula: see text] be a finitely generated graded [Formula: see text]-module. For [Formula: see text] set [Formula: see text]. Denote by [Formula: see text] the set of all prime ideals of [Formula: see text] containing [Formula: see text]. For [Formula: see text], let [Formula: see text] be the set of all [Formula: see text] such that [Formula: see text] In this paper, we prove that the sets [Formula: see text] and [Formula: see text] do not depend on [Formula: see text] for [Formula: see text]. We show that the annihilators [Formula: see text], [Formula: see text] are eventually stable, where [Formula: see text] for [Formula: see text]. As an application, we prove the asymptotic stability of some loci contained in the non-Cohen–Macaulay locus of [Formula: see text].

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.

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