scholarly journals Cohomological dimensions with respect to sum and intersection of ideals

2017 ◽  
Vol 102 (116) ◽  
pp. 115-120
Author(s):  
Alireza Vahidi
Keyword(s):  
Noetherian Ring ◽  
Projective Dimension ◽  
Equivalent Condition ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Finite Projective Dimension

Let R be a commutative Noetherian ring with non-zero identity, a and b proper ideals of R,M a finitely generated R-module with finite projective dimension, and X a finitely generated R-module. We study the cohomological dimensions of M and X with respect to a + b and a ? b. We show that the inequality cda+b(M,X) ? cda(M,X) + cdb(X) holds true and we find an equivalent condition for it to be equality.

FINITENESS DIMENSION AND BASS NUMBERS OF GENERALIZED LOCAL COHOMOLOGY MODULES

2013 ◽  
Vol 12 (07) ◽  
pp. 1350036 ◽  
Author(s):  
HERO SAREMI ◽  
AMIR MAFI
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Bass Numbers ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M, N two nonzero finitely generated R-modules. Let t be a non-negative integer. It is shown that dim Supp [Formula: see text] for all i < t if and only if there exists an ideal 𝔟 of R such that dim R/𝔟 ≤ 1 and [Formula: see text] for all i < t. As a consequence all Bass numbers and all Betti numbers of generalized local cohomology modules [Formula: see text] are finite for all i < t, provided that the projective dimension pd (M) is finite.

Primary decomposition of homogeneous ideal in idealization of a module

2018 ◽  
Vol 55 (3) ◽  
pp. 345-352
Author(s):  
Tran Nguyen An
Keyword(s):  
Noetherian Ring ◽  
Primary Decomposition ◽  
Associated Primes ◽  
Homogeneous Ideal ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Irreducible Decomposition

Let R be a commutative Noetherian ring, M a finitely generated R-module, I an ideal of R and N a submodule of M such that IM ⫅ N. In this paper, the primary decomposition and irreducible decomposition of ideal I × N in the idealization of module R ⋉ M are given. From theses we get the formula for associated primes of R ⋉ M and the index of irreducibility of 0R ⋉ M.

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings

1991 ◽  
Vol 34 (1) ◽  
pp. 155-160 ◽  
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

LetEbe an injective module over the commutative Noetherian ringA, and letabe an ideal ofA. TheA-module (0:Eα) has a secondary representation, and the finite set AttA(0:Eα) of its attached prime ideals can be formed. One of the main results of this note is that the sequence of sets (AttA(0:Eαn))n∈Nis ultimately constant. This result is analogous to a theorem of M. Brodmann that, ifMis a finitely generatedA-module, then the sequence of sets (AssA(M/αnM))n∈Nis ultimately constant.

scholarly journals Presentations for Subrings and Subalgebras of Finite CO-Rank

2019 ◽  
Vol 71 (1) ◽  
pp. 53-71
Author(s):  
Peter Mayr ◽  
Nik Ruškuc
Keyword(s):  
Lie Algebra ◽  
Noetherian Ring ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Commutative Noetherian Ring ◽  
Rank 2 ◽  
Finitely Presented ◽  
Algebra Over A Field

Abstract Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on $K$ and show that for finite generation it can be replaced by a weaker condition that the module $A/B$ be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension $1$ of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.

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.

On the Finiteness Dimension of Local Cohomology Modules

2014 ◽  
Vol 21 (03) ◽  
pp. 517-520 ◽  
Author(s):  
Hero Saremi ◽  
Amir Mafi
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Negative Integer ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M a non-zero finitely generated R-module. Let t be a non-negative integer. In this paper, it is shown that [Formula: see text] for all i < t if and only if there exists an ideal 𝔟 of R such that dim R/𝔟 ≤ 1 and [Formula: see text] for all i < t. Moreover, we prove that [Formula: see text] for all i.

Remarks on the local-global principle for a subcategory consisting of extension modules

2019 ◽  
Vol 18 (12) ◽  
pp. 1950236
Author(s):  
Takeshi Yoshizawa
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Global Principles ◽  
Finitely Generated Modules ◽  
Cohomology Modules ◽  
Global Principle

Faltings presented the local-global principle for the finiteness dimension of local cohomology modules. This paper deals with the local-global principle for an extension subcategory over a commutative Noetherian ring. We prove that finitely generated modules satisfy the local-global principles for certain extension subcategories. Additionally, we provide a generalization of Faltings’ local-global principle, which also includes the local-global principles for the Artinianness and Minimaxness of local cohomology modules.

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 An Euler characteristic for modules of finite G-dimension

2008 ◽  
Vol 102 (2) ◽  
pp. 206 ◽  
Author(s):  
Sean Sather-Wagstaff ◽  
Diana White
Keyword(s):  
Euler Characteristic ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Finitely Generated ◽  
Finite Projective Dimension ◽  
Extremal Behavior ◽  
Classical Invariant ◽  
Category Of Modules ◽  
Finitely Generated Modules

We extend Auslander and Buchsbaum's Euler characteristic from the category of finitely generated modules of finite projective dimension to the category of modules of finite G-dimension using Avramov and Martsinkovsky's notion of relative Betti numbers. We prove analogues of some properties of the classical invariant and provide examples showing that other properties do not translate to the new context. One unexpected property is in the characterization of the extremal behavior of this invariant: the vanishing of the Euler characteristic of a module $M$ of finite G-dimension implies the finiteness of the projective dimension of $M$. We include two applications of the Euler characteristic as well as several explicit calculations.

scholarly journals Stable under specialization sets and cofiniteness

2019 ◽  
Vol 18 (01) ◽  
pp. 1950015 ◽  
Author(s):  
K. Divaani-Aazar ◽  
H. Faridian ◽  
M. Tousi
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring

Let [Formula: see text] be a commutative noetherian ring, and [Formula: see text] a stable under specialization subset of [Formula: see text]. We introduce a notion of [Formula: see text]-cofiniteness and study its main properties. In the case [Formula: see text], or [Formula: see text], or [Formula: see text] is semilocal with [Formula: see text], we show that the category of [Formula: see text]-cofinite [Formula: see text]-modules is abelian. Also, in each of these cases, we prove that the local cohomology module [Formula: see text] is [Formula: see text]-cofinite for every homologically left-bounded [Formula: see text]-complex [Formula: see text] whose homology modules are finitely generated and every [Formula: see text].

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