ARITHMETIC RANK, COHOMOLOGICAL DIMENSION AND FILTER REGULAR SEQUENCES

2009 ◽  
Vol 08 (06) ◽  
pp. 855-862 ◽  
Author(s):  
ALI AKBAR MEHRVARZ ◽  
KAMAL BAHMANPOUR ◽  
REZA NAGHIPOUR
Keyword(s):  
Noetherian Ring ◽  
Cohomological Dimension ◽  
Regular Sequence ◽  
Commutative Noetherian Ring ◽  
Regular Sequences ◽  
Filter Regular Sequence ◽  
Arithmetic Rank

Let I be an ideal of a commutative Noetherian ring R such that ara (I) = t ≥ 2. The purpose of this article is to show that there exists an I-filter regular sequence y1, …, yt for R such that Rad (I) = Rad (y1, …, yt) and cd ((y1, …, yi), R) = i for all 1 ≤ i < t. Also, it is shown that ara (I) ≤ dim R + 1, which is a generalization of a nice result of Kronecker [14]. In addition, some applications are included.

Filter regular sequences and endomorphisms of local cohomology modules

2018 ◽  
Vol 17 (11) ◽  
pp. 1850213
Author(s):  
Ebrahim Zangoiezadeh ◽  
Kazem Khashyarmanesh
Keyword(s):  
Exact Sequence ◽  
Noetherian Ring ◽  
Closed Subset ◽  
Local Cohomology ◽  
Cohomological Dimension ◽  
Commutative Noetherian Ring ◽  
Regular Sequences ◽  
Ring Of Fractions ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let [Formula: see text] be a commutative Noetherian ring and [Formula: see text] be an ideal of [Formula: see text] such that [Formula: see text], where [Formula: see text] is the cohomological dimension of [Formula: see text] with respect to [Formula: see text] and [Formula: see text] is the grade of [Formula: see text]. We show that whenever, [Formula: see text] for all [Formula: see text] and all integers [Formula: see text] with [Formula: see text] and [Formula: see text], then there exists an exact sequence [Formula: see text] of endomorphisms of local cohomology modules, where [Formula: see text] and, for [Formula: see text], [Formula: see text] is the ring of fractions of [Formula: see text] with respect to multiplicatively closed subset [Formula: see text] of [Formula: see text].

scholarly journals On Regular Sequences

1966 ◽  
Vol 27 (1) ◽  
pp. 355-356 ◽  
Author(s):  
J. Dieudonné
Keyword(s):  
Algebraic Geometry ◽  
Noetherian Ring ◽  
Regular Sequence ◽  
Local Rings ◽  
Regular Sequences

The concept of regular sequence of elements of a ring A (first introduced by Serre under the name of A-sequence [2]), has far-reaching uses in the theory of local rings and in algebraic geometry. It seems, however, that it loses much of its importance when A is not a noetherian ring, and in that case, it probably should be superseded by the concept of quasi-regular sequence [1].

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 Filtrations in Module Categories, Derived Categories, and Prime Spectra

2020 ◽  
Author(s):  
Hiroki Matsui ◽  
Ryo Takahashi
Keyword(s):  
Noetherian Ring ◽  
Commutative Diagram ◽  
Closed Subset ◽  
Cohomological Dimension ◽  
Derived Categories ◽  
Commutative Noetherian Ring

Abstract Let $R$ be a commutative noetherian ring. The notion of $n$-wide subcategories of ${\operatorname{\mathsf{Mod}}}\ R$ is introduced and studied in Matsui–Nam–Takahashi–Tri–Yen in relation to the cohomological dimension of a specialization-closed subset of ${\operatorname{Spec}}\ R$. In this paper, we introduce the notions of $n$-coherent subsets of ${\operatorname{Spec}}\ R$ and $n$-uniform subcategories of $\mathsf{D}({\operatorname{\mathsf{Mod}}}\ R)$ and explore their interactions with $n$-wide subcategories of ${\operatorname{\mathsf{Mod}}}\ R$. We obtain a commutative diagram that yields filtrations of subcategories of ${\operatorname{\mathsf{Mod}}}\ R$, $\mathsf{D}({\operatorname{\mathsf{Mod}}}\ R)$ and subsets of ${\operatorname{Spec}}\ R$ and complements classification theorems of subcategories due to Gabriel, Krause, Neeman, Takahashi, and Angeleri Hügel–Marks–Šťovíček–Takahashi–Vitória.

The first non-isomorphic local cohomology modules with respect to their ideals

2018 ◽  
Vol 17 (12) ◽  
pp. 1850230
Author(s):  
Ali Fathi
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Regular Sequences ◽  
Maximal Ideals ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let [Formula: see text] be ideals of a commutative Noetherian ring [Formula: see text] and [Formula: see text] be a finitely generated [Formula: see text]-module. By using filter regular sequences, we show that the infimum of integers [Formula: see text] such that the local cohomology modules [Formula: see text] and [Formula: see text] are not isomorphic is equal to the infimum of the depths of [Formula: see text]-modules [Formula: see text], where [Formula: see text] runs over all prime ideals of [Formula: see text] containing only one of the ideals [Formula: see text]. In particular, these local cohomology modules are isomorphic for all integers [Formula: see text] if and only if [Formula: see text]. As an application of this result, we prove that for a positive integer [Formula: see text], [Formula: see text] is Artinian for all [Formula: see text] if and only if, it can be represented as a finite direct sum of [Formula: see text] local cohomology modules of [Formula: see text] with respect to some maximal ideals in [Formula: see text] for any [Formula: see text]. These representations are unique when they are minimal with respect to inclusion.

FINITENESS PROPERTIES OF LOCAL COHOMOLOGY MODULES AND GENERALIZED REGULAR SEQUENCES

2010 ◽  
Vol 09 (02) ◽  
pp. 315-325
Author(s):  
KAMAL BAHMANPOUR ◽  
SEADAT OLLAH FARAMARZI ◽  
REZA NAGHIPOUR
Keyword(s):  
Nonnegative Integer ◽  
Local Cohomology ◽  
Regular Sequence ◽  
Associated Primes ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Regular Sequences ◽  
Local Cohomology Modules ◽  
Finiteness Properties ◽  
Cohomology Modules

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M an R-module. The purpose of this paper is to show that if M is finitely generated and dim M/𝔞M > 1, then the R-module ∪{N|N is a submodule of [Formula: see text] and dim N ≤ 1} is 𝔞-cominimax and for some x ∈ R is Rx + 𝔞-cofinite, where t ≔ gdepth (𝔞, M). For any nonnegative integer l, it is also shown that if R is semi-local and M is weakly Laskerian, then for any submodule N of [Formula: see text] with dim N ≤ 1 the associated primes of [Formula: see text] are finite, whenever [Formula: see text] for all i < l. Finally, we show that if (R, 𝔪) is local, M is finitely generated, [Formula: see text] for all i < l, and [Formula: see text] then there exists a generalized regular sequence x1, …, xl ∈ 𝔞 on M such that [Formula: see text].

scholarly journals Cohomological dimension with respect to the linked ideals

2020 ◽  
pp. 2150104
Author(s):  
Maryam Jahangiri ◽  
Khadijeh Sayyari
Keyword(s):  
Noetherian Ring ◽  
Cohomological Dimension ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
The Ideal

Let [Formula: see text] be a commutative Noetherian ring. Using the new concept of linkage of ideals over a module, we show that if [Formula: see text] is an ideal of [Formula: see text] which is linked by the ideal [Formula: see text], then [Formula: see text] where [Formula: see text]. Also, it is shown that for every ideal [Formula: see text] which is geometrically linked with [Formula: see text] [Formula: see text] does not depend on [Formula: see text].

On Grothendieck's local duality theorem

1979 ◽  
Vol 85 (3) ◽  
pp. 431-437 ◽  
Author(s):  
M. H. Bijan-Zadeh ◽  
R. Y. Sharp
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Noetherian Ring ◽  
Duality Theorem ◽  
Great Emphasis ◽  
Triangulated Category ◽  
Elementary Approach ◽  
Commutative Noetherian Ring ◽  
Local Duality ◽  
Dualizing Complexes

In (11) and (12), a comparatively elementary approach to the use of dualizing complexes in commutative algebra has been developed. Dualizing complexes were introduced by Grothendieck and Hartshorne in (2) for use in algebraic geometry; the approach to dualizing complexes in (11) and (12) differs from that of Grothendieck and Hartshorne in that it avoids use of the concepts of triangulated category, derived category, and localization of categories, and instead places great emphasis on the concept of quasi-isomorphism of complexes of modules over a commutative Noetherian ring.

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 Proregular sequences, local cohomology, and completion

2003 ◽  
Vol 92 (2) ◽  
pp. 161 ◽  
Author(s):  
Peter Schenzel
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Čech Complex ◽  
Regular Sequences ◽  
Local Cohomology Modules ◽  
Derived Functors ◽  
Adic Completion ◽  
Cech Complex ◽  
Cohomology Modules ◽  
The Ideal

As a certain generalization of regular sequences there is an investigation of weakly proregular sequences. Let $M$ denote an arbitrary $R$-module. As the main result it is shown that a system of elements $\underline x$ with bounded torsion is a weakly proregular sequence if and only if the cohomology of the Čech complex $\check C_{\underline x} \otimes M$ is naturally isomorphic to the local cohomology modules $H_{\mathfrak a}^i(M)$ and if and only if the homology of the co-Čech complex $\mathrm{RHom} (\check C_{\underline x}, M)$ is naturally isomorphic to $\mathrm{L}_i \Lambda^{\mathfrak a}(M),$ the left derived functors of the $\mathfrak a$-adic completion, where $\mathfrak a$ denotes the ideal generated by the elements $\underline x$. This extends results known in the case of $R$ a Noetherian ring, where any system of elements forms a weakly proregular sequence of bounded torsion. Moreover, these statements correct results previously known in the literature for proregular sequences.

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