scholarly journals Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters

2010 ◽  
Vol 199 ◽  
pp. 95-105 ◽  
Author(s):  
Shiro Goto ◽  
Kazuho Ozeki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Local Rings ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Hilbert Coefficients

AbstractLet (A,m) be a Noetherian local ring with d = dim A ≥ 2. Then, if A is a Buchsbaum ring, the first Hilbert coefficients of A for parameter ideals Q are constant and equal to where hi(A) denotes the length of the ith local cohomology module of A with respect to the maximal ideal m. This paper studies the question of whether the converse of the assertion holds true, and proves that A is a Buchsbaum ring if A is unmixed and the values are constant, which are independent of the choice of parameter ideals Q in A. Hence, a conjecture raised by [GhGHOPV] is settled affirmatively.

scholarly journals Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters

2010 ◽  
Vol 199 ◽  
pp. 95-105 ◽  
Author(s):  
Shiro Goto ◽  
Kazuho Ozeki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Local Rings ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Hilbert Coefficients

AbstractLet (A,m) be a Noetherian local ring withd= dimA≥ 2. Then, ifAis a Buchsbaum ring, the first Hilbert coefficientsofAfor parameter idealsQare constant and equal towherehi(A)denotes the length of theith local cohomology moduleofAwith respect to the maximal ideal m. This paper studies the question of whether the converse of the assertion holds true, and proves thatAis a Buchsbaum ring ifAis unmixed and the valuesare constant, which are independent of the choice of parameter idealsQinA. Hence, a conjecture raised by [GhGHOPV] is settled affirmatively.

scholarly journals Krull dimension and generalized fractions

1985 ◽  
Vol 28 (3) ◽  
pp. 349-353 ◽  
Author(s):  
M. A. Hamieh ◽  
R. Y. Sharp
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Image Position ◽  
System Of Parameters

Let R be a (commutative Noetherian) local ring (with identity) having maximal ideal and dimension d≧l. It is shown in [5,3.6rsqb; that the local cohomology module may be described as a module of generalized fractions: if x1…,xd is a system of parameters for R, then , where U(x)d+1 is the triangular subset [4,2.1] of Rd+1 given by

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].

Hilbert–Samuel multiplicity and Northcott’s inequality relative to an Artinian module

2019 ◽  
Vol 30 (02) ◽  
pp. 379-396
Author(s):  
V. H. Jorge Pérez ◽  
T. H. Freitas
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Local Cohomology Module ◽  
Artinian Modules ◽  
Dimension Formula ◽  
Hilbert Coefficients ◽  
Artinian Module ◽  
Samuel Multiplicity

Let [Formula: see text] be a commutative quasi-local ring (with identity [Formula: see text]), and let [Formula: see text] be an [Formula: see text]-ideal such that [Formula: see text]. For [Formula: see text] an Artinian [Formula: see text]-module of N-dimension [Formula: see text], we introduce the notion of Hilbert-coefficients of [Formula: see text] relative to [Formula: see text] and give several properties. When [Formula: see text] is a co-Cohen–Macaulay [Formula: see text]-module, we establish the Northcott’s inequality for Artinian modules. As applications, we show some formulas involving the Hilbert coefficients and we investigate the behavior of these multiplicities when the module is the local cohomology module.

Idealizations of maximal Buchsbaum modules over a Buchsbaum ring

1988 ◽  
Vol 104 (3) ◽  
pp. 451-478 ◽  
Author(s):  
Kikumichi Yamagishi
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Noetherian Local Ring

Throughout this paper A denotes a Noetherian local ring with maximal ideal m and M denotes a finitely generated A-module. Moreover stands for the ith local cohomology functor with respect to m (cf. [10]). We refer to [15] for unexplained terminolog.

scholarly journals The structure of sally modules and Buchsbaumness of associated graded rings

2013 ◽  
Vol 212 ◽  
pp. 97-138 ◽  
Author(s):  
Kazuho Ozeki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Rings ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Hilbert Coefficients ◽  
Associated Graded Rings

AbstractLet A be a Noetherian local ring with the maximal ideal m, and let I be an m-primary ideal in A. This paper examines the equality on Hilbert coefficients of I first presented by Elias and Valla, but without assuming that A is a Cohen–Macaulay local ring. That equality is related to the Buchsbaumness of the associated graded ring of I.

scholarly journals The structure of sally modules and Buchsbaumness of associated graded rings

2013 ◽  
Vol 212 ◽  
pp. 97-138 ◽  
Author(s):  
Kazuho Ozeki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Rings ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Hilbert Coefficients ◽  
Associated Graded Rings

AbstractLetAbe a Noetherian local ring with the maximal ideal m, and letIbe an m-primary ideal inA. This paper examines the equality on Hilbert coefficients ofIfirst presented by Elias and Valla, but without assuming thatAis a Cohen–Macaulay local ring. That equality is related to the Buchsbaumness of the associated graded ring ofI.

scholarly journals ATTACHED PRIMES OF THE TOP GENERALIZED LOCAL COHOMOLOGY MODULES

2009 ◽  
Vol 79 (1) ◽  
pp. 59-67 ◽  
Author(s):  
YAN GU ◽  
LIZHONG CHU
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

AbstractLet (R,𝔪) be a commutative Noetherian local ring, letIbe an ideal ofRand letMandNbe finitely generatedR-modules. Assume that$\mathrm {pd} (M)=d\lt \infty $,$\dim N=n\lt \infty $. First, we give the formula for the attached primes of the top generalized local cohomology moduleHId+n(M,N); later, we prove that if Att(HId+n(M,N))=Att(HJd+n(M,N)), thenHId+n(M,N)=HJd+n(M,N).

Co-Cohen-Macaulay Modules and Generalized Local Cohomology

2011 ◽  
Vol 18 (spec01) ◽  
pp. 807-813
Author(s):  
Amir Mafi
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Finite Dimension ◽  
Cohomological Dimension ◽  
Projective Dimension ◽  
Proper Ideal ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Finite Projective Dimension

Let (R,𝔪) be a Noetherian local ring, 𝔞 a proper ideal of R, and M, N two finitely generated R-modules of finite projective dimension m and of finite dimension n, respectively. It is shown that if n ≤ 2, then the generalized local cohomology module [Formula: see text] is a co-Cohen-Macaulay module. Additionally, we show that [Formula: see text] for all i > m+s and [Formula: see text], where s is the cohomological dimension of N with respect to 𝔞.

scholarly journals Local rings with quasi-decomposable maximal ideal

2018 ◽  
Vol 168 (2) ◽  
pp. 305-322 ◽  
Author(s):  
SAEED NASSEH ◽  
RYO TAKAHASHI
Keyword(s):  
Direct Summand ◽  
Local Ring ◽  
Maximal Ideal ◽  
Projective Dimension ◽  
Complete Classification ◽  
Local Rings ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Direct Sum

AbstractLet (R, 𝔪) be a commutative noetherian local ring. In this paper, we prove that if 𝔪 is decomposable, then for any finitely generated R-module M of infinite projective dimension 𝔪 is a direct summand of (a direct sum of) syzygies of M. Applying this result to the case where 𝔪 is quasi-decomposable, we obtain several classifications of subcategories, including a complete classification of the thick subcategories of the singularity category of R.

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