The maximal dimension of formal fibers of local rings of an algebraic scheme of finite type

2019 ◽  
Vol 18 (06) ◽  
pp. 1950120
Author(s):  
Đoàn Trung Cu’ò’ng
Keyword(s):  
Finite Type ◽  
Local Ring ◽  
Local Algebra ◽  
Local Rings ◽  
Maximal Dimension ◽  
Algebraic Varieties ◽  
Noetherian Local Ring ◽  
Closed Point ◽  
Dimension One ◽  
Algebraic Scheme

For a scheme [Formula: see text] of finite type over a Noetherian local ring [Formula: see text] with a closed point [Formula: see text] of the special fiber, we show that the maximal dimension of the formal fibers of the local algebra [Formula: see text] equals to [Formula: see text] provided that either [Formula: see text] is complete of dimension one or the dimensions of the formal fibers of [Formula: see text] are less than [Formula: see text]. This extends Matsumura’s theorem for algebraic varieties.

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].

Graded Complexes Over Power Series Rings

1986 ◽  
Vol 38 (1) ◽  
pp. 158-178 ◽  
Author(s):  
Paul Roberts
Keyword(s):  
Power Series ◽  
Local Ring ◽  
Simple Structure ◽  
Free Resolution ◽  
Koszul Complex ◽  
Local Rings ◽  
Power Series Ring ◽  
Series Ring ◽  
Noetherian Local Ring ◽  
Power Series Rings

A common method in studying a commutative Noetherian local ring A is to find a regular subring R contained in A so that A becomes a finitely generated R-module, and in this way one can obtain some information about the original ring by applying what is known about regular local rings. By the structure theorems of Cohen, if A is complete and contains a field, there will always exist such a subring R, and R will be a power series ring k[[X1, …, Xn]] = k[[X]] over a field k. In this paper we show that if R is chosen properly, the ring A (or, more generally, an A-module M), will have a comparatively simple structure as an R-module. More precisely, A (or M) will have a free resolution which resembles the Koszul complex on the variables (X1, …, Xn) = (X); such a complex will be called an (X)-graded complex and will be given a precise definition below.

PSEUDO BUCHSBAUMNESS FOR IDEALIZATIONS OF NOETHERIAN MODULES OVER LOCAL RINGS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350084
Author(s):  
NGUYEN THI HONG LOAN
Keyword(s):  
Local Ring ◽  
Local Rings ◽  
Noetherian Local Ring ◽  
Noetherian Modules

Let (R, 𝔪) be a Noetherian local ring with dim R = d and M an R-module with dim M < d. We prove in this paper that the idealization R ⋉ M of M over R is a pseudo Buchsbaum ring if and only if so is R.

scholarly journals Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free

2011 ◽  
Vol 148 (1) ◽  
pp. 145-152 ◽  
Author(s):  
Hailong Dao
Keyword(s):  
Abelian Group ◽  
Local Ring ◽  
Complete Intersection ◽  
Homological Algebra ◽  
Local Rings ◽  
Noetherian Local Ring ◽  
Picard Groups ◽  
Crepant Resolutions ◽  
Torsion Free

AbstractLet (R,m) be a Noetherian local ring and UR=Spec(R)−{m} be the punctured spectrum of R. Gabber conjectured that if R is a complete intersection of dimension three, then the abelian group Pic(UR) is torsion-free. In this note we prove Gabber’s statement for the hypersurface case. We also point out certain connections between Gabber’s conjecture, Van den Bergh’s notion of non-commutative crepant resolutions and some well-studied questions in homological algebra over local rings.

Noetherian Dimension and Co-localization of Artinian Modules over Local Rings

2014 ◽  
Vol 21 (04) ◽  
pp. 663-670 ◽  
Author(s):  
Le Thanh Nhan ◽  
Tran Do Minh Chau
Keyword(s):  
Local Ring ◽  
Local Rings ◽  
Finitely Generated ◽  
Artinian Modules ◽  
Noetherian Local Ring ◽  
Noetherian Dimension ◽  
Base Ring ◽  
Finitely Generated Modules

Let (R, 𝔪) be a Noetherian local ring. Denote by N-dim RA the Noetherian dimension of an Artinian R-module A. In this paper, we give some characterizations for the ring R to satisfy N-dim RA = dim (R/ Ann RA) for certain Artinian R-modules A. Then the existence of a co-localization compatible with Artinian R-modules is studied and it is shown that if it is compatible with local cohomologies of finitely generated modules, then the base ring is universally catenary and all of its formal fibers are Cohen-Macaulay.

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 The Cohen-Macaulayness of the Rees algebras of local rings

1983 ◽  
Vol 89 ◽  
pp. 47-63 ◽  
Author(s):  
Shin Ikeda
Keyword(s):  
Local Ring ◽  
Local Rings ◽  
Rees Algebra ◽  
Rees Algebras ◽  
Noetherian Local Ring

Let (A, m, k) be a Noetherian local ring. We defineand call it the Rees algebra of A. Let X be an indeterminate over A, then R(A) can be identified with the A-subalgebra .

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.

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 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.

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