INTEGRAL CLOSURE OF STRONGLY GOLOD IDEALS

2019 ◽  
pp. 1-13 ◽  
Author(s):  
CĂTĂLIN CIUPERCĂ
Keyword(s):  
Polynomial Ring ◽  
Characteristic Zero ◽  
Integral Closure ◽  
Homogeneous Ideal ◽  
Rational Power

We prove that the integral closure of a strongly Golod ideal in a polynomial ring over a field of characteristic zero is strongly Golod, positively answering a question of Huneke. More generally, the rational power $I_{\unicode[STIX]{x1D6FC}}$ of an arbitrary homogeneous ideal is strongly Golod for $\unicode[STIX]{x1D6FC}\geqslant 2$ and, if $I$ is strongly Golod, then $I_{\unicode[STIX]{x1D6FC}}$ is strongly Golod for $\unicode[STIX]{x1D6FC}\geqslant 1$ . We also show that all the coefficient ideals of a strongly Golod ideal are strongly Golod.

Analytic spread and non-vanishing of asymptotic depth

2017 ◽  
Vol 163 (2) ◽  
pp. 289-299 ◽  
Author(s):  
CLETO B. MIRANDA–NETO
Keyword(s):  
Prime Ideal ◽  
Polynomial Ring ◽  
Prime Divisor ◽  
Characteristic Zero ◽  
Monomial Ideal ◽  
Integral Closure ◽  
Prime Ideals ◽  
Sufficient Condition ◽  
Analytic Spread

AbstractLetSbe a polynomial ring over a fieldKof characteristic zero and letM⊂Sbe an ideal given as an intersection of powers of incomparable monomial prime ideals (e.g., the case whereMis a squarefree monomial ideal). In this paper we provide a very effective, sufficient condition for a monomial prime idealP⊂ScontainingMbe such that the localisationMPhasnon-maximal analytic spread. Our technique describes, in fact, a concrete obstruction forPto be an asymptotic prime divisor ofMwith respect to the integral closure filtration, allowing us to employ a theorem of McAdam as a bridge to analytic spread. As an application, we derive – with the aid of results of Brodmann and Eisenbud-Huneke – a situation where the asymptotic and conormal asymptotic depths cannot vanish locally at such primes.

Hilbert's function in a semi-lattice

1959 ◽  
Vol 55 (3) ◽  
pp. 239-243
Author(s):  
A. Learner
Keyword(s):  
Local Ring ◽  
Polynomial Ring ◽  
Hilbert Function ◽  
Homogeneous Ideal ◽  
Samuel 1

Samuel (1) introduced a generalized Hilbert function, written Xq(r, a) and defined for arbitrary ideals a in a local ring Q with maximal ideai m. where q is m-primary.Northcott(2) proved that for a homogeneous ideal ã in a polynomial ring A[X1, …, Xn], where A = Q/q, this is equal to the ordinary Hilbert function χ(r, ã).

On the reduction numbers of monomial ideals

2019 ◽  
Vol 19 (10) ◽  
pp. 2050201
Author(s):  
Ibrahim Al-Ayyoub
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Upper Bounds ◽  
Integral Closure ◽  
Monomial Ideals ◽  
Explicit Method ◽  
Generating Set ◽  
Ideal Formula ◽  
Reduction Number ◽  
The Ideal

Let [Formula: see text] be a monomial ideal in a polynomial ring with two indeterminates over a field. Assume [Formula: see text] is contained in the integral closure of some ideal that is generated by two elements from the generating set of [Formula: see text]. We produce sharp upper bounds for each of the reduction number and the Ratliff–Rush reduction number of the ideal [Formula: see text]. Under certain hypotheses, we give the exact values of these reduction numbers, and we provide an explicit method for obtaining these sharp upper bounds.

scholarly journals Endomorphisms Preserving Coordinates of Polynomial Algebras

2013 ◽  
Vol 20 (03) ◽  
pp. 523-526
Author(s):  
Yu-Chang Li ◽  
Jie-Tai Yu
Keyword(s):  
Finite Number ◽  
Polynomial Ring ◽  
Characteristic Zero ◽  
Infinite Field ◽  
Polynomial Algebras ◽  
Nonzero Constant

It is proved that the Jacobian of a k-endomorphism of k[x1,…,xn] over a field k of characteristic zero, taking every tame coordinate to a coordinate, must be a nonzero constant in k. It is also proved that the Jacobian of an R-endomorphism of A=:R[x1,…,xn] (where R is a polynomial ring in a finite number of variables over an infinite field k), taking every R-linear coordinate of A to an R-coordinate of A, is a nonzero constant in k.

scholarly journals NILPOTENT MODULES OVER POLYNOMIAL RINGS

2020 ◽  
pp. 20-25
Author(s):  
А. Petravchuk ◽  
Ie. Chapovskyi ◽  
I. Klimenko ◽  
M. Sidorov
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Characteristic Zero ◽  
Finite Dimension ◽  
Polynomial Rings ◽  
Closed Field ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Groups Of Automorphisms

Let K be an algebra ically closed field of characteristic zero, K[X ] the polynomial ring in n variables. The vector space Tn = K[X] is a K[X ] -module with the action i = xi 'x  v v for vTn . Every finite dimensional submodule V of Tn is nilpotent, i.e. every f  K[X ] acts nilpotently (by multiplication) on V . We prove that every nilpotent K[X ] -module V of finite dimension over K with one-dimensional socle can be isomorphically embedded in the module Tn . The groups of automorphisms of the module Tn and its finite dimensional monomial submodules are found. Similar results are obtained for (non-nilpotent) finite dimensional K[X ] -modules with one dimensional socle.

scholarly journals Graded Betti Numbers of Balanced Simplicial Complexes

2021 ◽  
Author(s):  
Martina Juhnke-Kubitzke ◽  
Lorenzo Venturello
Keyword(s):  
Polynomial Ring ◽  
Betti Numbers ◽  
Simplicial Complexes ◽  
Upper Bounds ◽  
Combinatorial Type ◽  
Homogeneous Ideal ◽  
Graded Rings ◽  
Number Of Generators ◽  
Graded Betti Numbers ◽  
Linear Syzygies

AbstractWe prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings S/I, where S is a polynomial ring and $I\subseteq S$ I ⊆ S is a homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner rings of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type.

scholarly journals On the Stanley Depth of Powers of Monomial Ideals

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 607 ◽  
Author(s):  
S. A. Seyed Fakhari
Keyword(s):  
Polynomial Ring ◽  
Upper Bound ◽  
Integral Closure ◽  
Monomial Ideals ◽  
The Other ◽  
Finitely Generated ◽  
Stanley Depth ◽  
Survey Article ◽  
Symbolic Powers ◽  
Other Hand

In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s conjecture, and their counterexample is a quotient of squarefree monomial ideals. On the other hand, there is evidence showing that Stanley’s inequality can be true for high powers of monomial ideals. In this survey article, we collect the recent results in this direction. More precisely, we investigate the Stanley depth of powers, integral closure of powers, and symbolic powers of monomial ideals.

scholarly journals Asymptotic Behavior of the Length of Local Cohomology

2005 ◽  
Vol 57 (6) ◽  
pp. 1178-1192 ◽  
Author(s):  
Steven Dale Cutkosky ◽  
Huy Tài Hà ◽  
Hema Srinivasan ◽  
Emanoil Theodorescu
Keyword(s):  
Asymptotic Behavior ◽  
Polynomial Ring ◽  
Polynomial Growth ◽  
Local Cohomology ◽  
Irrational Number ◽  
Homogeneous Ideal ◽  
Image Position ◽  
Primary Ideals

AbstractLet k be a field of characteristic 0, R = k[x1, … , xd] be a polynomial ring, and m its maximal homogeneous ideal. Let I ⊂ R be a homogeneous ideal in R. Let λ(M) denote the length of an Rmodule M. In this paper, we show thatalways exists. This limit has been shown to be e(I)/d! form-primary ideals I in a local Cohen–Macaulay ring, where e(I) denotes the multiplicity of I. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extension modules may not have polynomial growth.

scholarly journals Actions of the additive group Ga on certain noncommutative deformations of the plane

2021 ◽  
Vol 29 (2) ◽  
pp. 269-279
Author(s):  
Ivan Kaygorodov ◽  
Samuel A. Lopes ◽  
Farukh Mashurov
Keyword(s):  
Automorphism Group ◽  
Polynomial Ring ◽  
Characteristic Zero ◽  
Weyl Algebra ◽  
Additive Group ◽  
Prime Characteristic ◽  
Arbitrary Characteristic ◽  
Arbitrary Polynomial ◽  
Comodule Algebra ◽  
The Family

Abstract We connect the theorems of Rentschler [18] and Dixmier [10] on locally nilpotent derivations and automorphisms of the polynomial ring A 0 and of the Weyl algebra A 1, both over a field of characteristic zero, by establishing the same type of results for the family of algebras A h = 〈 x , y | y x − x y = h ( x ) 〉 , {A_h} = \left\langle {x,y|yx - xy = h\left( x \right)} \right\rangle , , where h is an arbitrary polynomial in x. In the second part of the paper we consider a field 𝔽 of prime characteristic and study 𝔽[t]-comodule algebra structures on Ah . We also compute the Makar-Limanov invariant of absolute constants of Ah over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of Ah .

scholarly journals Cohomological and projective dimensions

2013 ◽  
Vol 149 (7) ◽  
pp. 1203-1210 ◽  
Author(s):  
Matteo Varbaro
Keyword(s):  
Polynomial Ring ◽  
Local Cohomology ◽  
The Other ◽  
Homogeneous Ideal ◽  
Other Hand ◽  
Local Cohomology Modules ◽  
Cohomology Modules

AbstractLet $\mathfrak{a}$ be a homogeneous ideal of a polynomial ring $R$ in $n$ variables over a field $\mathbb{k}$. Assume that $\mathrm{depth} (R/ \mathfrak{a})\geq t$, where $t$ is some number in $\{ 0, \ldots , n\} $. A result of Peskine and Szpiro says that if $\mathrm{char} (\mathbb{k})\gt 0$, then the local cohomology modules ${ H}_{\mathfrak{a}}^{i} (M)$ vanish for all $i\gt n- t$ and all $R$-modules $M$. In characteristic $0$, there are counterexamples to this for all $t\geq 4$. On the other hand, when $t\leq 2$, by exploiting classical results of Grothendieck, Lichtenbaum, Hartshorne and Ogus it is not difficult to extend the result to any characteristic. In this paper we settle the remaining case; specifically, we show that if $\mathrm{depth} (R/ \mathfrak{a})\geq 3$, then the local cohomology modules ${ H}_{\mathfrak{a}}^{i} (M)$ vanish for all $i\gt n- 3$ and all $R$-modules $M$, whatever the characteristic of $\mathbb{k}$ is.

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