A study of a family of monomial ideals

2021 ◽  
Author(s):  
J. William Hoffman ◽  
Haohao Wang
Keyword(s):  
Structural Properties ◽  
Monomial Ideal ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Symmetric Algebra ◽  
Rees Algebra ◽  
Implicit Equation ◽  
Ideal Formula ◽  
Moving Planes

In this paper, we study a family of rational monomial parametrizations. We investigate a few structural properties related to the corresponding monomial ideal [Formula: see text] generated by the parametrization. We first find the implicit equation of the closure of the image of the parametrization. Then we provide a minimal graded free resolution of the monomial ideal [Formula: see text], and describe the minimal graded free resolution of the symmetric algebra of [Formula: see text]. Finally, we provide a method to compute the defining equations of the Rees algebra of [Formula: see text] using three moving planes that follow the parametrization.

scholarly journals ON THE CONJECTURE OF VASCONCELOS FOR ARTINIAN ALMOST COMPLETE INTERSECTION MONOMIAL IDEALS

2019 ◽  
pp. 1-15
Author(s):  
KUEI-NUAN LIN ◽  
YI-HUANG SHEN
Keyword(s):  
Complete Intersection ◽  
Monomial Ideal ◽  
Short Note ◽  
Monomial Ideals ◽  
Rees Algebra

In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen–Macaulay.

Implosive graphs: Square-free monomials on symbolic Rees algebras

2016 ◽  
Vol 16 (08) ◽  
pp. 1750145 ◽  
Author(s):  
A. Flores-Méndez ◽  
I. Gitler ◽  
E. Reyes
Keyword(s):  
Structural Properties ◽  
Monomial Ideal ◽  
Minimal System ◽  
Rees Algebra ◽  
Rees Algebras ◽  
Vertex Set ◽  
Minimal System Of Generators

Let [Formula: see text] be the edge monomial ideal of a graph [Formula: see text], whose vertex set is [Formula: see text]. [Formula: see text] is implosive if the symbolic Rees algebra [Formula: see text] of [Formula: see text] has a minimal system of generators [Formula: see text] where [Formula: see text] are square-free monomials. We give some structural properties of implosive graphs and we prove that they are closed under clique-sums and odd subdivisions. Furthermore, we prove that universally signable graphs are implosive. We show that odd holes, odd antiholes and some Truemper configurations (prisms, thetas and even wheels) are implosive. Moreover, we study excluded families of subgraphs for the class of implosive graphs. In particular, we characterize which Truemper configurations and extensions of odd holes and antiholes are minimal nonimplosive.

scholarly journals Linear syzygies of Stanley-Reisner ideals

2001 ◽  
Vol 89 (1) ◽  
pp. 117 ◽  
Author(s):  
V Reiner ◽  
V Welker
Keyword(s):  
Simplicial Complex ◽  
Lower Bounds ◽  
Monomial Ideal ◽  
Independent Sets ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Minimal Free Resolution ◽  
Linear Syzygies ◽  
Alexander Dual ◽  
Syzygy Modules

We give an elementary description of the maps in the linear strand of the minimal free resolution of a square-free monomial ideal, that is, the Stanley-Reisner ideal associated to a simplicial complex $\Delta$. The description is in terms of the homology of the canonical Alexander dual complex $\Delta^*$. As applications we are able to prove for monomial ideals and $j=1$ a conjecture of J. Herzog giving lower bounds on the number of $i$-syzygies in the linear strand of $j^{th}$-syzygy modules show that the maps in the linear strand can be written using only $\pm 1$ coefficients if $\Delta^*$ is a pseudomanifold exhibit an example where multigraded maps in the linear strand cannot be written using only $\pm 1$ coefficients compute the entire resolution explicitly when $\Delta^*$ is the complex of independent sets of a matroid

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 Free resolution of powers of monomial ideals and Golod rings

2017 ◽  
Vol 120 (1) ◽  
pp. 59 ◽  
Author(s):  
N. Altafi ◽  
N. Nemati ◽  
S. A. Seyed Fakhari ◽  
S. Yassemi
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Monomial Order ◽  
Golod Ring ◽  
Linear Quotients ◽  
The Ideal

Let $S = \mathbb{K}[x_1, \dots, x_n]$ be the polynomial ring over a field $\mathbb{K}$. In this paper we present a criterion for componentwise linearity of powers of monomial ideals. In particular, we prove that if a square-free monomial ideal $I$ contains no variable and some power of $I$ is componentwise linear, then $I$ satisfies the gcd condition. For a square-free monomial ideal $I$ which contains no variable, we show that $S/I$ is a Golod ring provided that for some integer $s\geq 1$, the ideal $I^s$ has linear quotients with respect to a monomial order.

scholarly journals Betti Numbers of Transversal Monomial Ideals

2011 ◽  
Vol 18 (spec01) ◽  
pp. 925-936
Author(s):  
Rahim Zaare-Nahandi
Keyword(s):  
Monomial Ideal ◽  
Betti Numbers ◽  
Free Resolution ◽  
Circulant Matrix ◽  
Monomial Ideals ◽  
Minimal Free Resolution ◽  
Ground Field ◽  
Initial Ideal ◽  
The Matrix ◽  
Generic Singularities

In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of t-minors of a generic pluri-circulant matrix is a transversal monomial ideal. Using a Gröbner basis for this ideal, it is shown that the initial ideal of a generic pluri-circulant matrix is a stable monomial ideal when the matrix has two square blocks. By means of the Eliahou-Kervaire resolution for stable monomial ideals, the Betti numbers of this initial ideal is computed and it is proved that for some significant values of t, this ideal has the same Betti numbers as the corresponding transversal monomial ideal. The ideals treated in this paper naturally arise in the study of generic singularities of algebraic varieties.

scholarly journals Generalized Newton complementary duals of monomial ideals

2020 ◽  
pp. 2150021
Author(s):  
Katie Ansaldi ◽  
Kuei-Nuan Lin ◽  
Yi-Huang Shen
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Free Resolutions ◽  
Stable Ideal ◽  
Generalized Newton ◽  
Strongly Stable Ideals ◽  
The Given ◽  
Linear Quotients

Given a monomial ideal in a polynomial ring over a field, we define the generalized Newton complementary dual of the given ideal. We show good properties of such duals including linear quotients and isomorphism between the special fiber rings. We construct the cellular free resolutions of duals of strongly stable ideals generated in the same degree. When the base ideal is generated in degree two, we provide an explicit description of cellular free resolution of the dual of a compatible generalized stable ideal.

scholarly journals A family of monomial ideals with the persistence property

2019 ◽  
Vol 18 (05) ◽  
pp. 1950093
Author(s):  
Somayeh Moradi ◽  
Masoomeh Rahimbeigi ◽  
Fahimeh Khosh-Ahang ◽  
Ali Soleyman Jahan
Keyword(s):  
Monomial Ideal ◽  
Independent Set ◽  
Monomial Ideals ◽  
Prime Ideals ◽  
Stable Set ◽  
Ideal Formula ◽  
Path Graph ◽  
Associated Prime Ideals ◽  
Persistence Property ◽  
Positive Integers

In this paper, we introduce a family of monomial ideals with the persistence property. Given positive integers [Formula: see text] and [Formula: see text], we consider the monomial ideal [Formula: see text] generated by all monomials [Formula: see text], where [Formula: see text] is an independent set of vertices of the path graph [Formula: see text] of size [Formula: see text], which is indeed the facet ideal of the [Formula: see text]th skeleton of the independence complex of [Formula: see text]. We describe the set of associated primes of all powers of [Formula: see text] explicitly. It turns out that any such ideal [Formula: see text] has the persistence property. Moreover, the index of stability of [Formula: see text] and the stable set of associated prime ideals of [Formula: see text] are determined.

scholarly journals Stanley’s conjecture for critical ideals

2011 ◽  
Vol 48 (2) ◽  
pp. 220-226
Author(s):  
Azeem Haider ◽  
Sardar Khan
Keyword(s):  
Polynomial Ring ◽  
Hilbert Function ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Stanley Depth

Let S = K[x1,…,xn] be a polynomial ring in n variables over a field K. Stanley’s conjecture holds for the modules I and S/I, when I ⊂ S is a critical monomial ideal. We calculate the Stanley depth of S/I when I is a canonical critical monomial ideal. For non-critical monomial ideals we show the existence of a Stanley ideal with the same depth and Hilbert function.

scholarly journals Stanley depth and symbolic powers of monomial ideals

2017 ◽  
Vol 120 (1) ◽  
pp. 5 ◽  
Author(s):  
S. A. Seyed Fakhari
Keyword(s):  
Monomial Ideal ◽  
Monomial Ideals ◽  
Limit Behavior ◽  
Stanley Depth ◽  
Symbolic Powers

The aim of this paper is to study the Stanley depth of symbolic powers of a squarefree monomial ideal. We prove that for every squarefree monomial ideal $I$ and every pair of integers $k, s\geq 1$, the inequalities $\mathrm{sdepth} (S/I^{(ks)}) \leq \mathrm{sdepth} (S/I^{(s)})$ and $\mathrm{sdepth}(I^{(ks)}) \leq \mathrm{sdepth} (I^{(s)})$ hold. If moreover $I$ is unmixed of height $d$, then we show that for every integer $k\geq1$, $\mathrm{sdepth}(I^{(k+d)})\leq \mathrm{sdepth}(I^{{(k)}})$ and $\mathrm{sdepth}(S/I^{(k+d)})\leq \mathrm{sdepth}(S/I^{{(k)}})$. Finally, we consider the limit behavior of the Stanley depth of symbolic powers of a squarefree monomial ideal. We also introduce a method for comparing the Stanley depth of factors of monomial ideals.

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