scholarly journals Symbolic Powers of Monomial Ideals

2016 ◽  
Vol 60 (1) ◽  
pp. 39-55 ◽  
Author(s):  
Susan M. Cooper ◽  
Robert J. D. Embree ◽  
Huy Tài Hà ◽  
Andrew H. Hoefel
Keyword(s):  
Monomial Ideal ◽  
Monomial Ideals ◽  
Symbolic Powers ◽  
Positive Integers

AbstractWe investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal I ⊆ 𝕜[x0, … , xn] we show that for all positive integers m, t and r, where e is the big-height of I and . This captures two conjectures (r = 1 and r = e): one of Harbourne and Huneke, and one of Bocci et al. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.

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.

Partial Castelnuovo–Mumford regularities of sums and intersections of powers of monomial ideals

2010 ◽  
Vol 149 (2) ◽  
pp. 229-246 ◽  
Author(s):  
LÊ TUÂN HOA ◽  
TRÂN NAM TRUNG
Keyword(s):  
Asymptotic Behavior ◽  
Linear Function ◽  
Polynomial Ring ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Symbolic Powers ◽  
Integral Closures

AbstractLet I, I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp be monomial ideals of a polynomial ring R = K[X1,. . ., Xr] and Ln = I+∩jIn1j + ⋅ ⋅ ⋅ + ∩jIpjn. It is shown that the ai-invariant ai(R/Ln) is asymptotically a quasi-linear function of n for all n ≫ 0, and the limit limn→∞ad(R/Ln)/n exists, where d = dim(R/L1). A similar result holds if I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp are replaced by their integral closures. Moreover all limits $\lim_{n\to\infty} a_i(R/(\cap_j \overline{I_{1j}^n} + \cdots + \cap_j \overline{I_{pj}^n}))/n $ also exist.As a consequence, it is shown that there are integers p > 0 and 0 ≤ e ≤ d = dim R/I such that reg(In) = pn + e for all n ≫ 0 and pn ≤ reg(In) ≤ pn + d for all n > 0 and that the asymptotic behavior of the Castelnuovo–Mumford regularity of ordinary symbolic powers of a square-free monomial ideal is very close to a linear function.

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.

On the Normality of One-fibered Monomial Ideals

2016 ◽  
Vol 23 (03) ◽  
pp. 501-506
Author(s):  
Charef Beddani ◽  
Wahiba Messirdi
Keyword(s):  
Monomial Ideal ◽  
Monomial Ideals ◽  
Higher Dimensional ◽  
Positive Integers

This paper presents a generalization to the higher-dimensional situation of the main results of the first author about the normality of one-fibered monomial ideals [2, Théorèmes 2.4 and 3.8]. Precisely, we show that if I is a monomial ideal of R = k[x1,x2,…,xd], then I is normal one-fibered if and only if for all positive integers n and all x, y in R such that xy ∈ I2n, either x or y belongs to In.

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 On Monomial Golod Ideals

2020 ◽  
Author(s):  
Hailong Dao ◽  
Alessandro De Stefani
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Complete Characterization ◽  
Monomial Ideals

Abstract We study ideal-theoretic conditions for a monomial ideal to be Golod. For ideals in a polynomial ring in three variables, our criteria give a complete characterization. Over such rings, we show that the product of two monomial ideals is Golod.

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.

ON THE STANLEY DEPTH AND SIZE OF MONOMIAL IDEALS

2017 ◽  
Vol 59 (3) ◽  
pp. 705-715
Author(s):  
S. A. SEYED FAKHARI
Keyword(s):  
Lower Bound ◽  
Polynomial Ring ◽  
Monomial Ideal ◽  
Recursive Formula ◽  
Monomial Ideals ◽  
Stanley Depth

AbstractLet $\mathbb{K}$ be a field and S = ${\mathbb{K}}$[x1, . . ., xn] be the polynomial ring in n variables over the field $\mathbb{K}$. For every monomial ideal I ⊂ S, we provide a recursive formula to determine a lower bound for the Stanley depth of S/I. We use this formula to prove the inequality sdepth(S/I) ≥ size(I) for a particular class of monomial ideals.

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