scholarly journals Gin and lex of certain monomial ideals

2006 ◽  
Vol 99 (1) ◽  
pp. 76 ◽  
Author(s):  
Satoshi Murai ◽  
Takayuki Hibi
Keyword(s):  
Polynomial Ring ◽  
Hilbert Function ◽  
Monomial Ideal ◽  
Lexicographic Order ◽  
Monomial Ideals ◽  
Initial Ideal ◽  
Generic Initial Ideal ◽  
Lexsegment Ideal

Let $A = K[x_1,\ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ of characteristic $0$ with each $\deg x_i = 1$. Given arbitrary integers $i$ and $j$ with $2 \leq i \leq n$ and $3 \leq j \leq n$, we will construct a monomial ideal $I \subset A$ such that (i) $\beta_k(I) < \beta_k(\mathrm{Gin}(I))$ for all $k < i$, (ii) $\beta_i(I)= \beta_i(\mathrm{Gin}(I))$, (iii) $\beta_\ell((\mathrm{Gin}(I)) < \beta_\ell((\mathrm{Lex}(I))$ for all $\ell < j$ and (iv) $\beta_j(\mathrm{Gin}(I)) = \beta_j(\mathrm{Lex}(I))$, where $\mathrm{Gin}(I)$ is the generic initial ideal of $I$ with respect to the reverse lexicographic order induced by $x_1 > \cdots > x_n$ and where $\mathrm{Lex}(I)$ is the lexsegment ideal with the same Hilbert function as $I$.

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.

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.

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.

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.

The Stanley regularity of complete intersections and ideals of mixed products

2016 ◽  
Vol 16 (07) ◽  
pp. 1750122
Author(s):  
Lizhong Chu ◽  
V. H. Jorge Pérez
Keyword(s):  
Complete Intersection ◽  
Polynomial Ring ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Complete Intersections

Let [Formula: see text] be a polynomial ring over a field [Formula: see text] and [Formula: see text] a monomial ideal. We give some inequalities on Stanley regularity of monomial ideals. As consequences, we prove that [Formula: see text] and [Formula: see text] hold in the following cases: (1) [Formula: see text] is a complete intersection; (2) [Formula: see text] is an ideal of mixed products.

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 Monomial ideals and the failure of the Strong Lefschetz property

2021 ◽  
Author(s):  
Nasrin Altafi ◽  
Samuel Lundqvist
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Hilbert Function ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Hilbert Functions ◽  
Lefschetz Property ◽  
Strong Lefschetz Property ◽  
Sharp Lower Bound

AbstractWe give a sharp lower bound for the Hilbert function in degree d of artinian quotients $$\Bbbk [x_1,\ldots ,x_n]/I$$ k [ x 1 , … , x n ] / I failing the Strong Lefschetz property, where I is a monomial ideal generated in degree $$d \ge 2$$ d ≥ 2 . We also provide sharp lower bounds for other classes of ideals, and connect our result to the classification of the Hilbert functions forcing the Strong Lefschetz property by Zanello and Zylinski.

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 ON A CLASS OF MONOMIAL IDEALS

2013 ◽  
Vol 87 (3) ◽  
pp. 514-526 ◽  
Author(s):  
KEIVAN BORNA ◽  
RAHELEH JAFARI
Keyword(s):  
Prime Ideal ◽  
Polynomial Ring ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Total Degree ◽  
Associated Primes ◽  
Minimal Set ◽  
Irreducible Decomposition ◽  
Maximal Height

AbstractLet $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that $I$ is MHC (that is, $I$ satisfies the maximal height condition for the associated primes of $I$) if there exists a prime ideal $\mathfrak{p}\in {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I$ for which $\mathrm{ht} (\mathfrak{p})$ equals the number of indeterminates that appear in the minimal set of monomials generating $I$. Let $I= { \mathop{\bigcap }\nolimits}_{i= 1}^{k} {Q}_{i} $ be the irreducible decomposition of $I$ and let $m(I)= \max \{ \vert Q_{i}\vert - \mathrm{ht} ({Q}_{i} ): 1\leq i\leq k\} $, where $\vert {Q}_{i} \vert $ denotes the total degree of ${Q}_{i} $. Then it can be seen that when $I$ is primary, $\mathrm{reg} (S/ I)= m(I)$. In this paper we improve this result and show that whenever $I$ is MHC, then $\mathrm{reg} (S/ I)= m(I)$ provided $\vert {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I\vert \leq 2$. We also prove that $m({I}^{n} )\leq n\max \{ \vert Q_{i}\vert : 1\leq i\leq ~k\} - \mathrm{ht} (I)$, for all $n\geq 1$. In addition we show that if $I$ is MHC and $w$ is an indeterminate which is not in the monomials generating $I$, then $\mathrm{reg} (S/ \mathop{(I+ {w}^{d} S)}\nolimits ^{n} )\leq \mathrm{reg} (S/ I)+ nd- 1$ for all $n\geq 1$ and $d$ large enough. Finally, we implement an algorithm for the computation of $m(I)$.

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