On Stanley Depths of Certain Monomial Factor Algebras

2015 ◽  
Vol 58 (2) ◽  
pp. 393-401
Author(s):  
Zhongming Tang
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Primary Decomposition ◽  
Factor Algebra ◽  
Stanley Depth ◽  
Stanley Conjecture ◽  
Stanley Decomposition

AbstractLet S = K[x1 , . . . , xn] be the polynomial ring in n-variables over a ûeld K and I a monomial ideal of S. According to one standard primary decomposition of I, we get a Stanley decomposition of the monomial factor algebra S/I. Using this Stanley decomposition, one can estimate the Stanley depth of S/I. It is proved that sdepthS(S/I) ≤ sizeS(I). When I is squarefree and bigsizeS(I) ≤ 2, the Stanley conjecture holds for S/I, i.e., sdepthS(S/I) ≥ depthS(S/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 Combinatorial Reductions for the Stanley Depth of $I$ and $S/I$

10.37236/6783 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Mitchel T. Keller ◽  
Stephen J. Young
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Strict Inequality ◽  
Polynomial Rings ◽  
Computer Search ◽  
Stanley Depth ◽  
The Relationship

We develop combinatorial tools to study the relationship between the Stanley depth of a monomial ideal $I$ and the Stanley depth of its compliment, $S/I$. Using these results we are able to prove that if $S$ is a polynomial ring with at most 5 indeterminates and $I$ is a square-free monomial ideal, then the Stanley depth of $S/I$ is strictly larger than the Stanley depth of $I$. Using a computer search, we are able to extend this strict inequality up to polynomial rings with at most 7 indeterminates. This partially answers questions asked by Propescu and Qureshi as well as Herzog.

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.

scholarly journals Values and bounds for the Stanley depth

2011 ◽  
Vol 27 (2) ◽  
pp. 217-224
Author(s):  
MUHAMMAD ISHAQ ◽  
Keyword(s):  
Prime Ideal ◽  
Polynomial Algebra ◽  
Monomial Ideal ◽  
Prime Ideals ◽  
Stanley Depth ◽  
Stanley Conjecture ◽  
Associated Prime Ideals ◽  
Disjoint Sets ◽  
Associated Prime

We give different bounds for the Stanley depth of a monomial ideal I of a polynomial algebra S over a field K. For example we show that the Stanley depth of I is less than or equal to the Stanley depth of any prime ideal associated to S/I. Also we show that the Stanley conjecture holds for I and S/I when the associated prime ideals of S/I are generated by disjoint sets of variables.

scholarly journals Stanley depth of monomial ideals with small number of generators

2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Mircea Cimpoeaş
Keyword(s):  
Monomial Ideal ◽  
Monomial Ideals ◽  
Stanley Depth ◽  
Number Of Generators ◽  
Stanley Conjecture

AbstractFor a monomial ideal I ⊂ S = K[x 1...,x n], we show that sdepth(S/I) ≥ n − g(I), where g(I) is the number of the minimal monomial generators of I. If I =νI′, where ν ∈ S is a monomial, then we see that sdepth(S/I) = sdepth(S/I′). We prove that if I is a monomial ideal I ⊂ S minimally generated by three monomials, then I and S/I satisfy the Stanley conjecture. Given a saturated monomial ideal I ⊂ K[x 1,x 2,x 3] we show that sdepth(I) = 2. As a consequence, sdepth(I) ≥ sdepth(K[x 1,x 2,x 3]//I) +1 for any monomial ideal in I ⊂ K[x 1,x 2,x 3].

On Characteristic Poset and Stanley Decomposition of S/I

2015 ◽  
Vol 22 (spec01) ◽  
pp. 739-744
Author(s):  
Sarfraz Ahmad ◽  
Imran Anwar
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Finite Poset ◽  
Stanley Decomposition

Let K be a field and S = K[x1,…,xn] be the polynomial ring in n variables. Let I ⊂ S be a monomial ideal such that S/I is Cohen-Macaulay. By associating a finite poset [Formula: see text] to S/I, we show that if S/I is a Stanley ideal then T/Ĩ is also a Stanley ideal, where T = K[x11,…,x1a1,…,xn1,…,xnan] and Ĩ is the polarization of I.

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.

scholarly journals Stanley depth and size of a monomial ideal

2012 ◽  
Vol 140 (2) ◽  
pp. 493-504 ◽  
Author(s):  
Jürgen Herzog ◽  
Dorin Popescu ◽  
Marius Vladoiu
Keyword(s):  
Monomial Ideal ◽  
Stanley Depth

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 How to compute the Stanley depth of a monomial ideal

2009 ◽  
Vol 322 (9) ◽  
pp. 3151-3169 ◽  
Author(s):  
Jürgen Herzog ◽  
Marius Vladoiu ◽  
Xinxian Zheng
Keyword(s):  
Monomial Ideal ◽  
Stanley Depth
Sign in / Sign up

Export Citation Format

Share Document