scholarly journals Monomial Ideals of Graphs with Loops

2014 ◽  
Vol 60 (2) ◽  
pp. 321-336 ◽  
Author(s):  
Maurizio Imbesi ◽  
Monica La Barbiera
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideals ◽  
Linear Type ◽  
Connected Graphs ◽  
Edge Ideals ◽  
Linear Resolution ◽  
Algebraic Invariants ◽  
Linear Quotients

Abstract We investigate, using the notion of linear quotients, significative classes of connected graphs whose monomial edge ideals, not necessarily squarefree, have linear resolution, in order to compute standard algebraic invariants of the polynomial ring related to these graphs modulo such ideals. Moreover we are able to determine the structure of the ideals of vertex covers for the edge ideals associated to the previous classes of graphs which can have loops on any vertex. Lastly, it is shown that these ideals are of linear type.

scholarly journals Componentwise linear ideals

1999 ◽  
Vol 153 ◽  
pp. 141-153 ◽  
Author(s):  
Jürgen Herzog ◽  
Takayuki Hibi
Keyword(s):  
Simplicial Complex ◽  
Polynomial Ring ◽  
Betti Numbers ◽  
Monomial Ideals ◽  
Homogeneous Polynomials ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Alexander Dual ◽  
Componentwise Linear Ideals ◽  
The Ideal

AbstractA componentwise linear ideal is a graded ideal I of a polynomial ring such that, for each degree q, the ideal generated by all homogeneous polynomials of degree q belonging to I has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal IΔ arising from a simplicial complex Δ is componentwise linear if and only if the Alexander dual of Δ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.

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 On A Class of Vertex Cover Ideals

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Maurizio Imbesi ◽  
Monica La Barbiera
Keyword(s):  
Vertex Cover ◽  
Connected Graphs ◽  
Algebraic Invariants ◽  
Linear Quotients

Abstract We investigate vertex cover ideals associated to a significative class of connected graphs. These ideals are stated to be Cohen-Macaulay and, using the notion of linear quotients, standard algebraic invariants for them are computed.

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 Regularity and Free Resolution of Ideals Which Are Minimal To $d$-Linearity

2016 ◽  
Vol 118 (2) ◽  
pp. 161 ◽  
Author(s):  
M. Morales ◽  
A. A. Yazdan Pour ◽  
R. Zaare-Nahandi
Keyword(s):  
Polynomial Ring ◽  
Free Resolution ◽  
Free Resolutions ◽  
Edge Ideals ◽  
Homology Manifold ◽  
Linear Resolution ◽  
Vertex Set ◽  
Minimal Free Resolutions ◽  
Positive Integers ◽  
The Ideal

For given positive integers $n\geq d$, a $d$-uniform clutter on a vertex set $[n]=\{1,\dots,n\}$ is a collection of distinct $d$-subsets of $[n]$. Let $\mathscr{C}$ be a $d$-uniform clutter on $[n]$. We may naturally associate an ideal $I(\mathscr{C})$ in the polynomial ring $S=k[x_1,\dots,x_n]$ generated by all square-free monomials \smash{$x_{i_1}\cdots x_{i_d}$} for $\{i_1,\dots,i_d\}\in\mathscr{C}$. We say a clutter $\mathscr{C}$ has a $d$-linear resolution if the ideal \smash{$I(\overline{\mathscr{\mathscr{C}}})$} has a $d$-linear resolution, where \smash{$\overline{\mathscr{C}}$} is the complement of $\mathscr{C}$ (the set of $d$-subsets of $[n]$ which are not in $\mathscr C$). In this paper, we introduce some classes of $d$-uniform clutters which do not have a linear resolution, but every proper subclutter of them has a $d$-linear resolution. It is proved that for any two $d$-uniform clutters $\mathscr{C}_1$, $\mathscr{C}_2$ the regularity of the ideal $I(\overline{\mathscr{C}_1 \cup \mathscr{C}_2})$, under some restrictions on their intersection, is equal to the maximum of the regularities of $I(\overline{\mathscr{C}}_1)$ and $I(\overline{\mathscr{C}}_2)$. As applications, alternative proofs are given for Fröberg's Theorem on linearity of edge ideals of graphs with chordal complement as well as for linearity of generalized chordal hypergraphs defined by Emtander. Finally, we find minimal free resolutions of the ideal of a triangulation of a pseudo-manifold and a homology manifold explicitly.

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.

On the regularity of product of irreducible monomial ideals

2021 ◽  
Author(s):  
Yubin Gao
Keyword(s):  
Finite Number ◽  
Polynomial Ring ◽  
Monomial Ideals

Let [Formula: see text] be a polynomial ring in [Formula: see text] variables over a field [Formula: see text]. When [Formula: see text], [Formula: see text] and [Formula: see text] are monomial ideals of [Formula: see text] generated by powers of the variables [Formula: see text], it is proved that [Formula: see text]. If [Formula: see text], the same result for the product of a finite number of ideals as above is proved.

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 Arithmetical Rank of the Edge Ideals of Some Graphs

2012 ◽  
Vol 19 (spec01) ◽  
pp. 797-806 ◽  
Author(s):  
Fatemeh Mohammadi ◽  
Dariush Kiani
Keyword(s):  
Projective Dimension ◽  
Monomial Ideals ◽  
Common Vertex ◽  
Edge Ideals ◽  
Arithmetical Rank

In this paper, we compute the projective dimension of the edge ideals of graphs consisting of some cycles and lines which are joint in a common vertex. Moreover, we show that for such graphs, the arithmetical rank equals the projective dimension. As an application, we can compute the arithmetical rank for some homogenous monomial ideals.

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