k-Decomposable Monomial Ideals

2015 ◽  
Vol 22 (spec01) ◽  
pp. 745-756 ◽  
Author(s):  
Rahim Rahmati-Asghar ◽  
Siamak Yassemi
Keyword(s):  
Simplicial Complex ◽  
Monomial Ideals ◽  
Alexander Dual ◽  
Linear Quotients ◽  
Dimensional Simplicial Complex

In this paper we introduce a class of monomial ideals, called k-decomposable ideals. It is shown that the class of k-decomposable ideals is contained in the class of monomial ideals with linear quotients, and when k is large enough, the class of k-decomposable ideals is equal to the class of ideals with linear quotients. In addition, it is shown that a d-dimensional simplicial complex is k-decomposable if and only if the Stanley-Reisner ideal of its Alexander dual is a k-decomposable ideal, where k ≤ d. Moreover, it is shown that every k-decomposable ideal is componentwise k-decomposable.

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

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 Simplicial Complexes of Whisker Type

10.37236/4894 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mina Bigdeli ◽  
Jürgen Herzog ◽  
Takayuki Hibi ◽  
Antonio Macchia
Keyword(s):  
Simplicial Complex ◽  
Short Proof ◽  
Monomial Ideal ◽  
Simplicial Complexes ◽  
Linear Resolution ◽  
Cw Complex ◽  
Alexander Dual ◽  
Vertex Decomposable ◽  
The Ideal

Let $I\subset K[x_1,\ldots,x_n]$ be  a zero-dimensional monomial ideal, and $\Delta(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $\Delta(I)$ is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that  $\Delta(I)$ is even vertex decomposable. The ideal $L(I)$, which is defined to be the Stanley--Reisner ideal of the Alexander dual of $\Delta(I)$, has a linear resolution which is cellular and supported on a regular CW-complex. All powers of $L(I)$ have a linear resolution. We compute $\mathrm{depth}\ L(I)^k$ and show that $\mathrm{depth}\ L(I)^k=n$ for all $k\geq n$.

scholarly journals Golodness and polyhedral products for two-dimensional simplicial complexes

2018 ◽  
Vol 30 (2) ◽  
pp. 527-532
Author(s):  
Kouyemon Iriye ◽  
Daisuke Kishimoto
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Two Dimensional ◽  
Dimensional Simplicial Complex

AbstractGolodness of two-dimensional simplicial complexes is studied through polyhedral products, and combinatorial and topological characterizations of Golodness of surface triangulations are given. An answer to the question of Berglund is also given so that there is a two-dimensional simplicial complex which is rationally Golod but not Golod over{\mathbb{Z}/p}.

scholarly journals Local-to-global Urysohn width estimates

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexey Balitskiy ◽  
Aleksandr Berdnikov
Keyword(s):  
Riemannian Manifold ◽  
Simplicial Complex ◽  
Metric Space ◽  
Low Dimensional ◽  
Dimensional Simplicial Complex

Abstract The notion of the Urysohn d-width measures to what extent a metric space can be approximated by a d-dimensional simplicial complex. We investigate how local Urysohn width bounds on a Riemannian manifold affect its global width. We bound the 1-width of a Riemannian manifold in terms of its first homology and the supremal width of its unit balls. Answering a question of Larry Guth, we give examples of n-manifolds of considerable ( n - 1 ) {(n-1)} -width in which all unit balls have arbitrarily small 1-width. We also give examples of topologically simple manifolds that are locally nearly low-dimensional.

scholarly journals Shellability and the Strong gcd-Condition

10.37236/67 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Alexander Berglund
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Flag Complex ◽  
Alexander Dual ◽  
Golod Ring

Shellability is a well-known combinatorial criterion on a simplicial complex $\Delta$ for verifying that the associated Stanley-Reisner ring $k[\Delta]$ is Cohen-Macaulay. A notion familiar to commutative algebraists, but which has not received as much attention from combinatorialists as the Cohen-Macaulay property, is the notion of a Golod ring. Recently, Jöllenbeck introduced a criterion on simplicial complexes reminiscent of shellability, called the strong gcd-condition, and he together with the author proved that it implies Golodness of the associated Stanley-Reisner ring. The two algebraic notions were earlier tied together by Herzog, Reiner and Welker, who showed that if $k[\Delta^\vee]$ is sequentially Cohen-Macaulay, where $\Delta^\vee$ is the Alexander dual of $\Delta$, then $k[\Delta]$ is Golod. In this paper, we present a combinatorial companion of this result, namely that if $\Delta^\vee$ is (non-pure) shellable then $\Delta$ satisfies the strong gcd-condition. Moreover, we show that all implications just mentioned are strict in general but that they are equivalences if $\Delta$ is a flag complex.

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 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 On Generalization of Cycles and Chordality to Clutters from an Algebraic Viewpoint

2017 ◽  
Vol 24 (04) ◽  
pp. 611-624 ◽  
Author(s):  
Ashkan Nikseresht ◽  
Rashid Zaare-Nahandi
Keyword(s):  
Simplicial Complex ◽  
Commutative Algebra ◽  
Point Of View ◽  
Algebraic Point ◽  
Linear Resolution ◽  
Vertex Decomposable Simplicial Complex ◽  
Vertex Decomposable ◽  
The Relationship ◽  
Linear Quotients

In this paper, we study the notion of chordality and cycles in clutters from a commutative algebraic point of view. The corresponding concept of chordality in commutative algebra is having a linear resolution. We mainly consider the generalization of chordality proposed by Bigdeli et al. in 2017 and the concept of cycles introduced by Cannon and Faridi in 2013, and study their interrelations and algebraic interpretations. In particular, we investigate the relationship between chordality and having linear quotients in some classes of clutters. Also, we show that if [Formula: see text] is a clutter such that 〈[Formula: see text]〉 is a vertex decomposable simplicial complex or I([Formula: see text]) is squarefree stable, then [Formula: see text] is chordal.

Simplicial Complexes and Open Subsets of Non-Separable LF-Spaces

2011 ◽  
Vol 63 (2) ◽  
pp. 436-459 ◽  
Author(s):  
Kotaro Mine ◽  
Katsuro Sakai
Keyword(s):  
Simplicial Complex ◽  
Open Subset ◽  
Simplicial Complexes ◽  
Locally Finite ◽  
Direct Sum ◽  
Open Set ◽  
Finite Dimensional ◽  
Metric Topology ◽  
Dimensional Simplicial Complex

Abstract Let F be a non-separable LF-space homeomorphic to the direct sum , where . It is proved that every open subset U of F is homeomorphic to the product |K| × F for some locally finite-dimensional simplicial complex K such that every vertex v ∈ K(0) has the star St(v, K) with card St(v, K)(0) < 𝒯 = sup 𝒯n (and card K(0) ≤ 𝒯 ), and, conversely, if K is such a simplicial complex, then the product |K| × F can be embedded in F as an open set, where |K| is the polyhedron of K with the metric topology.

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