scholarly journals On Vertex Decomposable Simplicial Complexes and Their Alexander Duals

2016 ◽  
Vol 118 (1) ◽  
pp. 43 ◽  
Author(s):  
Somayeh Moradi ◽  
Fahimeh Khosh-Ahang
Keyword(s):  
Simplicial Complex ◽  
Vertex Cover ◽  
Betti Numbers ◽  
Recursive Formula ◽  
Projective Dimension ◽  
Graded Betti Numbers ◽  
Alexander Dual ◽  
Special Cases ◽  
Vertex Decomposable Simplicial Complex ◽  
Vertex Decomposable

In this paper we study the Alexander dual of a vertex decomposable simplicial complex. We define the concept of a vertex splittable ideal and show that a simplicial complex $\Delta$ is vertex decomposable if and only if $I_{\Delta^{\vee}}$ is a vertex splittable ideal. Moreover, the properties of vertex splittable ideals are studied. As the main result, it is proved that any vertex splittable ideal has a Betti splitting and the graded Betti numbers of such ideals are explained with a recursive formula. As a corollary, recursive formulas for the regularity and projective dimension of $R/I_{\Delta}$, when $\Delta$ is a vertex decomposable simplicial complex, are given. Moreover, for a vertex decomposable graph $G$, a recursive formula for the graded Betti numbers of its vertex cover ideal is presented. In special cases, this formula is explained, when $G$ is chordal or a sequentially Cohen-Macaulay bipartite graph. Finally, among the other things, it is shown that an edge ideal of a graph is vertex splittable if and only if it has linear resolution.

scholarly journals Homological invariants of the Stanley–Reisner ring of a k-decomposable simplicial complex

2017 ◽  
Vol 10 (03) ◽  
pp. 1750061
Author(s):  
Somayeh Moradi
Keyword(s):  
Simplicial Complex ◽  
Upper Bound ◽  
Betti Numbers ◽  
Recursive Formula ◽  
Projective Dimension ◽  
Simplicial Complexes ◽  
Monomial Ideals ◽  
Edge Ideals ◽  
Graded Betti Numbers ◽  
Shellable Simplicial Complex

In this paper, we study the regularity and the projective dimension of the Stanley–Reisner ring of a [Formula: see text]-decomposable simplicial complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of decomposable monomial ideals which is the dual concept for [Formula: see text]-decomposable simplicial complexes are studied and an inductive formula for the Betti numbers is given. As a corollary, for a shellable simplicial complex [Formula: see text], a formula for the regularity of the Stanley–Reisner ring of [Formula: see text] is presented. Finally, for a chordal clutter [Formula: see text], an upper bound for [Formula: see text] is given in terms of the regularities of edge ideals of some chordal clutters which are minors of [Formula: see text].

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 $k\geq n$.

Cover Product and Betti Polynomial of Graphs

2015 ◽  
Vol 58 (2) ◽  
pp. 320-333
Author(s):  
Aurora Llamas ◽  
Josá Martínez–Bernal
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Betti Numbers ◽  
Recursive Formula ◽  
Bipartite Graphs ◽  
Matching Number ◽  
Edge Ideal ◽  
Graded Betti Numbers ◽  
Vertex Set ◽  
Chordal Bipartite Graphs

AbstractThe cover product of disjoint graphs G and H with fixed vertex covers C(G) and C(H), is the graphwith vertex set V(G) ∪ V(H) and edge setWe describe the graded Betti numbers of GeH in terms of those of. As applications we obtain: (i) For any positive integer k there exists a connected bipartite graph G such that reg R/I(G) = μS(G) + k, where, I(G) denotes the edge ideal of G, reg R/I(G) is the Castelnuovo–Mumford regularity of R/I(G) and μS(G) is the induced or strong matching number of G; (ii)The graded Betti numbers of the complement of a tree depends only upon its number of vertices; (iii)The h-vector of R/I(G e H) is described in terms of the h-vectors of R/I(G) and R/I(H). Furthermore, in a diòerent direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs.

scholarly journals Projective dimension and regularity of path ideals of cycles

2018 ◽  
Vol 17 (10) ◽  
pp. 1850188 ◽  
Author(s):  
Guangjun Zhu
Keyword(s):  
Betti Numbers ◽  
Projective Dimension ◽  
New Class ◽  
Graded Betti Numbers

In this paper, we study the generalized path ideals, which is a new class of path ideals of cycle graphs. These ideals naturally generalize the standard path ideals of cycles, as studied by Alilooee and Faridi [On the resolution of path ideals of cycles, Comm. Algebra 43 (2015) 5413–5433]. We give some formulas to compute all the top degree graded Betti numbers of these path ideals of cycle graphs. As a consequence, we can give some formulas to compute their projective dimension and regularity.

scholarly journals Average Betti Numbers of Induced Subcomplexes in Triangulations of Manifolds

10.37236/8564 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Giulia Codenotti ◽  
Jonathan Spreer ◽  
Francisco Santos
Keyword(s):  
Simplicial Complex ◽  
Lower Bounds ◽  
Commutative Algebra ◽  
Betti Numbers ◽  
Upper Bounds ◽  
Weighted Sums ◽  
Weighted Averages ◽  
Connected Sums ◽  
Graded Betti Numbers ◽  
Strongly Connected

We study a variation of Bagchi and Datta's $\sigma$-vector of a simplicial complex $C$, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of $C$. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated manifolds and spheres such as connected sums and bistellar flips. In the language of commutative algebra, the invariants are weighted sums of graded Betti numbers of the Stanley-Reisner ring of $C$. This interpretation implies, by a result of Adiprasito, that the Billera-Lee sphere maximizes these invariants among triangulated spheres with a given $f$-vector. For the first entry of $\sigma$, we extend this bound to the class of strongly connected pure complexes. As an application, we show how upper bounds on $\sigma$ can be used to obtain lower bounds on the $f$-vector of triangulated $4$-manifolds with transitive symmetry on vertices and prescribed vector of Betti numbers.

scholarly journals Minimal Cellular Resolutions of the Edge Ideals of Forests

10.37236/8810 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Margherita Barile ◽  
Antonio Macchia
Keyword(s):  
Morse Theory ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Explicit Construction ◽  
Discrete Morse Theory ◽  
Edge Ideals ◽  
Cellular Resolutions ◽  
Graded Betti Numbers ◽  
Free Modules

We present an explicit construction of minimal cellular resolutions for the edge ideals of forests, based on discrete Morse theory. In particular, the generators of the free modules are subsets of the generators of the modules in the Lyubeznik resolution. This procedure allows us to ease the computation of the graded Betti numbers and the projective dimension.

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.

scholarly journals A Construction of Sequentially Cohen–Macaulay Graphs

2021 ◽  
Vol 28 (03) ◽  
pp. 399-414
Author(s):  
Aming Liu ◽  
Tongsuo Wu
Keyword(s):  
Simplicial Complex ◽  
Betti Numbers ◽  
Simple Graph ◽  
Ideal Formula ◽  
Vertex Decomposable ◽  
Linear Quotients ◽  
Facet Ideal

For every simple graph [Formula: see text], a class of multiple clique cluster-whiskered graphs [Formula: see text] is introduced, and it is shown that all such graphs are vertex decomposable; thus, the independence simplicial complex [Formula: see text] is sequentially Cohen–Macaulay. The properties of the graphs [Formula: see text] and [Formula: see text] constructed by Cook and Nagel are studied, including the enumeration of facets of the complex [Formula: see text] and the calculation of Betti numbers of the cover ideal [Formula: see text]. We also prove that the complex[Formula: see text] is strongly shellable and pure for either a Boolean graph [Formula: see text] or the full clique-whiskered graph [Formula: see text] of [Formula: see text], which is obtained by adding a whisker to each vertex of [Formula: see text]. This implies that both the facet ideal [Formula: see text] and the cover ideal [Formula: see text] have linear quotients.

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