scholarly journals On the Betti Numbers of Shifted Complexes of Stable Simplicial Complexes

2006 ◽  
Vol 13 (01) ◽  
pp. 47-56
Author(s):  
Zhongming Tang ◽  
Guifen Zhuang
Keyword(s):  
Simplicial Complex ◽  
Polynomial Ring ◽  
Betti Numbers ◽  
Simplicial Complexes ◽  
Base Field ◽  
Arbitrary Base

Let Δ be a stable simplicial complex on n vertexes. Over an arbitrary base field K, the symmetric algebraic shifted complex Δs of Δ is defined. It is proved that the Betti numbers of the Stanley-Reisner ideals in the polynomial ring K[x1, x2, …, xn] of the symmetric algebraic shifted complex, exterior algebraic shifted complex and combinatorial shifted complex of Δ are equal.

scholarly journals Random simplicial complexes, duality and the critical dimension

2019 ◽  
pp. 1-31
Author(s):  
Michael Farber ◽  
Lewis Mead ◽  
Tahl Nowik
Keyword(s):  
Simplicial Complex ◽  
Knot Theory ◽  
Betti Numbers ◽  
Critical Dimension ◽  
Simplicial Complexes ◽  
Alexander Duality ◽  
Random Simplicial Complexes

In this paper, we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behavior of the Betti numbers in the lower model is characterized by the notion of critical dimension, which was introduced by Costa and Farber in [Large random simplicial complexes III: The critical dimension, J. Knot Theory Ramifications 26 (2017) 1740010]: random simplicial complexes in the lower model are homologically approximated by a wedge of spheres of dimension equal the critical dimension. In this paper, we study the Betti numbers in the upper model and introduce new notions of critical dimension and spread. We prove that (under certain conditions) an upper random simplicial complex is homologically approximated by a wedge of spheres of the critical dimension.

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 Graded Betti Numbers of Balanced Simplicial Complexes

2021 ◽  
Author(s):  
Martina Juhnke-Kubitzke ◽  
Lorenzo Venturello
Keyword(s):  
Polynomial Ring ◽  
Betti Numbers ◽  
Simplicial Complexes ◽  
Upper Bounds ◽  
Combinatorial Type ◽  
Homogeneous Ideal ◽  
Graded Rings ◽  
Number Of Generators ◽  
Graded Betti Numbers ◽  
Linear Syzygies

AbstractWe prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings S/I, where S is a polynomial ring and $I\subseteq S$ I ⊆ S is a homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner rings of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type.

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].

Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes

10.37236/1245 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Art M. Duval
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Pure Case ◽  
Definition Of ◽  
Algebraic Shifting

Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the $h$-triangle, a doubly-indexed generalization of the $h$-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the $h$-triangle of a simplicial complex $K$ if and only if $K$ is sequentially Cohen-Macaulay. This generalizes a result of Kalai's for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible $h$-triangles.

Regular Hom-algebras admitting a multiplicative basis

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Antonio J. Calderón Martín
Keyword(s):  
Arbitrary Dimension ◽  
Base Field ◽  
Direct Sum ◽  
Multiplicative Basis ◽  
Arbitrary Base ◽  
The Family

AbstractLet {({\mathfrak{H}},\mu,\alpha)} be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field {{\mathbb{F}}}. A basis {{\mathcal{B}}=\{e_{i}\}_{i\in I}} of {{\mathfrak{H}}} is called multiplicative if for any {i,j\in I}, we have that {\mu(e_{i},e_{j})\in{\mathbb{F}}e_{k}} and {\alpha(e_{i})\in{\mathbb{F}}e_{p}} for some {k,p\in I}. We show that if {{\mathfrak{H}}} admits a multiplicative basis, then it decomposes as the direct sum {{\mathfrak{H}}=\bigoplus_{r}{{\mathfrak{I}}}_{r}} of well-described ideals admitting each one a multiplicative basis. Also, the minimality of {{\mathfrak{H}}} is characterized in terms of the multiplicative basis and it is shown that, in case {{\mathcal{B}}}, in addition, it is a basis of division, then the above direct sum is composed by means of the family of its minimal ideals, each one admitting a multiplicative basis of division.

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 Optimal Decision Trees on Simplicial Complexes

10.37236/1900 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Decision Tree ◽  
Decision Trees ◽  
Simplicial Complex ◽  
Elementary Theory ◽  
Simplicial Complexes ◽  
Optimal Decision ◽  
Property A ◽  
Recursive Definition ◽  
Topological Combinatorics ◽  
Definition Of

We consider topological aspects of decision trees on simplicial complexes, concentrating on how to use decision trees as a tool in topological combinatorics. By Robin Forman's discrete Morse theory, the number of evasive faces of a given dimension $i$ with respect to a decision tree on a simplicial complex is greater than or equal to the $i$th reduced Betti number (over any field) of the complex. Under certain favorable circumstances, a simplicial complex admits an "optimal" decision tree such that equality holds for each $i$; we may hence read off the homology directly from the tree. We provide a recursive definition of the class of semi-nonevasive simplicial complexes with this property. A certain generalization turns out to yield the class of semi-collapsible simplicial complexes that admit an optimal discrete Morse function in the analogous sense. In addition, we develop some elementary theory about semi-nonevasive and semi-collapsible complexes. Finally, we provide explicit optimal decision trees for several well-known simplicial complexes.

scholarly journals On the Structure of Graded Leibniz Algebras

2015 ◽  
Vol 22 (01) ◽  
pp. 83-96 ◽  
Author(s):  
Antonio J. Calderón Martín ◽  
José M. Sánchez Delgado
Keyword(s):  
Arbitrary Dimension ◽  
Unit Element ◽  
Homogeneous Component ◽  
Maximal Length ◽  
Base Field ◽  
Leibniz Algebras ◽  
Arbitrary Base

We study the structure of graded Leibniz algebras with arbitrary dimension and over an arbitrary base field 𝕂. We show that any of such algebras 𝔏 with a symmetric G-support is of the form [Formula: see text] with U a subspace of 𝔏1, the homogeneous component associated to the unit element 1 in G, and any Ij a well described graded ideal of 𝔏, satisfying [Ij, Ik]=0 if j ≠ k. In the case of 𝔏 being of maximal length, we characterize the gr-simplicity of the algebra in terms of connections in the support of the grading.

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