Graded Betti Numbers Research Articles

Graded betti numbers and shifts of the symbolic power of a general star configuration in ℙ n

In [37] the author finds a graded minimal free resolution of the 2-nd order symbolic power of a general star configuration in of any codimension r, which generalizes the result of Galetto et al. in [18, Theorem 5.3], and of Geramita et al. in [20, Theorem 3.2]. In [33] we find a graded minimal free resolution of the m-th order symbolic power of a general star configuration in of codimension r for m = 3, 4 with and also find them for codimension 2 with any In this paper, we show how to find a closed formula of graded Betti numbers and shifts of the symbolic power of a general star configuration in of any codimension r based on representation theory and matroid configurations for which generalizes the result of Biermann et al. in [3, Corollary 5.7] and [31, Proposition 7.9]. We also show an example of our methods to find a closed formula of their graded Betti numbers and shifts for any codimension r with

Properties of the resolutions of almost Gorenstein algebras

We study properties of the resolution of almost Gorenstein artinian algebras R / I , i.e. algebras defined by ideals I such that I = J + ( f ) , with J Gorenstein ideal and f ∈ R . Such algebras generalize the well known almost complete intersection artinian algebras. Then we give a new explicit description of the resolution and of the graded Betti numbers of almost complete intersection ideals of codimension 3 and we characterize the ideals whose graded Betti numbers can be achieved using artinian monomial ideals.

Graded Betti numbers of some families of circulant graphs

Let G be the circulant graph Cn(S) with S⊆{1,2,…,⌊n2⌋}, and let I(G) denote the edge ideal in the polynomial ring R=𝕂[x0,x1,…⁡,xn−1] over a field 𝕂. In this paper we compute the ℕ-graded Betti numbers of the edge ideals of three families of circulant graphs Cn(1,2,…,j^,…,⌊n2⌋), Clm(1,2,…,2l^,…,3l^,…,⌊lm2⌋) and Clm(1,2,…,l^,…,2l^,…,3l^,…,⌊lm2⌋). Other algebraic and combinatorial properties like regularity, projective dimension, induced matching number and when such graphs are well-covered, Cohen–Macaulay, sequentially Cohen–Macaulay, Buchsbaum and S2 are also discussed.

On some algebraic properties of edge ideals of ladder graphs

For any finite simple graph G one can associate a simplicial complex and an edge ideal I(G). We attempt to use the combinatorial properties of the given graph to understand some of the algebraic properties of the associated edge ideal. In particular, we study some important algebraic properties like graded Betti numbers, Castelnuovo–Mumford regularity and projective dimension of the edge ideal associated to the given ladder graph. We will introduce the new concept of A-type subgraphs and edge-disjoint A-type subgraphs to reduce the complexity of computing the regularity of a ladder graph. We are also able to find the projective dimension of a ladder graph using the Nguyen–Vu and Kimura’s works on the projective dimension of edge ideals.

Multigraded Betti numbers of Möbius domino ideals

We enumerate graded Betti numbers of the minimal free resolutions of ideals generated by domino tilings of Möbius bands by using combinatorial information from the domino tilings in two ways: first using a splitting function to find a recursive formula and second using a bijection to enumerate in terms of binomial coefficients and Fibonacci numbers.

Graded Betti Numbers of Balanced Simplicial Complexes

We 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 Isubseteq 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.

On surfaces of minimal degree in [formula omitted

There are exactly four surfaces of minimal degree in P5, up to projective equivalence. And they have the same graded Betti numbers. So it is a natural question to ask how to recognize them by their defining equations. In this paper we provide an answer to this question in terms of the rank loci of quadratic equations of those four surfaces. We show that the sets of rank 3 and rank 4 quadratic equations distinguish them.

Rationality of equivariant Hilbert series and asymptotic properties

AnFI\operatorname {FI}- or anOI\operatorname {OI}-moduleM\mathbf {M}over a corresponding noetherian polynomial algebraP\mathbf {P}may be thought of as a sequence of compatible modulesMn\mathbf {M}_nover a polynomial ringPn\mathbf {P}_nwhose number of variables depends linearly onnn. In order to study invariants of the modulesMn\mathbf {M}_nin dependence ofnn, an equivariant Hilbert series is introduced ifM\mathbf {M}is graded. IfM\mathbf {M}is also finitely generated, it is shown that this series is a rational function. Moreover, if this function is written in reduced form rather precise information about the irreducible factors of the denominator is obtained. This is key for applications. It follows that the Krull dimension of the modulesMn\mathbf {M}_ngrows eventually linearly innn, whereas the multiplicity ofMn\mathbf {M}_ngrows eventually exponentially innn. Moreover, for any fixed degreejj, the vector space dimensions of the degreejjcomponents ofMn\mathbf {M}_ngrow eventually polynomially innn. As a consequence, any graded Betti number ofMn\mathbf {M}_nin a fixed homological degree and a fixed internal degree grows eventually polynomially innn. Furthermore, evidence is obtained to support a conjecture that the Castelnuovo-Mumford regularity and the projective dimension ofMn\mathbf {M}_nboth grow eventually linearly innn. It is also shown that modulesM\mathbf {M}whose widthnncomponentsMn\mathbf {M}_nare eventually Artinian can be characterized by their equivariant Hilbert series. Using regular languages and finite automata, an algorithm for computing equivariant Hilbert series is presented.

Unitary representations of the Cherednik algebra: $$V^*$$-homology

We give a non-negative combinatorial formula, in terms of Littlewood-Richardson numbers, for the $$V^*$$ -homology of the unitary representations of the cyclotomic rational Cherednik algebra, and as a consequence, for the graded Betti numbers for the ideals of a class of subspace arrangements arising from the reflection arrangements of complex reflection groups.

Algebraic Properties of Edge Ideals and Cover Ideals of Unbalanced Crown Graphs

In this paper, we introduce the unbalanced crown graphs $${\mathcal {C}}_{m,n}$$ , and compute all graded Betti numbers of edge ideals of these graphs in terms of associated numerical data. As a consequence, we determine the regularity and the projective dimension of edge ideals of these graphs. Also, the Hilbert series of the cover ideals of these graphs is obtained.

Subspace Arrangements and Cherednik Algebras

Abstract The purpose of this article is to study the relationship between numerical invariants of certain subspace arrangements coming from reflection groups and numerical invariants arising in the representation theory of Cherednik algebras. For instance, we observe that knowledge of the equivariant graded Betti numbers (in the sense of commutative algebra) of any irreducible representation in category ${\mathscr{O}}$ is equivalent to knowledge of the Kazhdan–Lusztig character of the irreducible object (we use this observation in joint work with Fishel–Manosalva). We then explore the extent to which Cherednik algebra techniques may be applied to ideals of linear subspace arrangements: we determine when the radical of the polynomial representation of the Cherednik algebra is a radical ideal and, for the cyclotomic rational Cherednik algebra, determine the socle of the polynomial representation and characterize when it is a radical ideal. The subspace arrangements that arise include various generalizations of the $k$-equals arrangement. In the case of the radical, we apply our results with Juteau together with an idea of Etingof–Gorsky–Losev to observe that the quotient is Cohen–Macaulay for positive choices of parameters. In the case of the socle (in cyclotomic type), we give an explicit vector space basis in terms of certain specializations of nonsymmetric Jack polynomials, which in particular determines its minimal generators and Hilbert series and answers a question posed by Feigin and Shramov.

Splittings of toric ideals

Let I⊆R=K[x1,…,xn] be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal I can be “split” into the sum of two smaller toric ideals. For a general toric ideal I, we give a sufficient condition for this splitting in terms of the integer matrix that defines I. When I=IG is the toric ideal of a finite simple graph G, we give additional splittings of IG related to subgraphs of G. When there exists a splitting I=I1+I2 of the toric ideal, we show that in some cases we can describe the (multi-)graded Betti numbers of I in terms of the (multi-)graded Betti numbers of I1 and I2.

Elder-Rule-Staircodes for Augmented Metric Spaces

An augmented metric space is a metric space $(X, d_X)$ equipped with a function $f_X: X \to \mathbb{R}$. This type of data arises commonly in practice, e.g, a point cloud $X$ in $\mathbb{R}^d$ where each point $x\in X$ has a density function value $f_X(x)$ associated to it. An augmented metric space $(X, d_X, f_X)$ naturally gives rise to a 2-parameter filtration $\mathcal{K}$. However, the resulting 2-parameter persistent homology $\mathrm{H}_{\bullet}(\mathcal{K})$ could still be of wild representation type, and may not have simple indecomposables. In this paper, motivated by the elder-rule for the zeroth homology of 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode $\mathrm{H}_0(\mathcal{K})$. Specifically, if $n = |X|$, the elder-rule-staircode consists of $n$ number of staircase-like blocks in the plane. We show that if $\mathrm{H}_0(\mathcal{K})$ is interval decomposable, then the barcode of $\mathrm{H}_0(\mathcal{K})$ is equal to the elder-rule-staircode. Furthermore, regardless of the interval decomposability, the fibered barcode, the dimension function (a.k.a. the Hilbert function), and the graded Betti numbers of $\mathrm{H}_0(\mathcal{K})$ can all be efficiently computed once the elder-rule-staircode is given. Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in $O(n^2\log n)$ time, which can be improved to $O(n^2\alpha(n))$ if $X$ is from a fixed dimensional Euclidean space $\mathbb{R}^d$, where $\alpha(n)$ is the inverse Ackermann function.

Lovász–Saks–Schrijver ideals and parity binomial edge ideals of graphs

Let G be a simple graph on n vertices. Let LG and IG denote the Lovász–Saks–Schrijver(LSS) ideal and parity binomial edge ideal of G in the polynomial ring S=K[x1,…,xn,y1,…,yn] respectively. We classify graphs whose LSS ideals and parity binomial edge ideals are complete intersections. We also classify graphs whose LSS ideals and parity binomial edge ideals are almost complete intersections, and we prove that their Rees algebra is Cohen–Macaulay. We compute the second graded Betti number and obtain a minimal presentation of LSS ideals of trees and odd unicyclic graphs. We also obtain an explicit description of the defining ideal of the symmetric algebra of LSS ideals of trees and odd unicyclic graphs.

Almost complete intersection binomial edge ideals and their Rees algebras

Let G be a simple graph on n vertices and J G denote the binomial edge ideal of G in the polynomial ring S = K [ x 1 , … , x n , y 1 , … , y n ] . In this article, we compute the second graded Betti numbers of J G , and we obtain a minimal presentation of it when G is a tree or a unicyclic graph. We classify all graphs whose binomial edge ideals are almost complete intersection, prove that they are generated by a d -sequence and that the Rees algebra of their binomial edge ideal is Cohen-Macaulay. We also obtain an explicit description of the defining ideal of the Rees algebra of those binomial edge ideals.

On graded Betti numbers of edge rings of deficient complete bipartite graphs

In this paper, we introduce deficient complete bipartite graphs and deduce a formula for graded Betti numbers of their edge rings in terms of associated combinatorial data. As a consequence, we obtain the graded Betti numbers of edge rings of complete bipartite graphs and the n-crown graph Also, the projective dimension and the regularity of edge rings of these graphs are obtained.

Multigraded minimal free resolutions of simplicial subclutters

This paper concerns the study of a class of clutters called simplicial subclutters. Given a clutter C and its simplicial subclutter D, we compare some algebraic properties and invariants of the ideals I,J associated to these two clutters, respectively. We give a formula for computing the (multi)graded Betti numbers of J in terms of those of I and some combinatorial data about D. As a result, we see that if C admits a simplicial subclutter, then there exists a monomial u∉I such that the (multi)graded Betti numbers of I+(u) can be computed through those of I. It is proved that the Betti sequence of any graded ideal with linear resolution is the Betti sequence of an ideal associated to a simplicial subclutter of the complete clutter. These ideals turn out to have linear quotients. However, they do not form all the equigenerated square-free monomial ideals with linear quotients. If C admits ∅ as a simplicial subclutter, then I has linear resolution over all fields. Examples show that the converse is not true.

Average Betti Numbers of Induced Subcomplexes in Triangulations of Manifolds

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.

On Betti numbers in the linear strand and regularity of triangular graphs

To every finite simple graph G, we associate the so-called edge ideal I(G), which is a square-free quadratic monomial ideal generated by the monomials corresponding to the edges of G. In this paper, we determine all the initial graded Betti numbers of edge ideals of triangular graphs and alternate triangular graphs in terms of the underlying graph. We also compute the regularity of edge ideals of triangular and alternate triangular graphs.

Regularity and ℎ-polynomials of toric ideals of graphs

For all integers 4 ≤ r ≤ d 4 \leq r \leq d , we show that there exists a finite simple graph G = G r , d G= G_{r,d} with toric ideal I G ⊂ R I_G \subset R such that R / I G R/I_G has (Castelnuovo–Mumford) regularity r r and h h -polynomial of degree d d . To achieve this goal, we identify a family of graphs such that the graded Betti numbers of the associated toric ideal agree with its initial ideal, and, furthermore, that this initial ideal has linear quotients. As a corollary, we can recover a result of Hibi, Higashitani, Kimura, and O’Keefe that compares the depth and dimension of toric ideals of graphs.