Betti Table Research Articles

The eventual shape of the Betti table of 𝔪kM

Let [Formula: see text] be a standard graded polynomial ring over a field [Formula: see text] in a finite set of variables, and let [Formula: see text] be the graded maximal ideal of [Formula: see text]. It is known that for a finitely generated graded [Formula: see text]-module [Formula: see text] and all integers [Formula: see text], the module [Formula: see text] is componentwise linear. For large [Formula: see text] we describe the pattern of the Betti table of [Formula: see text] when [Formula: see text]. Moreover, we show that for any [Formula: see text], [Formula: see text] has linear quotients if [Formula: see text] is a monomial ideal.

The regularity of almost all edge ideals

A fruitful contemporary paradigm in graph theory is that almost all graphs that do not contain a certain subgraph have common structural characteristics. An example is the well-known result saying that almost all triangle-free graphs are bipartite. The “almost” is crucial, without it such theorems do not hold. In this paper we transfer this paradigm to commutative algebra and make use of deep graph theoretic results. A key tool is the critical graphs introduced relatively recently by Balogh and Butterfield, who proved that almost all graphs not containing a critical subgraph have common structural characteristics analogous to being bipartite.For a graph G, let IG denote its edge ideal, the monomial ideal generated by xixj for every edge ij of G. In this paper we study the graded Betti numbers of IG, which are combinatorial invariants that measure the complexity of a minimal free resolution of IG. The Betti numbers of the form βi,2i+2 constitute the “main diagonal” of the Betti table. It is well known that for edge ideals any Betti number to the left of this diagonal is always zero. We identify a certain “parabola” inside the Betti table and call parabolic Betti numbers the entries of the Betti table bounded on the left by the main diagonal and on the right by this parabola. Let βi,j be a parabolic Betti number on the r-th row of the Betti table, for r≥3. We prove that almost all graphs G with βi,j(IG)=0 can be partitioned into r−2 cliques and one independent set. In particular, for almost all graphs G with βi,j(IG)=0, the regularity of IG is r−1.

Random Subcomplexes and Betti Numbers of Random Edge Ideals

Abstract We study homological properties of random quadratic monomial ideals in a polynomial ring $R = {\mathbb {K}}[x_1, \dots x_n]$, utilizing methods from the Erd̋s–Rényi model of random graphs. Here, for a graph $G \sim G(n, p),$ we consider the coedge ideal $I_G$ generated by monomials corresponding to the missing edges of $G$ and study Betti numbers of $R/I_G$ as $n$ tends to infinity. Our main results involve setting the edge probability $p = p(n)$ so that asymptotically almost surely the Krull dimension of $R/I_G$ is fixed. Under these conditions, we establish various properties regarding the Betti table of $R/I_G$, including sharp bounds on regularity and projective dimension and distribution of nonzero normalized Betti numbers. These results extend work of Erman and Yang who studied such ideals in the context of conjectured phenomena in the nonvanishing of asymptotic syzygies. Along the way, we establish results regarding subcomplexes of random clique complexes as well as notions of higher-dimensional vertex $k$-connectivity that may be of independent interest.

Boij–Söderberg decompositions of lexicographic ideals

Boij–Soderberg theory describes the Betti diagrams of graded modules over a polynomial ring as a linear combination of pure diagrams with positive coefficients. In this paper, we focus on the Betti diagrams of lexicographic ideals. Mainly, we characterize the Boij–Soderberg decomposition of the Betti table of a lexicographic ideal in the polynomial ring with three variables, and show a nice connection between its Boij–Soderberg decomposition and the ones of other related lexicographic ideals.

Componentwise linearity of projective varieties with almost maximal degree

The degree of a projective subscheme has an upper bound in term of the codimension and the reduction number. If a projective variety has an almost maximal degree, that is, the degree equals to the upper bound minus one, then its Betti table has been described explicitly. We build on this work by showing that for most of such varieties, the defining ideals are componentwise linear and in particular the componentwise linearity is suitable for classifying the Betti tables of such varieties. As an application, we compute the Betti table of all varieties with almost maximal degree and componentwise linear resolution.

On syzygies of Segre embeddings of

We construct a nonminimal graded free resolution of Segre embeddings of although we don’t compute all maps. We use this to prove an explicit formula for certain nonzero entries in the graded Betti table, at the end of the first row. We work over an arbitrary field k.

Trimming complexes and applications to resolutions of determinantal facet ideals

We produce a family of complexes called trimming complexes and explore its applications. We demonstrate how trimming complexes can be used to deduce the Betti table for the minimal free resolution of the ideal generated by certain subsets of a generating set for an arbitrary ideal I. In particular, we compute the Betti table of the ideal obtained by removing an arbitrary generator from the ideal of submaximal pfaffians of a generic skew symmetric matrix M. We also explicitly compute the Betti table for the ideal generated by certain subsets of the generating set of the ideal of maximal minors of a generic n × m matrix. Such ideals are a subset of a class of ideals called determinantal facet ideals, whose higher degree Betti numbers had not previously been computed.

On the Betti numbers of edge ideals of skew Ferrers graphs

We prove that [Formula: see text] for any staircase skew Ferrers graph [Formula: see text], where [Formula: see text] and [Formula: see text]. As a consequence, Ene et al. conjecture is confirmed to hold true for the Betti numbers in the last column of the Betti table in a particular case. An explicit formula for the unique extremal Betti number of the binomial edge ideal of some closed graphs is also given.

On the (non-)vanishing of syzygies of Segre embeddings

We analyze the vanishing and non-vanishing behavior of the graded Betti numbers for Segre embeddings of products of projective spaces. We give lower bounds for when each of the rows of the Betti table becomes non-zero, and prove that our bounds are tight for Segre embeddings of products of P^1. This generalizes results of Rubei concerning the Green-Lazarsfeld property N_p for Segre embeddings. Our methods combine the Kempf-Weyman geometric technique for computing syzygies, the Ein-Erman-Lazarsfeld approach to proving non-vanishing of Betti numbers, and the theory of algebras with straightening laws.

Computing graded Betti tables of toric surfaces

We present various facts on the graded Betti table of a projectively embedded toric surface, expressed in terms of the combinatorics of its defining lattice polygon. These facts include explicit formulas for a number of entries, as well as a lower bound on the length of the quadratic strand that we conjecture to be sharp (and prove to be so in several special cases). We also present an algorithm for determining the graded Betti table of a given toric surface by explicitly computing its Koszul cohomology and report on an implementation in SageMath. It works well for ambient projective spaces of dimension up to roughly 25 25 , depending on the concrete combinatorics, although the current implementation runs in finite characteristic only. As a main application we obtain the graded Betti table of the Veronese surface ν 6 ( P 2 ) ⊆ P 27 \nu _6(\mathbb {P}^2) \subseteq \mathbb {P}^{27} in characteristic 40 009 40\,009 . This allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary Veronese surface ν d ( P 2 ) \nu _d(\mathbb {P}^2) .

Canonical syzygies of smooth curves on toric surfaces

In a first part of this paper, we prove constancy of the canonical graded Betti table among the smooth curves in linear systems on Gorenstein weak Fano toric surfaces. In a second part, we show that Green's canonical syzygy conjecture holds for all smooth curves of genus at most 32 or Clifford index at most 6 on arbitrary toric surfaces. Conversely we use known results on Green's conjecture (due to Lelli-Chiesa) to obtain new facts about graded Betti tables of projectively embedded toric surfaces.

Extremal rays of Betti cones

We study the Betti cone of modules of a standard graded algebra over a field, and find a class of modules whose Betti diagrams span extremal rays of the Betti cone. We also identify a class of one-dimensional Cohen–Macaulay rings with certain Hilbert series, for which the Betti cone is spanned by these extremal rays. These results lead to an algorithm for the decomposition of Betti table of modules, into the extremal rays, over such rings, and also help to obtain bounds for the multiplicity of the given module.

On curves lying on a rational normal surface scroll

In this paper, we study the minimal free resolution of non-ACM divisors X of a smooth rational normal surface scroll S=S(a1,a2)⊂Pr. Our main result shows that for a2≥2a1−1, there exists a nice decomposition of the Betti table of X as a sum of much simpler Betti tables. As a by-product of our results, we obtain a complete description of the graded Betti numbers of X for the cases where S=S(1,r−2) for all r≥3 and S=S(2,r−3) for all r≥6.

On the Lex-plus-powers Conjecture

Let S be a polynomial ring over a field and I⊆S a homogeneous ideal containing a regular sequence of forms of degrees d1,…,dc. In this paper we prove the Lex-plus-powers Conjecture when the field has characteristic 0 for all regular sequences such that di≥∑j=1i−1(dj−1)+1 for each i; that is, we show that the Betti table of I is bounded above by the Betti table of the lex-plus-powers ideal of I.

On the n-th row of the graded Betti table of an n-dimensional toric variety

We prove an explicit formula for the first non-zero entry in the n-th row of the graded Betti table of an n-dimensional projective toric variety associated to a normal polytope with at least one interior lattice point. This applies to Veronese embeddings of projective space where we prove a special case of a conjecture of Ein and Lazarsfeld. We also prove an explicit formula for the entire n-th row when the interior of the polytope is one-dimensional. All results are valid over an arbitrary field k.

Computing the Betti table of a monomial ideal: A reduction algorithm

In this paper we develop a new technique to compute the Betti table of a monomial ideal. We present a prototype implementation of the resulting algorithm and we perform some numerical experiments. As a major byproduct, we also prove new constraints on the shape of the possible Betti tables of a monomial ideal.

On vanishing patterns in j-strands of edge ideals

We consider two problems regarding vanishing patterns in the Betti table of edge ideals I over any fixed field. First, we show that the j-strand is connected if \(j=3\) (for \(j=2\) this is easy and known), and give examples where the j-strand is not connected for any \(j>3\). Next, we apply our result on strand connectivity to establish the subadditivity conjecture for edge ideals, \(t_{a+b}(I)\le t_a(I)+t_b(I)\), in case \(b=2,3\) (the case \(b=1\) is known). Here \(t_i(I)\) denote the maximal shifts in the minimal free resolution of S / I over its polynomial algebra.

Squarefree Monomial Modules and Extremal Betti Numbers

Let K be a field and let S=K[x1,…,xn] be a polynomial ring over K. Let [Formula: see text] be a finitely generated graded free S-module with basis {e1,…,er} in degrees f1,…,fr such that f1 ≤ f2 ≤ ⋯ ≤ fr. We examine some classes of squarefree monomial submodules of F. Hence, we focalize our attention on the Betti table of such classes in order to analyze the behavior of their extremal Betti numbers.

On Decomposing Betti Tables and O-Sequences

The Boij–Söderberg characterization decomposes a Betti table into a unique positive integral linear combination of pure diagrams. Given a module with a pure resolution, we describe explicit formulae for computing the decomposition of the Betti table of the module given the decomposition of the truncation of the Betti table, and vice versa.Nagel and Sturgeon described the decomposition of Betti tables of ideals with d-linear resolutions; indeed, the coefficients are precisely finite O-sequences. Using the extension formulae, we provide an explicit description of the coefficients of the decomposition of the Betti table of the quotient ring of such an ideal. Following from this, we describe the closed convex simplicial cone of O-sequences.

Bounds for the multiplicity of Gorenstein algebras

We prove upper bounds for the Hilbert-Samuel multiplicity of standard graded Gorenstein algebras. The main tool that we use is Boij-Soderberg theory to obtain a decomposition of the Betti table of a Gorenstein algebra as the sum of rational multiples of symmetrized pure tables. Our bound agrees with the one in the quasi-pure case obtained by Srinivasan [J. Algebra, vol.~208, no.~2, (1998)].