Sean Jacques, Mordechai Katzman
This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs.
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We prove a tight lower bound on the Betti numbers of tree and forest ideals and a tight upper bound on certain graded Betti numbers of squarefree monomial ideals.
Let $k$ be a field and $R=k[x_1,\ldots,x_n]/I=S/I$ a graded ring. Then $R$ has a $t$-linear resolution if $I$ is generated by homogeneous elements of degree $t$, and all higher syzygies are linear. Thus $R$ has a $t$-linear resolution if ${\rm Tor}^S_{i,j}(S/I,k)=0$ if $j\ne i+t-1$. For a simplicial complex $Δ$ on $[{\bf n}]=\{1,\ldots,n\}$ and a field $k$, the Stanley-Reisner ring $k[Δ]$ is $k[x_1,\ldots,x_n]/I$, where $I$ is generated by those squarefree monomials $x_{i_1}\cdots x_{i_k}$ for which $\{ i_1,\ldots,i_k\}$ does not belong to $Δ$. In \cite{Fr} the Stanley-Reisner rings with 2-lin…
We prove that multigraded Betti numbers of a simplicial forest are always either 0 or 1. Moreover a nonzero multidegree appears exactly at one homological degree in the resolution. Our work generalizes work of Bouchat on edge ideals of graph forests.
In this thesis we investigate certain types of monomial ideals of polynomial rings over fields. We are interested in minimal free resolutions of these ideals (or equivalently the quotients of the polynomial ring by the ideals) considered as modules over the polynomial ring. There is no simple method of finding such resolutions but in the case of Stanley-Reisner ideals Hochster's formula and its variants provide a way to compute the Betti numbers of these resolutions. Even with these formulae it is not in general possible to find especially explicit or useful descriptions of the Betti numbers. …
Given finite simple graph one can associate the edge ideal. In this paper we discuss the non-vanishingness of the graded Betti numbers of edge ideals in terms of the original graph. In particular, we give a necessary and sufficient condition for a chordal graph on which the graded Betti number does not vanish and characterize the graded Betti number for a forest. Moreover we characterize the projective dimension for a chordal graph.
We present an explicit construction of a minimal cellular resolution 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 to ease the computation of the graded Betti numbers and the projective dimension.
We prove the subadditivity property for the maximal degrees of the syzygies of facet ideals simplicial forests. For such an ideal $I$, if the $i$-th Betti number is nonzero and $i=a+b$, we show that there are monomials in the lcm lattice of $I$ that are complements in part of the lattice, each supporting a nonvanishing $a$-th and $b$-th Betti numbers. The subadditivity formula follows from this observation.
Matchings, Squarefree Powers and Betti Splittings
本文通过Betti分裂研究有限简单图边理想的无平方幂，证明了森林图规范化深度函数的非递增性，并显式计算其正则函数，证实了Erey和Hibi的猜想。
We give a sufficient condition for a monomial ideal to have a nonzero Betti number in each multidegree. In the case of facet ideals of simplicial forests, this condition becomes a necessary one and it allows us to characterize Betti numbers, projective dimension and regularity of such ideals combinatorially. Our condition is expressed in terms of minimal facet covers of simplicial complexes.
We determine (multi)graded Betti numbers of path ideals of lines and star graphs.
We compute one of the distinguished extremal Betti number of the binomial edge ideal of a block graph, and classify all block graphs admitting precisely one extremal Betti number.
We study the equality of the extremal Betti numbers of the binomial edge ideal $J_G$ and those of its initial ideal ${\rm in}(J_G)$ of a closed graph $G$. We prove that in some cases there is an unique extremal Betti number for ${\rm in}(J_G)$ and as a consequence there is an unique extremal Betti number for $J_G$ and these extremal Betti numbers are equal
Let $G$ be a cycle or a complete bipartite graph. We show that the binomial edge ideal $J_{G}$ and its initial ideal with respect to the lexicographic order have the same extremal Betti number.
The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope. The proof allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard graded K-algebra in terms of its Hilbert function.
Given integers $r$ and $b$ with $1 \leq b \leq r$, a finite simple connected graph $G$ for which ${\rm reg}(S/I(G)) = r$ and the number of extremal Betti numbers of $S/I(G)$ is equal to $b$ will be constructed.
Let $S$ be a polynomial ring in $n$ variables over a field $K$ of characteristic $0$. A numerical characterization of all possible extremal Betti numbers of any graded submodule of a finitely generated graded free $S$-module is given.
Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$ via upper-Koszul simplicial complexes, Betti splittings and the mapping cone construction. In particular, we provide recursive formulas for the Betti numbers of edge ideals of several classes of weighted oriented graphs. We also identify classes of weighted oriented graphs whose edge ideals have a unique extremal Betti number which allows us to compute the regularity and projective dimension for the identified classes. Furthermore, we charac…
We study the dependence of graded Betti numbers of monomial ideals on the characteristic of the base field. The examples we describe include bipartite ideals, Stanley--Reisner ideals of vertex-decomposable complexes and ideals with componentwise linear resolutions. We give a description of bipartite graphs and, using discrete Morse theory, provide a way of looking at the homology of arbitrary simplicial complexes through bipartite ideals. We also prove that the Betti table of a monomial ideal over the field of rational numbers can be obtained from the Betti table over any field by a sequence o…
We give a bijective proof that the Betti numbers of a minimal free resolution of the Stanley-Reisner ring of a cycle graph (viewed as a one-dimensional simplicial complex) are given by the number of standard Young tableaux of a given shape.
In this paper we study some algebraic properties of hypergraphs, in particula their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge, are not previously considered in the literature. Also, in a natural way, we define a product on hypergraphs, which in a sense is dual to the join operation on simplicial complexes. For such product, we give a general formula for the Betti numbers, which specializes neatly in case of linear resolutions.
This paper examines the one-to-one-to-one correspondence between threshold graphs, Betti numbers of quotients of polynomial rings by $2$-linear ideals, and anti-lecture hall compositions. In particular, we establish new explicit combinatorial mappings between each of these classes of objects and calculate the expected values of the Betti numbers and anti-lecture hall composition corresponding to a random threshold graph.
We give a bound for the Betti numbers of the Stanley-Reisner ring of a stellar subdivision of a Gorenstein* simplicial complex by applying unprojection theory. From this we derive a bound for the Betti numbers of iterated stellar subdivisions of the boundary complex of a simplex. The bound depends only on the number of subdivisions, and we construct examples which prove that it is sharp.
We define a set of invariants of a homogeneous ideal $I$ in a polynomial ring called the symmetric iterated Betti numbers of $I$. For $I_Γ$, the Stanley-Reisner ideal of a simplicial complex $Γ$, these numbers are the symmetric counterparts of the exterior iterated Betti numbers of $Γ$ introduced by Duval and Rose. We show that the symmetric iterated Betti numbers of an ideal $I$ coincide with those of a particular reverse lexicographic generic initial ideal $\Gin(I)$ of $I$, and interpret these invariants in terms of the associated primes and standard pairs of $\Gin(I)$. We verify that for an…
In this paper we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first six Betti numbers do not depend on the characteristic of the ground field. We also show that, if the number of variables $n$ is at most 10, all Betti numbers are independent of the ground field. For $n=11$, there exists precisely 4 examples in which the Betti numbers depend on the ground field. This is equivalent to the statement that the homology of flag complexes with at most 10 vertices is torsion free and that there exists precisely 4 non-isomorphic flag complexes with 11 vertic…
Multigraded Betti numbers of Veronese embeddings
本文研究非常面嵌入的多格Betti数，利用Hochster公式和离散Morse理论推导其消失与非消失结果。
We prove a conjectured lower bound of Nagel and Reiner on Betti numbers of edge ideals of bipartite graphs.
Betti splittings and multigraded Betti numbers of cover ideals of bipartite graphs
该研究探讨二部图覆盖理想的Betti分裂，证明二部图覆盖理想的任意$x_i$-划分为Betti分裂，且任意无平方项单项式理想的多重分次Betti数可出现在某二部图覆盖理想的对应分次Betti数特定部分中。
We provide a simple method to compute the Betti numbers if the Stanley-Reisner ideal of a simplicial tree and its Alexander dual.
We study the extremal Betti numbers of the class of $t$--spread strongly stable ideals. More precisely, we determine the maximal number of admissible extremal Betti numbers for such ideals, and thereby we generalize the known results for $t\in \{1,2\}$.
We express the multigraded Betti numbers of monomial ideals in 4 variables in terms of the multigraded Betti numbers of 66 squarefree monomial ideals, also in 4 variables. We use this class of 66 ideals to prove that monomial resolutions in 4 variables are independent of the base field. In addition, we give a formula for the Betti numbers of an arbitrary monomial ideal in 4 variables.
The asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field has recently been reviewed. We extend quasi polynomial behavior of graded Betti numbers of powers of homogenous ideals to Z-graded algebra over Notherian local ring. Furthermore our main result treats the Betti table of filtrations which is finite or integral over the Rees algebra.
We characterize componentwise linear monomial ideals with minimal Taylor resolution and consider the lower bound for the Betti numbers of componentwise linear ideals.
We compute the Betti numbers of the resolution of a special class of square-free monomial ideals, the ideals of mixed products. Moreover when these ideals are Cohen-Macaulay we calculate their type.
We show that Kimura's necessary and sufficient condition for the non-vanishingness of multigraded Betti numbers of chordal graphs can be extended to the wider class of weakly chordal graphs.
We prove that $β_p(I(G)) = β_{p,p+r}(I(G))$ for skew Ferrers graph $G$, where $p:=\pd(I(G))$ and $r:=\reg(I(G))$. As a consequence, we confirm that Ene, Herzog and Hibi's conjecture is true for the Betti numbers in the last columm of Betti table. We also give an explicit formula for the unique extremal Betti number of binomial edge ideal for some closed graphs.
Betti numbers of normal edge rings (\bf{I})
本文提出诱导子图方法，计算双耳图、紧图对应的正规边环的多重分次Betti数，揭示不同类型图的Betti数特性，预期该方法可推广至其他正规边环。
Betti Numbers of Edge Ideals of Weighted Oriented Crown Graphs
本文通过诱导子图方法计算加权有向冠图边理想的多分次Betti数，并证明其总Betti数与权函数无关。
We describe an algorithm for finding sharp upper bounds for the total Betti numbers of a saturated ideal given certain constraints on its Hilbert function. This algorithm is implemented in the Macaulay2 package, MaxBettiNumbers, along with variations that produce ideals with maximal total Betti numbers.
Bresinsky defined a class of monomial curves in $\mathbb{A}^{4}$ with the property that the minimal number of generators or the first Betti number of the defining ideal is unbounded above. We prove that the same behaviour of unboundedness is true for all the Betti numbers and construct an explicit minimal free resolution for this class.
Comparability of the total Betti numbers of toric ideals of graphs
本文研究了简单图环面理想的总Betti数对图微小变化的高敏感性，探讨了使其可预测变化的组合操作，并重点关注一种保持这些不变量的过程。
We give conditions for determining the extremal behavior for the (graded) Betti numbers of squarefree monomial ideals. For the case of non-unique minima, we give several conditions which we use to produce infinite families, exponentially growing with dimension, of Hilbert functions which have no smallest (graded) Betti numbers among squarefree monomial ideals and all ideals. For the case of unique minima, we give two families of Hilbert functions, one with exponential and one with linear growth as dimension grows, that have unique minimal Betti numbers among squarefree monomial ideals.
Let K be a field and S a polynomial ring in a finite number of variables over K. Let F be a finitely generated graded free S-module. 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.
Let I be a monomial ideal of height c in a polynomial ring S over a field k. If I is not generated by a regular sequence, then we show that the sum of the betti numbers of S/I is at least 2^c + 2^{c-1} and characterize when equality holds. Lower bounds for the individual betti numbers are given as well.
In the present paper we prove that all homogeneous ideals with linear quotients are componentwise linear. Moreover we establish an extended version of Eliahou-Kervaire formula for graded Betti numbers.
On Frobenius Betti numbers of graded rings of finite Cohen-Macaulay type
针对环的Frobenius Betti数显式计算场景极少的问题，本文聚焦于有限Cohen-Macaulay型Cohen-Macaulay分次环，开展其Frobenius Betti数的显式计算研究，拓展该奇异性不变量的计算范围。
We study the Betti numbers of binomial edge ideal associated to some classes of graphs with large Castelnuovo-Mumford regularity. As an application we give several lower bounds of the Castelnuovo-Mumford regularity of arbitrary graphs depending on induced subgraphs.
We show that there exists a saturated graded ideal in a standard graded polynomial ring which has the largest total Betti numbers among all saturated graded ideals for a fixed Hilbert polynomial.
Let $K$ be a field and $S = K[x_1,\dots,x_n]$ be a polynomial ring over $K$. We discuss the behaviour of the extremal Betti numbers of the class of squarefree strongly stable ideals. More precisely, we give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of such a class of squarefree monomial ideals.
Betti and Bass numbers of relative Frobenius maps
该研究在局部环同态满足适当假设下，得到同调有限复形相对弗罗贝尼乌斯推前的Betti数与Bass数的增长界，将Betti数研究扩展到系数情形并通过Bass数提供对偶视角，助力环同态信息的探究。
We use purely combinatorial arguments to give a formula to compute all graded Betti numbers of path ideals of line graphs and cycles. As a consequence we can give new and short proofs for the known formulas of regularity and projective dimensions of path ideals of line graphs.
On the initial Betti numbers
本文研究了局部环中模块的初始和终端贝蒂数，特别关注极小数模块的贝蒂数计算及特定情况下的 $β_0(ω_{\frac{R}{I}})$ 值。
This paper aims to provide several relations between Bass and Betti numbers of a given module and its deficiency modules. Such relations and the tools used throughout allow us to generalize some results of Foxby, characterize Cohen-Macaulay modules in equidimensionality terms and furnish a case for the Auslander-Reiten conjecture.
Let $S=\mathbb{K}[x_1,\ldots,x_n]$ the polynomial ring over a field $\mathbb{K}$. In this paper for some families of monomial ideals $I \subset S$ we study the minimal number of generators of $I^k$. We use this results to find some other Betti numbers of these families of ideals for special choices of $n$, the number of variables.
This paper gives a sharp upper bound for the Betti numbers of a finitely generated multigraded $R$-module, where $R=\Bbbk [x_{1},...,x_{m}]$ is the polynomial ring over a field $\Bbbk$ in $m$ variables. The bound is given in terms of the rank and the first two Betti numbers of the module. An example is given which achieves these bounds simultaneously in each homological degree. Using Alexander duality, a bound is established for the total multigraded Bass numbers of a finite multigraded module in terms of the first two total multigraded Bass numbers.
We show that the minimal number of generators and the Cohen-Macaulay type of a family of numerical semigroups generated by concatenation of arithmetic sequences is unbounded.
We have observed a gap in one of the arguments in the main theorem. We choose to withdraw the paper until we rectify the gap.
We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones, and we show the extremal Betti numbers of some classes of Cohen-Macaulay binomial edge ideals: Cohen-Macaulay bipartite and fan graphs. In addition, we compute the Hilbert-Poincaré series of the binomial edge ideals of some Cohen-Macaulay bipartite graphs.
For any numerical semigroup $S$, there are infinitely many numerical symmetric semigroups $T$ such that $S=\frac{T}{2}$ is their half. We are studying the Betti numbers of the numerical semigroup ring $K[T]$ when $S$ is a 3-generated numerical semigroup or telescopic. We also consider 4-generated symmetric semigroups and the so called 4-irreducible numerical semigroups.
We study how Betti numbers of ideals in a local ring change under small perturbations. Given $p\in\mathbb N$ and given an ideal $I$ of a Noetherian local ring $(R,\mathfrak m)$, our main result states that there exists $N>0$ such that if $J$ is an ideal with $I\equiv J\bmod \mathfrak m^N$ and with the same Hilbert function as $I$, then the Betti numbers $β_i^R(R/I)$ and $β_i^R(R/J)$ coincide for $0\le i\le p$. Moreover, we present several cases in which an ideal $J$ such that $I \equiv J \bmod \mathfrak m^N$ is forced to have the same Hilbert function as $I$, and therefore the same Betti numbe…
We give conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The conjectures are verified in the cases where the structure of resolutions are known, i.e., for modules of codimension two, for Gorenstein algebras of codimension three and for complete intersections. The motivation for the conjectures comes from the Multiplicity conjecture of Herzog, Huneke and Srinivasan.
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