Alfio Ragusa, Giuseppe Zappalà
We investigate the standard generalized Gorenstein algebras of homological dimension three, giving a structure theorem for their resolutions. Moreover in many cases we are able to give a complete description of their graded Betti numbers.
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In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of $t$-minors of a generic pluri-circulant matrix is a transversal monomial ideal . Using a Gröbner basis for this ideal, it is shown that the initial ideal of a generic pluri-circulant matrix is a stable monomial ideal when the matrix has two square blocks. By means of the Eliahou-Kervair resolution, the Betti numbers of this ini…
We introduce the notion of Betti category for graded modules over suitably graded polynomial rings, and more generally for modules over certain small categories. Our categorical approach allows us to treat simultaneously many important cases, such as monomial ideals and toric rings. We prove that in these cases the Betti category is a finite combinatorial object that completely determines the structure of the minimal free resolution. For monomial ideals, the Betti category is the same as the Betti poset that we studied in a previous article. We describe in detail and with examples how the theo…
The graded Betti numbers are not tropical invariants
该研究证明分次Betti数不是齐次理想或赋值域上射影代数概形热带化的不变量，通过构造一组对应同一热带理想但Betti表不同的饱和理想完成论证。
We introduce the notion of a Betti-linear monomial ideal, which generalizes the notion of lattice-linear monomial ideal introduced by Clark. We provide a characterization of Betti-linearity in terms of Tchernev's poset construction. As an application we obtain an explicit canonical construction for the minimal free resolutions of monomial ideals having pure resolutions.
Let $G$ be a finitely generated abelian group, and let $S = A[x_1, ..., x_n]$ be a $G$-graded polynomial ring over a commutative ring $A$. Let $I_1, ..., I_s$ be $G$-homogeneous ideals in $S$, and let $M$ be a finitely generated $G$-graded $S$-module. We show that, when $A$ is Noetherian, the nonzero $G$-graded Betti numbers of $MI_1^{t_1} ... I_s^{t_s}$ exhibit an asymptotic linear behavior as the $t_i$s get large.
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 $R=k[x_1, ..., x_n]$ be a polynomial ring and let $I\subset R$ be a graded ideal. In \cite{R}, Römer asked whether under the Cohen-Macaulay assumption the $i$-th Betti number $β_{i}(R/I)$ can be bounded above by a function of the maximal shifts in the minimal graded free $R$-resolution of $R/I$ as well as bounded below by a function of the minimal shifts. The goal of this paper is to establish such bounds for graded Cohen-Macaulay algebras $k[x_1, ..., x_n]/I$ when $I$ is a standard determinantal ideal of arbitrary codimension. We also discuss other examples as well as when these bounds ar…
In this article, we compute Betti numbers for a Cohen-Macaulay tangent cone of a monomial curve in the affine $4$-space corresponding to a pseudo symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties.
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 consider the open problem of determining the graded Betti numbers for fat point subschemes supported at general points of the projective plane. We relate this problem to the open geometric problem of determining the splitting type of the pullback of the cotangent bundle on the plane to the normalization of certain rational plane curves. We give a conjecture for the graded Betti numbers which would determine them in all degrees but one for every fat point subscheme supported at general points of the plane. We also prove our Betti number conjecture in a broad range of cases. An appendix discu…
We prove a conjectured lower bound of Nagel and Reiner on Betti numbers of edge ideals of bipartite graphs.
We determine the codimension of the Betti strata of the family G(H) parametrizing graded Artinian quotients A of the polynomial ring R in two variables, having Hilbert function H. The Betti stratum G(B,H) parametrizes all such quotients having the graded Betti numbers determined by the relation degrees B. Our method is to identify G(B,H) as the product of determinantal varieties. We then use a result of M. Boij that reduces the calculation of codimension to showing that the most special stratum has the expected codimension. We also show that the closure of the Betti stratum is Cohen-Macaulay, …
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.
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 provide a simple method to compute the Betti numbers if the Stanley-Reisner ideal of a simplicial tree and its Alexander dual.
We characterize componentwise linear monomial ideals with minimal Taylor resolution and consider the lower bound for the Betti numbers of componentwise linear ideals.
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 state a conjecture on the stability of Betti diagrams of powers of monomial ideals.
Let X be a smooth projective variety with torsion-free Picard group. We introduce complexes of vector spaces whose homology determines the structure of the minimal free resolution of the Cox ring of X over the polynomial ring and show how the homology of these complexes can be studied by purely geometric methods. As an application of these techniques we give a simple new proof of a characterization of the Cox rings of Del Pezzo surfaces (of degree >1) conjectured by Batyrev and Popov.
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 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.
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.
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.
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。
Boij-Söderberg Theory views the Betti diagrams of graded modules over polynomial rings as vectors in a rational vector space, and studies the cone that these vectors generate (called a 'Betti Cone'). The objects of study in this paper are the Betti cones generated by edge ideals; in particular this paper will present and prove a formula for the dimensions of these cones, and for the subcones generated by edge ideals of specific heights.
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.
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.
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.
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 numerical experiments suggesting a very promising efficiency. On the way of describing the method, we also prove new constraints on the shape of the possible Betti tables of a monomial ideal.
Let $S=K[x_1,\dots,x_n]$ be a polynomial ring in $n$ variables with coefficients over a field $K$. A $t$-spread lexsegment ideal $I$ of $S$ is a monomial ideal generated by a $t$-spread lexsegment set. We determine all $t$-spread lexsegment ideals with linear resolution by means of Betti splittings. As applications we provide formulas for the Betti numbers of such a class of ideals and furthermore we characterize all incompletely $t$-spread lexsegment ideals with linear quotients.
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 study monomial cut ideals associated to a graph $G$, which are a monomial analogue of toric cut ideals as introduced by Sturmfels and Sullivant. Primary decompositions, projective dimensions, and Castelnuovo-Mumford regularities are investigated if the graph can be decomposed as $0$-clique sums and disjoint union of subgraphs. The total Betti numbers of a cycle are computed. Moreover, we classify all Freiman ideals among monomial cut ideals.
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.
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.
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…
We extend a result of Caviglia and Sbarra to a polynomial ring with base field of any characteristic. Given a homogeneous ideal containing both a piecewise lex ideal and an ideal generated by powers of the variables, we find a lex ideal with the following property: the ideal in the polynomial ring generated by the piecewise lex ideal, the ideal of powers, and the lex ideal has the same Hilbert function and Betti numbers at least as large as those of the original ideal.
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.
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.
Let $S={\sf k}[X_1,\dots, X_n]$ be a polynomial ring, where ${\sf k}$ is a field. This article deals with the defining ideal of the Rees algebra of squarefree monomial ideal generated in degree $n-2$. As a consequence, we prove that Betti numbers of powers of the cover ideal of the complement graph of a tree do not depend on the choice of tree. Further, we study the regularity and Betti numbers of powers of cover ideals associated to certain graphs.
For an ideal $I$ of a Noetherian local ring $(R,\fm,k)$ we show that $\bt_1^R(I)-\bt_0^R(I)\geq -1$. It is demonstrated that some residual intersections of an ideal $I$ for which $\bt_1^R(I)-\bt_0^R(I)= -1\;\text{or}\;0$ are perfect. Some relations between Betti numbers and Bass numbers of the canonical module are studied.
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
Restrictions on the Betti tables of licci ideals
该研究针对licci理想的Betti表限制提出相关猜想，聚焦齐次licci理想生成元数量与分次自由分解最大末移的问题，并在大量情形下完成猜想证明。
Let $G$ be a finite simple graph on $n$ vertices. Let $J_G \subset K[x_1, \ldots, x_n]$ be the cover ideal of $G$. In this article, we obtain syzygies, Betti numbers and Castelnuovo-Mumford regularity of $J_G^s$ for all $s \geq 1$ for certain classes of graphs $G$.
A monomial ideal $I$ admits a Betti splitting $I=J+K$ if the Betti numbers of $I$ can be determined in terms of the Betti numbers of the ideals $J,K$ and $J \cap K$. Given a monomial ideal $I$, we prove that $I=J+K$ is a Betti splitting of $I$, provided $J$ and $K$ are componentwise linear, generalizing a result of Francisco, Hà and Van Tuyl. If $I$ has a linear resolution, the converse also holds. We apply this result recursively to the Alexander dual of vertex-decomposable, shellable and constructible simplicial complexes and to determine the graded Betti numbers of the defining ideal of thr…
Let I be a homogeneous ideal in a polynomial ring P over a field. By Macaulay's Theorem, there exists a lexicographic ideal L=Lex(I) with the same Hilbert function as I. Peeva has proved that the Betti numbers of P/I can be obtained from the graded Betti numbers of P/L by a suitable sequence of consecutive cancellations. We extend this result to any ideal I in a regular local ring (R,m) by passing through the associated graded ring. To this purpose it will be necessary to enlarge the list of the allowed cancellations. Taking advantage of Eliahou-Kervaire's construction, several applications ar…
An ideal $I \subset \mathbb{k}[x_1, \ldots, x_n]$ is said to have linear powers if $I^k$ has a linear minimal free resolution, for all $k$. In this paper we study the Betti numbers of $I^k$, for ideals $I$ with linear powers. The Betti numbers are computed explicitly, as polynomials in $k$, for the ideal generated by all square free monomials of degree $d$, for $d=2, 3$ or $n-1$, and the product of all ideals generated by $s$ variables, for $s=n-1$ or $n-2$. We also study the generators of the Rees ideal, for ideals with linear powers. Especially, we are interested in ideals for which the Rees…
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.
Several authors investigating the asymptotic behaviour of the Betti diagrams of the graded system obtained by taking powers of an ideal have shown that the shape of the nonzero entries in the diagrams stabilizes when $I$ is a homogeneous ideal with generators of the same degree. In this paper, we study the Betti diagrams of graded systems of ideals built by taking the initial ideals or generic initial ideals of powers, and discuss the stabilization of additional collections of Betti diagrams. Our main result shows that when I has generators of the same degree, the entries in the Betti diagrams…
Given an homogeneous monomial ideal $I$, we provide a question- and example-based investigation of the stabilization patterns of the Betti tables shapes of $I^d$ as we vary $d$. We build off Whieldon's definition of the stabilization index of $I$, Stab$(I)$, to define the stabilization sequence of $I$, StabSeq$(I)$, and use it to explore changes in the shapes of the Betti tables of $I^d$ as we vary $d$. We also present the stabilization indices and sequences of the collection of ideals $\{I_{n}\}$ where $I_{n}=(a^{2n}b^{2n}c^{2n},b^{4n}c^{2n},a^{3n}c^{3n},a^{6n-1}b)\subseteq \Bbbk[a,b,c]$.
We give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of any homogeneous ideal in a polynomial ring over a field.
We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certain classes of ideals that generalize (in two different directions) the edge ideals of threshold graphs and Ferrers graphs. In the process, we produce particularly simple cellular linear resolutions for strongly stable and squ…
Partial Betti splittings with applications to binomial edge ideals
本文提出齐次理想的偏 Betti 分裂概念，并将其用于二项式边理想，推广割边相关结果、给出通用分裂描述且算出树情形的总第二 Betti 数。
Let $S_n$ be a polynomial ring with $n$ variables over a field and $\{I_n\}_{n \geq 1}$ a chain of ideals such that each $I_n$ is a monomial ideal of $S_n$ fixed by permutations of the variables. In this paper, we present a way to determine all nonzero positions of Betti tables of $I_n$ for all large intergers $n$ from the $\mathbb Z^m$-graded Betti table of $I_m$ for some integer $m$. Our main result shows that the projective dimension and the regularity of $I_n$ eventually become linear functions on $n$, confirming a special case of conjectures posed by Le, Nagel, Nguyen and Römer.
Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. Our main results state that if the polynomial ring is equipped with a positive $\ZZ$-grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials. More precisely, in the case of $\ZZ$-grading, $\ZZ^2$ can be splitted into a finite number of regions such that each region corresponds to a polynomial that depending to the degree $(μ, t)$, $\dim_k \left(\tor_i^S(I^t, k)_μ \right)$ is equal …
In this short note we introduce a notion of extremality for Betti numbers of a minimal free resolution, which can be seen as a refinement of the notion of Mumford-Castelnuovo regularity. We show that extremal Betti numbers of an arbitrary submodule of a free S-module are preserved when taking the generic initial module. We relate extremal multigraded Betti numbers in the minimal resolution of a square free monomial ideal with those of the monomial ideal corresponding to the Alexander dual simplicial complex and generalize theorems of Eagon-Reiner and Terai. As an application we give easy (alte…
Betti Tables and the Licci Property
研究理想的 licci 性质与贝蒂表的关系，通过实例表明含变量立方的理想，其 licci 性质不能仅由贝蒂表确定；而含变量平方的理想，其 licci 性质由贝蒂表前两列完全确定。
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…
Let $K$ be a field and let $S=K[x_1,\dots,x_n]$ be a standard polynomial ring over a field $K$. We characterize the extremal Betti numbers, values as well positions, of a $t$-spread strongly stable ideal of $S$. Our approach is constructive. Indeed, given some positive integers $a_1,\dots,a_r$ and some pairs of positive integers $(k_1,\ell_1),\dots,(k_r,\ell_r)$, we are able to determine under which conditions there exist a $t$-spread strongly stable ideal $I$ of $S$ with $β_{k_i, k_i\ell_i}(I)=a_i$, $i=1, \ldots, r$, as extremal Betti numbers, and then to construct it.
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.
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