We introduce the $k$-strong Lefschetz property ($k$-SLP) and the $k$-weak Lefschetz property ($k$-WLP) for graded Artinian $K$-algebras, which are generalizations of the Lefschetz properties. The main results obtained in this paper are as follows:
1. Let $I$ be a graded ideal of $R=K[x_1, x_2, x_3]$ whose quotient ring $R/I$ has the SLP. Then the generic initial ideal of $I$ is the unique almost revlex ideal with the same Hilbert function as $R/I$.
2. Let $I$ be a graded ideal of $R=K[x_1, x_2, ..., x_n]$ whose quotient ring $R/I$ has the $n$-SLP. Suppose that all $k$-th differences of the Hilbert function of $R/I$ are quasi-symmetric. Then the generic initial ideal of $I$ is the unique almost revlex ideal with the same Hilbert function as $R/I$.
3. We give a sharp upper bound on the graded Betti numbers of Artinian $K$-algebras with the $k$-WLP and a fixed Hilbert function.
The authors T.Harima, J.C.Migliore, U.Nagel and J.Watanabe characterized the Hilbert function of algbebras with the Lefschetz property. We extend this characterization to algebras with the Lefschetz property m times. We also give upper bounds for the Betti numbers of Artinian algebras with a given Hilbert function and with the Lefschetz property m times and describe the cases in which these bounds are reached.
We give an alternate proof for a theorem of Migliore and Nagel. In particular, we show that if $\mathcal H$ is an SI-sequence, then the collection of Betti diagrams for all Artinian Gorenstein $k$-algebras with the weak Lefschetz property and Hilbert function $\mathcal H$ has a unique largest element.
For a standard Artinian $k$-algebra $A=R/I$, we give equivalent conditions for $A$ to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of the generic initial ideal $\mathrm{gin}(I)$ of $I$ under the reverse lexicographic order. Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of $\mathrm{gin}(I)$ are determined just by the the Hilbert function of $I$ if $A$ has the weak Lefschetz property. Furthermore, for the case that $A$ is a standard Artinian $k$-algebra of codimens…
In this paper we study the Weak Lefschetz property of two classes of standard graded Artinian Gorenstein algebras associated in a natural way to the Apéry set of numerical semigroups. To this aim we also prove a general result about the transfer of Weak Lefschetz property from an Artinian Gorenstein algebra to its quotients modulo a colon ideal.
It is known that graded cyclic modules over $S=K[x,y]$ have the Weak Lefschetz Property (WLP). This is not true for non-cyclic modules over $S$. The purpose of this note is to study which conditions on $S$-modules ensure the WLP. We give an algorithm to test the WLP for graded modules with fixed Hilbert function. In particular, we prove that indecomposable graded modules over $S$ with the Hilbert function $(h_0,h_1)$ have the WLP.
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.
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 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.
We deal with the Weak Lefschetz property (WLP) for Artinian standard graded Gorenstein algebras of codimension $3.$ We prove that many Gorenstein sequences force the WLP for such algebras. Moreover for every Gorenstein sequence $H$ of codimension 3 we found several Gorenstein Betti sequences compatible with $H$ which again force the WLP. Finally we show that for every Gorenstein Betti sequence the general Artinian standard graded Gorenstein algebra with such Betti sequence has the WLP.
We study $h$-vectors and graded Betti numbers of level modules up to multiplication by a rational number. Assuming a conjecture on the possible graded Betti numbers of Cohen-Macaulay modules we get a description of the possible $h$-vectors of level modules up to multiplication by a rational number. We also determine, again up to multiplication by a rational number, the cancellable $h$-vectors and the $h$-vectors of level modules with the weak Lefschetz property. Furthermore, we prove that level modules of codimension three satisfy the upper bound of the Multiplicity conjecture of Herzog, Hunek…
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.
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.
The basic sequence of a homogeneous ideal $I\sset R=k[\seq{x}{1}{r}]$ defining an Artinian graded ring $A=R/I$ not having the weak Lefschetz property has the property that the first term of the last part is less than the last term of the penultimate part. For a general linear form $\ell$ in $\seq{x}{1}{r}$, this fact affects in a certain way the behavior of the $r-1$ square matrices in $k[\ell]$ which represent the multiplications of the elements of $A$ by $\seq{x}{1}{r-1}$ through a minimal free presentation of $A$ over $k[\ell]$. Taking advantage of it, we consider some modules over an algeb…
We define the strong Lefschetz property for finite graded modules over graded Artinian algebras whose grading is not necessarily standard. We show that most results which have been obtained for Artinian algebras with standard grading can be extended for non-standard grading. Our results on the strong Lefschetz property for non-standard grading can be used to prove that certain Artinian complete intersections with standard grading have the strong Lefschetz property.
We give a necessary and sufficient condition for a standard graded Artinian ring defined by an m-full ideal, to have the weak Lefschetz property in terms of graded Betti numbers. This is a generalization of a theorem of Wiebe for componentwise linear ideals. We also prove that the class of componentwise linear ideals and that of completely m-full ideals coincide in characteristic zero and in positive characteristic, with the assumption that the generic initial ideal w.r.t. the graded reverse lexicographic order is stable.
Let S=K[X_1,...,X_n] be the polynomial ring over a field K. For bounded below Z^n-graded S-modules M and N we show that if Tor^S_p(M,N) is nonzero, then for every i between 0 and p, the dimension of the K-vector space Tor^S_i(M,N) is at least as big as the binomial coefficient (p,i). In particular, we get lower bounds for the total Betti numbers. These results are related to a conjecture of Buchsbaum and Eisenbud.
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 paper we prove parts of a conjecture of Herzog giving lower bounds on the rank of the free modules appearing in the linear strand of a graded $k$-th syzygy module over the polynomial ring. If in addition the module is $\mathbb{Z}^n$-graded we show that the conjecture holds in full generality. Furthermore, we give lower and upper bounds for the graded Betti numbers of graded ideals with a linear resolution and a fixed number of generators.
We determine the class of Hilbert series H so that if M is a finitely generated zero-dimensional R-graded module with the strong Lefschetz property, then the tensor product of M and k[y]/(y^m) has the strong Lefschetz property for y an indeterminate and all positive integers m if and only if the Hilbert series of M is in H. Given two finite graded R-modules M and N with the strong Lefschetz property, we determine sufficient conditions in order that the tensor product of M and N has the strong Lefschetz property.
We give a complete description of the cone of Betti diagrams over a standard graded hypersurface ring of the form k[x,y]/<q>, where q is a homogeneous quadric. We also provide a finite algorithm for decomposing Betti diagrams, including diagrams of infinite projective dimension, into pure diagrams. Boij--Soederberg theory completely describes the cone of Betti diagrams over a standard graded polynomial ring; our result provides the first example of another graded ring for which the cone of Betti diagrams is entirely understood.
For standard graded Artinian $K$-algebras defined by componentwise linear ideals and Gotzmann ideals, we give conditions for the weak Lefschetz property in terms of numerical invariants of the defining ideals.
An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of Artinian Gorenstein algebras with the Weak Lefschetz Property, a property shared by most Artinian Gorenstein algebras. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration $G$ of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the Weak Lefschetz Property. Furthermore, we s…
The main goal of this paper is to size up the minimal graded free resolution of a homogeneous ideal in terms of its generating degrees. By and large, this is too ambitious an objective. As understood, sizing up means looking closely at the two available parameters: the shifts and the Betti numbers. Since, in general, bounds for the shifts can behave quite steeply, we filter the difficulty by the subadditivity of the syzygies. The method we applied is hopefully new and sheds additional light on the structure of the minimal free resolution. We use the Boij-Söderberg techniques for the Betti numb…
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.
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.
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 A be a standard graded Artinian algebra over a field of characteristic zero and let z be a linear form in A. We define the central simple modules for each such pair (A, z). Assume that A is Gorenstein. Then we prove that A has the strong Lefschetz property if and only if there exists a linear form z in A such that all central simple modules of the pair (A,z) have the strong Lefschetz property. In the course of proof we need to extend the definition of the strong Lefschetz property to finite graded modules over graded Artinian algebra, which previously was defined only for standard graded A…
We use the results by Eisenbud and Schreyer to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination Betti diagrams of modules with a pure resolution. This implies the Multiplicity Conjecture of Herzog, Huneke and Srinivasan for modules that are not necessarily Cohen-Macaulay. We give a combinatorial proof of the convexity of the simplicial fan spanned by the pure diagrams.
We provide a structure theorem for all almost complete intersection ideals of depth three in any Noetherian local ring. In particular, we find that the minimal generators are the pfaffians of suitable submatrices of an alternating matrix. From the graded version of the previous result, we characterize the graded Betti numbers of all 3-codimensional almost complete intersection schemes of P^r.
We study inequalities between graded Betti numbers of ideals in a standard graded algebra over a field and their images under embedding maps, defined earlier by us in [Math. Z. 274, (2013), no. 3-4, pp. 809-819; arXiv:1009.4488]. We show that if graded Betti numbers do not decrease when we replace ideals in an algebra by their embedded versions, then the same behaviour is carried over to ring extensions. As a corollary we give alternative inductive proofs of earlier results of Bigatti, Hulett, Pardue, Mermin-Peeva-Stillman and Murai. We extend a hypersurface restriction theorem of Herzog-Popes…
Let $Q\in \K[x_1,...,x_n] = S$ be a homogeneous polynomial of degree $d$. The freeness of the logarithmic derivation module, $D(Q)$, and of its natural generalizations, has been widely studied. In the free case, $D(Q) \simeq \bigoplus_{i=1}^n S(-d_i)$ where the $d_i$'s are the exponents of the module; and as a direct consequence of the Saito-Ziegler criterion, the formula $d = \sum_i d_i$ holds. In this paper we give a generalization of this formula in the non-free case. Moreover, we show that an equivalent formula is also true in the quasi-homogeneous case, and show to what extent it can be g…
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.
Graded Betti numbers of graded Möbius algebras of uniform matroids
New Mexico State University（新墨西哥州立大学）；Iowa State University（爱荷华州立大学）；Bryn Mawr College（布林莫尔学院）；University of Nebraska-Lincoln（内布拉斯加大学林肯分校）；Marshall Univeristy（马歇尔大学）；Western University（韦仕敦大学）
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…
A connected sum construction for local rings was introduced in a paper by H. Ananthnarayan, L. Avramov, and W.F. Moore. In the graded Artinian Gorenstein case, this can be viewed as an algebraic analogue of the topological construction of the same name. We give two alternative description of this algebraic connected sum: the first uses algebraic analogues of Thom classes of vector bundles and Gysin homomorphisms, the second is in terms of Macaulay dual generators. We also investigate the extent to which the connected sum construction preserves the weak or strong Lefschetz property, thus provid…
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 $I\subseteq S$ is an 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 of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they on…
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.
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…
We first describe a situation in which every graded Betti number in the tail of the resolution of $\frac RJ$ may be read from the socle degrees of $\frac RJ$. Then we apply the above result to the ideals $J$ and $J^{[q]}$; and thereby describe a situation in which the graded Betti numbers in the tail of the resolution of $R/J^{[q]}$ are equal to the graded Betti numbers in the tail of a shift of the resolution of $R/J$.
In this paper, we give the necessary and sufficient conditions for the Cohen-Macaulayness of the associated graded ring of a simplicial affine semigroups using Gröbner basis. We generalize the concept of homogeneous numerical semigroup for the simplicial affine semigroup and show that the Betti numbers of the corresponding semigroup ring matches with the Betti numbers of the associated graded ring. We also define the nice extension for simplicial affine semigroups, motivated by the notion of a nice extension of the numerical semigroups.
Let $Γ$ be a rooted tree and let $t$ be a positive integer. We study algebraic invariants and properties of the path ideal generated by monomial corresponding to paths of length $(t-1)$ in $Γ$. In particular, we give a recursive formula to compute the graded Betti numbers, a general bound for the regularity, an explicit computation of the linear strand, and we characterize when this path ideal has a linear resolution.
Let $S = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $°x_i = 1$. Let $Δ$ be a simplicial complex on $[n] = \{1, ..., n \}$ and $I_Δ\subset S$ its Stanley--Reisner ideal. We write $Δ^e$ for the exterior algebraic shifted complex of $Δ$ and $Δ^c$ for a combinatorial shifted complex of $Δ$. Let $β_{ii+j}(I_Δ) = \dim_K \Tor_i(K, I_Δ)_{i+j}$ denote the graded Betti numbers of $I_Δ$. In the present paper it will be proved that (i) $β_{ii+j}(I_{Δ^e}) \leq β_{ii+j}(I_{Δ^c})$ for all $i$ and $j$, where the base field is infinite, and (ii) $β_{ii+j}(I_Δ) \leq…
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 study the weak Lefschetz property of a class of graded Artinian Gorenstein algebras of codimension three associated in a natural way to the Apéry set of a numerical semigroup generated by four natural numbers. We show that these algebras have the weak Lefschetz property whenever the initial degree of their defining ideal is small.
We find a sufficient condition that $\H$ is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function $\H=(h_0,h_1,..., h_{d-1}>h_d=h_{d+1})$ cannot be level if $h_d\le 2d+3$, and that there exists a level O-sequence of codimension 3 of type $\H$ for $h_d \ge 2d+k$ for $k\ge 4$. Furthermore, we show that $\H$ is not level if $β_{1,d+2}(I^{\rm lex})=β_{2,d+2}(I^{\rm lex})$, and also prove that any codimension 3 Artinian graded algebra $A=R/I$ cannot be level if $β_{1,d+2}(\Gin(I))=β_{2,d+2}(\Gin(I))$. In this case, the H…
Graded Betti numbers of the Jacobian algebra and total Tjurina numbers of plane curves
Université Côte d’Azur（蔚蓝海岸大学）；CNRS（法国国家科学研究中心）；Simion Stoilow Institute of Mathematics（西奥多·斯托伊洛数学研究所）；Faculty of Mathematics and Informatics, Ovidius University（奥维迪乌斯大学数学与信息学院）
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.
