Stanley proved that, in characteristic zero, all artinian monomial complete intersections have the strong Lefschetz property. We provide a positive characteristic complement to Stanley's result in the case of artinian monomial complete intersections generated by monomials all of the same degree, and also for arbitrary artinian monomial complete intersections in characteristic two.
To establish these results, we first prove an a priori lower bound on the characteristics that guarantee the Lefschetz properties. We then use a variety of techniques to complete the classifications.
In this paper we classify the monomial complete intersection algebras, in two variables, and of positive characteristic, which has the strong Lef- schetz property. Together with known results, this gives a complete classi- fication of the monomial complete intersections with the strong Lefschetz property.
We introduce a general technique for decomposing monomial algebras which we use to study the Lefschetz properties. We apply our technique to various classes of algebras, including monomial almost complete intersections and Gorenstein algebras. In particular, we prove that Gorenstein codimension three algebras arising from numerical semigroups have the strong Lefschetz property. We also study the reverse of the splitting operation -- a gluing operation -- which gives a way to construct monomial algebras with the Lefschetz properties.
We study the WLP and SLP of artinian monomial ideals in $S=\mathbb{K}[x_1,\dots ,x_n]$ via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of $S/I$ is linear for at least $n-2$ steps. We give an affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial ideals with almost linear resolutions.
In [MMR], Michałek--Miró-Roig give a beautiful geometric characterization of Artinian quotients by ideals generated by quadratic or cubic monomials, such that the multiplication map by a general linear form fails to be injective in the first nontrivial degree. Their work was motivated by conjectures of Ilardi and Mezzetti-Miró-Roig-Ottaviani, connecting the failure to Laplace equations and classical results of Togliatti on osculating planes. We study quotients by quadratic monomial ideals, explaining failure of the Weak Lefschetz Property for some cases not covered by [MMR].
We determine a sharp lower bound for the Hilbert function in degree $d$ of a monomial algebra failing the weak Lefschetz property over a polynomial ring with $n$ variables and generated in degree $d$, for any $d\geq 2$ and $n\geq 3$. We consider artinian ideals in the polynomial ring with $n$ variables generated by homogeneous polynomials of degree $d$ invariant under an action of the cyclic group $\mathbb{Z}/d\mathbb{Z}$, for any $n\geq 3$ and any $d\geq 2$. We give a complete classification of such ideals in terms of the weak Lefschetz property depending on the action.
In this paper we study the Lefschetz properties of monomial complete intersections in positive characteristic. We give a complete classification of the strong Lefschetz property when the number of variables is at least three, which proves a conjecture by Cook II. We also extend earlier results on the weak Lefschetz property by dropping the assumption on the residue field being infinite, and by giving new sufficient criteria.
Every artinian quotient of $K[x,y]$ has the strong Lefschetz property if $K$ is a field of characteristic zero or is an infinite field whose characteristic is greater than the regularity of the quotient. We improve this bound in the case of monomial ideals. Using this we classify when both bounds are sharp. Moreover, we prove that the artinian quotient of a monomial ideal in $K[x,y]$ always has the strong Lefschetz property, regardless of the characteristic of the field, exactly when the ideal is lexsegment. As a consequence we describe a family of non-monomial complete intersections that alwa…
Many algebras are expected to have the Weak Lefschetz property though this is often very difficult to establish. We illustrate the subtlety of the problem by studying monomial and some closely related ideals. Our results exemplify the intriguing dependence of the property on the characteristic of the ground field, and on arithmetic properties of the exponent vectors of the monomials.
We study the weak Lefschetz property and the Hilbert function of level Artinian monomial almost complete intersections in three variables. Several such families are shown to have the weak Lefschetz property if the characteristic of the base field is zero or greater than the maximal degree of any minimal generator of the ideal. Two of the families have an interesting relation to tilings of hexagons by lozenges. This lends further evidence to a conjecture by Migliore, Miro-Roig, and the second author. Finally, using our results about the weak Lefschetz property, we show that the Hilbert function…
The weak and strong Lefschetz properties are two basic properties that Artinian algebras may have. Both Lefschetz properties may vary under small perturbations or changes of the characteristic. We study these subtleties by proposing a systematic way of deforming a monomial ideal failing the weak Lefschetz property to an ideal with the same Hilbert function and the weak Lefschetz property. In particular, we lift a family of Artinian monomial ideals to finite level sets of points in projective space with the property that a general hyperplane section has the weak Lefschetz property in almost all…
We prove the strong Lefschetz property for certain complete intersections defined by products of linear forms, using a characterization of the strong Lefschetz property in terms of central simple modules.
In this paper, we study the strong Lefschetz property of artinian complete intersection ideals generated by products of linear forms. We prove the strong Lefschetz property for a class of such ideals with binomial generators.
In this work a combinatorial approach towards the weak Lefschetz property is developed that relates this property to enumerations of signed perfect matchings as well as to enumerations of signed families of non-intersecting lattice paths in certain triangular regions. This connection is used to study Artinian quotients by monomial ideals of a three-dimensional polynomial ring. Extending a main result in the recent memoir [Boij, Migliore, Miró-Roig, Nagel, Zanello], we completely classify the quotients of type two that have the weak Lefschetz property in characteristic zero. We also derive resu…
We compute the generic initial ideal of a complete intersection of embedding dimension three with strong Lefschetz property and we show that it is an almost reverse lexicographic ideal. This enable us to give a proof for Moreno's conjecture in the case $n=3$.
In this paper, we study the dependence of the weak Lefschetz property of algebras defined by a special class of monomials ideals in a polynomial ring with coefficient in a field, to the characteristic of the base field.
We use Macaulay's inverse system to study the Hilbert series for almost complete intersections generated by uniform powers of general linear forms. This allows us to give a classification of the Weak Lefschetz property for these algebras, settling a conjecture by Migliore, Miró-Roig, and Nagel.
Let K be an algebraically closed field of characteristic p > 0. We apply a theorem of C. Han to give an explicit description for the weak Lefschetz property of the monomial Artinian complete intersection A = K[X,Y,Z]/(X^d,Y^d,Z^d) in terms of d and p. This answers a question of J. Migliore, R. M. Miro-Roig and U. Nagel and, equivalently, characterizes for which characteristics the rank-2 syzygy bundle Syz(X^d,Y^d,Z^d) on PP^2 satisfies the Grauert-Muelich theorem. As a corollary we obtain that for p=2 the algebra A has the weak Lefschetz property if and only if d=(2^t+1)/3 or d=(2^t-1)/3 for s…
Deciding the presence of the weak Lefschetz property often is a challenging problem. In this work an in-depth study is carried out in the case of Artinian monomial ideals with four generators in three variables. We use a connection to lozenge tilings to describe semistability of the syzygy bundle of such an ideal, to determine its generic splitting type, and to decide the presence of the weak Lefschetz property. We provide results in both characteristic zero and positive characteristic.
Given an ideal $I=(f_1,\ldots,f_r)$ in $\mathbb C[x_1,\ldots,x_n]$ generated by forms of degree $d$, and an integer $k>1$, how large can the ideal $I^k$ be, i.e., how small can the Hilbert function of $\mathbb C[x_1,\ldots,x_n]/I^k$ be? If $r\le n$ the smallest Hilbert function is achieved by any complete intersection, but for $r>n$, the question is in general very hard to answer. We study the problem for $r=n+1$, where the result is known for $k=1$. We also study a closely related problem, the Weak Lefschetz property, for $S/I^k$, where $I$ is the ideal generated by the $d$'th powers of the v…
We compute the Stanley depth of irreducible monomial ideals and we show that the Stanley depth of a monomial complete intersection ideal is the same as the Stanley depth of it's radical. Also, we give some bounds for the Stanley depth of a monomial complete intersection ideal.
We characterize the monomial complete intersections in three variables satisfying the Weak Lefschetz Property (WLP), as a function of the characteristic of the base field. Our result presents a surprising, and still combinatorially obscure, connection with the enumeration of plane partitions. It turns out that the rational primes p dividing the number, M(a,b,c), of plane partitions contained inside an arbitrary box of given sides a,b,c are precisely those for which a suitable monomial complete intersection (explicitly constructed as a bijective function of a,b,c) fails to have the WLP in chara…
In 2012, Migliore, the first author, and Nagel conjectured that, for all $n\geq 4$, the artinian ideal $I=(L_0^d,\ldots,L_{2n+1}^d) \subset R=k[x_0,\ldots,x_{2n}]$ generated by the $d$-th powers of $2n+2$ general linear forms fails to have the weak Lefschetz property if and only if $d>1$. This paper is entirely devoted to prove partially this conjecture. More precisely, we prove that $R/I$ fails to have the weak Lefschetz property, provided $4\leq n\leq 8,\ d\geq 4$ or $d=2r,\ 1\leq r\leq 8,\ 4\leq n\leq 2r(r+2)-1$.
Stanley showed that monomial complete intersections have the strong Lefschetz property. Extending this result we show that a simple extension of an Artinian Gorenstein algebra with the strong Lefschetz property has again the strong Lefschetz property.
We show that the Stanley's Conjecture holds for an intersection of four monomial prime ideals of a polynomial algebra $S$ over a field and for an arbitrary intersection of monomial prime ideals $(P_i)_{i\in [s]}$ of $S$ such that $P_i\not\subset Σ_{1=j\not =i}^s P_j$ for all $i\in [s]$.
The properties of the intersection algebra of two principal monomial ideals in a polynomial ring are investigated in detail. Results are obtained regarding the Hilbert series and the canonical ideal of the intersection algebra using methods from the theory of diophantine linear equations with integer coefficients.
In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen-Macaulay.
Let $I$ be an $m$-generated complete intersection monomial ideal in $S=K[x_1,...,x_n]$. We show that the Stanley depth of $I$ is $n-\floor{\frac{m}{2}}$. We also study the upper-discrete structure for monomial ideals and prove that if $I$ is a squarefree monomial ideal minimally generated by 3 elements, then the Stanley depth of $I$ is $n-1$.
We determine the Hochschild cohomology algebras of the square-free monomial complete intersections. In particular, we provide a formula for the cup product which gives the cohomology module an algebra structure and then we describe this structure in terms of generators and relations. In addition, we compute the Hilbert series of the Hochschild cohomology of these algebras.
We prove that any quadratic complete intersection with certain action of the symmetric group has the strong Lefschetz property over a field of characteristic zero. As a consequence of it we construct a new class of homogeneous complete intersections with generators of higher degrees which have the strong Lefschetz property.
We classify all unmixed monomial ideals I of codimension 2 which are generically a complete intersection and which have the property that the symbolic power algebra A(I) is standard graded. We give a lower bound for the highest degree of a generator of A(I) in the case that I is a modification of the vertex cover ideal of a bipartite graph, and show that this highest degree can be any given number. We furthermore give an upper bound for the highest degree of a generator of the integral closure of A(I) in the case that I is a monomial ideal which is generically a complete intersection.
A standard graded artinian monomial complete intersection algebra $A=\Bbbk[x_1,x_2,\ldots,x_n]/(x_1^{a_1},x_2^{a_2},\ldots,x_n^{a_n})$, with $\Bbbk$ a field of characteristic zero, has the strong Lefschetz property due to Stanley in 1980. In this paper, we give a new proof for this result by using only the basic properties of linear algebra. Furthermore, our proof is still true in the case where the characteristic of $\Bbbk$ is greater than the socle degree of $A$, namely $a_1+a_2+\cdots+a_n - n$.
Lefschetz properties for monomial complete intersections
Busitema University（布西泰马大学）；Università degli Studi di Genova（热那亚大学）；CIMAT - Centro de Investigación en Matemáticas（墨西哥数学研究中心）；University of Notre Dame（圣母大学）；Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西蒙·斯托伊洛数学研究所）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
We continue the study of intersection algebras $\mathcal B = \mathcal B_R(I, J)$ of two ideals $I, J$ in a commutative Noetherian ring $R$. In particular, we exploit the semigroup ring and toric structures in order to calculate various invariants of the intersection algebra when $R$ is a polynomial ring over a field and $I,J$ are principal monomial ideals. Specifically, we calculate the $F$-signature, divisor class group, and Hilbert-Samuel and Hilbert-Kunz multiplicities, sometimes restricting to certain cases in order to obtain explicit formulæ. This provides a new class of rings where formu…
Let $A=\pmb k[x_1,...,x_n]/{(x_1^d,...,x_n^d)}$, where $\pmb k$ is an infinite field. If $\pmb k$ has characteristic zero, then Stanley proved that $A$ has the Weak Lefschetz Property (WLP). Henceforth, $\pmb k$ has positive characteristic $p$. If $n=3$, then Brenner and Kaid have identified all $d$, as a function of $p$, for which $A$ has the WLP. In the present paper, the analogous project is carried out for $4\le n$. If $4\le n$ and $p=2$, then $A$ has the WLP if and only if $d=1$. If $n=4$ and $p$ is odd, then we prove that $A$ has the WLP if and only if $d=kq+r$ for integers $k,q,d$ with …
We introduce a natural correspondence between artinian monomial almost complete intersections in three variables and punctured hexagonal regions. We use this correspondence to investigate the algebras for the presence of the weak Lefschetz property. In particular, we relate the field characteristics in which such an algebra fails to have the weak Lefschetz property to the prime divisors of the enumeration of signed lozenge tilings of the associated punctured hexagonal region. On the one side this allows us to establish the weak Lefschetz property in many new cases. For numerous classes of punc…
Let $R = K[x_1, x_2, x_3, x_4]$ be the polynomial ring over a field of characteristic zero. For the ideal $(x_1^a, x_2^b, x_3^c, x_4^d) \subset R$, where at least one of $a$, $b$, $c$ and $d$ is equal to two, we prove that its generic initial ideal with respect to the reverse lexicographic order is the almost revlex ideal corresponding to the same Hilbert function.
In this article we give explicit descriptions of the multiplicities of some classes of monomial ideals. For instance, we give a formula for the multiplicities of all codimension 1 monomial ideals, and another formula for the multiplicities of almost complete intersections. We also introduce a new class of ideals that extends the family of monomial complete intersections, and give a formula for their multiplicity, as well as a visual interpretation of this invariant.
We generalize some properties related to Hilbert series and Lefschetz properties of Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal $J$ of dimension one. When the saturation $I$ of $J$ is a complete intersection, we get explicit formulas for a number of related invariants. New examples of hypersurfaces $V:f=0$ in $P^n$ whose Jacobian ideal $J_f$ satisfies this property and with explicit nontrivial Alexander polynomials are given in any dimension. A Lefschetz type property for the graded quotient $I/J$ is …
We study the weak Lefschetz property of artinian Gorenstein algebras and in particular of artinian complete intersections. In codimension four and higher, it is an open problem whether all complete intersections have the weak Lefschetz property. For a given artinian Gorenstein algebra $A$ we ask what linear forms are Lefschetz elements for this particular algebra, i.e., which linear forms $\ell$ give maximal rank for all the multiplication maps $\times \ell: [A]_i \longrightarrow [A]_{i+1}$. This is a Zariski open set and its complement is the \emph{non-Lefschetz locus}. For monomial complete …
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.
