Related papers: Enumerations deciding the weak Lefschetz property
A graded Artinian algebra $A$ has the Weak Lefschetz Property if there exists a linear form $\ell$ such that the multiplication map by $\ell:[A]_i\to [A]_{i+1}$ has maximum rank in every degree. The linear forms satisfying this property…
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.
Given a base field $\Bbbk$ of characteristic zero, for each graph $G$, we associate the artinian algebra $A(G)$ defined by the edge ideal of $G$ and the squares of the variables. We study the weak Lefschetz property of $A(G)$. We classify…
We study the weak Lefschetz property of a class of graded Artinian Gorenstein algebras of codimension three associated in a natural way to the Ap\'ery set of a numerical semigroup generated by four natural numbers. We show that these…
Much progress has been made in classifying when the weak Lefschetz property holds for $A=\mathbb{F}[x,y,z]/I$ where $\text{char}(\mathbb{F})=0$ and $I=(x_{1}^{d_{1}},y^{d_{2}},z^{d_{3}},x^{a_{1}}y^{a_{2}}z^{a_{3}})$ is a monomial almost…
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 Ap\'ery set of numerical semigroups. To this aim we also prove a general result about the…
Given a simple graph $G$, the artinian monomial algebra associated to $G$, denoted by $A(G)$, is defined by the edge ideal of $G$ and the squares of the variables. In this article, we classify some tadpole graphs $G$ for which $A(G)$ has or…
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…
We characterize a binomial such that the Artinian algebra whose Macaulay dual generator is the binomial is a complete intersection. As an application, we prove that the Artinian algebra with a binomial Macaulay dual generator has the strong…
In this paper we give a new family of complete intersections which have the strong Lefschetz property. The family consists of (Artinian algebras defined by) ideals generated by power sum symmetric polynomials of consecutive degrees and of…
We show a number of properties of the commutator algebra of a nilpotent matrix over a field. In particular we determine the simple modules of the commutator algebra. Then the results are applied to prove that certain Artinian 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…
We give a characterization of the Lefschetz elements in Artinian Gorenstein rings over a field of characteristic zero in terms of the higher Hessians. As an application, we give new examples of Artinian Gorenstein rings which do not have…
In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree $1$ for a complete intersection standard Artinian Gorenstein algebra of codimension $6$ presented by quadrics. We prove also…
We study almost complete intersection ideals in a polynomial ring, generated by powers of all the variables together with a power of their sum. Our main result is an explicit description of the reduced Gr\"obner bases for these ideals under…
Consider a standard graded artinian $k$-algebra $B$ and an extension of $B$ by a new variable, $A=B\otimes_k k[x]/(x^d)$ for some $d\geq 1$. We will show how maximal rank properties for powers of a general linear form on $A$ can be…
We consider artinian algebras $A=\mathbb{C}[x_0,\ldots,x_m]/I$, with $I$ generated by a regular sequence of homogeneous forms of the same degree $d\geq 2$. We show that the multiplication by a general linear form from $A_{d-1}$ to $A_d$ is…
This paper initiates a systematic study for key properties of Artinian Gorenstein \(K\)-algebras having binomial Macaulay dual generators. In codimension 3, we demonstrate that all such algebras satisfy the strong Lefschetz property, can be…
We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The complete intersection property of numerical semigroup algebras is investigated using factorizations of monomials into minimal ones. The goal is to…
We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle…