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 Artinian algebras.
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.
We prove that a sequence $h$ of non-negative integers is the Hilbert function of some Artinian Gorenstein algebra with the strong Lefschetz property if and only if it is an SI-sequence. This generalizes the result by T. Harima which characterizes the Hilbert functions of Artinian Gorenstein algebras with the weak Lefschetz property. We also provide classes of Artinian Gorenstein algebras obtained from the ideal of points in $\mathbb{P}^n$ such that some of their higher Hessians have non-vanishing determinants. Consequently, we provide families of such algebras satisfying the SLP.
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 study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We show that there is a 1-1 correspondence between rank matrices and Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices that occur for linear forms with vanishing third power. As a consequence, we show for such algebras that the possible Jordan types with parts of length at most four are uniquely determine…
We introduce the cohomological blow up of a graded Artinian Gorenstein (AG) algebra along a surjective map, which we term BUG (Blow Up Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blow up of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersecti…
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.
We study the general Jordan type of standard graded Artinian Gorenstein algebras, it is a finer invariant than Weak and Strong Lefschetz properties for those algebras. We prove that their Jordan types are determined by the rank of certain Mixed Hessians. We give a description of the possible Jordan types for algebras of low socle degree and low codimension.
We introduce Artinian Gorenstein algebras defined by the face posets of regular polyhedra. We consider the strong Lefschetz property and Hodge--Riemann relation for the algebras. We show the strong Lefschetz property of the algebras for all Platonic solids. On the other hand, for some Platonic solids, we show that the algebras do not satisfy the Hodge--Riemann relation with respect to some strong Lefschetz elements.
T. Harima and J. Watanabe studied the Lefschetz properties of free extension Artinian algebras $C$ over a base $A$ with fibre $B$. The free extensions are deformations of the usual tensor product, when $C$ is also Gorenstein, so are $A$ and $B$, and it is natural to ask for the relation among the Macaulay dual generators for the algebras. Writing a dual generator $F$ for $C$ as a homogeneous "polynomial" in $T$ and the dual variables for $B$, and given the dual generator for $B$, we give sufficient conditions on $F$ that ensure that $C$ is a free extension of $A={\sf k}[t]/(t^n)$ with fiber $B…
The Jordan type of an Artinian algebra is the Jordan block partition associated to multiplication by a generic element of the maximal ideal. We study the Jordan type for Artinian Gorenstein (AG) local algebras A, and the interaction of Jordan type with the symmetric decomposition of the Hilbert function H(A). We give examples of Gorenstein sequences H for which the family Gor(H) of AG algebras having Hilbert function H has several irreducible components, each corresponding to a symmetric decomposition of H. The component structure results from the intersection of two opposing filtrations of th…
Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra $A$ having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is grea…
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 study properties of the resolution of almost Gorenstein artinian algebras $R/I,$ i.e. algebras defined by ideals $I$ such that $I=J+(f),$ with $J$ Gorenstein ideal and $f\in R.$ Such algebras generalize the well known almost complete intersection artinian algebras. Then we give a new explicit description of the resolution and of the graded Betti numbers of almost complete intersection ideals of codimension $3$ and we characterize the ideals whose graded Betti numbers can be achieved using artinian monomial ideals.
Let $k$ be a field of characteristic $0$. Using the method of idealization, we show that there is a non-Koszul, quadratic, Artinian, Gorenstein, standard graded $k$-algebra of regularity $3$ and codimension $8$, answering a question of Mastroeni, Schenck, and Stillman. We also show that this example is minimal in the sense that no other idealization that is non-Koszul, quadratic, Artinian, Gorenstein algebra, with regularity $3$ has smaller codimension. We also construct an infinite family of graded, quadratic, Artinian, Gorenstein algebras $A_m$, indexed by an integer $m \ge 2$, with the foll…
Given a set of distinct points $X=\{P_1, \dots,P_r\} $ in $ \mathbb P^n_{\res}, $ in this paper we characterize being $X$ arithmetically Gorenstein through the ``special" structure of the inverse system of the defining ideal $I(X) \subseteq R= \res[X_0, \dots, X_n].$ We describe the corresponding Zariski locally closed subset of $\mathbb (\mathbb P_{\res}^{n})^r$ parametrized by the coordinates of the points. Several examples are given to show the effectiveness of the results. As a consequence of the main result we characterize the Artinian Gorenstein rings which are the Artinian reduction of …
A quaternary quartic form, a quartic form in four variables, is the dual socle generator of an Artinian Gorenstein ring of codimension and regularity 4. We present a classification of quartic forms in terms of rank and powersum decompositions which corresponds to the classification by the Betti tables of the corresponding Artinian Gorenstein rings. This gives a stratification of the space of quaternary quartic forms which we compare with the Noether-Lefschetz stratification. We discuss various phenomena related to this stratification. We study the geometry of powersum varieties for a general f…
