Artinian Gorenstein Algebras Research Articles

The weak Lefschetz property for Standard graded, Artinian Gorenstein algebras of embedding dimension four and socle degree three

Let k be an arbitrary field and A be a standard graded Artinian Gorenstein k-algebra of embedding dimension four and socle degree three. Then, except for exactly one exception, A has the weak Lefschetz property. Furthermore, the exception occurs only in characteristic two.

A theorem of Gordan and Noether via Gorenstein rings

Gordan and Noether proved in their fundamental theorem that an hypersurface X=V(F)subseteq {{mathbb {P}}}^n with nle 3 is a cone if and only if F has vanishing hessian (i.e. the determinant of the Hessian matrix). They also showed that the statement is false if nge 4, by giving some counterexamples. Since their proof, several others have been proposed in the literature. In this paper we give a new one by using a different perspective which involves the study of standard Artinian Gorenstein {{mathbb {K}}}-algebras and the Lefschetz properties. As a further application of our setting, we prove that a standard Artinian Gorenstein algebra R={{mathbb {K}}}[x_0,dots ,x_4]/J with J generated by a regular sequence of quadrics has the strong Lefschetz property. In particular, this holds for Jacobian rings associated to smooth cubic threefolds.

Artinian Gorenstein algebras of embedding dimension four and socle degree three

We prove that in the polynomial ring Q=k[x,y,z,w], with k an algebraically closed field of characteristic zero, all Gorenstein homogeneous ideals I satisfying the inclusions (x,y,z,w)4⊆I⊆(x,y,z,w)2 can be obtained by doubling from a grade three perfect ideal J⊂I such that Q/J is a locally Gorenstein ring. Moreover, a graded minimal free resolution of the Q-module Q/I can be completely described in terms of a graded minimal free resolution of the Q-module Q/J and a homogeneous embedding of a shift of the canonical module ωQ/J into Q/J.

On the Lefschetz property for quotients by monomial ideals containing squares of variables

Let Δ be an (abstract) simplicial complex on n vertices. One can define the Artinian monomial algebra A ( Δ ) = k [ x 1 , … , x n ] / 〈 x 1 2 , … , x n 2 , I Δ 〉 , where k is a field of characteristic 0 and I Δ is the Stanley-Reisner ideal associated to Δ . In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of A ( Δ ) in terms of the simplicial complex Δ . We are able to completely analyze when the WLP holds in degree 1, complementing work by Migliore, Nagel and Schenck on the WLP for quotients by quadratic monomials. We give a complete characterization of all 2-dimensional pseudomanifolds Δ such that A ( Δ ) satisfies the WLP. We also construct Artinian Gorenstein algebras that fail the WLP by combining our results and the standard technique of Nagata idealization.

Lefschetz properties of some codimension three Artinian Gorenstein algebras

Codimension two Artinian algebras 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 Artinian Gorenstein algebras. Despite much work, the strong Lefschetz property for codimension three Artinian Gorenstein algebra has remained largely mysterious; our results build on and strengthen some of the previous results. We here show that every standard-graded codimension three Artinian Gorenstein algebra A having maximum value of the Hilbert function at most six has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is greater than the socle degree of A, we show that A is almost strong Lefschetz, they are strong Lefschetz except in the extremal pair of degrees.

Perazzo 3-folds and the weak Lefschetz property

We deal with Perazzo 3-folds in P4, i.e. hypersurfaces X=V(f)⊂P4 of degree d defined by a homogeneous polynomial f(x0,x1,x2,u,v)=p0(u,v)x0+p1(u,v)x1+p2(u,v)x2+g(u,v), where p0,p1,p2 are algebraically dependent but linearly independent forms of degree d−1 in u,v, and g is a form in u,v of degree d. Perazzo 3-folds have vanishing hessian and, hence, the associated graded Artinian Gorenstein algebra Af fails the strong Lefschetz Property. In this paper, we determine the maximum and minimum Hilbert function of Af and we prove that if Af has maximal Hilbert function it fails the weak Lefschetz Property while it satisfies the weak Lefschetz Property when it has minimum Hilbert function. In addition, we classify all Perazzo 3-folds in P4 such that Af has minimum Hilbert function.

Artinian Gorenstein algebras that are free extensions over k[t]/(tn), and Macaulay duality

T. Harima and J. Watanabe studied the Lefschetz properties of free extension Artinian algebras C over a base A with fiber 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=k[t]∕(tn) with fiber B. We give examples exploring the sharpness of the statements. We also consider a special set of coinvariant algebras C which are free extensions of A, but which do not satisfy the sufficient conditions of our main result.

Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

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 some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of “special” Artinian algebras: for them we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree 1 and the “lifting” of a weak Lefschetz property to an algebra from one of its quotients.

Jordan types with small parts for Artinian Gorenstein algebras of codimension three

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 determined by at most three parameters.

Cohomological Blowups of Graded Artinian Gorenstein Algebras along Surjective Maps

Abstract We introduce the cohomological blowup of a graded Artinian Gorenstein algebra along a surjective map, which we term BUG (blowup Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blowup 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 intersections. We have included many examples throughout this manuscript.

Waring and cactus ranks and strong Lefschetz property for annihilators of symmetric forms

In this note we show that the complete symmetric polynomials are dual generators of compressed artinian Gorenstein algebras satisfying the Strong Lefschetz Property. This is the first example of an explicit dual form with these properties. For complete symmetric forms of any degree in any number of variables, we provide an upper bound for the Waring rank by establishing an explicit power sum decomposition. Moreover, we determine the Waring rank, the cactus rank, the resolution and the Strong Lefschetz Property for any Gorenstein algebra defined by a symmetric cubic form. In particular, we show that the difference between the Waring rank and the cactus rank of a symmetric cubic form can be made arbitrarily large by increasing the number of variables. We provide upper bounds for the Waring rank of generic symmetric forms of degrees four and five.

The weak Lefschetz property of graded Gorenstein algebras associated to the Apéry set of a numerical semigroup

It has been conjectured that all graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras A of the Apéry set of M-pure symmetric numerical semigroups generated by four natural numbers. These algebras are graded Artinian Gorenstein algebras of codimension three.

Connected sums of graded Artinian Gorenstein algebras and Lefschetz properties

In their paper [1], H. Ananthnarayan, L. Avramov, and W.F. Moore introduced a connected sum construction for local Gorenstein rings A,B over a local Gorenstein ring T, which, in the graded Artinian case, can be viewed as an algebraic analogue of the topological construction of the same name. We give two alternative descriptions 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 of A,B over an Artinian Gorenstein algebra T preserves the weak or strong Lefschetz property, thus providing new classes of rings which satisfy these properties.

The eigenvalues of the Hessian matrices of the generating functions for trees with k components

Let us consider a truncated matroid MΓr of rank r of a graphic matroid of a graph Γ. The basis for MΓr is the set of the forests with r edges in Γ. We consider this basis generating function and compute its Hessian. In this paper, we show that the Hessian of the basis generating function of the truncated matroid of the graphic matroid of the complete or complete bipartite graph does not vanish by calculating the eigenvalues of the Hessian matrix. Moreover, we show that the Hessian matrix of the basis generating function of the truncated matroid of the graphic matroid of the complete or complete bipartite graph has exactly one positive eigenvalue. As an application, we show the strong Lefschetz property for the Artinian Gorenstein algebra associated to the truncated matroid.

Hilbert functions of Artinian Gorenstein algebras with the strong Lefschetz property

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 generali ...

The eigenvalues of Hessian matrices of the complete and complete bipartite graphs

In this paper, we consider the Hessian matrices $$H_{\Gamma }$$ of the complete and complete bipartite graphs, and the special value of $${\tilde{H}}_{\Gamma }$$ at $$x_{i}=1$$ for all $$x_{i}$$ . We compute the eigenvalues of $${\tilde{H}}_{\Gamma }$$ . We show that one of them is positive and that the others are negative. In other words, the metric with respect to the symmetric matrix $$\tilde{H}_{\Gamma }$$ is Lorentzian. Hence those Hessian $$\det (H_{\Gamma })$$ are not identically zero. As an application, we show the strong Lefschetz property for the Artinian Gorenstein algebra associated to the graphic matroids of the complete and complete bipartite graphs with at most five vertices.

Strict log-concavity of the Kirchhoff polynomial and its applications to the strong Lefschetz property

Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are “strictly” log-concave for simple matroids. In this paper, we show this strictness for simple graphic matroids, that is, we show that Kirchhoff polynomials of simple graphs are strictly log-concave. Our key observation is that the Kirchhoff polynomial of a complete graph can be seen as the (irreducible) relative invariant of a certain prehomogeneous vector space, which may be independently interesting in its own right. Furthermore, we prove that for any ai∈R>0, a1x1+⋯+anxn∈RM1 satisfies the strong Lefschetz property (moreover, Hodge–Riemann bilinear relation) at degree one of the Artinian Gorenstein algebra RM⁎ associated to a graphic matroid M, which is defined by Maeno and Numata for all matroids.

High order weak Lefschetz properties, higher Hessians and Jordan types for standard graded Artinian Gorenstein algebras

High order weak Lefschetz properties, higher Hessians and Jordan types for standard graded Artinian Gorenstein algebras

From subcategories to the entire module categories

Abstract In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander–Reiten quiver of the subcategories is obtained only by the Auslander–Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, when Λ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the 2 × 2 {2\times 2} lower triangular matrix algebra over Λ and the stable Auslander algebra of Λ can be estimated by the category of modules over the stable Cohen–Macaulay Auslander algebra of Λ.

Symmetric decomposition of the associated graded algebra of an Artinian Gorenstein algebra

We study the symmetric subquotient decomposition of the associated graded algebras A⁎ of a non-homogeneous commutative Artinian Gorenstein (AG) algebra A. This decomposition arises from the stratification of A⁎ by a sequence of ideals A⁎=CA(0)⊃CA(1)⊃⋯ whose successive quotients Q(a)=C(a)/C(a+1) are reflexive A⁎ modules. These were introduced by the first author [47,48], developed in the Memoir [49], and have been used more recently by several groups, especially those interested in short Gorenstein algebras, and in the scheme length (cactus rank) of forms.For us a Gorenstein sequence is an integer sequence H occurring as the Hilbert function H=H(A) for an AG algebra A, that is not necessarily homogeneous. Such a Hilbert function H(A) is the sum of symmetric non-negative sequences HA(a)=H(QA(a)), each having center of symmetry (j−a)/2 where j is the socle degree of A: we call these the symmetry conditions, and the decomposition D(A)=(HA(0),HA(1),…) the symmetric decomposition of H(A) (Theorem 1.4). We here study which sequences may occur as the summands HA(a): in particular we construct in a systematic way examples of AG algebras A for which HA(a) can have interior zeroes, as HA(a)=(0,s,0,…,0,s,0). We also study the symmetric decomposition sets D(A), and in particular determine which sequences HA(a) can be non-zero when the dual generator is linear in a subset of the variables (Theorem 4.1).Several groups have studied “exotic summands” of the Macaulay dual generator F: these are summands that involve more successive variables than would be expected from the symmetric decomposition of the Hilbert function H(A). Studying these, we recall a normal form for the Macaulay dual generator of an AG algebra that has no “exotic” summands (Theorem 2.7). We apply this to Gorenstein algebras that are connected sums (Section 2.4).We give throughout many examples and counterexamples, and conclude with some open questions about symmetric decomposition.