This article concerns commutative algebras over a field $k$ of characteristic zero which are finite dimensional as vectorspaces, and particularly those of such algebras which are graded. Here the term graded is applied to non-negatively graded algebras $A$ with $A_0$ reduced and finite dimensional. Thus the trivial grading $A=A_0$ is only allowed if $A$ is a product of finite field extensions of $k$. It has been conjectured (G. Cortiñas, S. Geller, C. Weibel; The Artinian Berger Conjecture. Math. Zeitschrift {\bf 228} 3 (1998) 569-588) that for all finite dimensional algebras $A$ which are not principal ideal algebras (i.e. have at least one nonprincipal ideal), the following submodule of the Kähler differentials is nonzero: $$\bigcap{\ker(Ω_A @>>>Ω_B)}$$ Here the intersection is taken over all principal ideal algebras $B$ and all homomorphisms $A @>>>B$. In this paper we prove that the conjecture holds for both Gorenstein graded and standard graded algebras.
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 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.
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…
In this paper we classify, up to analytic isomorphism, the family of almost stretched Artinian complete intersection A=R/I with a given Hilbert function, in the case R is a power series ring with an arbitrary number of variables.
We introduce a family of standard bigraded binomial Artinian Gorenstein algebras, whose combinatoric structure characterizes the ones presented by quadrics. These algebras provide, for all socle degree grater than two and in sufficiently large codimension with respect to the socle degree, counter-examples to Migliore-Nagel conjectures, see \cite{MN1} and \cite{MN2}. One of them predicted that Artinian Gorenstein algebras presented by quadratics should satisfy the weak Lefschetz property. We also prove a generalization of a Hessian criterion for the Lefschetz properties given by Watanabe, see \…
The classification of local Artinian Gorenstein algebras is equivalent to the study of orbits of a certain non-reductive group action on a polynomial ring. We give an explicit formula for the orbits and their tangent spaces. We apply our technique to analyse when an algebra is isomorphic to its associated graded algebra. We classify algebras with Hilbert function (1, 3, 3, 3, 1), obtaining finitely many isomorphism types, and those with Hilbert function (1, 2, 2, 2, 1, 1, 1). We consider fields of arbitrary, large enough, characteristic.
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…
We classify the orbits of nets of conics under the action of the projective linear group and we determine the specializations of these orbits, using geometric and algebraic methods. We study related geometric questions, as the parametrization of planar cubics. We show that Artinian algebras of Hilbert function H=(1,3,3,0) determined by nets, can be smoothed - deformed to a direct sum of fields; and that algebras of Hilbert function H=(1,r,2,0), determined by pencils of quadrics, can also be smoothed. In a Historical Appendix A we describe related work prior to 1977. In an Update Appendix B we …
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…
Let A = bigoplus_{i >= 0} A_i be a standard graded Artinian K-algebra, where char K = 0. Then A has the Weak Lefschetz property if there is an element ell of degree 1 such that the multiplication times ell : A_i --> A_{i+1} has maximal rank, for every i, and A has the Strong Lefschetz property if times ell^d : A_i --> A_{i+d} has maximal rank for every i and d. The main results obtained in this paper are the following. 1) EVERY height three complete intersection has the Weak Lefschetz property. (Our method, surprisingly, uses rank two vector bundles on P^2 and the Grauert-Mulich theorem.) 2) W…
The purpose of this paper is to study families of Artinian or one dimensional quotients of a polynomial ring $R$ with a special look to level algebras. Let $\GradAlg^H(R)$ be the scheme parametrizing graded quotients of $R$ with Hilbert function $H$. Let $B \to A$ be any graded surjection of quotients of $R$ with Hilbert function $H_B$ and $H_A$, and h-vectors $h_B=(1,h_1,...,h_j,...)$ and $h_A$, respectively. If $\depth A = \dim A \leq 1$ and $A$ is a ``truncation'' of $B$ in the sense that $h_A=(1,h_1,...,h_{j-1},α,0,0,...)$ for some $α\leq h_j$, then we show there is a close relationship be…
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.
In a recent paper it has been established that over an Artinian ring R all two-dimensional polynomial automorphisms having Jacobian determinant one are tame if R is a Q-algebra. This is a generalization of the famous Jung-Van der Kulk Theorem, which deals with the case that R is a field (of any characteristic). Here we will show that for tameness over an Artinian ring, the Q-algebra assumption is really needed: we will give, for local Artinian rings with square-zero principal maximal ideal, a complete description of the tame automorphism subgroup. This will lead to an example of a non-tame aut…
Hilbert functions and Jordan type of Perazzo Artinian algebras
Borås University（博罗斯大学）；Queen’s University（女王大学）；KTH Royal Institute of Technology（皇家理工学院）；Università degli Studi di Udine（乌迪内大学）；Università di Genova（热那亚大学）；Northeastern University（东北大学）；Universidade de Évora（埃武拉大学）；Università di Trieste（的里雅斯特大学）
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 degree. In the first open case, namely Hilbert function (1,3,6,6,3,1), we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from $\mathbb P^2$ to $\mathbb P^3$, and Hesse configurations in $\mathbb P^2$.
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 …
This paper can be seen as a continuation of the works contained in the recent preprints [Za], of the second author, and [Mi], of Juan Migliore. Our results are: 1). There exist codimension three artinian level algebras of type two which do not enjoy the Weak Lefschetz Property (WLP). In fact, for $e\gg 0$, we will construct a codimension three, type two $h$-vector of socle degree $e$ such that {\em all} the level algebras with that $h$-vector do not have the WLP. We will also describe the family of those algebras and compute its dimension, for each $e\gg 0$. 2). There exist reduced level sets …
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 $A^*=C_A(0)\supset C_A(1)\supset\cdots$ whose successive quotients $Q(a)=C(a)/C(a+1)$ are reflexive $A^*$ modules. These were introduced by the first author, 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 integ…
We introduce a new type of Hessian matrix, that we call Mixed Hessian. The mixed Hessian is used to compute the rank of a multiplication map by a power of a linear form in a standard graded Artinian Gorenstein algebra. In particular we recover the main result of \cite{MW} for identifying Strong Lefschetz elements, generalizing it also for Weak Lefschetz elements. This criterion is also used to give a new proof that Boolean algebras have the Strong Lefschetz Property (SLP). We also construct new examples of Artinian Gorenstein algebras presented by quadrics that does not satisfy the Weak Lefsch…
It is known that all complete intersection Artinian standard graded algebras of codimension 3 have the Weak Lefschetz Property. Unfortunately, this property does not continue to be true when you increase the number of minimal generators for the ideal defining the algebra. For instance, it is not more valid for almost complete intersection Artinian standard graded algebras of codimension 3. On the other hand, the Hilbert functions of all Weak Lefschetz Artinian graded algebras are unimodal and with the positive part of their first difference forming an $O$-sequence (i.e. are Weak Lefschetz sequ…
In this article, we study the $k$-Lefschetz properties for non-Artinian algebras, proving that several known results in the Artinian case can be generalized in this setting. Moreover, we describe how to characterize the graded algebras having the $k$-Lefschetz properties using sectional matrices. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements, with particular attention to the class of free arrangements.
Given a multi-index sequence $$σ$$, we present a new efficient algorithm to compute generators of the linear recurrence relations between the terms of $$σ$$. We transform this problem into an algebraic one, by identifying multi-index sequences, multivariate formal power series and linear functionals on the ring of multivariate polynomials. In this setting, the recurrence relations are the elements of the kerne l$I$σ$$ of the Hankel operator $H$σ$$ associated to $$σ$$. We describe the correspondence between multi-index sequences with a Hankel operator of finite rank and Artinian Gorenstein Alge…
Let $K$ be an algebraically closed field of characteristic $0$, and let $A$ be an Artinian Gorenstein local commutative and Noetherian $K$--algebra, with maximal ideal $M$. In the present paper we prove a structure theorem describing such kind of $K$--algebras satisfying $M^4=0$. We use this result in order to prove that such a $K$--algebra $A$ has rational Poincaré series and it is always smoothable in any embedding dimension, if $\dim_K M^2/M^3 \le 4$. We also prove that the generic Artinian Gorenstein local $K$--algebra with socle degree three has rational Poincaré series, in spite of the f…
The purpose of this note is to characterize the finite Hilbert functions which force all of their artinian algebras to enjoy the Weak Lefschetz Property (WLP). Curiously, they turn out to be exactly those (characterized by Wiebe in $[Wi]$) whose Gotzmann ideals have the WLP. This implies that, if a Gotzmann ideal has the WLP, then all algebras with the same Hilbert function (and hence lower Betti numbers) have the WLP as well. However, we will answer in the negative, even in the case of level algebras, the most natural question that one might ask after reading the previous sentence: If $A$ is …
Let $R$ be the power series ring or the polynomial ring over a field $k$ and let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $Γ. $ The result is effective in the sense that any polynomial of degree $s$ produces an Artinian Gorenstein $k$-algebra of socle degree $s.$ In a recent paper, the authors extended Macaulay's correspondence characterizing the $R$-submodules of $Γ$ in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. However, th…
We introduce and study higher order Jacobian ideals, higher order and mixed Hessians, higher order polar maps, and higher order Milnor algebras associated to a reduced projective hypersurface. We relate these higher order objects to some standard graded Artinian Gorenstein algebras, and we study the corresponding Hilbert functions and Lefschetz properties.
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 injective. We prove that the Weak Lefschetz Property holds for artinian complete intersection algebras as above, with $d=2$ and $m\leq 4$. Apparently, this was previously known only for $m\leq 3$. Although we are proposing only very limited progress towards the WLP conjecture for complete intersections, we hope that the methods of the present article can ill…
It has been conjectured that {\it 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. In 2010, Bryant proved that these algebras are graded Artinian Gorenstein algebras of codimension three. In a recent article, Guerrieri showed that if $A$ is not a complete intersection, then $A$ is of form $A=R/I$ with $R=K[x,y,z]$ and \begin{align*} I…
Scuola Politecnica e delle Scienze di Base, Università degli Studi di Napoli “Federico II”（那不勒斯费德里科二世大学）；Università degli Studi di Udine（乌迪内大学）；Dipartimento di Matematica ed Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”（那不勒斯费德里科二世大学）
In this paper we study the isomorphism classes of local, Artinian, Gorenstein k-algebras A whose maximal ideal M satisfies dim_k(M^3/M^4)=1 by means of Macaulay's inverse system generalizing a recent result by J. Elias and M.E. Rossi. Then we use such results in order to complete the description of the singular locus of the Gorenstein locus of the punctual Hilbert scheme of degree 11.
We show that the image of a subshift $X$ under various injective morphisms of symbolic algebraic varieties over monoid universes with algebraic variety alphabets is a subshift of finite type, resp. a sofic subshift, if and only if so is $X$. Similarly, let $G$ be a countable monoid and let $A$, $B$ be Artinian modules over a ring. We prove that for every closed subshift submodule $Σ\subset A^G$ and every injective $G$-equivariant uniformly continuous module homomorphism $τ\colon Σ\to B^G$, a subshift $Δ\subset Σ$ is of finite type, resp. sofic, if and only if so is the image $τ(Δ)$. Generaliza…
Let $k$ be a field of characteristic zero, $\CO$ be a dg operad over $k$ and let $A$ be an $\CO$-algebra. In this note we define formal deformations of $A$, construct the deformation functor $$\Def_A:\dgar(k)\to\simpl$$ from the category of artinian local dg algebras to the category of simplicial sets. In the case $\CO$ and $A$ are non-positively graded, we prove that $\Def_A$ is governed by the tangent Lie algebra $T_A$ (defined as derivations of a cofibrant resolution of $A$). A very easy example shows that the result does not hold without this condition.
This paper is devoted to deformation theory of graded Lie algebras over $\Z$ or $\Z_l$ with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artin local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations. In the second part, we use a version of Schlessinger criteria for functors on the Artinian category of nilpotent Lie algebras which is formulated by Pridham, and explore arithmetic deformations usi…
