# Lefschetz properties and Jordan types of Artinian algebras

University dissertation from Stockholm, Sweden : KTH Royal Institute of Technology

Abstract:

This thesis contains six papers concerned with studying the Lefschetz properties and Jordan types of linear forms for graded Artinian algebras. Lefschetz properties and Jordan types carry information about the ranks of multiplication maps by linear forms on graded Artinian algebras. A graded Artinian algebra $A$ is said to have the weak Lefschetz property (WLP) if multiplication map by some linear form $\ell$ $\in$ $A_1$ on $A$ has maximal rank in all degrees. If it holds for all powers of $\ell$ the algebra is said to have the strong Lefschetz property (SLP). Jordan type $P_{\ell,A}$ is a partition determining the Jordan block decomposition for the multiplication map by $\ell$on $A$. It is a finer invariant than the WLP and SLP, which determines the ranks of multiplication maps by all powers of $\ell$ on $A$.

Papers A and B concern the study of the Lefschetz properties of Artinian algebras quotient of a polynomial ring in $n\geq 3$ variables over a field of characteristic zero by a monomial ideal $I$ generated in a single degree $d\geq 2$. Paper A studies the connection between the Lefschetz properties of such algebras and their minimal free resolutions, namely the number of linear steps in their resolutions. Paper B consists of two parts. The first part provides sharp lower bounds for the Hilbert function in degree $d$ of such algebras failing the WLP. The second part of Paper B deals algebras quotients of polynomial rings by ideals generated by forms of the same degree and invariant under action of a cyclic group. The main result of this part classifies such algebras satisfying the WLP in terms of the representation of the action.

Papers C and D both deal with determining the Jordan type partitions of linear forms for graded Artinian algebras with codimension two. Paper C concerns the problem for complete intersection Artinian algebras over an algebraically closed field of characteristic zero or large enough. The results of Paper C classify partitions of an integer $n$ that occur as Jordan type partitions for Artinian complete intersection algebras and some linear forms. Such classifications are provided in terms of the numerical conditions of the partitions. Also for a given Hilbert function of such algebras, the Jordan type partitions are completely determined by which higher Hessians vanish at the point corresponding to the linear form. Some combinatorial invariants of such partitions, namely branch label or hook code, have been studied in this paper as well.

Paper D concerns the generalization of the results of Paper C. The family $G_T$ of graded Artinian quotients $A = S/I$ of $S= \mathsf{k}[x,y]$, having arbitrary Hilbert function $h_A=T$ has been studied. The cell $\mathbb{V}(E_P)$ corresponding to a partition $P$ having diagonal lengths $T$ is comprised of all ideals $I$ in $S$ whose initial ideal is the monomial ideal $E_P$ determined by $P$. These cells give a decomposition of the variety $G_T$ into affine spaces. The main result of Paper D determines the generic number $\kappa(P)$ of generators for the ideals in each cell $\mathbb{V}(E_P)$; generalizing a result of Paper C. In particular, partitions having the generic number of generators for an ideal defining an algebra $A$ in $G_T$ are determined.

Paper E concerns the SLP of Artinian Gorenstein algebras via studying the higher Hessians of dual generators. The main result of this paper characterizes the Hilbert functions of Artinian Gorenstein algebras having arbitrary codimension satisfying the SLP. It proves that a sequence $h$ of non-negative integers is the Hilbert function of some Artinian Gorenstein algebra with the SLP if and only if $h$ is an SI-sequence. This is done by studying the higher Hessians of the dual generator of an Artinian Gorenstein quotient of the coordinate ring of a set of points in the projective space. Using this approach, we provide families of Artinian Gorenstein algebras obtained by points in the projective plane satisfying the SLP.

Paper F concerns the study of Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. This paper introduces rank matrices of linear forms for such algebras. There is a 1-1 correspondence between rank matrices and Jordan degree types. The main result of this paper classifies all matrices that occur as the rank matrix for some Artinian Gorenstein algebra $A$ of codimension three and a linear form $\ell$ such that $\ell^3=0$. As a consequence, we prove that Jordan types with parts of length at most four are uniquely determined by at most three parameters.