Ancestor ideals of vector spaces of forms, and level algebras

Let R be the polynomial ring in r variables over a field k, with maximal ideal M and let V denote a vector subspace of the space of degree-j homogeneous elements of R. We study three related algebras determined by V. The first is the ``ancestor algebra'' whose defining ideal is the largest graded ideal whose intersection with M^j is the ideal (V). The second is the ``level algebra'', whose defining ideal L(V) is the largest graded ideal of R such that the degree-j component is V; and third is the algebra R/(V). When r=2, we determine the possible Hilbert functions H for each of these algebras, and as well the dimension of each Hilbert function stratum. We characterize the graded Betti numbers of these algebras in terms of certain partitions depending only on H, and give the codimension of each stratum in terms of invariants of the partitions. When r=2 and k is algebraically closed the Hilbert function strata for each of the three algebras satisfy a frontier property that the closure of a stratum is the union of more special strata. The family G(H) of all graded quotients of R having the given Hilbert function is a natural desingularization of this closure.Comment: 41 pages, LateX file, to appear Journal of Algebr

Betti strata of height two ideals

We determine the codimension of the Betti strata of the family G(H) parametrizing graded Artinian quotients A of the polynomial ring R in two variables, having Hilbert function H. The Betti stratum G(B,H) parametrizes all such quotients having the graded Betti numbers determined by the relation degrees B. Our method is to identify G(B,H) as the product of determinantal varieties. We then use a result of M. Boij that reduces the calculation of codimension to showing that the most special stratum has the expected codimension. We also show that the closure of the Betti stratum is Cohen-Macaulay, and the union of lower strata. As application, we determine the Hilbert functions possible for A, given the socle type; and we determine the Hilbert function of the intersection of t general enough level algebra quotients of R, each of a given Hilbert function.Comment: 17 pages, minor corrections/emendatio

Reducible family of height three level algebras

Let $R=k[x_1,..., x_r]$ be the polynomial ring in $r$ variables over an infinite field $k$, and let $M$ be the maximal ideal of $R$. Here a \emph{level algebra} will be a graded Artinian quotient $A$ of $R$ having socle $Soc(A)=0:M$ in a single degree $j$. The Hilbert function $H(A)=(h_0,h_1,... ,h_j)$ gives the dimension $h_i=\dim_k A_i$ of each degree-$i$ graded piece of $A$ for $0\le i\le j$. The embedding dimension of $A$ is $h_1$, and the \emph{type} of $A$ is \dim_k \Soc (A), here $h_j$. The family \Levalg (H) of level algebra quotients of $R$ having Hilbert function $H$ forms an open subscheme of the family of graded algebras or, via Macaulay duality, of a Grassmannian. We show that for each of the Hilbert functions $H=H_1=(1,3,4,4)$ and $H=H_2=(1,3,6,8,9,3)$ the family $LevAlg (H)$ parametrizing level Artinian algebras of Hilbert function $H$ has several irreducible components. We show also that these examples each lift to points. However, in the first example, an irreducible Betti stratum for Artinian algebras becomes reducible when lifted to points. These were the first examples we obtained of multiple components for \Levalg(H) in embedding dimension three. We also show that the second example is the first in an infinite sequence of examples of type three Hilbert functions $H(c)$ in which also the number of components of LevAlg(H) gets arbitrarily large. The first case where the phenomenon of multiple components can occur (i.e. the lowest embedding dimension and then the lowest type) is that of dimension three and type two. Examples of this first case have been obtained by the authors and also by J.-O. Kleppe.Comment: 20 pages. Minor revisio

Artinian Gorenstein algebras of embedding dimension four: Components of PGor(H) for H=(1,4,7,..., 1)

We first determine all height four Gorenstein sequences beginning H=(1,4,7,...), and we show that their first differences satisfy $\Delta H_{\le j/2}$ is an O-sequence. We then study the family PGor(H) parametrizing all graded Artinian Gorenstein [AG] quotients A=R/I of the polynomial ring R=K[w,x,y,z] having a Hilbert function H as above. We give a structure theorem for such AG quotients with $I_2\cong$. For most H this subfamily forms an irreducible component of PGor(H), and for a slightly more restrictive set, PGor(H) has several irreducible components. M. Boij and others had already shown that PGor(T) is reducible for certain Gorenstein sequences T in codimensions at least four.Comment: 29 pages. To appear Vasconcelos special issue of JPA

Artinian algebras and Jordan type

The Jordan type of an element $\ell$ of the maximal ideal of an Artinian k-algebra A acting on an A-module M of k-dimension n, is the partition of n given by the Jordan block decomposition of the multiplication map $m_\ell$ on M. In general the Jordan type has more information than whether the pair $(\ell,M)$ is strong or weak Lefschetz. We develop basic properties of the Jordan type and their loci for modules over graded or local Artinian algebras. We as well study the relation of generic Jordan type of $A$ to the Hilbert function of $A$. We introduce and study a finer invariant, the Jordan degree type. In our last sections we give an overview of topics such as the Jordan types for Nagata idealizations, for modular tensor products, and for free extensions, including examples and some new results. We as well propose open problems.Comment: 53 pages. Added results, examples for Jordan degree type (Section 2.4) and Jordan type and initial ideal (Section 2.5