Reducible family of height three level algebras

Abstract

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 level algebra will be a graded Artinian quotient A of R having socleThe 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 1 = (1, 3, 4, 4) and H 2 = (1, 3, 6, 8, 9, 3) the family LevAlg(H) has several irreducible components (Theorems 2.3(A), 2.4). 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 (Theorem 2.3(B)). We 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 gets arbitrarily large (Theorem 2.10). 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 Examples of this first case have been obtained by the authors (unpublished) and also by J.O. Kleppe.

Artinian algebras and Jordan type

“…However, in codimension three, even for graded algebras the family Gr(H) of quotients of R = k[x, y, z] having Hilbert function H may be reducible. One example is given in [BoI1,Theorem 2.3], where H = (1, 3, 4, 4). Here the behavior of one component of Gr(H) with respect to Jordan type may be different than that of another.…”

The structure of the inverse system of level K-algebras

“…Their study was initiated by Stanley in [Sta77]. Since then they have been widely investigated, especially in the Artinian case, see for instance [Boi94], [Boi99], [Boi00], [Boi09], [Ber09], [GHM + 07], [Iar84], [Ste14]. However, there are also many examples of level rings in positive dimension: Stanley-Reisner rings of matroid simplicial complexes [Sta77], associated graded rings of semigroup rings corresponding to arithmetic sequences [MT95], determinantal rings corresponding to generic matrices [BV88] or generic symmetric matrices [Con94], [Got79].…”