A nonunimodal graded gorenstein artin algebra in codimension five

A characterization of Gorenstein Hilbert functions in codimension four with small initial degree

“…(2) Even assuming that the initial degree is 2, the unimodality conclusion of Theorem 3.1 is false in every codimension five or more. Indeed, Example 2 of [3] has codimension 5 and initial degree 2, and even the much earlier Example 4.3 of [23] has codimension 13 and initial degree 2. Moreover, from the non-unimodal Gorenstein h-vector of codimension 5 and initial degree 2, it is easy to construct (for instance by using inverse systems) non-unimodal Gorenstein h-vectors of any codimension r ≥ 6 and initial degree 2.…”

“…• (3, 3, 3), (3,3,2), (3, 3, 1), (3, 2, 2), (3, 1, 1), (2, 2, 2), (2, 2, 1), (2, 1, 1), (1, 1, 1). In all these cases, there is a GCD in degree e − i − 1 > e 2 , so Lemma 2.4 shows that h R/I is unimodal.…”

“…Recall that an h-vector is said to be unimodal if it is never strictly increasing after a strict decrease. Later on, in each codimension ≥ 5, examples of h-vectors were found that are not unimodal (see [3], [6], [5]). Since then the problem has been open whether non-unimodal Gorenstein h-vectors of codimension 4 exist (see, e.g., [3], [14], [21], [28,Problem 2.19], or [27, p. 66]).…”

Components of the space parametrizing graded Gorenstein Artin algebras with a given Hilbert function

“…The algebras in the families G n,c are generalizations of the examples given by R. Stanley [10] and by D. Bernstein and A. Iarrobino [2] showing the existence of nonunimodal Hilbert functions of Gorenstein algebras. In addition, the Hilbert functions of the algebras in G n,c are given by T n,c+2,c .…”

“…This is can be done by choosing a compressed Artin level algebra B in two variables with socle of dimension n − 1 in degree c − 1, i.e., B is a level algebra with Hilbert function H B (d) = min{d + 1, (c − d)(n − 1)}, and take A = B × Bˇ-the trivial extension of A by its dual.This Gorenstein algebra has the right Hilbert function, (cf. D. Bernstein and A. Iarrobino [2], or M. Boij and D. Laksov [3]), and the square of the ideal generated by the generators of Bˇis zero in A.…”