Mats Boij

Graded gorenstein artin algebras whose hilbert functions have a large number of valleys

In [4] Stanley showed that the Hilbert function of graded Goren-stein Artin algebras need not be unimodal. In this article we prove the exis¬tence of graded Gorenstein Artin algebras whose Hilbert functions have local maxima at any given set of points with the symmetry and the socle degree as the only restrictions.

Betti numbers of compressed level algebras

Abstract We present a conjecture on the generic Betti numbers of level algebras. We prove the conjecture in the case of Gorenstein Artin algebras of embedding dimension four, and in the case of Artin level algebras whose socle dimension is large. Furthermore, we present computational evidence for the conjecture.

Monomials as sums of powers: The real binary case

We generalize an example, due to Sylvester, and prove that any monomial of degree d in R[x(0), x(1)], which is not a power of a variable, cannot be written as a linear combination of fewer than d powers of linear forms.

Level algebras with bad properties

This paper can be seen as a continuation of the works contained in the recent article (J. Alg., 305 (2006), 949-956) of the second author, and those of Juan Migliore (math. AC/0508067). 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 >> 0, we will construct a codimension three, type two h- vector of socle degree e such that 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 >> 0. 2). There exist reduced level sets of points in P-3 of type two whose artinian reductions all fail to have theWLP. Indeed, the examples constructed here have the same h- vectors we mentioned in 1). 3). For any integer r >= 3, there exist non- unimodal monomial artinian level algebras of codimension r. As an immediate consequence of this result, we obtain another proof of the fact (first shown by Migliore in the abovementioned preprint, Theorem 4.3) that, for any r >= 3, there exist reduced level sets of points in P-r whose artinian reductions are non- unimodal.

The Cone of Betti Diagrams of Bigraded Artinian Modules of Codimension Two

We describe the positive cone generated by bigraded Betti diagrams of artinian modules of codimension two, whose resolutions become pure of a given type when taking total degrees. If the differences, p and q, of these total degrees are relatively prime, the extremal rays are parametrized by order ideals in ℕ2 contained in the region px+qy<(p−1)(q−1). We also consider some examples concerning artinian modules of codimension three.

Cones of Hilbert Functions

We study the closed convex hull of various collections of Hilbert functions. Working over a standard graded polynomial ring with modules that are generated in degree 0, we describe the supporting hyperplanes and extreme rays for the cones generated by the Hilbert functions of all modules, all modules with bounded alpha-invariant, and all modules with bounded Castelnuovo-Mumford regularity. The first of these cones is infinite-dimensional and simplicial, the second is finite-dimensional but neither simplicial nor polyhedral, and the third is finite-dimensional and simplicial.

Notes on diagonal coinvariants of the dihedral group

The bigraded Hilbert function and the minimal free resolutions for the diagonal coinvariants of the dihedral groups are exhibited, as well as for all their bigraded invariant Gorenstein quotients.