arXiv subjects

Mats Boij

Publications and source records attributed to Mats Boij.

24 records · Page 2Linked to original sources

arXiv2010

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 of these total degrees, p and q, are relatively prime, the extremal rays are parametrised by order ideals in N^2 contained in the region px + qy < (p-1)(q-1). We also consider some examples concerning artinian modules of codimension three.

math.AC↗

arXiv2009

Some algebraic consequences of Green's hyperplane restriction theorems

We discuss a paper of M. Green from a new algebraic perspective, and provide applications of its results to level and Gorenstein algebras, concerning their Hilbert functions and the weak Lefschetz property. In particular, we will determine a new infinite class of symmetric $h$-vectors that cannot be Gorenstein $h$-vectors, which was left open in a recent work of Migliore-Nagel-Zanello. This includes the smallest example previously unknown, $h=(1,10,9,10,1)$. As M. Green's results depend heavily on the characteristic of the base field, so will ours. The appendix will contain a new argument, kindly provided to us by M. Green, for Theorems 3 and 4 of his paper, since we had found a gap in the original proof of those results during the preparation of this manuscript.

math.AC↗

arXiv2008

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.

math.AC↗

arXiv2008

Betti numbers of graded modules and the Multiplicity Conjecture in the non-Cohen-Macaulay case

We use the results by Eisenbud and Schreyer to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination Betti diagrams of modules with a pure resolution. This implies the Multiplicity Conjecture of Herzog, Huneke and Srinivasan for modules that are not necessarily Cohen-Macaulay. We give a combinatorial proof of the convexity of the simplicial fan spanned by the pure diagrams.

math.AC↗

arXiv2007

Graded Betti numbers of Cohen-Macaulay modules and the Multiplicity conjecture

We give conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The conjectures are verified in the cases where the structure of resolutions are known, i.e., for modules of codimension two, for Gorenstein algebras of codimension three and for complete intersections. The motivation for the conjectures comes from the Multiplicity conjecture of Herzog, Huneke and Srinivasan.

math.AC↗

arXiv2006

Level algebras with bad properties

This paper can be seen as a continuation of the works contained in the recent preprints [Za], of the second author, and [Mi], of Juan Migliore. 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 $e\gg 0$, we will construct a codimension three, type two $h$-vector of socle degree $e$ such that {\em 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 $e\gg 0$. 2). There exist reduced level sets of points in ${\mathbf P}^3$ of type two whose artinian reductions all fail to have the WLP. Indeed, the examples constructed here have the same $h$-vectors we mentioned in 1). 3). For any integer $r\geq 3$, 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 [Mi], Theorem 4.3) that, for any $r\geq 3$, there exist reduced level sets of points in ${\mathbf P}^r$ whose artinian reductions are non-unimodal.

math.AC↗