228 research outputs found

### 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

### 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.Comment: 24 pages, references and examples adde

### 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.Comment: A few revisions. Final version to appear in JPA

### 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 zero, we describe the supporting hyperplanes and extreme rays for the
cones generated by the Hilbert functions of all modules, all modules with
bounded a-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.Comment: 20 pages, 2 figure

### Poset structures in Boij-S\"oderberg theory

Boij-S\"oderberg theory is the study of two cones: the cone of cohomology
tables of coherent sheaves over projective space and the cone of standard
graded minimal free resolutions over a polynomial ring. Each cone has a
simplicial fan structure induced by a partial order on its extremal rays. We
provide a new interpretation of these partial orders in terms of the existence
of nonzero homomorphisms, for both the general and the equivariant
constructions. These results provide new insights into the families of sheaves
and modules at the heart of Boij-S\"oderberg theory: supernatural sheaves and
Cohen-Macaulay modules with pure resolutions. In addition, our results strongly
suggest the naturality of these partial orders, and they provide tools for
extending Boij-S\"oderberg theory to other graded rings and projective
varieties.Comment: 23 pages; v2: Added Section 8, reordered previous section

### The cone of Betti diagrams over a hypersurface ring of low embedding dimension

We give a complete description of the cone of Betti diagrams over a standard
graded hypersurface ring of the form k[x,y]/, where q is a homogeneous
quadric. We also provide a finite algorithm for decomposing Betti diagrams,
including diagrams of infinite projective dimension, into pure diagrams.
Boij--Soederberg theory completely describes the cone of Betti diagrams over a
standard graded polynomial ring; our result provides the first example of
another graded ring for which the cone of Betti diagrams is entirely
understood.Comment: Minor edits, references update

### Three flavors of extremal Betti tables

We discuss extremal Betti tables of resolutions in three different contexts.
We begin over the graded polynomial ring, where extremal Betti tables
correspond to pure resolutions. We then contrast this behavior with that of
extremal Betti tables over regular local rings and over a bigraded ring.Comment: 20 page

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