228 research outputs found

    Reducible family of height three level algebras

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    Let R=k[x1,...,xr]R=k[x_1,..., x_r] be the polynomial ring in rr variables over an infinite field kk, and let MM be the maximal ideal of RR. Here a \emph{level algebra} will be a graded Artinian quotient AA of RR having socle Soc(A)=0:MSoc(A)=0:M in a single degree jj. The Hilbert function H(A)=(h0,h1,...,hj)H(A)=(h_0,h_1,... ,h_j) gives the dimension hi=dimkAih_i=\dim_k A_i of each degree-ii graded piece of AA for 0ij0\le i\le j. The embedding dimension of AA is h1h_1, and the \emph{type} of AA is \dim_k \Soc (A), here hjh_j. The family \Levalg (H) of level algebra quotients of RR having Hilbert function HH 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=H1=(1,3,4,4)H=H_1=(1,3,4,4) and H=H2=(1,3,6,8,9,3)H=H_2=(1,3,6,8,9,3) the family LevAlg(H)LevAlg (H) parametrizing level Artinian algebras of Hilbert function HH 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)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

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    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

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    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 hh-vectors that cannot be Gorenstein hh-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)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

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    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

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    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

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    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

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    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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