Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Betti numbers of ideals generated by $n+1$ powers of general linear forms

Eric Dannetun

Abstract

We study ideals generated by $n+1$ powers of general linear forms in $R= k[x_1,\dots,x_n]$. By generalizing the ideas in a recent paper of Diethorn et al., we determine the Betti numbers of such ideals when at least one generator is a square. It follows that all such ideals are level. As a consequence, we show that a generic ideal in $R$ generated by $n+1$ forms, with at least one quadric generator, is level. We also determine the Betti numbers of the Artinian Gorenstein algebras linked to these almost complete intersections. By describing the dual generators of these algebras, we obtain a family of forms, including the elementary symmetric polynomials, whose annihilator ideals have the strong Lefschetz property. Finally, we give explicit generators for the annihilator ideal of any elementary symmetric polynomial.

Betti numbers of ideals generated by $n+1$ powers of general linear forms

Abstract

We study ideals generated by powers of general linear forms in . By generalizing the ideas in a recent paper of Diethorn et al., we determine the Betti numbers of such ideals when at least one generator is a square. It follows that all such ideals are level. As a consequence, we show that a generic ideal in generated by forms, with at least one quadric generator, is level. We also determine the Betti numbers of the Artinian Gorenstein algebras linked to these almost complete intersections. By describing the dual generators of these algebras, we obtain a family of forms, including the elementary symmetric polynomials, whose annihilator ideals have the strong Lefschetz property. Finally, we give explicit generators for the annihilator ideal of any elementary symmetric polynomial.
Paper Structure (6 sections, 24 theorems, 70 equations, 1 figure)

This paper contains 6 sections, 24 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Preliminaries
  4. Resolutions of $R/I$ and $R/G$
  5. SLP for the linked Artinian Gorenstein algebra
  6. Generators of $G$

Key Result

Lemma 2.2

Let $\phi\colon M\to N$ be a module homomorphism inducing an isomorphism $M_{(\tau)}\cong N_{(\tau)}.$ If $F_\bullet \to M$ and $G_\bullet \to N$ are minimal free resolutions of $M$ and $N$, then any lift of $\phi$ to a morphism $F_\bullet \to G_\bullet$ restricts to an isomorphism $F_\bullet^{(\tau

Theorems & Definitions (50)

  • Remark 2.1
  • Lemma 2.2: Lemma 3.4 in Pardue-Richert
  • Remark 2.3
  • Theorem 2.4
  • Conjecture 2.5: Fröberg's conjecture Frobergs-formodan
  • Theorem 2.6: Complete-Intersections
  • Theorem 2.7
  • proof : Proof sketch
  • Corollary 2.8
  • proof
  • ...and 40 more