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Symmetry, Unimodality, and Lefschetz Properties for Graded Modules

Zachary Flores

Abstract

We investigate the Weak Lefschetz Properties for modules whose minimal free resolutions are given by generalized Kosuzl complexes in dimension three through a careful study of their Betti numbers and the symmetry and unimodality of their Hilbert functions. We also study the non-Lefschetz locus for finite length modules in arbitrary dimension, and are able to generalize several previous results on the non-Lefschetz locus in this setting. Along the way, we find several connections with a Gorenstein analogue for finite length modules and Artin level modules that are both interesting and useful throughout this paper.

Symmetry, Unimodality, and Lefschetz Properties for Graded Modules

Abstract

We investigate the Weak Lefschetz Properties for modules whose minimal free resolutions are given by generalized Kosuzl complexes in dimension three through a careful study of their Betti numbers and the symmetry and unimodality of their Hilbert functions. We also study the non-Lefschetz locus for finite length modules in arbitrary dimension, and are able to generalize several previous results on the non-Lefschetz locus in this setting. Along the way, we find several connections with a Gorenstein analogue for finite length modules and Artin level modules that are both interesting and useful throughout this paper.

Paper Structure

This paper contains 8 sections, 25 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. The Minimal Free Resolution of $M$
  3. The map $\varepsilon$
  4. The map $\delta$
  5. Computing the $c_j$ and $d_i$
  6. The Unimodality and Symmetry of the Hilbert Function of $M$
  7. Lefschetz Properties for $M$
  8. The non-Lefschetz Locus for Graded Modules

Key Result

Lemma 2.1

If $\varphi: \bigoplus_{j=1}^{n+2} R(-b_j) \longrightarrow \bigoplus_{i=1}^n R(-a_i)$ is as in eq: defM, then $b_i > a_i$ for $i = 1,\ldots, n$.

Theorems & Definitions (56)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 46 more