Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A hypergraph characterization of nearly complete intersections

Chiara Bondi, Courtney R. Gibbons, Yuye Ke, Spencer Martin, Shrunal Pothagoni, Andrew Stelzer

Abstract

Recently, nearly complete intersection ideals were defined by Boocher and Seiner to establish lower bounds on Betti numbers for monomial ideals (arXiv:1706.09866). Stone and Miller then characterized nearly complete intersections using the theory of edge ideals (arXiv:2101.07901). We extend their work to fully characterize nearly complete intersections of arbitrary generating degrees and use this characterization to compute minimal free resolutions of nearly complete intersections from their degree 2 part.

A hypergraph characterization of nearly complete intersections

Abstract

Recently, nearly complete intersection ideals were defined by Boocher and Seiner to establish lower bounds on Betti numbers for monomial ideals (arXiv:1706.09866). Stone and Miller then characterized nearly complete intersections using the theory of edge ideals (arXiv:2101.07901). We extend their work to fully characterize nearly complete intersections of arbitrary generating degrees and use this characterization to compute minimal free resolutions of nearly complete intersections from their degree 2 part.

Paper Structure

This paper contains 5 sections, 12 theorems, 10 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Hypergraphs and squarefree monomial ideals
  3. NCI-hypergraphs
  4. Resolutions of NCIs
  5. Appendix: Vertex-Weighted Graphs

Key Result

Theorem 1

Let $G$ be a connected graph with at least 3 vertices. The edge ideal $I$ of $G$ is not a nearly complete intersection (generated in degree 2) if and only if there exist vertices $v_1$, $v_2$, $v_3$, $v_4$, $v_5$ in $G$ such that $v_1$ is a leaf in their induced subgraph $H$ and $H$ has a spanning t

Figures (4)

  • Figure 1: Illustration of Lemma \ref{['lemma:commuting-operations']}.
  • Figure 2: Proof illustrations.
  • Figure 3: $G_4$ with the corresponding matching set in red.
  • Figure 4: A joinable hypergraph with its corresponding vertex-weighted graph representation.

Theorems & Definitions (35)

  • Theorem : Miller-Stone 2020
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Remark 2.8
  • ...and 25 more