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Betti Numbers of Edge Ideals of Weighted Oriented Crown Graphs

Zexin Wang, Dancheng Lu

TL;DR

We address the problem of determining the Betti numbers for edge ideals of weighted oriented crown graphs. The authors introduce an induced-subgraph approach that reduces global Betti data to information from induced subgraphs, combined with Betti splitting and dominant-ideal techniques to obtain exact formulas. The main results include a weight-independent closed form for the total Betti numbers $\beta_i(I_n)$, a complete multigraded description $\beta_{i,\alpha}(I_n)$, and a formula for the regularity $\mathrm{reg}(I_n)=\sum_{i=1}^n w_i - n + 3$, with the all-ones weight case recovering known crown-graph results RS1/RS2. This work provides a complete multigraded resolution for this class and demonstrates that induced-subgraph data governs the global algebraic invariants.

Abstract

We compute the multigraded Betti numbers of edge ideals for weighted oriented crown graphs using a novel approach. This approach, which we still call the \emph{induced subgraph approach}, originates from our prior work on computing Betti numbers of normal edge rings(see \cite{WL}). Notably, we prove that the total Betti numbers of edge ideals for weighted oriented crown graphs are independent of the weight function.

Betti Numbers of Edge Ideals of Weighted Oriented Crown Graphs

TL;DR

We address the problem of determining the Betti numbers for edge ideals of weighted oriented crown graphs. The authors introduce an induced-subgraph approach that reduces global Betti data to information from induced subgraphs, combined with Betti splitting and dominant-ideal techniques to obtain exact formulas. The main results include a weight-independent closed form for the total Betti numbers , a complete multigraded description , and a formula for the regularity , with the all-ones weight case recovering known crown-graph results RS1/RS2. This work provides a complete multigraded resolution for this class and demonstrates that induced-subgraph data governs the global algebraic invariants.

Abstract

We compute the multigraded Betti numbers of edge ideals for weighted oriented crown graphs using a novel approach. This approach, which we still call the \emph{induced subgraph approach}, originates from our prior work on computing Betti numbers of normal edge rings(see \cite{WL}). Notably, we prove that the total Betti numbers of edge ideals for weighted oriented crown graphs are independent of the weight function.
Paper Structure (6 sections, 18 theorems, 47 equations)

This paper contains 6 sections, 18 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Upper Bound
  4. Betti Numbers of Edge Ideals of Induced Subgraphs
  5. Induced Subgraphs
  6. Main results

Key Result

Lemma 1.1

Let $H$ be an induced subgraph of a weighted oriented graph $D$. Then, for $\mathbf{a} \in \mathbb{N}^{V(D)}$ with $\mathrm{supp}(\mathbf{a}) \subseteq V(H)$, where $\mathrm{supp}(\mathbf{a})$ is defined as $\mathrm{supp}(x^{\mathbf{a}})$, we have

Theorems & Definitions (33)

  • Lemma 1.1
  • Theorem 1.2: Proposition \ref{['Gs betti']} and Theorem \ref{['main2']}
  • Theorem 1.3: Theorem \ref{['main']} and Corollary \ref{['crown reg']}
  • Definition 2.1
  • Corollary 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • ...and 23 more