Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Depth of powers of edge ideals of edge-weighted integrally closed cycles

Guangjun Zhu, Jiaxin Li, Yijun Cui, Yi Yang

TL;DR

This work determines the depth of powers of the edge ideal for edge-weighted integrally closed cycles. Using depth lemmas, short exact sequences, and the integral-closure characterization via forbidden induced subgraphs, the authors derive explicit formulas that depend on how many edges carry non-trivial weights: three non-trivial edges give depth $2$ for all powers, two non-trivial edges give depth $\max\{\lceil (n-t)/3\rceil,1\}$, and a single non-trivial edge yields a piecewise depth that depends on $t$ and the parity of $n$. The base case depth$(S/I(C_{f w}^n))$ is shown to be $\lceil (n-1)/3\rceil$, and these results collectively extend known unweighted-cycle and weighted-path depth results, providing exact, computable homological invariants for a broad class of weighted cycles.

Abstract

This paper gives some exact formulas for the depth of powers of the edge ideal of an edge-weighted integrally closed cycle.

Depth of powers of edge ideals of edge-weighted integrally closed cycles

TL;DR

This work determines the depth of powers of the edge ideal for edge-weighted integrally closed cycles. Using depth lemmas, short exact sequences, and the integral-closure characterization via forbidden induced subgraphs, the authors derive explicit formulas that depend on how many edges carry non-trivial weights: three non-trivial edges give depth for all powers, two non-trivial edges give depth , and a single non-trivial edge yields a piecewise depth that depends on and the parity of . The base case depth is shown to be , and these results collectively extend known unweighted-cycle and weighted-path depth results, providing exact, computable homological invariants for a broad class of weighted cycles.

Abstract

This paper gives some exact formulas for the depth of powers of the edge ideal of an edge-weighted integrally closed cycle.
Paper Structure (6 sections, 30 theorems, 88 equations)

This paper contains 6 sections, 30 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The depth of the edge ideal
  4. The depth of powers of the edge ideal
  5. $C_{\bf w}^n$ has at least two edges with non-trivial weights
  6. $C_{\bf w}^n$ has only one edge with non-trivial weights

Key Result

Theorem 1.1

Let $C_{\bf w}^n$ be a non-trivially weighted integrally closed $n$-cycle. Then we have

Theorems & Definitions (47)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 37 more