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Minimal Cellular Resolutions of Path Ideals

Trung Chau, Selvi Kara, Kyle Wang

TL;DR

The paper proves that $p$-path ideals of paths admit minimal Barile-Macchia resolutions (bridge-friendly), and it derives explicit formulas for their projective dimension and graded Betti numbers via Morse resolutions. For cycles, it shows that $p$-path ideals admit minimal generalized Barile-Macchia resolutions, even when Barile-Macchia resolutions are not minimal (as in certain edge ideals of cycles). The approach relies on classifying critical cells through bridges, gaps, and true gaps and leveraging the bridge-friendly property, connecting to Pruned20’s framework. The results yield precise invariants and topological realizations (CW-complex support) for these resolutions. Overall, the work extends discrete Morse-theoretic constructions to a broad class of monomial ideals arising from graph-theoretic paths and cycles, providing both combinatorial formulas and topological interpretations.

Abstract

In this paper, we prove that the path ideals of both paths and cycles have minimal cellular resolutions. Specifically, these minimal free resolutions coincide with the Barile-Macchia resolutions for paths, and their generalized counterparts for cycles. Furthermore, we identify edge ideals of cycles as a class of ideals that lack a minimal Barile-Macchia resolution, yet have a minimal generalized Barile-Macchia resolution.

Minimal Cellular Resolutions of Path Ideals

TL;DR

The paper proves that -path ideals of paths admit minimal Barile-Macchia resolutions (bridge-friendly), and it derives explicit formulas for their projective dimension and graded Betti numbers via Morse resolutions. For cycles, it shows that -path ideals admit minimal generalized Barile-Macchia resolutions, even when Barile-Macchia resolutions are not minimal (as in certain edge ideals of cycles). The approach relies on classifying critical cells through bridges, gaps, and true gaps and leveraging the bridge-friendly property, connecting to Pruned20’s framework. The results yield precise invariants and topological realizations (CW-complex support) for these resolutions. Overall, the work extends discrete Morse-theoretic constructions to a broad class of monomial ideals arising from graph-theoretic paths and cycles, providing both combinatorial formulas and topological interpretations.

Abstract

In this paper, we prove that the path ideals of both paths and cycles have minimal cellular resolutions. Specifically, these minimal free resolutions coincide with the Barile-Macchia resolutions for paths, and their generalized counterparts for cycles. Furthermore, we identify edge ideals of cycles as a class of ideals that lack a minimal Barile-Macchia resolution, yet have a minimal generalized Barile-Macchia resolution.
Paper Structure (6 sections, 29 theorems, 38 equations, 1 figure, 1 algorithm)

This paper contains 6 sections, 29 theorems, 38 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Morse resolutions
  4. Barile-Macchia matchings and Barile-Macchia resolutions.
  5. Minimal free resolutions of path ideals of paths
  6. Minimal free resolutions of path ideals of cycles

Key Result

Theorem 2.2

BW02 Let $A$ be a homogeneous acyclic matching of $I$. Then $A$ induces a cellular resolution $\mathcal{F}_A$, where: The resulting (cellular) free resolution $\mathcal{F}_A$ is called the Morse resolution of $R/I$ associated to $A$. Furthermore, $\mathcal{F}_A$ is minimal if for any two $A$-critical cells $\sigma$ and $\sigma'$ with $|\sigma'|=|\sigma|-1$, we have $\mathop{\mathrm{lcm}}\nolimits

Figures (1)

  • Figure 1: Only possible orderings of $i,j,k$ along $C_n$: $j<i<k$ (left) and $j<k<i<j+n$ (right).

Theorems & Definitions (64)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Corollary 2.10
  • Definition 2.11
  • ...and 54 more