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Interval Decomposition of Infinite Persistence Modules over a Principal Ideal Domain

Jiajie Luo, Gregory Henselman-Petrusek

TL;DR

This work extends interval decomposition from finite to infinite indexing for pointwise free, finitely-generated persistence modules over a PID. The authors prove a necessary and sufficient condition: $f:I\to R\text{-Mod}$ decomposes into interval modules iff every structure map $f(a\le b)$ has free cokernel, leveraging consistent bases and a Zorn's lemma extension to build the decomposition. They provide a constructive framework via partitioning bases into birth/death blocks and an incremental extension argument, generalizing finite-index results and connecting to torsion-free persistent homology. The application to integer persistence links the decomposition to coefficient-field independence of persistence diagrams through universal coefficient-type arguments and a short exact sequence, highlighting practical implications for topological data analysis with non-field coefficients.

Abstract

We study pointwise free and finitely-generated persistence modules over a principal ideal domain, indexed by a (possibly infinite) totally-ordered poset category. We show that such persistence modules admit interval decompositions if and only if every structure map has free cokernel. We also show that, in torsion-free settings, the integer persistent homology module of a filtration of topological spaces admits an interval decomposition if and only if the associated persistence diagram is invariant to the choice of coefficient field. These results generalize prior work where the indexing category is finite.

Interval Decomposition of Infinite Persistence Modules over a Principal Ideal Domain

TL;DR

This work extends interval decomposition from finite to infinite indexing for pointwise free, finitely-generated persistence modules over a PID. The authors prove a necessary and sufficient condition: decomposes into interval modules iff every structure map has free cokernel, leveraging consistent bases and a Zorn's lemma extension to build the decomposition. They provide a constructive framework via partitioning bases into birth/death blocks and an incremental extension argument, generalizing finite-index results and connecting to torsion-free persistent homology. The application to integer persistence links the decomposition to coefficient-field independence of persistence diagrams through universal coefficient-type arguments and a short exact sequence, highlighting practical implications for topological data analysis with non-field coefficients.

Abstract

We study pointwise free and finitely-generated persistence modules over a principal ideal domain, indexed by a (possibly infinite) totally-ordered poset category. We show that such persistence modules admit interval decompositions if and only if every structure map has free cokernel. We also show that, in torsion-free settings, the integer persistent homology module of a filtration of topological spaces admits an interval decomposition if and only if the associated persistence diagram is invariant to the choice of coefficient field. These results generalize prior work where the indexing category is finite.

Paper Structure

This paper contains 11 sections, 19 theorems, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Persistence modules
  4. A brief discussion on finitely-indexed persistence modules
  5. Necessity
  6. Sufficiency
  7. Consistent bases and Zorn's condition
  8. Partitioning consistent bases with images and kernels
  9. Incremental Extension
  10. Application: Integer decomposition, relative homology, and field-choice independence in persistent homology
  11. A useful short exact sequence

Key Result

Theorem 1.1

Let $f:I\to R\mathrm{-Mod}$ be a pointwise free and finitely-generated persistence module, where $I$ may be infinite and $R$ is a PID. Then $f$ admits an interval decomposition if and only if every map $f(a\leq b)$ has free cokernel.

Figures (2)

  • Figure 1: The four cases for defining $\beta_a^{pq}$, for different values of $i_p$ and $i_q$.
  • Figure 2: Indices in the proof of Theorem \ref{['thm:matching_matrix_general']}. For $c,b\in J$ such that $c < a < b$, we choose $i_u,i_\ell \in S$ such that $\mathrm{Im}[c,a] = \mathrm{Im}[i_u,a]$ and $\ker[a,b] = \ker[a, i_\ell]$. The index $i_u$ (respectively, $i_\ell$) may be greater or less than $c$ (respectively, $b$).

Theorems & Definitions (46)

  • Theorem 1.1: Main result
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Lemma 2.9
  • ...and 36 more