What the study found
The study found a characterization of when pointwise free and finitely generated persistence modules over a principal ideal domain can be split into intervals. It also found a link, in torsion-free settings, between interval decompositions of integer persistent homology and whether the persistence diagram is invariant across coefficient fields.
Why the authors say this matters
The authors say these results generalize earlier work that only covered finite indexing categories. The study suggests this extends the framework for understanding persistent homology and persistence modules in more general settings.
What the researchers tested
The researchers studied pointwise free and finitely generated persistence modules over a principal ideal domain, indexed by a possibly infinite totally ordered poset category. They also examined integer persistent homology of filtrations of topological spaces in torsion-free settings and compared persistence diagrams across coefficient fields.
What worked and what didn't
They showed that such persistence modules admit interval decompositions if and only if every structure map has free cokernel. They also showed that, in torsion-free settings, integer persistent homology admits an interval decomposition if and only if the associated persistence diagram is invariant to the choice of coefficient field.
What to keep in mind
The abstract does not describe experimental data, examples, or numerical results. It also limits the second result to torsion-free settings, and the summary provided does not include further limitations beyond the scope described.
Key points
- Persistence modules over a principal ideal domain have interval decompositions exactly when every structure map has free cokernel.
- The second result applies to torsion-free integer persistent homology of filtrations of topological spaces.
- In that setting, interval decompositions correspond to persistence diagrams that do not change with the choice of coefficient field.
- The results extend prior work from finite indexing categories to possibly infinite totally ordered poset categories.
Disclosure
- Research title:
- Interval decompositions depend on free cokernels and field choice
- Authors:
- Jiajie Luo, Gregory Henselman‐Petrusek
- Publication date:
- 2026-04-24
- DOI:
- 10.1090/proc/17733
- OpenAlex record:
- View
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