Interval Decomposition of Infinite Persistence Modules over a Principal Ideal Domain and Field Choice in Persistent Homology

TL;DR

This paper characterizes when infinite persistence modules over a principal ideal domain admit interval decompositions, linking algebraic conditions to invariance of persistence diagrams across coefficient fields.

Contribution

It extends the theory of interval decompositions to infinite modules and relates diagram invariance to the algebraic structure of the modules.

Findings

Persistence modules admit interval decompositions iff structure maps have free cokernel.

Integer persistent homology admits interval decomposition iff persistence diagram is field-invariant.

Results generalize finite-indexing prior work.

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.