Interval Decomposition of Persistence Modules over a Principal Ideal Domain

TL;DR

This work addresses the problem of interval decompositions for persistence modules over a principal ideal domain (PID). It proves a necessary-and-sufficient condition: a pointwise free and finitely-generated module $f$ splits into interval submodules if and only if the cokernel of every structure map $f(a\le b)$ is free, and it provides a constructive matrix-algebra algorithm to compute the interval decomposition when it exists. The algorithm is finite and polynomial-time for coefficient rings like $\mathbb{Z}$ or $\mathbb{Q}[x]$, and the paper connects this result to established field-independence criteria (Obayashi–Yoshiwaki) as well as the concept of universal cycle bases via the saecular lattice and Smith normal form. These contributions extend persistence theory beyond fields, offering practical tools for PID-coefficient persistence and linking algebraic structure to topological invariants. Overall, the work furnishes both a rigorous criterion for when interval decompositions exist over PIDs and an explicit, efficient method to compute them, with broad implications for generalized persistence and related invariants.

Abstract

The study of persistent homology has contributed new insights and perspectives into a variety of interesting problems in science and engineering. Work in this domain relies on the result that any finitely-indexed persistence module of finite-dimensional vector spaces admits an interval decomposition -- that is, a decomposition as a direct sum of simpler components called interval modules. This result fails if we replace vector spaces with modules over more general coefficient rings. To address this problem, we introduce an algorithm to determine whether or not a persistence module of pointwise free and finitely-generated modules over a principal ideal domain (PID) splits as a direct sum of interval submodules. If one exists, our algorithm outputs an interval decomposition. When considering persistence modules with coefficients in $\Z$ or $\Q[x]$, our algorithm computes an interval decomposition in polynomial time. This is the first algorithm with these properties of which we are aware. We also show that a persistence module of pointwise free and finitely-generated modules over a PID splits as a direct sum of interval submodules if and only if the cokernel of every structure map is free. This result underpins the formulation of our algorithm. It also complements prior findings by Obayashi and Yoshiwaki regarding persistent homology, including a criterion for field independence and an algorithm to decompose persistence homology modules.