Interval Decomposition of Persistence Modules over a Principal Ideal Domain

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. We introduce an algorithm to determine whether 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. Our algorithm is finite (respectively, polynomial) time if the problem of computing Smith normal form over the chosen PID is finite (respectively, 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 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 of simplex-wise filtrations.