Interval Decompositions for Persistence Modules of Free Abelian Groups

The study of persistence rests largely 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 interval modules. This result fails if we replace vector spaces with modules over more general coefficient rings. For example, not every persistence module of finitely-generated free abelian groups admits an interval decomposition. Nevertheless, many interesting examples of such persistence modules have been empirically observed to decompose into intervals. Due to the prevalence of these modules in applied and theoretical settings, it is important to understand the conditions under which interval decomposition is possible. We provide a necessary and sufficient condition, and a polynomial-time algorithm to either (a) compute an interval decomposition of a persistence module of free abelian groups, or (b) certify that no such decomposition exists. This complements earlier work, which characterizes filtered topological spaces whose persistence diagrams are independent of the choice of ground field.