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The study of persistence relies heavily on the result that anyfinitely-indexed persistence module of finite-dimensional vector spaces admitsan interval decomposition -- that is, a decomposition as a direct sum ofinterval modules.  This result fails if we replace vector spaces with modules over more generalcoefficient rings. For example, not every persistence module offinitely-generated free abelian groups admits an interval decomposition.Nevertheless, many interesting examples of persistence modules of abeliangroups have been found to decompose into intervals, empirically.  Due to the prevalence of these modules in applied and theoretical settings,it is important to understand the conditions under which interval decompositionis possible. We provide a necessary and sufficient condition, and apolynomial-time algorithm to either (a) compute an interval decomposition of apersistence module of freely- and finitely-generated modules over PIDs, or (b)certify that no such decomposition exists. This complements earlier work, whichcharacterizes filtered topological spaces whose persistence diagrams areindependent of the choice of coefficient field.

Interval Decomposition for Persistence Modules Freely Generated over PIDs