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.

翻译：持续研究主要依赖于一个结论：有限索引的有限维向量空间持续模允许区间分解——即分解为区间模的直和。若将向量空间替换为更一般系数环上的模，该结论不再成立。例如，并非每个有限生成自由阿贝尔群的持续模都具有区间分解。然而，经验观察表明，许多此类持续模的有趣实例确实能分解为区间。由于这些模在应用和理论场景中的普遍性，理解区间分解成立的条件至关重要。我们给出了一个充要条件，并提出一种多项式时间算法，该算法能：（a）计算自由阿贝尔群持续模的区间分解，或（b）证明此类分解不存在。这补充了先前关于过滤拓扑空间持久图独立于基域选择特征的研究工作。