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.

翻译：持续同调的研究为科学与工程中诸多有趣问题提供了新的见解与视角。该领域的工作基于以下结论：有限索引的有限维向量空间持续性模允许区间分解——即分解为称作区间模的更简单分量的直和。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。我们提出一种算法，用于判定主理想整环（PID）上逐点自由且有限生成的持续性模是否能分裂为区间子模的直和。若存在这样的分解，则该算法可输出区间分解。若在所选PID上计算史密斯标准型的时间复杂度为有限（或多项式），则我们的算法为有限（或多项式）时间复杂度。据我们所知，这是首个具备这些特性的算法。我们还证明：PID上逐点自由且有限生成的持续性模能分裂为区间子模的直和，当且仅当每个结构映射的余核是自由的。该结论支撑了算法的构建，并补充了Obayashi与Yoshiwaki关于持续同调的先前发现，包括域无关性的判据以及单纯形过滤中持续同调模的分解算法。