We study pointwise free and finitely-generated persistence modules over a principal ideal domain, indexed by a (possibly infinite) totally-ordered poset category. We show that such persistence modules admit interval decompositions if and only if every structure map has free cokernel. We also show that, in torsion-free settings, the integer persistent homology module of a filtration of topological spaces admits an interval decomposition if and only if the associated persistence diagram is invariant to the choice of coefficient field. These results generalize prior work where the indexing category is finite.

翻译：研究主理想整环上逐点自由且有限生成的持续模，其指标集为（可能无限）全序偏序范畴。我们证明，此类持续模存在区间分解当且仅当每个结构映射都具有自由余核。同时证明，在无挠情形下，拓扑空间滤子的整数持续同调模存在区间分解当且仅当关联的持续图不依赖于系数域的选择。这些结果推广了指标集为有限情形的已有工作。