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：我们研究主理想整环上逐点自由且有限生成的持久模，其指标集为（可能无限的）全序偏序范畴。我们证明：此类持久模存在区间分解当且仅当每个结构映射具有自由余核。此外，在无挠条件下，拓扑空间过滤的整数持续同调模存在区间分解当且仅当关联的持久图对系数域的选择具有不变性。这些结论推广了先前局限于有限指标范畴的研究工作。