The paper introduces an algorithm to determine whether a persistence module of pointwise free and finitely-generated modules over a PID splits as a direct sum of interval submodules. If such a decomposition exists, the algorithm outputs it.
The key theoretical result is that a persistence module f over a PID splits into a direct sum of interval modules if and only if the cokernel of every structure map f(a ≤b) is free. This condition is equivalent to several other well-known conditions, such as the module fb splitting as a direct sum I ⊕C for some submodule C ⊆fb, where I is the image of f(a ≤b).
The algorithm relies on the saecular lattice of submodules of each module fa, which is generated by the images and kernels of the structure maps. The authors show that the direct image and inverse image operators induce homomorphisms between these saecular lattices. They then use the freeness of the cokernels to construct complementary submodules, which allows them to build an interval decomposition incrementally.
The 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 the authors are aware.
The results have applications in persistent homology, where interval decompositions provide important invariants like persistence diagrams. The authors show that the persistence diagram of a filtered topological space is independent of the choice of coefficient field if and only if the associated persistence module over the integers splits as a direct sum of interval submodules (subject to a torsion-freeness condition in the next lowest dimension).
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by Jiajie Luo,G... at arxiv.org 04-15-2024
https://arxiv.org/pdf/2310.07971.pdfDeeper Inquiries