Abstract
The Representation Theorem by Zomorodian and Carlsson has been the starting point of the study of persistent homology under the lens of representation theory. In this work, we give a more accurate statement of the original theorem and provide a complete and self-contained proof. Furthermore, we generalize the statement from the case of linear sequences of R-modules to R-modules indexed over more general monoids. This generalization subsumes the Representation Theorem of multidimensional persistence as a special case.
Highlights
Persistent homology, introduced by Edelsbrunner et al (2002), is a multi-scale extension of classical homology theory
The idea is to track how homological features appear and disappear in a shape when the scale parameter is increasing. This data can be summarized by a barcode where each bar corresponds to a homology class that appears in the process and represents the range of scales where the class is present
For G = Nk, we prove that this condition is equivalent to the property that all sequences in our persistence module are of finite type, see Fig. 2, but this equivalence fails for general monoids
Summary
Persistent homology, introduced by Edelsbrunner et al (2002), is a multi-scale extension of classical homology theory. In Zomorodian and Carlsson (2005), the authors assign an R[t]-module to a persistence module of finite type and state: The proof is the Artin–Rees theory in commutative algebra (Eisenbud 1995). The persistence module (M) as in Zomorodian’s proof is not of finitely generated type, because no inclusion Mi → Mi+1 is an isomorphism This counterexample raises the question: what are the requirements on the ring R to make the claimed correspondence valid? The statement implies the ZC-Representation Theorem for commutative Noetherian rings, because if R is commutative with unity and Noetherian, finitely generated modules are finitely presented. Our second main result is that finitely presented graded modules over R[G] correspond again to generalized persistence modules with a finiteness condition.
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