A refinement of multi-dimensional persistence

We study the multi-dimensional persistence of Carlsson and Zomorodian and obtain a finer classification based upon the higher tor-modules of a persistence module. We propose a variety structure on the set of isomorphism classes of these modules, and present several examples. We also provide a geometric interpretation for the higher tor-modules of homology modules of multi-filtered simplicial complexes.

💡 Research Summary

The paper revisits the theory of multi‑dimensional persistence introduced by Carlsson and Zomorodian and proposes a substantially finer invariant for classifying persistence modules. The authors observe that the classical approach—characterising a module over the polynomial ring R = 𝕜