Homological approximations in persistence theory

TL;DR

This paper introduces homological invariants for persistence modules over finite posets, showing their relation to existing invariants and proposing a new, finer invariant based on spread modules.

Contribution

It defines a new class of homological invariants for persistence modules and demonstrates their equivalence to known invariants, also introducing a finer invariant using spread modules.

Findings

Dimension vector and rank invariant are equivalent to homological invariants.

The new invariant based on single-source spread modules is finer than the rank invariant.

Homological invariants respect homological data and are generated by indecomposable modules.

Abstract

We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data and takes values in the free abelian group generated by a finite set of indecomposable modules. We focus in particular on groups generated by "spread modules", which are sometimes called "interval modules" in the persistence theory literature. We show that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules. We also show that the free abelian group generated by the "single-source" spread modules gives rise to a new invariant which is finer than the rank invariant. They are also thankful to an anonymous referee for their thorough reading of this paper and suggestions for improvement.