Homological approximations in persistence theory

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 exact structure 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 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.