  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 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.
