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. Contents 1. Introduction 1 1.1. Organization and main results 2 1.2. Acknowledgements 3 2. Relative homological algebra 3 3. Spread modules 6 3.1. Morphisms between spread modules 8 4. Homological invariants in persistence theory 10 4.1. Projectivisation 10 4.2. Homological invariants 12 5. Examples and comparison to other invariants 13 5.1. The dimension vector 13 5.2. The barcode 13 5.3. The rank invariant 13 5.4. Compressed multiplicities 15 References 15 2020 Mathematics Subject Classification. 55N31, 16E20 (primary); 16Z05, 18G35 (secondary). Key words and phrases. persistence modules, invariants, Grothendieck groups, relative homological algebra, exact structures. 1 There is an even more general notion of a persistence module over a small category, see [BCB20].

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Classes of groups and semigroups have been defined in regards to their computation complexity in the past, notably by Epstein et al. with their early work on automatic groups, see [1]. More general classes have followed. First, the asynchronous case for groups was defined and studied, notably by the same group of authors. Afterwards, monoids and semigroups have been investigated, notably by Campbell et al., see [4], and Duncan et al., see [5]. However, a key feature of automatic groups does not hold for automatic semigroups; while the notion of automaticity for finitely generated groups does not depend on the choice of generators, it does for finitely generated semigroups.Showing there exists no automatic structure for a given generating set is doable, but doing so for any generating set can be tedious, see for example [6], where Hoffmann and Thomas build a finitely generated commutative semigroup which is not automatic under any choice of generating set. This makes the feature extremely valuable. Recently, Blanchette et al. introduced quasi-automatic semigroups, a more general class of semigroups for which the feature actually holds, see [2].Several relations between these classes have been established. For instance, we know that the class of automatic groups sits strictly inside the class of asynchronously automatic groups, which itself is contained into the class of quasi-automatic groups. Geometric characterizations have also been made. Automatic groups are exactly those having the so-called Lipschitz property. Asynchronously automatic groups are characterized by the Lipschitz Hausdorff property of Epstein et al., along with a special function named a departure function. For quasi-automatic groups, we have a different weak form of the Lipschitz property that again characterizes the class geometrically.Our main result here is to establish a new link between those classes; we will show that for groups, being quasi-automatic is equivalent to being asynchronously automatic. This is particularly useful because quasi-automatic structures are somewhat natural, as opposed to the more complex asynchronously automatic structures. Context and definitionsLet A be a generating set of a semigroup S and p : A + ։ S be the canonical map which maps a word to the element of S it represents. We write u ∼ v if u and v represent the same element of S. A rational language L ⊆ A + that maps onto S is called a dictionary. A dictionary is called quasi-automatic if for each a in A ∪ {ε}, the relation R L a = {(u, v) ∈ L × L|ua ∼ v} is rational. If these relations are also recognizable by two-tape automata, this dictionary is called asynchronously automatic. We use a theorem of Nivat (see [3], proposition 4) to work with these rational relations.The theorem states that a relation R is rational if and only if there exist some finite B, some ra-1