2.1 Spread modules

In this section, we define a subclass of $I(\mathcal {P})$ -modules which we refer to as spread modules. As we explain in Remark 2.5, these are often referred to as “interval modules” in the persistence theory literature. We will study this class of modules in more details in Section 5.

Let $a\in \mathcal {P}$ , and let $B\subseteq \mathcal {P}$ be a set of incomparable elements. We say $a\leq B$ (resp. $a\geq B$ ) if there exists $b\in B$ such that $a\leq b$ ( resp. $a\geq b$ ). Sets of incomparable elements of $\mathcal {P}$ can then be partially ordered by the relation

$$ \begin{align*}A\leq B \iff \forall_{a \in A}\forall_{b \in B} \ a\leq B \text{ and } A\leq b.\end{align*} $$

Let Q be the Hasse quiver of $\mathcal {P}$ . We call $[a,c]:=\{b \mid a\leq b\leq c\}$ the interval from a to c. A subset S of $\mathcal {P}$ is called connected if it is a connected part of Q; that is, there exists a non-oriented path between any two points of S. A subset S of $\mathcal {P}$ is called convex if for any pair $(a,b)\in S$ , $[a,b]\subseteq S$ .

Clearly, intervals are connected and convex. We generalize the notion of interval with the following definition.

Definition 2.1 Let $A, B \subseteq \mathcal {P}$ be two sets of incomparable elements with $A \leq B$ . The spread from A to B is the subset

$$ \begin{align*}[A, B] := \{ x\; | \text{ there exists } a\in A, b\in B \text{ such that } a\leq x \leq b\}.\end{align*} $$

We refer to A and B as the sets of sources and targets of $[A,B]$ , respectively.

If $A = \{a\}$ contains only a single element, we call $[A,B]$ a single-source spread and write $[a,B]$ in place of $[\{a\},B]$ . Single-target spreads are defined and notated analogously. Note that spreads that are both single-source and single-target are precisely intervals (as in the classical poset-theory literature; see, e.g., [Reference StanleySta11]). Spreads are also convex. Indeed, if $x,z\in [A,B]$ , then there exist a and b such that $a\leq x$ and $z\leq b$ , so for any $y\in [x,z]$ , we have $a\leq x\leq y\leq z \leq b$ , showing that it is convex. In fact, all finite convex subsets are spreads, as made precise in the following proposition.

Proof It is clear that A and B are sets of incomparable elements and that $A \leq B$ . Now, take $y\in X$ to be neither maximal nor minimal. Then there exist elements in X that are, respectively, bigger and smaller than y, say $x_1\leq y \leq z_1$ . We then iteratively find bigger and smaller elements, obtaining a sequence

$$ \begin{align*}x_n\leq \cdots \leq x_1 \leq y \leq z_1 \leq \cdots \leq z_m,\end{align*} $$

where $x_n$ is minimal and $z_m$ is maximal. This sequence must end in a finite number of steps because $\mathcal {P}$ , and thus also X, is finite.

We also fix the following notation for convenience.

Definition 2.3 Let $X\subseteq \mathcal {P}$ be a set of incomparable elements. By Proposition 2.2, the sets

$$ \begin{align*}[-\infty,X]:=\{y \mid y \leq X\} \qquad [X,\infty]:=\{y \mid X \leq y\}\end{align*} $$

are spreads. We call them downsets and upsets, respectively. Note that downsets are sometimes called “order ideals,” and are characterized by being closed under taking predecessors. Likewise upsets are sometimes called “order filters,” and are characterized by being closed under taking successors.

Definition 2.4 Given a spread $[A,B]\subseteq \mathcal {P}$ , we call spread module the persistence module $M_{[A,B]}$ defined by

$$\begin{align*}M_{[A,B]}(p)= \begin{cases} K, & \text{ if } p\in [A,B],\\ 0, & \text{ if } p\not\in [A,B], \end{cases} \quad \quad \quad \quad M_{[A,B]}(p,q)= \begin{cases} 1_K, & \text{ if } p,q \in [A,B],\\ 0, & \text{ else.} \end{cases} \end{align*}$$

If in addition $|A| = 1 = |B|$ , we call $M_{[A,B]}$ an interval module. We likewise define single-source spread modules, single-target spread modules, upset modules, and downset modules in the natural way.

The following are some important examples of spread modules.

By construction, if M is a spread module, then $\dim M(a) \leq 1$ for all $a \in \mathcal {P}$ . Modules with this property are sometimes called thin. Another motivation for studying spread modules is the following.

As previously mentioned, if $\mathcal {P}$ is totally ordered, then all indecomposable modules are spread modules. Theorem 2.8 thus points toward a natural invariant: the multiplicity of thin modules in the direct sum decomposition. If the quiver of $\mathcal {P}$ is Dynkin type A, this invariant is equivalent to the barcode. More generally, however, this invariant loses all information about any indecomposable direct summand which is not thin. Nevertheless, in some applications, it looks like the proportion of non-thin factors is small, making this invariant somewhat valuable. See, e.g., [Reference Escolar and HiraokaEH16, Section 5] for further discussion.

An immediate consequence of Theorem 2.8 is that, over these particular posets, thin indecomposable modules are uniquely determined by their support. For more general posets, however, there are thin indecomposables which are neither isomorphic to spread modules nor determined by their support, as seen in the following example.

Example 2.9 As in [Reference MillerMil, Example 2.7], let $\mathcal {P}$ be the poset with Hasse diagram.

Note that the incidence algebra $I(\mathcal {P})$ is in this case a path algebra of type $\widetilde {A}_3$ . Now, for $\lambda \in K$ , let $N_\lambda $ be the following thin module.

It is well known that each $N_\lambda $ is indecomposable and that $N_\lambda \simeq N_{\lambda '}$ if and only if $\lambda = \lambda '$ . Moreover, we observe that $N_\lambda $ is a spread module if and only if $\lambda = 1$ (in which case we have $N_\lambda = M_{[\{1,2\},\{3,4\}]}$ ). This shows that $\mathsf {mod}I(\mathcal {P})$ contains thin indecomposables which are neither isomorphic to spread modules nor determined by their support.

3.1 The dimension vector/Hilbert function

We begin by overviewing many different ways that one may interpret the dimension vector. Recall that for a persistence module M, the dimension vector (or Hilbert function) is defined by

$$ \begin{align*}\underline{\mathrm{dim}} M = (\dim M(a))_{a \in \mathcal{P}}.\end{align*} $$

This invariant is often considered as taking values in the free abelian group with basis $\mathcal {P}$ ; however, we can also see $\underline {\mathrm {dim}}{M}$ as an element of the free abelian group with basis $\{[S_a] \mid a \in \mathcal {P}\}$ . (Precisely, we are treating $\underline {\mathrm {dim}} M$ as an element of the classical Grothendieck group of $\mathsf {mod} I(\mathcal {P})$ . See Section 4.) The second choice highlights the fact that given a short exact sequence

$$ \begin{align*}0 \rightarrow L \rightarrow M \rightarrow N \rightarrow 0\end{align*} $$

of persistence modules, one has that

$$ \begin{align*}\underline{\mathrm{dim}}(M) = \underline{\mathrm{dim}}(L) + \underline{\mathrm{dim}}(N).\end{align*} $$

In particular, the dimension vector $\underline {\mathrm {dim}} M$ and the multiset of composition factors for M uniquely determine one another. As a special case, we have that $\underline {\mathrm {dim}}(M\oplus N) = \underline {\mathrm {dim}}(M) + \underline {\mathrm {dim}}(N)$ ; that is, the dimension vector is an (additive) invariant.

We now turn toward understanding the dimension vector $\underline {\mathrm {dim}} M$ using the projective modules $P_a$ . To start with, recall the well-known fact that for $a \in \mathcal {P}$ , one has $\dim M(a) = \dim \operatorname {\mathrm {Hom}}_{I(\mathcal {P})}(P_a,M)$ . One may then see $\underline {\mathrm {dim}} M$ as counting the number of “test morphisms” from a set of well-understood modules (namely the indecomposable projectives). Moreover, if $I(\mathcal {P})$ has finite global dimension, then there is a change of basis $\sigma : \mathbb {Z}^{|\mathcal {P}|} \rightarrow \mathbb {Z}^{|\mathcal {P}|}$ which can be conceptualized as sending the free abelian group with basis $\{S_a \mid a \in \mathcal {P}\}$ to that with basis $\{P_a \mid a \in \mathcal {P}\}$ . Formally, $\sigma $ is defined as the inverse of the Cartan matrix of $I(\mathcal {P})$ . Since we are working over a poset algebra, this also coincides with the Möbius inversion formula. We will give further details in Section 4, but for the purpose of this section, the change of basis $\sigma $ works as follows. Given an arbitrary projective module P, there is a unique direct sum decomposition $P \simeq \bigoplus _{a \in \mathcal {P}} (P_a)^{r_a}$ . We then set $[P] = \sum _{a \in \mathcal {P}} r_a [P_a]$ , where $[P_a]$ is the basis element corresponding to $P_a$ . Now, given M an arbitrary persistence module, we choose a finite projective resolution

$$ \begin{align*}P_m \rightarrow \cdots \rightarrow P_1 \rightarrow P_0 \rightarrow M.\end{align*} $$

One then has $\sigma (\underline {\mathrm {dim}} M) = \sum _{j = 1}^m (-1)^j [P_j]$ . Taking this one step further, let us denote by $[M]_a$ the coefficient of $[P_a]$ in $\sigma (\underline {\mathrm {dim}} M)$ . Now, define

$$ \begin{align*}P_+ = \bigoplus_{a: [M]_a> 0} (P_a)^{[M]_a}, \qquad\qquad P_- = \bigoplus_{a: [M]_a < 0} (P_a)^{-[M]_a}.\end{align*} $$

One may then consider the pair $(P_+,P_-)$ as a “signed approximation” of M by the category of projective modules. Indeed, by applying $\sigma ^{-1}$ , we obtain the equation $\underline {\mathrm {dim}} M = \underline {\mathrm {dim}} P_+ - \underline {\mathrm {dim}} P_-$ .

To summarize, we have the following ways to interpret the dimension vector $\underline {\mathrm {dim}} M$ :

1. (1) as recording the dimensions of the vector spaces comprising M,

2. (2) as recording the number of “test morphisms” from the indecomposable projectives to M,

3. (3) as some data determined by a projective resolution of M,

4. (4) as a “signed approximation” of M by the category of projective modules.

As we will see in Section 4, “the homological invariants” we introduce in this paper are readily given interpretations in the spirit (2)–(4) above. In essence, we will enlarge the set of indecomposable projective modules to a larger set $\mathcal {X}$ of indecomposable $I(\mathcal {P})$ -modules, typically a subset of the connected spread modules. This instantly gives rise to the invariant $(\dim \operatorname {\mathrm {Hom}}_{I(\mathcal {P})}(R,M))_{R \in \mathcal {X}}$ , which can be seen as counting “test morphisms” from the objects in $\mathcal {X}$ to M. For many choices of $\mathcal {X}$ , this is equivalent to data coming from an “ $\mathcal {X}$ -resolution” of M. As each step of such a resolution is an “approximation” in some precise sense, the result is readily interpreted as a “signed approximation” of M by $\mathcal {X}$ . Finally, as made explicit in Proposition 5.1, it is also possible to understand our invariants more directly in the spirit of (1) above.

In the remainder of this section, we focus on many of the invariants which serve as motivation and background for our study. In particular, we emphasize existing works that yield interpretations in the spirt of (2)–(4) above. We also discuss alternative approaches to extracting data from a projective resolution, as these can also be readily applied to our new framework.

3.2 The rank invariant and its generalizations

Recall the definition of the rank invariant $\underline {\mathrm {rk}} M$ from equation (1.1). In this section, we explain various interpretations and generalizations of the rank invariant which have appeared in recent work.

3.2.2 Signed barcodes and Möbius inversion

In this section, we will discuss how Möbius inversion can be used to reinterpret the $\mathcal {R}$ -rank invariants. We refer to [Reference StanleySta11, Section 3.7] for the definition of Möbius inversion.

As previously mentioned, when $\mathcal {P}$ is totally ordered, the (classical) rank invariant is equivalent to the barcode (see also Section 7.1 for the formal statement). To unpack this fact, let us view the rank invariant and barcode as taking values in the free abelian groups whose bases are given by the sets of (i) interval subsets of $\mathcal {P}$ and (ii) interval modules over $I(\mathcal {P})$ . Then Möbius inversion is a change of basis $\sigma : \mathbb {Z}^{{|\mathcal {P}|}\choose {2}} \rightarrow \mathbb {Z}^{{|\mathcal {P}|}\choose {2}}$ which sends $\underline {\mathrm {rk}}(M)$ to the barcode of M. We emphasize that this change of basis is not the one that sends the interval $[a,b]$ to the interval module $M_{[a,b]}$ .

We now return to the case where $\mathcal {P}$ is not totally ordered. As in the preceding section, let $\mathcal {R}$ be a subset of the power set $2^{\mathcal {P}}$ consisting of sets X as in Definition 3.2. Then $\mathcal {R}$ is itself a poset under inclusion. We denote by $\delta (M,\mathcal {R})$ the result of applying the resulting Möbius inversion formula to the invariant $\operatorname {\mathrm {rk}}(M,\mathcal {R})$ . Invariants obtained in this way are sometimes referred to as “generalized persistence diagrams.” See, e.g., [Reference Asashiba, Escolar, Nakashima and YoshiwakiAENYb, Reference Betthauser, Bubenik and EdwardsBBE22, Reference Bubenik and ElchesenBE22, Reference Botnan, Oppermann and OudotBOO Reference Kim and MémoliKM21, Reference McCleary and PatelMP22, Reference PatelPat18].

In [Reference Asashiba, Escolar, Nakashima and YoshiwakiAENYb, Reference Botnan, Oppermann and OudotBOO, Reference Kim and MémoliKM21], the invariants $\delta (M,\mathcal {R})$ are considered for the choices of $\mathcal {R}$ (and generality of poset) discussed in the previous section. To better understand these invariants, given a set $\mathcal {R}$ of connected spreads, we denote $\mathcal {M}(\mathcal {R}) = \{[M_{[A,B]}] | [A,B] \in \mathcal {R}\}$ . In all three works, the invariant $\delta (M,\mathcal {R})$ is considered as taking values in the free abelian group which has $\mathcal {M}(\mathcal {R})$ as a basis.Footnote ³ It is then shown that for any $[A,B] \in \mathcal {R}$ one has $\delta (M_{[A,B]},\mathcal {R}) = [M_{[A,B]}]$ . Due to this fact, one can interpret these particular invariants as “signed approximations” by spread modules. The word “signed” here refers to the fact that some of the coefficients of $\delta (M,\mathcal {R})$ may be negative.

Finally, for $[A,B] \in \mathcal {R}$ , let us denote by $r_{[A,B]}$ the coefficient of $[M_{[A,B]}]$ in $\delta (M,\mathcal {R})$ . Now, define

$$ \begin{align*}M_+ = \bigoplus_{[A,B]: r_{[A,B]}> 0} (M_{[A,B]})^{r(M,[A,B])}, \qquad\qquad M_- = \bigoplus_{[A,B]: r_{[A,B]} < 0} (M_{[A,B]})^{-r(M,[A,B])}.\end{align*} $$

By applying the inverse Möbius inversion formula, one then obtains the equation $\operatorname {\mathrm {rk}}(M,\mathcal {R}) = \operatorname {\mathrm {rk}}(M_+,\mathcal {R}) - \operatorname {\mathrm {rk}}(M_-,\mathcal {R})$ . See [Reference Botnan, Oppermann and OudotBOO, Theorem 2.5], where the pair $(M_+,M_-)$ is referred to as a “signed rank decomposition.” In the special case that $\mathcal {R} = \mathcal {I}$ is the set of intervals, we recall that $\operatorname {\mathrm {rk}}(M,\mathcal {I}) = \underline {\mathrm {rk}}(M)$ . In this case, the pair $(M_+,M_-)$ admits a visual interpretation called the “signed barcode.” See [Reference Botnan, Oppermann and OudotBOO, Section 6].

To summarize, there has been a large amount of recent work aimed at turning the $\mathcal {R}$ -rank invariants $\operatorname {\mathrm {rk}}(M,\mathcal {R})$ into signed approximations $\delta (M,\mathcal {R})$ by spread modules. One of our goals is to examine if and when these coincide with the homological notion of a (signed) approximation by spread modules, as defined in Section 4. For the (classical) rank invariant, [Reference Botnan, Oppermann and OudotBOO] has already given an affirmative answer to this question, which we will now explain.

3.3 Generalized Betti numbers and g-vectors

As the dimension vector can be derived from the data of a projective resolution, so too can homological invariants be derived from the data of a “relative projective resolution.” In our discussion of the dimension vector, this is done by taking the signed sum of the terms which appear in the resolution. However, this is not the only way to turn the data of a resolution into a vector which is readily usable in machine learning. We will briefly explain some of the alternative vectorization methods, all of which can be used in a straightforward manner for our relative homological algebra framework.

Let M be an $I(\mathcal {P})$ -module and assume it admits a finite projective resolution

$$ \begin{align*}P_m \xrightarrow{f_m} \cdots \xrightarrow{f_2} P_1 \xrightarrow{f_1} P_0 \xrightarrow{f_0} M.\end{align*} $$

We will further assume that this projective resolution is taken to be “minimal,” meaning that each map $f_k$ is the projective cover of $\ker (f_{k-1})$ . For $a \in \mathcal {P}$ , and $k \in \{0,\ldots ,m\}$ , we denote by $r_{k,a}$ the multiplicity of $P_a$ as a direct summand of $P_k$ . Each $r_{k,a}$ can be seen as an invariant of M, and in particular each tuple $(r_{k,a})_{a \in \mathcal {P}}$ is referred to as the kth generalized Betti numbers of M. This is a straightforward analog of classical Betti numbers, where one records the multiplicities of free modules in a free resolution.

We recall that the dimension vector can be recovered from the generalized Betti numbers. Indeed, the dimension vector $\underline {\mathrm {dim}} M$ is then (equivalent to) $\sum _{k = 0}^m\sum _{a \in \mathcal {P}}r_{k,a} [P_a]$ , where $[P_a]$ is the basis element associated with $P_a$ . This means that the generalized Betti numbers contain more information than the dimension vector. On the other hand, we caution that, unlike the dimension vector, the generalized Betti numbers require the assumption that our chosen projective resolution is minimal. Without this, they will no longer be well defined.

Especially in cases where an algebra has infinite global dimension, one may also record only the first k generalized Betti numbers for some choice of k. This is related to the concept of “g-vectors” from representation theory. Given a module M, its g-vector is by definition $(r_{0,a}-r_{1,a})_{a \in \mathcal {P}}$ . This invariant is particularly important to the concepts of cluster algebras [Reference Fomin and ZelevinskyFZ07], $\tau $ -tilting theory [Reference Adachi, Iyama and ReitenAIR14], and stability conditions [Reference KingKin94].

3.4 Amplitudes and a view toward stability

We conclude our motivation section with a brief discussion of stability. In this paper, we primarily discuss invariants as a tool for determining whether two persistence modules belong to the same isomorphism class. However, in applications, one is often interested in whether two persistence modules are derived from similar data (or from similar filtrations of topological spaces). In order for invariants to be useful for this purpose, one turns to stability. Given an invariant p, the key idea is to define a meaningful notion of distance on the output space of p. Informally, one then says that p is stable if given two “similar” persistence modules M and N, one has that the distance between $p(M)$ and $p(N)$ is sufficiently small.

After the first version of this manuscript was posted to arXiv, Oudot and Scoccola proved a stability result for certain invariants coming from exact structures [Reference Oudot and ScoccolaOS, Theorem 26]. Their result is proved with respect to the “Bottleneck dissimilarity function” for finitely presented $\mathbb {R}^n$ -persistence modules. Another place one may turn to look for stability results is the recent work of Giunti, Nolan, Otter, and Waas on amplitudes and their stability [Reference Giunti, Nolan, Otter and WaasGNOW]. Amplitudes are $\mathbb {Z}$ -valued invariants which are defined for any abelian category, and are characterized by how they behave with respect to short exact sequences. One could readily adapt this definition to the generality of exact categories by considering only those short exact sequences which are deemed “admissible.” It remains an open question to determine whether such a modification could be used to prove stability results for the invariants introduced in this paper.

4.1 Approximations, exact structures, and Grothendieck groups

Let $\mathcal {X}$ be a finite set of indecomposable modules, and let M be an arbitrary module. We recall that $\mathrm {add}(\mathcal {X})$ is the subcategory of $\mathsf {mod}\Lambda $ consisting of all finite direct sums (including the empty direct sum) of modules in $\mathcal {X}$ . We say a morphism $f: R\rightarrow M$ , where $R \in \mathrm {add}(\mathcal {X})$ , is a (right) $\mathcal {X}$ -approximation if every morphism $R' \rightarrow M$ with $R' \in \mathrm {add}(\mathcal {X})$ factors through f. Since we have assumed $\mathcal {X}$ to be finite, such a morphism is guaranteed to exist. We call f a minimal approximation by $\mathcal {X}$ if for all $g: R \rightarrow R$ such that $f \circ g = f$ , the endomorphism g is an isomorphism. This is equivalent for asking the domain of f to have the minimum possible number of direct summands.

The following lemma is crucial to the remainder of our setup.

We now consider two examples. Note in particular that Example 4.3(2) shows that there may exist epimorphisms from $\mathrm {add}(\mathcal {X})$ to M which are not $\mathcal {X}$ -approximations.

We will assume from now on that $\mathcal {X}$ contains all of the indecomposable projectives. In light of the previous lemma, we can then consider an analog of a projective resolution of M, formed by taking approximations by $\mathcal {X}$ at each step. This yields the following diagram, where each $p_j$ is an approximation by $\mathcal {X}$ and $q_j=i_{j-1} \circ p_j$ for every j.

The first row of this diagram is exact by definition, and is called a resolution of M by  $\mathcal {X}$ or $\mathcal {X}$ -resolution. If every $p_i$ is a minimal approximation, then the resolution itself is called minimal. We then have the following definitions (see [Reference Enochs and JendaEJ00, Section 8.4]).

Now, let $\mathcal {E}_{\mathrm {max}}$ be the class of short exact sequences in $\mathsf {mod}\Lambda $ . We then denote

$$ \begin{align*}\mathcal{E}_{\mathcal{X}} = \left\{\left(0 \rightarrow M \xrightarrow{f} N \xrightarrow{g} L \rightarrow 0\right)\in \mathcal{E}_{max} \ \middle| \ (\forall R \in \mathcal{X}) \ \operatorname{\mathrm{Hom}}(R,g) \text{ is surjective}\right\}.\end{align*} $$

Equivalently, the short exact sequences in $\mathcal {E}_{\mathcal {X}}$ are precisely those on which the functor $\operatorname {\mathrm {Hom}}(R,-)$ is exact for all $R \in \mathcal {X}$ .

It is shown in [Reference Dräxler, Reiten, Smalø and SolbergDRSS99] that $\mathcal {E}_{\mathcal {X}}$ gives an exact structure on $\mathsf {mod}\Lambda $ . Readers are referred to [Reference BühlerBü11] for more information on exact structures. In the present paper, we will only use the following straightforward facts about $\mathcal {E}_{\mathcal {X}}$ .

Bearing this result in mind, we are prepared for the main definition of this section.

Before proceeding, we highlight two important examples of relative Grothendieck groups.

Proposition 4.9 If the $\mathcal {X}$ -dimension of $\Lambda $ is not properly infinite, the Grothendieck group $K_0(\Lambda ,\mathcal {X})$ is free abelian with basis $\{[R]\mid R\in \mathcal {X} \}$ . In particular, $K_0(\Lambda , \mathcal {X}) \simeq \mathbb {Z}^{|\mathcal {X}|}$ . Moreover, if

$$ \begin{align*}0\to R_n\xrightarrow{q_n}\cdots \xrightarrow{q_2} R_1\xrightarrow{q_1} R_0\xrightarrow{q_0} M\to 0\end{align*} $$

is a finite $\mathcal {X}$ -resolution of $M \in \mathsf {mod}\Lambda $ , then $[M] = \sum _{i = 0}^n (-1)^i [R_i]$ .

Proof Let $M\in \mathsf {mod}\Lambda $ and choose a finite $\mathcal {X}$ -resolution of M as in the statement. We can extract short exact sequences

$$ \begin{align*}0\to \text{ker}(q_i) {\hookrightarrow} R_i {\twoheadrightarrow} \text{ker}(q_{i+1})\to 0.\end{align*} $$

Each of these short exact sequences is in $\mathcal {E}_{\mathcal {X}}$ by Proposition 4.6. So, in the Grothendieck group, $[R_i]= [\text {ker}(q_i)] + [\text {ker}(q_{i+1})]$ for all i. This yields

$$ \begin{align*} [M]&= [R_0] - [\text{ker}(q_0)]\\ &= [R_0] - \big( [R_1] - [\text{ker}(q_1)]\big) \\ &= [R_0] - [R_1] + \big( [R_2] - [\text{ker}(q_2)]\big)\\ &=\cdots\\ &= \sum_{i=0}^n (-1)^i [R_i], \end{align*} $$

where each $R_i$ is in $\text {add}(\mathcal {X})$ . This shows $\{[R] \mid R\in \mathcal {X} \}$ generates the Grothendieck group.

It remains to show that $K_0(\Lambda ,\mathcal {X}) = \langle [R]\rangle _{R \in \mathcal {X}}$ is free abelian. To see this, we recall that the short exact sequences ending in an $\mathcal {X}$ -approximation generate the subgroup $H_{\mathcal {X}}$ defining $K_0(\Lambda ,\mathcal {X})$ . See Definition 4.7 and [Reference Auslander and SolbergAS93]. Moreover, given $R \in \mathrm {add}(\mathcal {X})$ , the identity $R\rightarrow R$ is a minimal $\mathcal {X}$ -approximation. Together, these facts imply that every exact sequence generating $H_{\mathcal {X}}$ on the subcategory $\mathrm {add}(\mathcal {X})$ is split. This concludes the proof.

4.2 Homological and dim-hom invariants

In this section, we define dim-hom invariants and homological invariants. In order to make these definitions precise, we first formalize what we mean by an invariant, as outlined in the introduction.

To study different types of invariants, we consider inclusions of exact structures (see [Reference Brüstle, Hassoun, Langford and RoyBHLR20]). We need in particular the following lemma.

We note that, even if $\mathsf {mod}\Lambda $ contains infinitely many indecomposable objects, the statement and proof of Lemma 4.11 can be adapted to allow one to take $\mathcal {X}$ to be the set of all indecomposable modules. Thus, for any finite set $\mathcal {Y}$ of indecomposable modules containing the indecomposable projectives, we have that $K_0(\Lambda , \mathcal {Y})$ is a quotient of $K_0^{\mathrm{split}}(\Lambda )$ . This leads to the following definition.

We will show that if a given $\mathcal {X}$ respects some assumptions extending (2b) above, then the homological and dim-hom invariants relative to $\mathcal {X}$ coincide. See Theorem 4.22.

The following example is well known.

We will explore homological invariants (and dim-hom invariants) in the context of persistence modules in Section 6.

4.3 Projectivization

Fix a finite set $\mathcal {X}$ of indecomposable modules which contains the indecomposable projectives. In this section, we use the theory of projectivization to construct an algebra with indecomposable projective modules corresponding to modules in $\mathcal {X}$ . In particular, this will allow us to show that, for many choices of $\mathcal {X}$ , the homological and dim-hom invariants relative to $\mathcal {X}$ coincide. It will also allow us to reduce the problem of computing $\mathcal {X}$ -resolutions to that of computing projective resolutions over an endomorphism algebra. The ideas outlined here are well known, and readers are referred to [Reference Auslander, Reiten and SmaløARS95, Section II.2] for details.

We recall that a morphism $f: M \rightarrow N$ is called a section (resp. retraction) if there exists a morphism $g: N \rightarrow M$ such that $g\circ f$ (resp. $f \circ g$ ) is the identity.

Now, consider the module $T = T_{\mathcal {X}}: = \bigoplus _{R\in \mathcal {X}} R$ . Then $\Gamma = \Gamma _{\mathcal {X}} := \operatorname {\mathrm {End}}_{\Lambda }(T)^{op}$ is a finite-dimensional K-algebra under the composition of morphisms and the standard vector space structure. Denote by Q the Gabriel quiver of $\Gamma $ . The vertices of Q can be identified with $\mathcal {X}$ . Moreover, for $R, R' \in \mathcal {X}$ , the number of arrows $R \rightarrow R'$ in Q coincides with the number of linearly independent $\mathcal {X}$ -irreducible morphisms $R' \rightarrow R$ in $\operatorname {\mathrm {Hom}}_{\Lambda }(X,Y)$ .

The association $M\mapsto \operatorname {\mathrm {Hom}}_{\Lambda }(T,M)$ yields a functor from $\mathsf {mod}\Lambda $ to $\mathsf {mod} \Gamma $ . The action of $\Gamma $ on $\operatorname {\mathrm {Hom}}_{\Lambda }(T,M)$ is given by precomposition. This functor induces an additive equivalence between $\mathrm {add}(\mathcal {X})$ and the category of projective $\Gamma $ -modules. That is, given an (indecomposable) module $R\in \mathcal {X}$ , the $\Gamma $ -module $\operatorname {\mathrm {Hom}}_{\Lambda }(T,R)$ is isomorphic to the projective module (over $\Gamma $ ) at the vertex corresponding to R in the quiver Q.

The following is critical.

Proposition 4.16 [Reference Auslander, Reiten and SmaløARS95, Proposition 2.1] Let $R \in \mathrm {add}(\mathcal {X})$ and $M \in \mathsf {mod}\Lambda $ . Then the functor $\operatorname {\mathrm {Hom}}_\Lambda (T,-)$ induces an isomorphism

$$ \begin{align*}\operatorname{\mathrm{Hom}}_\Lambda(R,M) \simeq \operatorname{\mathrm{Hom}}_\Gamma(\operatorname{\mathrm{Hom}}_\Lambda(T,R),\operatorname{\mathrm{Hom}}_\Lambda(T,M)).\end{align*} $$

Putting this together, we have the following.

While the global dimension of $\Gamma $ gives an upper bound on the $\mathcal {X}$ -dimension, we demonstrate with the following example that this bound will in general not be tight.

Example 4.18 Let $\Lambda $ be as in Example 4.3. Then $\Gamma = \operatorname {\mathrm {End}}_\Lambda (P_1\oplus S_1\oplus P_2)^{\mathrm{op}}$ is the so-called Auslander algebra of $\Lambda $ . The Gabriel quiver of $\Gamma $ is

and we can write $\Gamma \simeq KQ/(\iota \pi )$ . It is well known that the global dimension of $\Gamma $ is 2, whereas the $\mathcal {X}$ -dimension of $\Lambda $ is clearly 0 since $\mathrm {add}(\mathcal {X}) = \mathsf {mod}\Lambda $ .

As a consequence of Propositions 4.17 and 4.9, we have the following.

We are now prepared to prove the main results of this section.

Proposition 4.21 Let $\mathcal {X}$ be a finite set of indecomposable $\Lambda $ -modules which contains the indecomposable projectives, and suppose that the global dimension of $\Gamma = \Gamma _{\mathcal {X}}$ is finite. Given $M \in \mathsf {mod}\Lambda $ , write $[M] \in K_0(\Lambda ,\mathcal {X})$ in the form

$$ \begin{align*}[M] = \sum_{R \in \mathcal{X}} c_R[R],\end{align*} $$

where each $c_R \in \mathbb {Z}$ . Then, for $R' \in \mathcal {X}$ , we have

$$ \begin{align*}\dim\operatorname{\mathrm{Hom}}_\Lambda(R',M) = \sum_{R \in \mathcal{X}}c_R\dim\operatorname{\mathrm{Hom}}_\Lambda(R',R).\end{align*} $$

Proof Let n be the $\mathcal {X}$ -dimension of M. We proceed by induction of n.

If $n = 0$ , then $M \in \mathrm {add}(\mathcal {X})$ . The result then follows immediately from the additivity of the functors $\operatorname {\mathrm {Hom}}(R',-)$ .

Now, let $n> 0$ and suppose that the result holds for $n - 1$ . Let $f: R \twoheadrightarrow M$ be a minimal $ \mathcal {X}$ -approximation of M. Note that f is surjective by Lemma 4.2 and the short exact sequence

$$ \begin{align*}0 \rightarrow \ker f \rightarrow R \rightarrow M \rightarrow 0\end{align*} $$

is in $\mathcal {E}_{\mathcal {X}}$ by Proposition 4.6(2). So we have $[M] = [R] - [\ker f]$ . Moreover, for $R' \in \mathcal {X}$ , Proposition 4.6(1) implies that

$$ \begin{align*}\dim\operatorname{\mathrm{Hom}}_\Lambda(R', M) = \dim\operatorname{\mathrm{Hom}}_\Lambda(R', R) - \dim\operatorname{\mathrm{Hom}}_\Lambda(R',\ker f).\end{align*} $$

Since the $\mathcal {X}$ -dimensions of R and $\ker f$ are 0 and $n-1$ , respectively, the result then follows from the induction hypothesis.

5.1 Single-source spreads

The single-source spread modules defined in Section 2.1 are particularly relevant for persistence theory. To be precise, take an arbitrary module $M \in \mathsf {mod}I(\mathcal {P})$ , some vector $v\in M(a)$ , and consider the set of vertices where v persists:

$$ \begin{align*}X(v)=\{x\in \mathcal{P} | \; M(a,x)(v)\neq 0\}.\end{align*} $$

We observe that $X(v)$ is precisely a single-source spread. The following proposition thus ties the notions of morphism and persistence together.

Proof If the map $M_{[a,B]}(a)\ni 1_K\mapsto v$ induces a morphism $f:M_{[a,B]}\to M$ , then for each vertex $x\in [a,B]$ , $f_x\circ M_{[a,B]}(a,x)(1_K)=M(a,x)(v)$ . In particular, if $x\in X(v)$ , $f_x$ cannot be zero, so its domain cannot be zero. Then $[a,B]\supseteq X(v)$ . Conversely, if this containment is respected, the described map is indeed a morphism.

Now, recall that $P_a$ , the indecomposable projective module at a, is isomorphic to $M_{[a,\infty ]}$ . Clearly, $X(v)\subseteq [a,\infty ]$ , so the map $P_a(a)\ni 1_K\mapsto v$ induces a morphism $g:P_a\to M$ . By definition, $g_x\neq 0$ if and only if $x\in X(v)$ . This means that $\text {im}(g)$ is a thin module of support $X(v)$ . By taking $f_x(1_K)$ as the basis vector for the image at each vertex x, we get

$$\begin{align*}\text{im}(g)(x,y)= \begin{cases} 1_K, & \text{ if } x,y\in X(v),\\ 0, & \text{otherwise.} \end{cases} \end{align*}$$

So $\text {im}(g)\simeq M_{X(v)}$ as expected.

Single-source spread modules also admit other characterizations.

Proof We first note that given $a, b \in \mathcal {P}$ , we have that $P_b \subseteq P_a$ if and only if $a \leq b$ .

(1 $\implies $ 2): Write $M = M_{[a,B]}$ . Define

$$ \begin{align*}B' = \{x\mid x> a, \ x \not\leq B, \text{ and }\forall y \in \mathcal{P}: y < x \implies y \leq B\}.\end{align*} $$

That is, $B'$ consists of those elements of $\mathcal {P}$ which cover elements of $[a,B]$ , but do not themselves lie in $[a,B]$ . We conclude that $P_x \subset P_a$ for all $x \in B'$ . Moreover, we have $M_{[a,B]} \simeq P_a/(\sum _{x \in B'} P_x)$ .

(2 $\implies $ 3): Under the assumption of (2), the projective cover of M is the quotient map $P\rightarrow M$ .

(3 $\implies $ 1): Let $p: P \rightarrow M$ be the projective cover of M. By assumption, there exists $a \in \mathcal {P}$ such that $P = P_a$ . In particular, we have $\dim P_a(x) \leq 1$ for all $x \in \mathcal {P}$ . Now, let B be the set of elements which are maximal in

$$ \begin{align*}\{x \in \mathcal{P} \mid P_a(x) \neq 0 \text{ and } (\ker p)(x) = 0\}.\end{align*} $$

It is clear that B is a set of incomparable elements of $\mathcal {P}$ and that $M \simeq M_{[a,B]}$ . This completes the proof.

5.2 Morphisms between spread modules

We now turn our attention toward computing the Hom-spaces between spread modules. We begin with an example.

Example 5.4 Let $\mathcal {P} = \{1,2,3,4,5\}\times \{1,2,3\}$ be the $3\times 5$ grid. We will denote elements of $\mathcal {P}$ using concatenation rather than ordered pairs, e.g., 13 will be used in place of $(1,3)$ . Consider the spreads $[\{13,41\},43]$ (drawn below in dashed red) and $[11,\{23,51\}]$ (drawn below in dotted blue).

We want to compute the space of morphisms between the spread representations $M := M_{[\{13,41\},43]}$ and $N := M_{[11,\{23,51\}]}$ . Since the supports of M and N only intersect at $13, 23$ , and $41$ , any $f: M_{[\{13,41\},43]} \rightarrow M_{[11,\{23,51\}]}$ must satisfy $f_v = 0$ for $v \notin \{13,23,41\}$ . We then have a pair of commutative diagrams:

These diagrams imply that $f_{13} = f_{23}$ and that $f_{41} = 0$ . Thus, choosing a morphism $f:M \rightarrow N$ is equivalent to choosing a scalar $\lambda \in K$ and setting $f_{13}(1) = \lambda $ . So, $\text {Hom} \left (M_{[\{13,41\},43]},M_{[11,\{23,51\}]}\right )\simeq K$ .

Generalizing this example, we have the following explicit description of the Hom-space between spread modules. See also [Reference MillerMil, Proposition 3.10].

Proposition 5.5 Let $[A,B]$ and $[C,D]$ be spreads. Denote by $X_1,\ldots ,X_n$ the connected components of $[A,B] \cap [C,D]$ which satisfy

(⋆) $$ \begin{align} \{a\in A \; | a\leq X_i \} \subseteq X_i \text{ and } \{d \in D \; | \; X_i \leq d\} \subseteq X_i.\end{align} $$

Then there are inverse isomorphisms $\Phi : K^n \rightleftarrows \operatorname {\mathrm {Hom}}(M_{[A,B]},M_{[C,D]}): \Psi $ given as follows.

• • Let $(\lambda _1,\ldots ,\lambda _n) \in K^n$ . We identify each $\lambda _i$ with the linear map $K \rightarrow K$ given as scalar multiplication by $\lambda _i$ . Then, for each $x \in \mathcal {P}$ , set

  $$ \begin{align*}\Phi(\lambda_1,\ldots,\lambda_n)_x = \begin{cases} \lambda_i, & x \in X_i,\\0, & \text{otherwise.}\end{cases}\end{align*} $$

• • For each i, choose some $x_i \in X_i$ . Then, for $f \in \operatorname {\mathrm {Hom}}(M_{[A,B]},M_{[C,D]})$ , set

  $$ \begin{align*}\Psi(f) = (f_{x_1}(1),\ldots,f_{x_n}(1)).\end{align*} $$

Proof We will first show that $\Psi $ is well defined. Let $f: M_{[A,B]}\rightarrow M_{[C,D]}$ , and let $x, y \in \mathcal {P}$ lie in the same connected component Y of $[A,B] \cap [C,D]$ . Then x and y are connected by a path in the Hasse quiver of Y. Moreover, if $x \leq z \in Y$ , then $M_{[A,B]}(x,z) = 1_K = M_{[C,D]}(x,z)$ . These two observations together imply that $f_x = f_y$ , and therefore $\Psi $ is well defined.

We will next show that $\Psi $ is injective. Let $f: M_{[A,B]}\rightarrow M_{[C,D]}$ , and suppose $\Psi (f) \,{=}\, 0$ . We wish to show that $f = 0$ , or equivalently that $f_x = 0$ for $x \notin \bigcup _{i = 1}^n X_i$ . This is clear if $x \notin [A,B]$ or $x \notin [C,D]$ ; thus, assume that $x \in [A,B] \cap [C,D]$ . By assumption, the connected component Y containing x does not satisfy ( $\star $ ). We then have two cases to consider. Suppose first that there exists $a \in A$ with $a \leq Y$ and $a \notin Y$ . Then there exists $y \in Y$ with $a \leq y$ . By the previous paragraph, this means $f_a = f_y = f_x$ . However, we cannot have $a \in [C,D]$ , because otherwise the interval $[a,y]$ would be contained in Y. We conclude that $f_x = 0$ . The case where there exists $d \in D$ with $Y \leq d$ and $d \notin Y$ is analogous.

We will now show that $\Phi $ is well defined. Let $(\lambda _1,\ldots ,\lambda _n) \in K^n$ . Obviously, $\Phi (\lambda _1,\ldots ,\lambda _n)$ satisfies the morphism condition within each connected component of $[A,B] \cap [C,D]$ . The only other morphism conditions to check are at the boundaries of the components $X_1,\ldots ,X_n$ . Since these components satisfy ( $\star $ ), there is no diagram of either of the following shapes:

Therefore, the only patterns that could make the morphism condition fail never happen. We conclude that $\Phi (\lambda _1,\ldots ,\lambda _n)$ is well defined.

It is clear that $\Psi \circ \Phi $ is the identity on $K^n$ . In particular, this means $\Psi $ is surjective. Since we have already shown that $\Psi $ is injective, we conclude that $\Psi $ is an isomorphism with inverse $\Phi $ .

To conclude this section, we tabulate several consequences of Proposition 5.5.

As a corollary, we also directly get the following characterization of morphism between intervals, as is well known for type A quivers.

Next, we show the following necessary conditions to have nonzero morphisms between spreads of the same type, which will be crucial to show Theorem 6.2. We note that Lemma 5.9(2) is an immediate consequence of [Reference MillerMil, Proposition 3.10], but we provide a proof here for the convenience of the reader.

Proof (1) Suppose $\operatorname {\mathrm {Hom}}(M_{[a,B]},M_{[c,D]}) \neq 0$ . By Corollary 5.7, we have $c \leq a$ . Thus, suppose $c = a$ . It follows that $X:= [a,B] \cap [c,D]$ is connected and must therefore satisfy ( $\star $ ). It is clear that $c \in X$ , so it remains only to show that $D \subseteq X$ . Since $c \in X$ and $c \leq d$ for all $d \in D$ , we have $\{d\in D| X \leq d\}= D$ . As X verifies ( $\star $ ), we thus have $D\leq X\leq B$ . So, $D\leq B$ , and $[c,D]\subseteq [a,B]$ .

(2) We prove the contrapositive. Suppose $[A,\infty ] \not \subseteq [B,\infty ]$ , and let $f: M_{[A,\infty ]} \rightarrow B_{[B,\infty ]}$ be a morphism. Because $[A,\infty ]$ and $[B,\infty ]$ are upsets, the assumption that $[A, \infty ] \not \subseteq [B,\infty ]$ means there exists $c \in A \setminus [B,\infty ]$ . In particular, we have $f_c = 0$ . Now, let $a \in A$ be arbitrary. Since $[A,\infty ]$ is connected, it follows that there exists a finite set $\{x_0,\ldots ,x_k\} \subseteq [A,\infty ]$ such that

$$ \begin{align*}c = x_0 \leq x_1 \geq x_2 \leq \cdots \geq x_k = a.\end{align*} $$

It follows by induction that $f_{x_i} = 0$ for each $x_i$ . In particular, we have that $f_a = 0$ for all $a \in A$ . By Proposition 5.5, we conclude that $f = 0$ .

Finally, Proposition 5.5 can be used to show the following generalization of the well-known fact that irreducible morphisms (in the traditional sense) are either monomorphisms or epimorphisms (see, for example, [Reference Auslander, Reiten and SmaløARS95, Lemma 5.1]). Our proof follows similar arguments to the traditional case, which arise as a special case of the following by setting $\mathrm {add}(X)= \mathsf {mod}I(\mathcal {P})$ .

6.1 Homological invariants

We now turn our attention to showing that many interesting sets of spread modules give rise to endomorphism algebras of finite global dimension, and thus homological invariants. The basis for these results is the following well-known lemma. Note that this lemma holds more generally, but we have stated it in the generality of the other results of this section. Thus, for the duration of this section, $\mathcal {X}$ denotes an arbitrary set of spread modules which contains the indecomposable projectives.

Proof Let Q be the Gabriel quiver of $\Gamma = \operatorname {\mathrm {End}}_{I(\mathcal {P})}(T)^{op}$ . By the discussion following Definition 4.15, there is a directed path from X to Y in Q if and only if there is a sequence of nonzero morphisms

$$ \begin{align*}Y = Z_0 \xrightarrow{f_1} Z_1 \xrightarrow{f_2} \cdots \xrightarrow{f_m} Z_m = X\end{align*} $$

with each $Z_i \in \mathcal {X}$ . That is, if $\preceq $ denotes the transitive closure of $\rightarrow $ , then there is a directed path from X to Y if and only if $Y \preceq X$ . The assumption that $\preceq $ is a partial order then implies that the quiver Q contains no oriented cycles. It is well-known consequence that $\mathsf {gl.dim}\Gamma < \infty $ .

We are now prepared to prove our first main theorem.

In case $\mathcal {X}$ is the set of all single-source spread modules, we refer to the quotient map $K_0^{\mathrm{split}}(I(\mathcal {P})) \rightarrow K_0(I(\mathcal {P}),\mathcal {X})$ as the single-source homological spread invariant. We defer further discussion of this invariant, and others which fit into the framework of Theorem 6.2(1), to Section 7.

It is an interesting problem to characterize precisely which sets of spread modules yield homological invariants. Unfortunately, over arbitrary posets, we cannot hope for an extension of Theorem 6.2 to arbitrary choices of $\mathcal {X}$ , as the following example shows.

Example 6.4 Let $\mathcal {P}$ be the poset with Hasse diagram

Note that the incidence algebra $I(\mathcal {P})$ is in this case a path algebra of type $\widetilde {A}_5$ . Now, set

$$ \begin{align*}\mathcal{X} = \{\text{indecomposable projectives}\} \cup \{M_{[\{1,3\},\{5,6\}]}, M_{[\{1,2\},\{4,6\}]}, M_{[\{2,3\},\{4,5\}]}\}.\end{align*} $$

The Gabriel quiver of $\operatorname {\mathrm {End}}_{I(\mathcal {P})}(T)^{op}$ is then

The oriented 3-cycle at the center of this quiver has the relation that the composition of any two arrows is 0. Moreover, the three oriented pentagons (for example, from $M_{\{[1,3\},\{5,6\}]}$ to $M_{[6,6]}$ ) all have commutativity relations. We claim that the global dimension of $\Gamma $ is infinite. To see this, given a spread module $M_{[A,B]} \in \mathcal {X}$ , we denote by $P_{[A,B]}$ the projective $\Gamma $ -module at the vertex labeled by $M_{[A,B]}$ . Now, let N be the indecomposable $\Gamma $ -module supported at the vertices labeled by $M_{[\{1,3\},\{5,6\}]}$ , $M_{[3,\{5,6\}]}$ , and $M_{[5,5]}$ . Then N has a minimal projective resolution

$$ \begin{align*}\cdots\rightarrow P_{[\{1,2\},\{4,6\}]}\rightarrow P_{[\{1,3\},\{5,6\}]}\rightarrow P_{[\{2,3\},\{4,5\}]}\rightarrow P_{[\{1,2\},\{4,6\}]}\rightarrow N\rightarrow 0.\end{align*} $$

Indeed, the kernel of the first map is the $\Gamma $ -module supported at the vertices labeled by $M_{[\{1,2\},\{4,6\}]}, M_{[1,\{4,6\}]}$ , and $M_{[6,6]}$ . The symmetry of the quiver then makes it clear that this will be the projective resolution of N.

In addition, we can see directly that the $\mathcal {X}$ -dimension of $(\mathsf {mod}I(\mathcal {P}),\mathcal {E}_{\mathcal {X}})$ is properly infinite. Indeed, the following is readily seen to be a minimal $\mathcal {X}$ -resolution of the spread representation $M_{[1,6]}$ :

$$ \begin{align*}\cdots\rightarrow M_{[\{1,2\},\{4,6\}]}\rightarrow M_{[\{1,3\},\{5,6\}]}\rightarrow M_{[\{2,3\},\{4,5\}]}\rightarrow M_{[\{1,2\},\{4,6\}]}\rightarrow M_{[1,6]}\rightarrow 0.\end{align*} $$

Note that there are connected spread modules which are not included in set $\mathcal {X}$ in Example 6.4. Thus, while Example 6.4 shows that we cannot fully generalize Theorem 1.2 to arbitrary sets of spread modules, one natural question is what happens when one asks that $\mathcal {X}$ be the set of all spread modules.

6.2 Algorithms and extensions to infinite posets

Proposition 4.17 can be leveraged into an algorithm for computing the value of the invariant $[M]_{\mathcal {X}}$ when $\Gamma $ has finite global dimension. Indeed, one can first compute a minimal projective resolution of the $\Gamma $ -module $\operatorname {\mathrm {Hom}}_{I(\mathcal {P})}(T,M)$ . By Proposition 4.19, the class of M in the Grothendieck group can then be computed by alternating sum of the terms of this resolution. The advantage of this approach is that one avoids needing to compute an $\mathcal {X}$ -resolution directly in favor of the more straightforward process of computing a projective resolution.

Another way to help compute approximations more effectively is to reduce the size of $\mathcal {X}$ in function of the module we are approximating. In the following proposition, we show how this can be done when all quotients of the modules of $\mathcal {X}$ lie in $\mathrm {add}(\mathcal {X})$ . We emphasize that this is different than saying $\mathrm {add}(\mathcal {X})$ is closed under quotients. For example, if we take $\mathcal {X}$ to be the set of all single-source spread modules, then all nonzero quotients of the modules in $\mathcal {X}$ will lie in $\mathcal {X}$ (hence also in $\mathrm {add}(\mathcal {X})$ ) by Proposition 5.2. On the other hand, the category $\mathrm {add}(\mathcal {X})$ will not be closed under arbitrary quotients, as this would force $\mathrm {add}(\mathcal {X}) = \mathsf {mod}I(\mathcal {P})$ . Indeed, every module is a quotient of its projective cover, which will lie in $\mathrm {add}(\mathcal {X})$ .

Proof It suffices to show that all morphisms from $\mathrm {add}(\mathcal {X})$ to M factor through morphisms from $\mathrm {add}(\mathcal {Y})$ . Let $R \in \mathrm {add}(\mathcal {X})$ , that is, $R=R_1\oplus \cdots \oplus R_n$ for some $R_i\in \mathcal {X}$ , and take some morphism $f:R\to M$ . Then f admits a factorization of the form

$$ \begin{align*}R\rightarrow \bigoplus_{i = 1}^n f(R_i) \rightarrow \sum_{i = 1}^n f(R_i) = f(R)\hookrightarrow M.\end{align*} $$

As each $f(R_i)$ is a quotient of a module in $\mathcal {X}$ and satisfies $\mathrm {supp}(f(R_i)) \subseteq \mathrm {supp}(M)$ , this shows that f factors through $\mathrm {add}(\mathcal {Y})$ .

7.1 The barcode

In this section, we assume that the Hasse quiver of $\mathcal {P} = (\mathcal {P}, \preceq )$ is Dynkin type A. More precisely, as a set we identify $\mathcal {P}$ with $\{1,2,\ldots ,|\mathcal {P}|\}$ . We denote by $\preceq $ the partial order defining $\mathcal {P}$ and by $\leq $ the standard linear order on $\mathcal {P} = \{1,2,\ldots ,|\mathcal {P}|\}$ . The assumption that the Hasse quiver of $\mathcal {P}$ is Dynkin type A then amounts to the assumption that all cover relations in $\mathcal {P}$ are either of the form $i \preceq i + 1$ or $i + 1 \preceq i$ .

Under this setup, it is well known that the indecomposable $I(\mathcal {P})$ -modules are precisely the connected spread modules. Moreover, the connected spreads under the order $\preceq $ correspond directly to the intervals under the order $\leq $ . For example, if the quiver of $\mathcal {P}$ is $1 \rightarrow 2 \leftarrow 3$ , it is common to identify the spread module $M_{[\{1,3\},2]}$ with the interval $[1,3]$ . More generally, let $F \simeq \mathbb {Z}^{|\mathcal {P}|\choose 2}$ be the free abelian group generated by the closed intervals of $(\mathcal {P}, \leq )$ . Then the identification between indecomposable $I(\mathcal {P})$ -modules and closed intervals of $(\mathcal {P}, \leq )$ induces an isomorphism ${\underline {\mathrm {bar}}:K_0^{\mathrm{split}}(I(\mathcal {P})) \rightarrow F}$ . Given a persistence module $M \in \mathsf {mod}I(\mathcal {P})$ , the data $\underline {\mathrm {bar}}[M]$ are often referred to as the barcode of M.

7.2 The rank invariant

We return to $\mathcal {P}$ denoting an arbitrary finite poset.

Denote by F the free abelian group generated by the (closed) intervalsFootnote ⁴ of $\mathcal {P}$ . Then the rank invariant (equation (1.1)) can be considered as a map $p: K_0^{\mathrm{split}}(I(\mathcal {P}))\rightarrow F$ . It is shown in [Reference Botnan, Oppermann and OudotBOO, Theorem 2.5] that p is surjective, and is therefore an invariant in the sense of Definition 4.10. In order to show that this invariant is homological, we recall a special collection of single-source spreads following [Reference Botnan, Oppermann and OudotBOO]. We append an extra element $\infty $ to $\mathcal {P}$ , and we set $a < \infty $ for all $a\in \mathcal {P}$ . For any $a < b \in \mathcal {P}\cup \{\infty \}$ , the persistence module $N_{\langle a , b\langle }$ , defined by

$$ \begin{align*}N_{\langle a, b\langle}(c) = \begin{cases} K, & a \leq c \text{ and } b \not\leq c,\\0, & \text{otherwise,}\end{cases}\qquad \qquad N_{\langle a, b\langle}(c,d) = \begin{cases} 1_K, & N_{\langle a, b\langle}(c) = K = N_{\langle a, b\langle}(d)\\0, & \text{otherwise}\end{cases}\end{align*} $$

is called the hook module from a to b. Note that, if $b \neq \infty $ , then $N_{\langle a,b\langle }$ is precisely the quotient of the projective at a by the projective at b. Likewise, if $b = \infty $ , then $N_{\langle a,b\langle }$ is precisely the projective at a. In particular, hook modules are single source spread modules by Proposition 5.2.

One of the main results of [Reference Botnan, Oppermann and OudotBOO], specialized to finite posets and restated in the language of this paper, is the following.

For the convenience of the reader, we give a proof of this result using the language of our framework.

Proof We first note that the quotient map $K_0^{\mathrm{split}}(I(\mathcal {P})) \rightarrow K_0(I(\mathcal {P}),\mathcal {H})$ is homological as a consequence of Theorem 1.2(1) and the fact that every hook module is also a single-source spread module. In particular, this means that this quotient map is equivalent to the invariant $p: K_0^{\mathrm{split}}(I(\mathcal {P})) \rightarrow \mathbb {Z}^{|\mathcal {H}|}$ given by $p(M) = (\dim \operatorname {\mathrm {Hom}}(M,H))_{H \in \mathcal {H}}$ . It therefore suffices to show that the rank invariant is equivalent to p.

To see this, first let $N_{\langle a,b\langle } \in \mathcal {H}$ with $b \neq \infty $ , and let v be a nonzero vector in $N_{\langle a,b\langle }(a)$ . Then, for any $M \in \mathsf {mod}I(\mathcal {P})$ , the morphisms from $N_{\langle a,b\langle }$ to M can be identified with the vectors in $\ker M(a,b)$ . Indeed, for $w \in M(a)$ , there is a morphism $f: N_{\langle a,b\langle } \rightarrow M$ with $f_a(v) = w$ if and only if $M(a,b)(w) = 0$ . As $N_{\langle a,\infty \langle } = P_a$ , this means the following all hold:

$$ \begin{align*} \operatorname{\mathrm{rk}}(M(a,b)) &= \dim\operatorname{\mathrm{Hom}}(N_{\langle a,\infty\langle},M) - \dim\operatorname{\mathrm{Hom}}(N_{\langle a,b\langle},M),\\ \dim\operatorname{\mathrm{Hom}}(N_{\langle a,b\langle},M) &= \operatorname{\mathrm{rk}}(M(a,a)) - \operatorname{\mathrm{rk}}(M(a,b)),\\ \operatorname{\mathrm{rk}}(M(a,a)) &= \dim\operatorname{\mathrm{Hom}}(N_{\langle a,\infty\langle},M). \end{align*} $$

This shows that $\underline {\mathrm {rk}}(M)$ is determined by $p(M)$ and vice versa. Thus, $\underline {\mathrm {rk}}$ and p are equivalent invariants.

We are now ready to prove our second main theorem, which shows that the single-source spread invariant is finer than the rank invariant.

Remark 7.5 The hypotheses of Theorem 1.3 include the case where $\mathcal {P}$ is totally ordered, but are slightly more general. Indeed, we only require that for every $a \in \mathcal {P}$ , the upset $[a,\infty ]$ is totally ordered. Such subsets are sometimes called “principle upsets.” For example, consider the poset with Hasse quiver

The principle upsets of this poset are $\{1,2,4,5\}, \{2,4,5\}, \{3,4,5\}, \{4,5\}$ , and $\{5\}$ , all of which are totally ordered. Thus, the rank invariant and the homological spread invariant coincide for this particular poset.

In the spirit of the counterexample used to prove Theorem 7.4, we consider the following.

Example 7.6 Let $\mathcal {P}$ be a finite poset which contains a unique maximal element. Let $\mathcal {H}$ be the set of hook modules, and let $\mathcal {I}$ be the set of interval modules. Since $\mathcal {P}$ contains a unique maximal element, we note that $\mathcal {I}$ contains all of the indecomposable projectives. Since every interval is a single-source spread, the quotient map $K_0^{\mathrm{split}}(I(\mathcal {P}))\rightarrow K_0(I(\mathcal {P}),\mathcal {I})$ is therefore a homological invariant by Theorem 6.2(1). We also know that the quotient map $K_0^{\mathrm{split}}(I(\mathcal {P}))\rightarrow K_0(I(\mathcal {P}),\mathcal {H})$ is equivalent to the rank invariant and is also homological by Theorem 7.2.

These two invariants have the same rank (given by the number of pairs $a \leq b \in \mathcal {P}$ ), but are not equivalent. For example, let $\mathcal {P} = \{(0,0), (0,1), (1,0), (1,1)\}$ be the $2\times 2$ grid. We consider the modules

We note that $\underline {\mathrm {dim}}(M) = \underline {\mathrm {dim}}(M')$ and that $\underline {\mathrm {rk}}(M) = \underline {\mathrm {rk}}(M')$ . On the other hand, we see that M is a direct sum of interval modules, whereas $M'$ has an $\mathcal {I}$ -resolution

$$ \begin{align*}0 \rightarrow M_{[(1,1),(1,1)]} \rightarrow M_{[(0,0),(1,1)]}\oplus M_{[(0,0),(0,0)]} \rightarrow M' \rightarrow 0.\end{align*} $$

Since this resolution has a nonzero term in degree 1, this means that $[M]_{\mathcal {I}} \neq [M']_{\mathcal {I}}$ .

We conclude this section with a detailed explanation of why the signed barcode of [Reference Botnan, Oppermann and OudotBOO] and the homological invariant with respect to $\mathcal {I}$ are not equivalent.

7.3 Generalized rank invariants

We conclude by discussing the generalizations of the rank invariant appearing in [Reference Asashiba, Escolar, Nakashima and YoshiwakiAENYb, Reference Botnan, Oppermann and OudotBOO, Reference Kim and MémoliKM21]. Again $\mathcal {P}$ denotes an arbitrary finite poset in this section. Given a set $\mathcal {R}$ of connected spread modules, we recall the “ $\mathcal {R}$ -rank invariant” $\operatorname {\mathrm {rk}}(-,\mathcal {R})$ from Section 3. We also recall that a “generalized persistence diagram” $\delta (-,\mathcal {R})$ can be obtained by applying Möbius inversion to the $\mathcal {R}$ -rank invariant. We emphasize that while $\delta (-,\mathcal {R})$ and $\operatorname {\mathrm {rk}}(-,\mathcal {R})$ are equivalent as invariants, $\delta (-,\mathcal {R})$ has the advantage of yielding a “signed approximation” of a given persistence module by the spreads in $\mathcal {R}$ .

To understand the connection between the $\mathcal {R}$ -rank invariants, dim-hom invariants, and homological invariants, we first consider the following example.

Proof Let M be the following module:

Over this poset, indecomposable modules are uniquely determined by their dimension vector. For example, we identify M with the vector $\begin {bmatrix}1 & 1 & 0\\1 & 2 & 1\end {bmatrix}$ .

Seeking a contradiction, we compute the “generalized persistence diagram” obtained by applying Möbius inversion to the invariant $\operatorname {\mathrm {rk}}(M,\mathcal {R})$ . Following [Reference Kim and MémoliKM21, Proposition 3.19] or [Reference Asashiba, Escolar, Nakashima and YoshiwakiAENYb, Definition 5.9], this yields

$$ \begin{align*}\delta(M,\mathcal{R}) = \begin{bmatrix}1&0&0\\1&1&1 \end{bmatrix} + \begin{bmatrix}1&1&0\\1&1&0 \end{bmatrix} + \begin{bmatrix}0&0&0\\0&1&0 \end{bmatrix} - \begin{bmatrix}1&0&0\\1&1&0 \end{bmatrix}.\end{align*} $$

Now, denote

$$ \begin{align*}N := M \oplus \begin{bmatrix}1&0&0\\1&1&0 \end{bmatrix},\qquad\qquad L := \begin{bmatrix}1&0&0\\1&1&1 \end{bmatrix} \oplus \begin{bmatrix}1&1&0\\1&1&0 \end{bmatrix} \oplus \begin{bmatrix}0&0&0\\0&1&0 \end{bmatrix}.\end{align*} $$

Recalling that $\delta (X,\mathcal {R}) = X$ whenever X is a direct sum of spread modules (see [Reference Kim and MémoliKM21, Theorem 3.14]), we conclude that $\delta (N,\mathcal {R}) = \delta (L,\mathcal {R})$ . On the other hand, denote $X = \begin {bmatrix} 1 & 0 & 0\\1 & 1 & 0\end {bmatrix}$ and observe that

$$ \begin{align*}\dim\operatorname{\mathrm{Hom}}_{I(\mathcal{P})}(X,N) = 1, \qquad\qquad \dim\operatorname{\mathrm{Hom}}_{I(\mathcal{P})}(X,L) = 0.\end{align*} $$

Thus, the dim-hom invariant induced by $\mathcal {X}$ can distinguish N and L, whereas the invariant $\delta (-,\mathcal {R})$ cannot. Since the latter is equivalent to the generalized rank invariant $\operatorname {\mathrm {rk}}(-,\mathcal {R})$ , this proves the result.

As an immediate consequence, we obtain the following.

To summarize, let $\mathcal {P}, \mathcal {R}$ , and $\mathcal {X}$ be as in Corollary 7.10. We have shown that the generalized rank invariant $\operatorname {\mathrm {rk}}(-,\mathcal {R})$ , and thus also the signed barcode $\delta (-,\mathcal {R})$ , is not a dim-hom invariant relative to $\mathcal {X}$ . This choice of $\mathcal {X}$ is particularly natural (see Question 6.5). Indeed, if the quotient map $p: K_0^{\mathrm{split}}(I(\mathcal {P})) \rightarrow K_0(I(\mathcal {P}),\mathcal {X})$ is a homological invariant, then both p and $\operatorname {\mathrm {rk}}(-,\mathcal {R})$ assign a persistence module M to a “signed approximation” of M by the set of spread modules. Corollary 7.10 would then imply that these approximations do not coincide, even if one allows for a change of basis.

While restricting $\mathcal {X}$ to be composed of only spread modules is a natural choice, it remains an open question whether the generalized rank invariant is either homological or a dim-hom invariant relative to some other set $\mathcal {Y}$ . We have shown with a brute-force calculation that this is not the case when $\mathcal {P}$ is the $2\times 3$ grid, which leads to the following conjecture.