Bigraded Betti numbers and Generalized Persistence Diagrams

Commutative diagrams of vector spaces and linear maps over $\mathbb{Z}^2$ are objects of interest in topological data analysis (TDA) where this type of diagrams are called 2-parameter persistence modules. Given that quiver representation theory tells us that such diagrams are of wild type, studying informative invariants of a 2-parameter persistence module $M$ is of central importance in TDA. One of such invariants is the generalized rank invariant, recently introduced by Kim and M\'emoli. Via the M\"obius inversion of the generalized rank invariant of $M$, we obtain a collection of connected subsets $I\subset\mathbb{Z}^2$ with signed multiplicities. This collection generalizes the well known notion of persistence barcode of a persistence module over $\mathbb{R}$ from TDA. In this paper we show that the bigraded Betti numbers of $M$, a classical algebraic invariant of $M$, are obtained by counting the corner points of these subsets $I$s. Along the way, we verify that an invariant of 2-parameter persistence modules called the interval decomposable approximation (introduced by Asashiba et al.) also encodes the bigraded Betti numbers in a similar fashion. We also show that the aforementioned results are optimal in the sense that they cannot be extended to $d$-parameter persistence modules for $d \geq 3$.


Introduction
Multiparameter persistent homology.Theoretical foundations of persistent homology, one of the main protagonists in topological data analysis (TDA), have been rapidly developed in the last two decades, allowing a large number of applications.Persistent homology is obtained by applying the homology functor to an R (or Z) -indexed increasing family of topological spaces [12,28].This parametrized family of topological spaces, for example, often arises as either a sublevel set filtration of a real-valued map on a topological space, or the Vietories-Rips simplicial filtration of a metric space.
With more complex input data, we obtain R d -indexed increasing families (d > 1) of topological spaces, e.g. a sublevel set filtration of a topological space that is filtered by multiple realvalued functions, or a Vietories-Rips-sublevel simplicial filtration of a metric space equipped with a map [12,16].By applying the homology functor (with coefficients in a fixed field k) to such a multiparameter filtration, we obtain a d -parameter persistence module R d (or Z d ) → vec, a functor from the poset R d (or Z d ) to the category vec of finite dimensional vector spaces and linear maps over the field k.In contrast to the case of d = 1, there is no discrete and complete invariant for R d (or Z d ) → vec for d > 1 [16].In quiver representation theory, functors R d (or Z d ) → vec (d > 1) are of wild type, implying that there is no simple invariant which completely encodes the isomorphism type of R d (or Z d ) → vec [23,32].Nevertheless, there have been many studies on the invariants of d -parameter persistence modules, e.g.[16,18,33,39,47,53,55].

Multigraded Betti numbers.
Multigraded Betti numbers encode important information about the algebraic structure of a multigraded module over the polynomial ring in n variables [30,48].For multiparameter persistence modules that arise from data, multigraded Betti numbers provide insight about the coarse-scale topological features of the data (cf.[11]).For 2-parameter persistence modules, the multigraded Betti numbers are also called the bigraded Betti numbers.RIVET [42] represents the bigraded Betti numbers of a 2-parameter persistence module as a collection of colored dots in the plane.More interestingly, RIVET employs the bigraded Betti numbers to implement an interactive visualization of the fibered barcode.Recently, Lesnick-Wright [43] and Kerber-Rolle [37] developed efficient algorithms for computing minimal presentations and the bigraded Betti numbers of 2-parameter persistence modules.
Persistence diagram and its generalizations.In most applications of 1-parameter persistent homology, the notion of persistence diagram [29,40] (or equivalently barcode [17]; cf.Definition 2.3) plays a central role.The persistence diagram of any M : R → vec is not only a visualizable topological summary of M , but also a stable and complete invariant of M [21].In contrast, as mentioned before, there is no simple complete invariant for d -parameter persistence modules when d > 1.
Patel introduced the notion of generalized persistence diagram for constructible functors R → C , in which C satisfies certain properties [51].Construction of the generalized persistence diagram is based on the observation that the persistence diagram of M : R → vec [29] is an instance of the Möbius inversion of the rank invariant [16] of M .McCleary and Patel showed that the generalized persistence diagram is stable when C is a skeletally small abelian category [45].Kim and Mémoli further extended Patel's generalized persistence diagram to the setting of functors P → C in which P is a essentially finite poset such as a finite d -dimensional grid [38].The generalized persistence diagram of P → C is defined as the Möbius inversion of the generalized rank invariant of P → C .The generalized persistence diagram is not only a complete invariant of interval decomposable persistence modules P → vec (Theorem 2.20), but is also well-defined regardless of the interval decomposability.The generalized rank invariant of R d (or Z d ) → vec is proven to be stable with respect to a certain generalization of the erosion distance [51] and the interleaving distance [41] (see the latest version of the arXiv preprint of [38]).
Our contributions.Assume that a given M : Z 2 → vec is finitely generated.We establish a combinatorial formula for extracting the bigraded Betti numbers of M from the generalized persistence diagram of M (Theorem 3.5).More interestingly, the formula we found is a generalization of a well-known formula for extracting the bigraded Betti numbers from interval decomposable persistence modules (Theorem 2.9).
Namely, for any finitely generated interval decomposable M : Z 2 → vec, there is a visually intuitive way to find the bigraded Betti numbers of M from the indecomposable summands of M .An example of this process is shown in Fig. 1 (A)-(C).For any finitely generated N : Z 2 → vec, which may not be interval decomposable, we utilize a similar process to find the bigraded Betti numbers of N from the (Int-)generalized persistence diagram of N .This process is shown in Fig. 1 (A')-(C').In a sense, Theorem 3.5 thus reinforces the viewpoint that the (Int-)generalized persistence diagram is a proxy for the barcode (Definition 2.3) of persistence modules [2,38].
One implication of Theorem 3.5 is that all invariants of 2-parameter persistence modules that are computed by the software RIVET [42] are encoded by the generalized persistence diagram.In other words, we obtain the following hierarchy of invariants for any finitely generated M : Z 2 → vec, where invariant A is placed above invariant B if invariant B can be recovered from invariant A:

Hilbert function
We remark that the generalized persistence diagram is equivalent to the generalized rank invariant (Definitions 2.17 and 2.18).Also, the fibered barcode is equivalent to the (standard) rank invariant [16].Hence, in the diagram above, generalized persistence diagram and fibered barcode can be replaced by generalized rank invariant and rank invariant, respectively.In the course of establishing Theorem 3.5, we verify that the interval decomposable approximation and multirank invariant of 2-parameter persistence modules (introduced by Asashiba et al. [2] and Thomas [54], respectively) also encode the bigraded Betti numbers (Remark 2.19 and Corollary A.10).
Remark 1.1.It should not be construed that Theorem 3.5 provides a practically efficient way to compute the bigraded Betti numbers.Rather, we hope that the aforementioned efficient Black dots, red stars and blue squares indicate three different corner types of the expanded intervals (see Fig. 2).The bigraded Betti numbers of M can be read from these corner types; for each p ∈ Z 2 , β j (M )(p) is equal to the number of black dots, red stars, and blue squares at p when j = 0, 1, 2, respectively.(A') Another Z 2 -indexed persistence module N whose support is contained in a 3×4 grid.N is not interval decomposable.(B') The Int-generalized persistence diagram of N (Definition 2.18) is shown, where the multiplicity of the red interval is -1 and the multiplicity of each blue interval is 1. (C') is similarly interpreted as in (C), where corner points of the red interval negatively contribute to the counting of the bigraded Betti numbers.More details are provided in Example 3.6.algorithms to compute the bigraded Betti numbers could be useful for approximating the generalized persistence diagram.
We also show that for d ≥ 3, the generalized persistence diagram does not determine the multigraded Betti numbers of d -parameter persistence modules (Theorem 4.1).Lastly, we show that, in general, the generalized persistence diagram and the multirank invariant do not determine each other (Examples A.11 and A.12).

Other related work.
McCleary and Patel utilized the Möbius inversion formula for establishing a functorial pipeline to summarize simplicial filtrations over finite lattices into persistence diagrams [46].Botnan et al. introduced notions of signed barcode and rank decomposition for encoding the rank invariant of multiparameter persistence modules as a linear combination of rank invariants of indicator modules [10].In their paper, Möbius inversion was utilized for computing the rank decomposition, characterizing the generalized persistence diagram in terms of rank decompositions.Asashiba et al. provided a criterion for determining whether or not a given multiparameter persistence module is interval decomposable without having to explicitly compute indecomposable decompositions [1].Dey and Xin proposed an efficient algorithm for decomposing multiparameter persistence modules and introduced a notion of persistent graded Betti numbers, a refined version of the graded Betti numbers [27].Dey et al. reduced the problem of computing the generalized rank invariant of a given 2-parameter persistence module to computing the indecomposable decompositions of zigzag persistence modules [24].Blanchette et al. developed a theoretical framework for building new invariants of a persistence module over a poset using homological algebra [6].
Organization.In Section 2, we review the notions of persistence modules, multigraded Betti numbers, and generalized persistence diagrams.In Section 3, we show that the bigraded Betti numbers can be recovered from the generalized persistence diagram.In Section 5, we discuss open questions.In the appendix, we prove that (a) the multirank invariant (introduced in [54]) also determines the bigraded Betti numbers, and that (b) in general, the generalized persistence diagram and the multirank invariant do not determine each other.

Preliminaries
In Section 2.1, we review the notions of persistence modules and interval decomposability.In Section 2.2, we recall the notion of multigraded Betti numbers (an invariant of multiparameter persistence modules).In Section 2.3, we review the Möbius inversion formula in combinatorics.In Section 2.4, we review the notions of generalized rank invariant and generalized persistence diagram.In Section 2.5, we provide a formula of the (Int-)generalized persistence diagram in a certain setting, which will be useful in the next section.

Persistence modules and their interval decomposability
Let P be a poset.We regard P as the category that has points of P as its objects and for p, q ∈ P there is a unique morphism p → q if and only if p ≤ q in P. For d ∈ N, let R d and subsets of R d (such as Z d ) be given the partial order defined by (a 1 , a 2 , . . ., Every vector space in this paper is over some fixed field k.Let vec denote the category of finite dimensional vector spaces and linear maps over k. A P-indexed persistence module, or simply a P-module, refers to a functor M : P → vec.In other words, to each p ∈ P, a vector space M (p) is associated, and to each pair p ≤ q in P, a linear map ϕ M (p, q) : M (p) → M (q) is associated.Importantly, whenever p ≤ q ≤ r in P, it is required that ϕ M (p, r ) = ϕ M (q, r )•ϕ M (p, q).When P = R d or Z d , M is also called a d -parameter persistence module.
Consider a zigzag poset of n points, where ↔ stands for either ≤ or ≥.A functor from a zigzag poset (of n points) to vec is called a zigzag module (of length n) [13].
A morphism between P-modules M and N is a natural transformation f : M → N between M and N .That is, f is a collection { f p : M (p) → N (p)} p∈P of linear maps such that for every pair p ≤ q in P, the following diagram commutes: The kernel of f , denoted by ker( f ) : P → vec, is defined as follows: For p ∈ P, ker( f )(p) := ker( f p ) ⊆ M (p).For p ≤ q in P, ϕ ker( f ) (p, q) is the restriction of ϕ M (p, q) to ker( f p ). Two Pmodules M and N are (naturally) isomorphic, denoted by M ∼ = N , if there exists a natural transformation { f p } p∈P from M to N where each f p is an isomorphism.
The direct sum M N of M , N : P → vec is the P-module where (M N )(p) = M (p) N (p) for p ∈ P and ϕ M N (p, q) = ϕ M (p, q) ϕ N (p, q) for p ≤ q in P. A nonzero P-module M is indecomposable if whenever M = M 1 M 2 for some P-modules M 1 and M 2 , either M 1 = 0 or M 2 = 0.In what follows, we review the notion of interval decomposability.Definition 2.2.Let P be a poset.An interval of P is a subset I ⊆ P such that: (i) I is nonempty.(ii) If p, q ∈ I and p ≤ r ≤ q, then r ∈ I .(iii) I is connected, i.e. for any p, q ∈ I , there is a sequence p = p 0 , p 1 , • • • , p ℓ = q of elements of I with either p i ≤ p i +1 or p i +1 ≤ p i for each i ∈ [0, ℓ − 1]. 1 By Int(P), we denote the set of all intervals of P.
For example, any interval of a zigzag poset in (1) is a set of consecutive points in {• 1 , • 2 , . . ., • n }.For an interval I of a poset P, the interval module V I : P → vec is defined as V I .We call barc(M ) the barcode of M .
Theorem 2.4 ([3, 22, 32]).For d = 1, any M : R d (or Z d ) → vec is interval decomposable and thus admits a (unique) barcode.However, for d ≥ 2, M may not be interval decomposable.Lastly, any zigzag module is interval decomposable and thus admits a (unique) barcode.
The following notation is useful in the rest of the paper.
Notation 2.5.Assume that a P-module M is isomorphic to the direct sum i ∈I M i for some indexing set I where each M i is indecomposable.For I ∈ Int(P), we define mult(I , M ) as the cardinality of the set {i ∈ I : M i ∼ = V I }.In words, mult(I , M ) is the number of those summands M i which are isomorphic to the interval module V I .

Multigraded Betti numbers
In this section we review the notion of multigraded Betti numbers [30].Fix any p ∈ Z d .Then, the upper set p The collection {v 1 , . . ., v n } is called a (homogeneous) generating set for M .
Let us assume that {v 1 , . . ., v n } is a minimal homogeneous generating set for M , i.e. there is no homogeneous generating set for M that includes fewer than n elements.Let 1 This definition of interval is not the standard definition of interval used in order theory but it is often used in the literature concerned with persistence modules over posets; e.g.[7].In order theory language, I is a nonempty convex connected subset of P.
Then, the set {1 p 1 , . . ., 1 p n } generates F 0 and the morphism . ., n is surjective.Let K 0 := ker(η 0 ) ⊆ F 0 and let ı 0 : K 0 → F 0 be the inclusion map.Iterate this process using K 0 in place of M . 2amely, identify a minimal homogeneous generating set {v ′ 1 , . . ., v ′ m } for K 0 where v ′ j ∈ (K 0 ) p ′ j for some p ′ 1 , . . ., p ′ m ∈ Z d and consider the free module and the surjection η 1 : By repeating this process, we obtain a minimal free resolution of M : This resolution is unique up to isomorphism [30,Theorem 1.6].Hilbert's Syzygy Theorem guarantees that F j = 0 for j > d [34].
We will see that for an interval decomposable Z 2 -module M , its bigraded Betti numbers can be extracted from barc(M ).To this end, we will make use of a certain regions that arise by "blowing-up" intervals from barc(M ): Definition 2.7.Given any I ∈ Int(Z 2 ), the subset of R 2 will be referred to as the region corresponding to I in R 2 .
(i) For any finitely generated M , N : . For the interval module V I : Z 2 → vec, the j th bigraded Betti number β j (V I )(p) is equal to 1 if p is a j th type corner point of I + and is equal to 0 otherwise; see Fig. 2.
Remark 2.8 directly implies: Theorem 2.9.Given any finitely generated interval decomposable module M : Z 2 → vec, the bigraded Betti numbers of M can be extracted from barc(M ).More specifically, the bigraded Betti numbers of M can be extracted from the corner points of the elements in the multiset In Theorem 3.5, we remove the assumption that M be interval decomposable and generalize Theorem 2.9 to the setting of any finitely generated Z 2 -modules.Points on the upper boundary (dashed lines) do not belong to I + , while points on the lower boundary (solid lines) belong to I + .Points which lie on both boundaries do not belong to I + .

The Möbius inversion formula in combinatorics
In this section, we briefly review the Möbius inversion formula, a fundamental concept in combinatorics [5,52].
A poset A is said to be locally finite if for all p, q ∈ A with p ≤ q, the set [p, q] := {r ∈ A : p ≤ r ≤ q} is finite.Let A be a locally finite poset.The Möbius function µ For q 0 ∈ A, consider the principal ideal q ↓ 0 := {q ∈ A : q ≤ q 0 }.Note that if we assume that q ↓ is finite for all q ∈ A, then A must be locally finite.To see this, note that, for any p, q ∈ A with p ≤ q, the set [p, q] is a subset of the finite set q ↓ .Theorem 2.10 (Möbius Inversion formula).Assume that q ↓ is finite for all q ∈ A. Let k be a field.For any pair of functions f , g : The function f is called the Möbius inversion of g .One interpretation of the Möbius inversion formula is that of a discrete analogue of the derivative of a real-valued map in elementary calculus, as explained in the following example: Example 2.11.Let [m] = {0, 1, . . ., m} with the usual order.Then, Hence, for any function g for a ̸ = 0 and f (0) = g (0).Hence, at each point a ̸ = 0, f (a) captures the rate of change of g around that point.

Generalized rank invariant and generalized persistence diagrams
In this section we review the notions of generalized rank invariant and generalized persistence diagram [38,51].
Throughout this subsection, let P denote a finite connected poset (Definition 2.2 (iii)).
Consider any P-module M .Then M admits a limit and a colimit of M : lim ; see the appendix for a review of the definitions of limits and colimits (Definitions A. 4 and A.6).This implies that, for every p ≤ q in P, Since P is connected, these equalities imply that ι p • π p = ι q • π q : L → C for any p, q ∈ P. In words, the composition ι p • π p is independent of p.The canonical limit-to-colimit map ψ M : lim ← − − M → lim − − → M is therefore defined to be the linear map ι p • π p where p is any point in P.

Definition 2.12 ([38]
).The rank of M : P → vec is defined as the rank of the canonical limit-tocolimit map ψ M : lim The rank of M : P → vec counts the multiplicity of the fully supported interval module V P in a direct sum decomposition of M into indecomposable modules: Theorem 2.13 ([19, Lemma 3.1]).For any M : P → vec, the rank of M is equal to mult(P, M ).
Let p, q ∈ P. We say that p covers q and write q ◁ p if q < p and there is no r ∈ P such that q < r < p.
A subposet I ⊆ P is said to be path-connected in P if for any p ̸ = q in I , there exists a sequence p = p 0 , p 1 , . . ., p n = q in I such that either p i ◁ p i +1 or p i +1 ◁ p i in P for i = 0, . . ., n − 1.For example, the set {0, 2} is a connected (Definition 2.2 (iii)) subposet of {0, 1, 2} equipped with the usual order, but is not path-connected in {0, 1, 2}.
By Con(P) we denote the poset of all path-connected subposets of P that is ordered by inclusions.We remark that, since P is finite, Con(P) is finite.For example, assume that P is the zigzag poset {• 1 < • 2 > • 3 }.Then, Con(P) consists of the six elements: { In fact, in order to define the generalized rank invariant, P does not need to be finite [38,Section 3].However, for this work, it suffices to consider the case when P is finite.
(ii) Let I , J ∈ Con(P) with J ⊇ I .Then rk(M )(J ) ≤ rk(M )(I ), i.e. rk(M ) is order-reversing.This is because the canonical limit-to-colimit map lim The following is a corollary of Theorem 2.13.Proposition 2.16 ([38, Proposition 3.17]).Let M : P → vec be interval decomposable.Then for any I ∈ Con(P), In words, rk(M )(I ) equals the total multiplicity of intervals J in barc(M ) that contain I .
For any poset A, let A op denote the opposite poset of A, i.e. p ≤ q in A if and only if q ≤ p in A op .By virtue of Theorem 2.10 we have: Definition 2.17.Let P be a finite connected poset.The generalized persistence diagram of M : P → vec is the unique function dgm(M ) : Con(P) → Z that satisfies, for any I ∈ Con(P), In other words, dgm(M ) is the Möbius inversion of rk(M ) over Con op (P).That is, for I ∈ Con(P), ( The function µ Con op (P) has been precisely computed in [38,Section 3].Next, we restrict the domain of rk(M ) and dgm(M ) to the collection Int(P) of all intervals of P. For M : P → vec, let rk I (M ) denote the restriction of rk(M ) : Con(P) → Z ≥0 to Int(P).We consider the Möbius inversion of rk I (M ) over the poset Int op (P).Again by virtue of Theorem 2.10 we have: Then, dgm I (M ) is equivalent to the interval decomposable approximation δ tot (M ) given in [2]; this is a direct corollary of Theorem 2.13.The Möbius function µ Int op ([m]×[n]) has been precisely computed in [2], which leads to Theorem 2.22 below.
Although we do not require M : P → vec to be interval decomposable in order to define dgm(M ) or dgm I (M ), these two diagrams generalize the notion of barcode (Definition 2. The equality given in Equation ( 7) was first proved in [38,Theorem 3.14], but we include a proof here for completeness.Theorem 2.20 implies that both dgm(M ) and dgm I (M ) are able to completely determine the isomorphism type of an interval decomposable persistence module M (which also implies that each of rk(M ) and rk I (M ) is strong enough to determine the isomorphism type of M ).However, in general, the generalized persistence diagram dgm(M ) is more discriminative than the Intgeneralized persistence diagram dgm I (M ); see Example A.2 in the appendix.
In Section 3, the case when P is a zigzag poset of length 3 will be useful.
Since M is a zigzag module, it is interval decomposable (Theorem 2.4).Thus, we have dgm(M )(I ) = mult(I , M ) for I ∈ Con(P), the multiplicity of I in barc(M ).Since Con(P) = Int(P), each dgm(M ) above can be replaced by dgm I (M ).  ) depicted as in Fig. 3 (A).Note that cov(I ) = {J 1 , J 2 , J 3 } where J 1 , J 2 and J 3 are depicted as in Fig. 3

(B). For any ([3] × [2]
)-module M , we have: The following remark will be useful in the next section.

Extracting the bigraded Betti numbers from the generalized persistence diagram
In this section we aim at establishing Theorem 3.5, as a generalization of Theorem 2.9.
Let M be a finitely generated Z 2 -module. 4We may assume that M (p) = 0 for p ̸ ≥ (0, 0).Then, all algebraic information of M can be recovered from the restricted module for some large enough positive integers m and n.We will show that the generalized persistence diagram of M ′ determines the bigraded Betti numbers of M .• If p ∈ Z 2 is not greater than equal to (0, 0), then M (p) = 0.
• For (0, 0) ≤ p in Z 2 , we have that M (p) = M ′ (q) where q is the maximal element of [m]× [n] such that q ≤ p (we write q = ⌊p⌋ m,n in this case).
• For (0, 0) we have β j (M )(p) = 0, j = 0, 1, 2. For p ∈ [m + 1] × [n + 1], we have: where each sum is taken over J ∈ Con([m] × [n]). 5Moreover, each dgm(M ′ ) above can be replaced by dgm I (M ′ ) where each sum is taken over ) can include p and thus the sum We defer the proof of Proposition 3.2 to the end of this section.Remark 3.3.In Proposition 3.2, the equation for β 1 (M ) with respect to dgm I (M ′ ) can be further simplified by removing the fourth, sixth, and seventh sums, i.e.This is because the connected sets J (Definition 2.2 (iii)) over which the fourth, sixth, and seventh sums are taken cannot be intervals of [m] × [n] (those connected sets J cannot satisfy Definition 2.2 (ii)).Similarly, if dgm(M ′ )(J ) = 0 for all non-intervals J ∈ Con([m] × [n]), then the fourth, sixth, and seventh sums can be eliminated in the equation for β 1 (M ). 6y virtue of Remark 3.3, Proposition 3.2 admits a simple pictorial interpretation which generalizes Remark 2.8 (ii) and Theorem 2.9.To state this interpretation, we introduce the following notation.Notation 3.4.Given any I ∈ Con(Z 2 ), let I + ⊂ R 2 be the corresponding region (cf.equation ( 2)).Then I + admits the 3 types of corner points depicted in Figure 4.For j = 0, 1, 2, we define functions τ j (I + ) : Z 2 → {0, 1, 2} as follows: for j = 0, 2, let τ j (I + )(p) := 1 if p is a j th type corner point of I + , and 0 otherwise.For j = 1, let 2, p is a 1 st -type corner point of I + with multiplicity 2 1, p is a 1 st -type corner point of I + with multiplicity 1 0, otherwise.
Our main theorem below says that the bigraded Betti numbers of a given Z 2 -module M encoded by an ([m] × [n])-module M ′ can be read off from the corner points of the elements in either of Then, for every j = 0, 1, 2 and for every p ∈ Z 2 , we have Also we have: Figure 4: The three different types of corner points in I + ⊂ R 2 and J + ⊂ R 2 .Note that two different 1 st type corner points of J are located at p. See Definition A.1 for a rigorous description of each of the three types of corner points.
Notice that, by Theorem 2.20, the theorem above is a generalization of Theorem 2.9.We prove Theorem 3.5 at the end of this section.(C') For i = 1, 2, 3, 4, expand I i to its corresponding region I + i in R 2 (cf.Definition 2.7).The corner points of each I + i are marked according to their types as described in Fig. 4. By Theorem 3.5, for each p ∈ Z 2 and j = 0, 1, 2, β j (N )(p) is equal to the number of black dots, red stars, and blue squares at p respectively, where the corner points of the red interval I + 4 negatively contribute to the counting.The net sum is illustrated in Fig. 6. (ii) Consider K := {(0, 1), (1, 1), (1, 0)} ∈ Int([3] × [2]) that is depicted in Fig. 5.We claim that for all J ⊇ K , dgm I (N ′ )(J ) = 0: By Remarks 2.15 (ii) and 2.24, it suffices to show that  rk I (N ′ )(K ) = 0.This follows from Theorem 2.13 and the fact that the zigzag module N ′ | K does not admit a summand that is isomorphic to the interval module V K : K → vec.An alternative way to prove rk

Details about
and π p : L → N ′ (p) are the canonical projections for p ∈ K .Then, we have: Remarks 2.15 and (ii), we have that rk I (N ′ )(J ) = 0. Therefore, by Theorem 2.22, we have: Lemma 3.7 ([49, 50]).Given any finitely generated Z 2 -module M , for every p ∈ Z 2 , we have: A combinatorial proof of this lemma can be found in [49,Corollary 2.3].This lemma can be also proved by utilizing machinery from commutative algebra as follows (see [30,Section 2A.3] for details): A finitely generated 2-parameter persistence module M can equivalently be considered as an N 2 graded module over k[x 1 , x 2 ].The bigraded Betti numbers of M can be defined using tensor products, after which Lemma 3.7 follows by tensoring M with the Koszul complex on x 1 and x 2 .
Proof of Proposition 3.2.We consider the case j = 1, as the other cases are similar.By Lemma 3.7, . Then, we claim that β 1 (M )(p) = 0.This fact can be shown by checking that 0 = n p = m p−e 1 −e 2 and We will now find a formula for each term in the right-hand side (RHS) of Equation ( 14) in terms of the generalized rank invariant of Next, consider M | {p−e 1 ≤p≥p−e 2 } , which is a zigzag module and thus it is interval decomposable (Theorem 2.4).Recall that n p is the multiplicity of {p} in the barcode of M | {p−e 1 ≤p≥p−e 2 } .From Example 2.21, we know that: Similarly, we have: Combining equations ( 14), ( 15), ( 16), ( 17) yields: , by invoking Equation (4), we obtain: dgm(M ′ )(J ) Let J ∈ Con([m] × [n]).The multiplicity of dgm(M ′ )(J ) in the RHS of Equation ( 18) is fully determined by the intersection of J and the four-point set {p −e 1 −e 2 , p −e 1 , p −e 2 , p}.For example, if p − e 1 , p − e 1 − e 2 ∈ J and p − e 2 , p ∉ J , then dgm(M ′ )(J ) occurs only in the fourth and sixth sums, and has an overall multiplicity of zero in the RHS of Equation (18).For another example, if p − e 1 ∈ J and p − e 1 − e 2 , p − e 2 , p ∉ J , then dgm(M ′ )(J ) occurs only in the fourth summand, which yields the first sum of the RHS in Equation ( 19) below.Considering all possible 2 4 combinations of the intersection of J and the four-point set {p − e 1 − e 2 , p − e 1 , p − e 2 , p} yields as claimed.
Proof of Theorem 3.5.We only prove equation ( 12) with j = 1, as the other cases are similar.By Remark 3.3, Otherwise, τ 1 (I + )(p) = 0.These four cases (i),(ii),(iii), and (iv) correspond to the four sums on the RHS of Equation (20) in order, and also correspond to the 1 st corner types (i),(ii),(iii), (iv) given in Figure 4. Therefore: To prove this theorem, we find a pair of Z 3 -modules that have the same generalized persistence diagram, but different multigraded Betti numbers.Since any Z d -module M can be trivially extended to the Z d +1 -module M × 0, the existence of such a pair proves the claim for arbitrary d ≥ 3.

Conclusions
The formula in Theorem 3.5 for computing the bigraded Betti numbers reinforces the fact that the (Int-)generalized persistence diagram and the interval decomposable approximation by Asashiba et al. (Remark 2.19) are a proxy for the "barcode" of M in a novel way.Some open questions follow.
(i) Note that when M is a finitely generated Z 2 -module, dgm(M ) can recover dgm I (M ) by construction while dgm I (M ) may not be able to recover dgm(M ).However, if M is interval decomposable, then both dgm(M ) and dgm I (M ) are equivalent to the barcode of M by Theorem 2.20.Are there other settings in which dgm I (M ) can recover dgm(M )?
For the interval modules V I j for j = 0, 1, 2, 3 given in the proof of Theorem 4.1, the figures above depict the multigraded Betti numbers for V I j in order from left to right.In these illustrations, a black dot at p indicates that β 0 (V I j )(p) = 1, a red star at p indicates that β 1 (V I j )(p) = 1, a blue square at p indicates that β 2 (V I j )(p) = 1, and a black triangle at p indicates that β 3 (V I j )(p) = 1.For i ≥ 3, j ∈ {0, 1, 2, 3}, and p ∈ Z 3 , β i (V I j )(p) = 0.
(ii) How can we utilize Theorem 3.5 alongside efficient algorithms for computing the bigraded Betti numbers [43,37], as a means to estimate or calculate the generalized persistence diagram of a 2-parameter persistence module (cf.Remark 1.1)?
Figure 8: An illustration for Example A.2 illustrating that, in general, dgm(M ) is a stronger invariant than dgm I (M ) .
Limits and colimits.We recall the notions of limit and colimit [44,Chapter V].In what follows, I stands for a small category, i.e.I has a set of objects and a set of morphisms.Let C be any category.It is possible that a functor does not have a limit at all.However, if a functor does have a limit then the terminal property of the limit guarantees its uniqueness up to isomorphism.For this reason, we sometimes refer to a limit as the limit of a functor.When I is a finite category and C = vec, any functor F : I → vec admits a limit in vec.
Cocones and colimits are defined in a dual manner: It is possible that a functor does not have a colimit at all.However, if a functor does have a colimit then the initial property of the colimit guarantees its uniqueness up to isomorphism.For this reason, we sometimes refer to a colimit as the colimit of a functor.When I is a finite category and C = vec, any functor F : I → vec admits a colimit in vec.
Multirank invariant.We review the notion of the multirank invariant for a persistence module [54], which is a natural generalization of the rank invariant and differs from the generalized rank invariant.Then, we demonstrate that the multirank invariant of a zigzag module M with a length of 3 completely determines the isomorphism type of M .This fact, along with Corollary 3.8, implies that the multirank invariant of any Z 2 -module N determines the bigraded Betti numbers of N .
Let P be a poset.For a P-module M and any s, t ∈ P, let us define the map where the map s∈S M (s) → t ∈T M (t ) is canonically defined by the map given in Equation (22) for each pair of s ∈ S and t ∈ T .The multirank invariant of M is the map that sends every pair of finite subsets S, T ⊂ P to multirk M (S, T ).
A list of useful properties of the multirank invariant follows.
Comparison between the multirank invariant and the generalized rank invariant.We show that neither the generalized rank invariant nor the multirank invariant is a strictly stronger invariant than the other.We do this by providing two pairs of persistence modules: The first pair is distinguishable by their multirank invariants but not by their generalized rank invariants where i 1 , i 2 : k → k 2 are the canonical inclusions into the first factor and the second factor of k 2 , respectively.Consider the interval modules V {a} , V {a,b} , V {a,c} , V {a,d } over P.
It is not hard to verify that the generalized rank invariants of the P-modules N 1 := M ⊕ V {a} and N 2 := V {a,b} ⊕V {a,c} ⊕V {a,d } are identical.However, for S := {b, c, d } and T := {a}, we have that multirk N 1 (S, T ) = 2 ̸ = 3 = multirk N 2 (S, T ).
On the other hand, we claim that for any pair S, T ⊂ P, multirk L 1 (S, T ) = multirk L 2 (S, T ).
Since dim((L 1 ) p ) = dim((L 2 ) p ) for each p ∈ P, Remark A.8 (vii) implies that one needs to check Equation ( 23) only for disjoint nonempty sets S, T ⊂ P. Let S, T ⊂ P be nonempty disjoint sets.Note that, unless a belongs to T , the both sides of Equation ( 23) are zero.Therefore, it is only necessary to verify Equation (23) in the case where S ⊂ {b, c, d } and T = {a}.This verification is straightforward.

Figure 1 :
Figure 1: (A) A Z 2 -indexed persistence module M whose support is contained in a 3 × 4 grid.(B) M is interval decomposable, and the barcode of M consists of the two blue intervals of Z 2 (Definitions 2.2 and 2.3).(C) Expand each of the blue intervals from (B) to intervals in R 2 as follows: Each point p = (p 1 , p 2 ) in the two intervals is expanded to the unit square [p 1 ,p 1 + 1) × [p 2 , p 2 +1) ⊂ R 2 .Black dots, red stars and blue squares indicate three different corner types of the expanded intervals (see Fig.2).The bigraded Betti numbers of M can be read from these corner types; for each p ∈ Z 2 , β j (M )(p) is equal to the number of black dots, red stars, and blue squares at p when j = 0, 1, 2, respectively.(A') Another Z 2 -indexed persistence module N whose support is contained in a 3×4 grid.N is not interval decomposable.(B') The Int-generalized persistence diagram of N (Definition 2.18) is shown, where the multiplicity of the red interval is -1 and the multiplicity of each blue interval is 1. (C') is similarly interpreted as in (C), where corner points of the red interval negatively contribute to the counting of the bigraded Betti numbers.More details are provided in Example 3.6.

Theorem 2 . 1 (
Krull-Remak-Schmidt-Azumaya[3]).Any P-module M has a direct sum decomposition M ∼ = i M i where each M i is indecomposable.Such a decomposition is unique up to isomorphism and reordering of the summands.

Figure 2 :
Figure 2: An interval I ∈ Int(Z 2 ) and its corresponding region I + ⊂ R 2 with its corner points.Points on the upper boundary (dashed lines) do not belong to I + , while points on the lower boundary (solid lines) belong to I + .Points which lie on both boundaries do not belong to I + .
Int(P) = Con(P) (Definition 2.2).In general, Int(P) is a subposet of Con(P).Definition 2.14.The generalized rank invariant of M : P → vec is the function rk(M ) : Con(P) → Z ≥0 which maps I ∈ Con(P) to the rank of the restriction M | I of M .

-
Generalized persistence diagram of an ([m] × [n])-module.In this section we review a formula of the Int-generalized persistence diagram of an ([m] × [n])module for any fixed integers m, n ≥ 0. Let us consider the poset Int([m] × [n]).Then, given any two distinct I , J ∈ Int([m] × [n]), we say that J covers I if J ⊋ I and there is no interval K such that J ⊋ K ⊋ I .For I ∈ Int([m] × [n]), let us define cov(I ) as the collection of all J ∈ Int([m] × [n]) that cover I .Given any nonempty S ⊆ Int([m] × [n]), by S, we denote the smallest interval J that contains all I ∈ S. The following theorem is established by invoking Remark 2.19 and finding an explicit formula for the Möbius function µ Int op (P) that appears in Equation (6) with P = [m] × [n].

Definition 3 . 1 .
A given Z 2 -module M is said to be encoded by M ′ : [m]×[n] → vec if the following hold:

5 .
Similarly, one can compute dgm I (N ′ )(I 3 ) = 1, dgm I (N ′ )(I 4 ) = −1, and dgm I (N ′ )(L) = 0 for any L ∈ Int([3] ×[2]) that has not been considered so far.Proofs of Proposition 3.2 and Theorem 3.Lemma 3.7 below will be used in the proof of Proposition 3.2.Let M be any finitely generated Z 2 -module.For any p ∈ Z 2 , consider the subposet {p −e 1 ≤ p ≥ p −e 2 } where e 1 = (1, 0) and e 2 = (0, 1).The restriction of M to {p −e 1 ≤ p ≥ p − e 2 } is a zigzag module and thus admits a barcode (Theorem 2.4).Let n p be the multiplicity of {p} in the barcode of M | {p−e 1 ≤p≥p−e 2 } .Similarly, we also consider the subposet {p + e 1 ≥ p ≤ p + e 2 } and define m p to be the multiplicity of {p} in the barcode of M | {p+e 1 ≥p≤p+e 2 } .

4Theorem 4 . 1 .
Impossibility of extending Theorem 3.5 to d -parameter persistence modules for d ≥ 3In this section, we show that Theorem 3.5 cannot be extended to Z d -modules for d ≥ 3.This impossibility claim directly follows from: For d ≥ 3, the generalized persistence diagram does not determine the multigraded Betti numbers of Z d -modules.In particular, there exists a pair of Z d -modules that have the same generalized persistence diagram, but different multigraded Betti numbers.

Definition A. 3 (FA
Cone).Let F : I → C be a functor.A over F is a pair L, (π x ) x∈ob(I ) consisting of an object L in C and a collection (π x ) x∈ob(I ) of morphisms π x : L → F (x) that commute with the arrows in the diagram of F , i.e. if g : x → y is a morphism in I , then π y = F (g ) • π x in C .Equivalently, the diagram below commutes: limit of F : I → C is a terminal object in the collection of all cones over F : Definition A.4 (Limit).Let F : I → C be a functor.A limit of F is a cone over F , denoted by lim ← − − F, (π x ) x∈ob(I ) or simply lim ← − − F , with the following terminal property: If there is another cone L ′ , (π ′ x ) x∈ob(I ) of F , then there is a unique morphism u : L ′ → lim ← − − F such that π ′ x = π x • u for all x ∈ ob(I ).

Definition A. 5 (A
Cocone).Let F : I → C be a functor.A cocone over F is a pair C , (i x ) x∈ob(I ) consisting of an object C in C and a collection (i x ) x∈ob(I ) of morphisms i x : F (x) → C that commute with the arrows in the diagram of F , i.e. if g : x → y is a morphism in I , then i x = i y • F (g ) in C , i.e. the diagram below commutes.colimit of a functor F : I → C is an initial object in the collection of cocones over F : Definition A.6 (Colimit).Let F : I → C be a functor.A colimit of F is a cocone, denoted by lim − − → F, (i x ) x∈ob(I ) or simply lim − − → F , with the following initial property: If there is another cocone C ′ , (i ′ x ) x∈ob(I ) of F , then there is a unique morphism u : lim − − → F → C ′ such that i ′ x = u • i x for all x ∈ ob(I ).

) Definition A. 7 .
For a P-module M and finite subsets S, T ⊂ P, the multirank from S to T is defined asmultirk M (S, T ) := rank s∈S M (s) → t ∈T M (t ) ,

11 .
. The second pair exhibits the opposite scenario.In what follows, let P := {a, b, c, d } be the poset equipped with the partial order ≤:= {(b, a), (c, a), (d , a)} ⊂ P × P. The Hasse diagram of P is given below.A P-module M is given by