Finite presentation of finitely determined modules

In this article we study certain notions of `tameness' for the persistence modules studied in topological data analysis. In particular, we show that after adding infinitary points the so called finitely determined modules become finitely presented.


Introduction
This article is motivated by topological data analysis, which is a recent field of mathematics studying the shape of data.One of the main methods of topological data analysis is persistent homology.In persistent homology one studies the data by associating a filtered topological space to it.By taking homology with coefficients in a field, one obtains a diagram of vector spaces and linear maps.This diagram is called a persistence module.In the standard case, the filtration is indexed by Z or R, but the indexing set can by any poset.Carlsson and Zomorodian realized that one can consider persistence modules indexed by Z n as Z n -graded modules over a polynomial ring of n variables (see [2,p. 78,Thm. 1]).This opened the way for methods of commutative algebra and algebraic geometry in topological data analysis.However, it is important to consider also more general indexing sets.More formally, a persistence module indexed by a poset C with coefficients in a field k is a functor from C, interpreted as a category, to the category of k-vector spaces.For the sake of generality, instead of a field k, we prefer in this article to work with any commutative ring R. Following the terminology of representation theory, we call a functor C → R-Mod an RC-module.In this terminology, a persistence module is then a kC-vector space.
Persistence modules need not be finitely presented.For computational reasons, one has therefore introduced several notions of 'tameness' for them.In this context, Miller defines in [7,p. 24,Def. 4.1] an encoding of an RC-module M by a poset D to be a poset morphism f : C → D with an RD-module N such that the restriction res f N ∼ = M .We define in [4, p. 22 for the upset generated and the downset cogenerated by T , respectively.This is a straightforward generalization of the notion of a 'positively a-determined' N ngraded module, where a ∈ N n , as defined in [6,p. 186,Def. 2.1].One can look at them as modules determined by their restriction to the interval [0, a] ⊆ N n .They are finitely generated.Positively a-determined modules have been much studied by commutative algebraists.See e.g.[1] and the references therein.Suppose now that S ⊆ C is a finite set.We will consider the set S of all minimal upper bounds of the subsets of S. We are going to define a functor α : C → S by mapping an element of C to the unique minimal upper bound of the elements of S below it.In our main result, Theorem 2.5, we will prove that M is S-determined for some finite S ⊆ C if and only if α is an encoding of M .
The definition given by Miller in [6, p. 186, Def.2.1] includes also the so called 'finitely determined' modules.They are further studied in the context of topological data-analysis in [7].Finitely determined modules are Z n -graded modules fully determined by their restriction to an interval [a, b] ⊆ Z n .They are not finitely presented in general.However, as a consequence of Theorem 2.5 we can show in Theorem 4.7 that after adding infinitary points to Z n , finitely determined modules in fact become finitely presented.We follow here an idea due to Perling (see [8, p. 16]).We also show in Proposition 4.13 that our terminology is compatible with that of admissible posets used in [8].

Preliminaries
Throughout this article we use the terminology of category theory.We will always assume that C is a small category and R a commutative ring.For any set X, we denote by R[X] the free R-module generated by X.An RC-module is a functor from C to the category of R-modules.A morphism between RC-modules is a natural transformation.For more details on RC-modules, we refer to [5] and [10].
Recall first that an RC-module M is called where I is a finite set, and where I and J are finite sets, and c i , d j ∈ C for all i ∈ I and j ∈ J. See, for example, [9].
Let ϕ : S → C be a functor between small categories.Recall that the restriction res ϕ : RC-Mod → RS-Mod is the functor defined by precomposition with ϕ, and the induction ind ϕ : RS-Mod → RC-Mod is its left Kan extension along ϕ.The induction is the left adjoint of the restriction.The counit of this adjunction gives us for every RC-module M the canonical morphism More explicitly, for any RC-module M and RS-module N , we have the pointwise formulas for all s ∈ S and c ∈ C.Here (ϕ/c) denotes the slice category.Its objects are pairs (s, u), where s ∈ S and u : ϕ(s) → c is a morphism in C. For (s, u), (t, v) ∈ Ob(ϕ/c), a morphism (s, u) → (t, v) is a morphism f : s → t in S with vϕ(f ) = u.We will typically assume that S is a full subcategory of C and that ϕ is the inclusion functor.In this case, we use the notations res S and ind S instead of res ϕ and ind ϕ .If C is also a poset, the latter formula yields Let C be a small category and S ⊆ C a full subcategory.An RC-module M is said to be S-generated if the natural morphism where the tensor product is taken pointwise.Since the morphism ρ M factors through the canonical morphism µ M , we see that M is S-generated if and only if µ M is an epimorphism.
Following [3, p. 13, Prop.2.14], we say that M is S-presented if it is S-generated and the following condition holds: Given an exact sequence of RC-modules where N is S-generated, then L is S-generated.It is shown in [3, p. 13, Prop.2.14], that M is S-presented if and only if µ M is an isomorphism.

Modules over strongly bounded posets
In order to prove Theorem 2.5, we need to recall some order theory.In the following, C always denotes a poset.Notation 2.1.Let S ⊆ C be a finite subset.We denote the set of minimal upper bounds of S by mub(S).If S is finite, we set In other words, Ŝ is the set of minimal upper bounds of non-empty subsets of S.
We say that the poset C is strongly bounded from above if every finite S ⊆ C has a unique minimal upper bound in C. If C is strongly bounded from above, then Ŝ is finite.The condition of C being strongly bounded from above is equivalent to C being a bounded join-semilattice.Also note that if C is strongly bounded from above, then C is weakly bounded from above and mub-complete, as defined in [4,p. 23,Def. 4.5,Def. 4.6].
Let C be strongly bounded from above, and let S ⊆ C be a finite set.From now on, we consider mub(S) as an element of C, and not as a (one element) set.
In particular, every element of Ŝ is then of the form mub(S ′ ), where S ′ ⊆ S is a non-empty subset.Viewing C as a join-semilattice, we have the join-operation Extending this operation to finite sets, we get an operation that coincides with taking minimal upper bounds.Lemma 2.2.Let C be strongly bounded from above, and let S ⊆ C be a finite subset.Then Ŝ = Ŝ.
Encouraged by Proposition 2.3, we will just write α instead of α S , if there is no risk of confusion.Before moving on to the main theorem of this section, we require one more lemma.Lemma 2.4.Let C be strongly bounded from above, and let S ⊆ C be a finite subset.
Let C be strongly bounded from above, let M be an RC-module, and let S ⊆ C be a finite subset.The morphism α gives rise to a natural transformation where for any c ∈ C, T α,c is the morphism We are now able to prove our main result Theorem 2.5.Let C be strongly bounded from above, and let M be an RC-module.Given a finite subset S ⊆ C, the following conditions are equivalent: 1) For all c ≤ d in C, If min(C) ∈ S, then condition 1) says that M is S-determined.
Proof.Suppose first that 1) holds.We can safely assume that S includes the minimum element of C, so that Supp(M ) ⊆ ↑S = C.This will not affect the sets Ŝ or S, nor the functor α.Therefore M is S-determined.We have proved in [4,p. 25,Cor. 4.13] that an S-determined module is Ŝ-presented.Lemma 2.2 now tells us that M is Ŝ-presented.So M ∼ = ind Ŝ res Ŝ M .This implies that for c ∈ C, Furthermore, by Lemma 2.4, we get colim d≤α(c), d∈ If 2) holds, we immediately see that the functor α with the R S-module res S M is an encoding of M .
Finally, suppose that 3) is true.Assume that c ≤ d and S ∩ ↓c = S ∩ ↓d.We need to show that M (c ≤ d) is an isomorphism.Since α is an encoding of M , there exists an R S-module N such that res α N ∼ = M .Here res so the morphism res α N (c ≤ d) is an isomorphism.Thus M (c ≤ d) is an isomorphism.Therefore 1) holds true.

Adding infinitary points
One approach to understand RZ n -modules better is to expand the set Z n to include points at infinity.This idea has been utilized by Perling in [8].Set Z := Z ∪ {−∞}.It is easy to see that Z n inherits a poset structure from Z n .Any RZ n -module M can be naturally extended to an RZ n -module M by setting More formally, this is the coinduction of M with respect to the inclusion Z n → Z n .The functor M → M establishes an equivalence of categories between the category RZ n -Mod and its essential image in RZ n -Mod.
Let S ⊆ Z n be a finite non-empty subset.We denote by mlb(S) the (unique) maximal lower bound of S. In this section, we will define a morphism β "dual" to α.The idea is to map an element to the maximal lower bound of the elements of S above it.The morphism β will play a crucial role in the proof of our main result, Theorem 4.7.
We restrict ourselves to cartesian subsets of Z n , i.e. subsets of the form S = where S 1 , . . ., S n are subsets of Z.In this situation, we can calculate α and β coordinatewise.We begin with the following observation.Proof.Since both 1) and 2) are proved in the same way, we will only present the proof of 1) here.Let i ∈ {1, . . ., n}.The existence of max(p i (S)) follows from the fact that p i (S) is non-empty, linearly ordered and finite.Write We will show that d i = max(p i (S)).First, since d is an upper bound of S and the canonical projection p i preserves order, we see that is an upper bound of S such that d ′ < d, contradicting the minimality of d.Thus we have We now have Proof.To prove 1), let p i be the canonical projection Z n → Z for all i ∈ {1, . . ., n}.
Next, the proof for 2) is done in the same way as 1), this time using Proposition 3.1 2).
Finally, for 3), we note that S is finite and cartesian, so 1) implies Ŝ = S. Since S already contains the minimum element of Z n , we get for all c ∈ S.Here the set S ∩ ↑c is always non-empty, because S is final in S.
We can now give coordinatewise formulas for α and β.
Proposition 3.4.We write α i := α Si and β i := β Si for all i ∈ {1, . . ., n}.For Proof.To prove 1), we will first show that where p i : Z n → Z is the canonical projection for all i ∈ {1, . . ., n}.Since p i (S) = S i and p i (↓c) = ↓c i , we see that p i (S ∩ ↓c) ⊆ S i ∩ ↓c i .For the other direction, suppose that d ∈ S i ∩ ↓c i .Then d ≤ c i , so we have an element Hence p i (S ∩ ↓c) = S i ∩ ↓c i .Now, using this result and Proposition 3.1 1), we get For 2), the proof is similar.Let c ∈ S. We will first show that From p i (S) = S i and p i (↑c) = ↑c i , we see that p i (S ∩ ↑c) ⊆ S i ∩ ↑c i .Next, suppose that d ∈ S i ∩ ↑c i .Since c ∈ S, there is an element s := (s 1 , . . ., s n ) ∈ S such that s ≥ c.Because d ≥ c i and S is cartesian, we again have an element To finish the proof, we use Proposition 3.1 2): We note that α and β • α are "continuous" in the following sense.1) If N is an RS-module, then follows: For any i ∈ {1, . . ., n}, we set a i = min(S i ∩ Z), if it exists, and This guarantees that we always have c ≤ c ′ and c ′ ∈ Z n .With the notation from Proposition 3.4, we may write and therefore Next, for 2), let Q be an RS-module.Now res β Q is an RS-module, so by 1), we have lim d≥c, d∈Z n (res On the other hand, by definition, for all e ∈ Z n , This means that we may write the above isomorphism as Corollary 3.6.Let N be an RZ n -module, and let c ∈ Z n .Then Proof.For 1), we note that res S N is an RS-module, where (res S N )(d) = N (d) for all d ∈ S. We may then apply Proposition 3.5 1) to get the result.For 2), we use Proposition 3.5 2) on the RS-module res S N .

Finitely determined modules
Let M be an RC-module.We say that M is pointwise finitely presented if M (c) is finitely presented for all c ∈ C. Slightly generalizing the definition of Miller in [7,p. 25,Ex. 4.5], where R = k is a field, we say that an RZ n -module M is finitely determined, if M is pointwise finitely presented, and for some a ≤ b in Z n , the convex projection π : for all i ∈ {1, . . ., n}.Note that a pointwise finitely presented RZ n -module M is finitely determined if and only if there exists It now follows from the case n = 1 that Remark 4.4.In an effort to keep the notation simpler, we only defined β for the elements in the image of α.Of course, we could have defined β in a fully dual fashion to α, starting from posets that are strongly bounded from below, adding the point ∞ to Z, and defining a set S dually to S. This would have resulted in the situation where for all c ∈ Z n .In other words, the same result would have been achieved.
We saw in Remark 4.
In preparation for the proof of Theorem 4.7, we will now show that a similar result applies to β in both cases.Coordinatewise, for i = {1, . . ., n}, this implies that either Thus M (c ≤ d) is an isomorphism, and M is [a + u, b]-determined.
To demonstrate Theorem 4.7 and Corollary 4.8, it is convenient to take the point of view of topological data analysis, and consider the births and deaths of elements of a module.Given an RZ n module M , one can track how an element x ∈ M (c), where c ∈ Z n , evolves when mapped with the homomorphisms M (c ≤ c ′ ), (c, c ′ ∈ Z n ).We say that the element x is born at c if it is not in the image of any morphism M (c ′ ≤ c), where c ′ < c.On the other hand, the element Consider now an RZ 2 -module M that is finitely determined, and let π : Z 2 → [a, b] be the accompanying convex projection.Note that no new elements are born or die in the leftmost edge or the bottom edge of the box [a, b].This follows from the fact that every element on these two edges has already appeared infinite times before, and was born at some infinitary point.Let us write a = (a 1 , a 2 ).For example, if an element, say x ∈ M ((a 1 , c)), maps to zero on the leftmost edge of [a, b], in M ((a 1 , c + 1)), then x ∈ M ((−∞, c)) will also map to zero in M ((−∞, c + 1)).Thus x does not "die" at the point (a In the next example, we will demonstrate how, for a finitely determined module M , the extension M has births and deaths at infinitary points that guarantee the existence of a finite presentation of M .In particular, we have M ((−∞, −∞)) = R, and Furthermore, by Theorem 4.7, M is now finitely presented.In more concrete terms, we have an exact sequence of RZ 2 -modules Here (−∞, −∞) is the only birth of M , while (1, −∞) and (−∞, 1) are the deaths.
Example 4.11.If k is a field, then it is well known that finitely generated kZ nmodules are finitely presented.This result, however, does not apply to kZ nmodules.For a counterexample, consider a kZ 2 -module M , where Clearly M is finitely generated with its only birth in (−∞, −∞).It is not finitely presented, since the deaths happen at points (n, −n) for all n ∈ Z.
Finally, we want to relate Theorem 4.7 to the work of Perling ([8]).Recall that a subset L ⊆ Z n is a join-sublattice if mub(S) ∈ L for every finite subset S ⊆ L.
Note that this is equivalent to the condition that L = L. Given a join-sublattice L ⊆ Z n , following Perling in [8, pp.16-19, Ch.Remark 4.12.It turns out that unzip L is essentially the same thing as res α , when L is finite and α := α L .There is the slight complication that unzip L is defined for RL-modules, while res α is defined for R L-modules.We may, however, extend an RL-module N to an R L-module Ñ by setting Ñ ((−∞, . . ., −∞)) = 0, if (−∞, . . ., −∞) / ∈ L, and Ñ (c) = N (c), otherwise.Having defined the module Ñ in this way, we see that unzip L N ∼ = res α Ñ .
Given an RZ n -module M , the join-sublattice L is called M -admissible in [8,p. 18,Def. 3.4] if the condition M ∼ = unzip L zip L M is satisfied.This leads us to the following proposition.
, Def. 4.1] an RC-module M to be S-determined if there exists a subset S ⊆ C such that Supp(M ) ⊆ ↑S, and for every c ≤ d in C the implication S ∩ ↓c = S ∩ ↓d ⇒ M (c ≤ d) is an isomorphism holds.For any T ⊆ C, we use the usual notations ↑T := {c ∈ C | t ≤ c for some t ∈ T } and ↓T := {c ∈ C | c ≤ t for some t ∈ T }

Proposition 2 . 3 .
which belongs to Ŝ by definition.Assume that C is strongly bounded from above.Then C has a minimum element min(C) = mub(∅).Let S ⊆ C be a finite subset.Denote S := Ŝ ∪ {min(C)}.We define a poset morphism α S : C → S by setting α S (c) = mub(S ∩ ↓c) for every c ∈ C. In other words, α S maps each c ∈ C to the minimal upper bound of the elements of S below it.To show that α S actually is a poset morphism, suppose that c ≤ d in C. Then S ∩ ↓c ⊆ S ∩ ↓d, which implies that α S (c) ≤ α S (d).Let C be strongly bounded from above, and let S ⊆ C be a finite subset.Then α S = α Ŝ = α S .Proof.Using Lemma 2.2, we first note that S = S and S = S. Let c ∈ C. We claim that mub(S ∩ ↓c) = mub( Ŝ ∩ ↓c) = mub( S ∩ ↓c).The latter equation follows from the fact that for all subsets T ⊆ C, we have mub(T ) = mub(T ∪ {min(C)}).In particular, mub(T ) = min(C), if T = ∅.For the first equation, since S ⊆ Ŝ, we have mub(S ∩ ↓c) ≤ mub( Ŝ ∩ ↓c).On the other hand, Ŝ ∩ ↓c is a subset of Ŝ.Thus mub( Ŝ ∩ ↓c) ∈ Ŝ = Ŝ, where the equation follows from Lemma 2.2.By the definition of Ŝ, we may now write mub( Ŝ ∩ ↓c) = mub(s 1 , . . ., s n ),
, b] gives M an encoding by the closed interval [a, b] ⊆ Z n .Here the convex projection π takes every point in Z n to its closest point in the interval [a, b].If a = (a 1 , . . ., a n ) and b = (b 1 , . . ., b n ), we have for any c

Corollary 4 . 8 .
If M is an RZ n -module and a, b ∈ Z n such that a ≤ b, then the following are equivalent: 1) M is encoded by the convex projection π : Z n → [a, b]; 2) M is [a + u, b]-determined, where u := (1, . . ., 1) ∈ Z n .Proof.We showed in the proof of Theorem 4.7 that 2) implies 1).Conversely, suppose that 1) holds.Let c ≤ d in Z n such that [a + u, b] ∩ ↓c = [a + u, b] ∩ ↓d.
3.1], we define the zip-functor zip L : RZ n -Mod → RL-Mod and the unzip-functor unzip L : RL-Mod → RZ n -Mod.Contrary to Perling, we do not assume that R is a field.The zip-functor maps an RZ n -module M to the RL-module res L M , whereas he unzip-functor maps an RL-module N to an RZ n -module unzip L N defined by(unzip L N )(c) = N (mub(L ∩ ↓c)), if L ∩ ↓c = ∅;0, otherwise for all c ∈ Z n .Note that Supp(unzip L N ) ⊆ ↑L.

Proposition 4 .
13. Let M be an RZ n -module, and L a finite join-sublattice.Then L is M -admissible if and only if M is L-determined.Proof.Let c ∈ Z n .With the earlier notation, we see thatunzip L zip L M = unzip L res L M ∼ = res α res L M ,where(res α res L M )(c) = (res α res L M )(c), if L ∩ ↓c ∅; 0, otherwise.Assume first that M ∼ = unzip L zip L M .If L ∩ ↓c = ∅,we have M (c) = 0 by the definition of the functor unzip L .But in this case α(c) ≤ c, so that L ∩ ↓α(c) = ∅.Using the definition of unzip L again, we get (res α res L M )(c) = M (α(c)) = 0. On the other hand, if there is an element d ∈ L ∩ ↓c, then, by the above formula, M (c) ∼ = (res α res L M )(c).Thus, M ∼ = res α res L M and Supp(M ) ⊆ ↑L, so M is L-determined by Theorem 2.5.Conversely, suppose that M is L-determined.By Theorem 2.5, we have M ∼ = res α res L M and Supp(M ) ⊆ ↑L.The above formula shows us that (unzip L zip L M )(c) = (res α res L M )(c) for all c ∈ ↑L.If c / ∈ ↑L, then c / ∈ Supp(M ), which means that M (c) = 0.In this case, we also have (unzip L zip L M )(c) = 0 by the definition of the functor unzip L .Thus we have an isomorphism M ∼ = unzip L zip L M.
1 , c + 1), but rather at the infinitary point (−∞, c + 1) ∈ [a + u, b].[4, p. 15, Def.3.6], we say that c is a birth if λ M,c is a non-epimorphism, and a death if λ M ,c is a non-monomorpism.Furthermore, suppose that M is [a + u, b]-determined, and the births are "well-behaved" enough.That is, for any birth c, the module M (c)/ Im λ M ,c is projective.The latter of course holds if R is a field.Then, as we discussed in [4, p. 21, Remark 3.27], births and deaths show the positions of the minimal generators and relations of M .