Generalized persistence and graded structures
aa r X i v : . [ m a t h . A T ] F e b GENERALIZED PERSISTENCE AND GRADED STRUCTURES
EERO HYRY AND MARKUS KLEMETTI
Abstract.
We investigate the correspondence between generalized persis-tence modules and graded modules in the case the indexing set has a monoidaction. We introduce the notion of an action category over a monoid gradedring. We show that the category of additive functors from this category to thecategory of Abelian groups is isomorphic to the category of modules gradedover the set with a monoid action, and to the category of unital modules over acertain smash product. Furthermore, when the indexing set is a poset, we pro-vide a new characterization for a generalized persistence module being finitelypresented. Introduction
One of the main methods of topological data analysis is persistent homology. Inthe simplest case, data is encoded in an increasing nested sequence of simplicialcomplexes. This filtration reflects the topological and geometric structure of thedata at different scales. By taking homology with coefficients in a field, one obtainsthe corresponding persistence module - a sequence of vector spaces and linear maps.Carlsson and Zomorodian [28, p. 259, Thm. 3.1 (Correspondence)] realized that onecan view persistence modules as modules over a polynomial ring of one variable.The variable acts on the module as a shift. Considering filtrations indexed by N n leads to the so called multipersistence. In [5, p. 78, Thm. 1], Carlsson andZomorodian showed that multipersistence modules now correspond to modules overa polynomial ring of n variables. More generally, one can start from a filtration ofa topological space indexed by a preordered set. However, the resulting generalizedpersistence modules do not necessarily have an immediate expression as a moduleover a graded ring.The correspondences by Carlsson and Zomorodian opened the graded perspectivein topological data analysis, leading many researchers to utilize graded module the-ory in their investigations (see, for example, [6], [18], [3], [12], [13], [15], [25]). Themost general cases of modules over a ring in this line of research are modules gradedover Abelian groups with monoids as their positive cones, and modules canonicallygraded over cancellative monoids. In this article, we want to propose a new generictheoretical framework for understanding generalized persistence modules under thelens of graded algebra by considering monoid actions on preordered sets. Secondly, Date : February 12, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Persistence modules, Graded modules, Action categories, Smash prod-ucts, Finitely presented.The second author was supported in part by Finnish Cultural Foundation. we want to investigate finitely presented generalized persistence modules. In partic-ular, we will give a certain subclass of preordered sets over which finite presentationcan be characterized by a suitable ’tameness’ condition.We now want to explain this in more detail. Using the language of categorytheory, it is convenient to define a generalized persistence module as a functorfrom a preordered set P to the category of k -vector spaces, where k is a field. Inrepresentation theory, given a commutative ring R and a small category C , a functor C → R - Mod is called an R C -module. In this terminology, a generalized persistencemodule is then a kP -vector space. Following Mitchell ([20]), we also regard a smallpreadditive category A as a ‘ring with several objects’, and an additive functor A → Ab as an A -module. The R C -modules may then be seen as modules over thelinearization R C , where R C is a preadditive category with the same objects as C and morphisms R [Mor C ( c, d )], where c, d ∈ Ob C (for any set S , we denote by R [ S ]the free R -module generated by S ).Suppose now that G is a monoid. To any G -act A , we can associate an actioncategory G ∫ A , whose objects are the elements of A and for any a, b ∈ A the mor-phisms a → b are pairs ( a, g ) where g ∈ G with b = ga . It is easy to see thatthe category of R ( G ∫ A )-modules is now equivalent to the category of A -graded R [ G ]-modules. Note that if the action of G on A is free, then simply G ∫ A = A .Given any G -graded ring S , this leads us is to investigate the relationship between A -graded S -modules and modules over A in general. We define the action categoryover S , denoted by G ∫ S A , with objects A and morphisms s ∈ M g ∈ G, ga = b S g , where a, b ∈ A . In the case S = R [ G ], G ∫ S A is just the linearization of G ∫ A . Ourfirst main result, Theorem 2.9, then says that the categories of A -graded S -modulesand G ∫ S A -modules are isomorphic.We can also look at the category algebra R [ G ∫ A ]. If C is any category, thenthe category algebra R [ C ] is defined as the free R -module with a basis consistingof the morphisms of C , and the product of two basis elements is given by theircomposition, if defined, and is zero otherwise. It now turns out in Proposition 2.14that the category algebra R [ G ∫ A ] coincides with the smash product R [ G ] A , whichhas been much studied in ring theory (see [22]). This leads us to Theorem 2.16,where we identify G ∫ S A -modules with the category of unital S A -modules i.e. thecategory of S A -modules M with M = ( S A ) M .We then turn to consider finitely presented generalized persistence modules.Note that being finitely presented is a categorical property, so an equivalence be-tween generalized persistence modules and graded modules preserves this property.Recall first that an R C -module M is finitely presented if there exists an exactsequence M j ∈ J R [Mor C ( d j , − )] → M i ∈ I R [Mor C ( c i , − )] → M → , where I and J are finite sets, and c i , d j ∈ C for all i ∈ I , j ∈ J . We will look at posets C which are weakly bounded from above and mub-complete. By ‘weakly bounded’we mean that every finite subset S ⊆ C has a finite number of minimal upperbounds in C , whereas C is mub-complete if given a finite non-empty subset S ⊆ C and an upper bound c of S , there exists a minimal upper bound s of S such that s ≤ ENERALIZED PERSISTENCE AND GRADED STRUCTURES 3 c . In our Theorem 4.15, we characterize finitely presented generalized persistencemodules in this situation. More precisely, we can show that an R C -module M isfinitely presented if and only if the R -modules M ( c ) are finitely presented for all c ∈ C , and M is S -determined for some finite set S ⊆ C .Given S ⊆ C , we call an R C -module M S -determined if Supp M ⊆ ↑ S and theimplication S ∩ ↓ c = S ∩ ↓ d ⇒ the morphism M ( c ≤ d ) is an isomorphismholds for every c ≤ d in C . Here Supp( M ) := { c ∈ C | M ( c ) = 0 } denotes thesupport of M , and 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 } for the upset generated and the downset cogenerated by T , respectively. Our intu-ition for this definition comes from topological data analysis, where one tracks howthe elements of each M ( c ) evolve in the morphisms M ( c ≤ c ′ ) ( c, c ′ ∈ C ). One saysthat an element m ∈ M ( c ) is born at c if it is not in the image of any morphism M ( c ′ ≤ c ), where c ′ < c , and dies at c ′′ if M ( c ≤ c ′′ )( m ) = 0 and M ( c ≤ c ′ )( m ) = 0for all c ≤ c ′ < c ′′ . Suppose that there exists a set S such that all births and deathsoccur inside S . The condition S ∩ ↓ c = S ∩ ↓ d then implies that looking downfrom both c and d , we see the same deaths and births. In particular, the morphism M ( c ≤ d ) must be an isomorphism.Our proof for Theorem 4.15 starts from the fact that an R C -module M is finitelypresented if and only if the R -modules M ( c ) are finitely presented for all c ∈ C and M is S -presented for some finite subset S ⊆ C . Here S -presented means theexistence of a set S ⊆ C and an exact sequence of the type M s ∈ S B s [Mor C ( s, − )] → M s ∈ S A s [Mor C ( s, − )] → M → , where A s and B s are R -modules for all s ∈ S . It is easily seen that if M is S -presented, then M is S -determined. We denote the set of minimal upper boundsof non-empty subsets of a finite set S ⊆ C byˆ S := [ ∅6 = S ′ ⊆ S mub C ( S ′ ) . In Corollary 4.13 we now make the crucial observation that M is ˆˆ S -presented if S ⊆ C is a finite set such that M is S -determined.As a useful tool we introduce the sets of births and and deaths relative to S by B S ( M ) := { c ∈ C | colim s Monoid action. In this section we recall some basic properties of monoidactions. Let G be a monoid and let A be a set. If there exists an operation · : G × A → A such that ( gh ) a = g ( ha ) and 1 G · a = a for all g, h ∈ G and a ∈ A ,we say that A is a (left) G -act . We then get a preorder on A by setting a ≤ b if b = ga for some g ∈ G . The action naturally gives rise to two categories having A as the set of objects.First, we have a small thin category A , where for all a, b ∈ A there exists aunique morphism a → b if a ≤ b in the preorder. By abuse of notation we write a ≤ b for this morphism. Recall that in general a category is thin if there exists atmost one morphism between any two objects.Secondly, there is an action category G ∫ A , where morphisms a → b are pairs( a, g ) such that b = ga for some g ∈ G . If there is no possibility of confusion, wesometimes denote the morphism ( a, g ) by g . Composition of morphisms in G ∫ A isdefined by the multiplication of G :( ga, h ) ◦ ( a, g ) = ( a, hg ) . ENERALIZED PERSISTENCE AND GRADED STRUCTURES 5 There is an obvious functor G ∫ A → A where a a and ( a, g ) ( a ≤ ga ) . This functor is an isomorphism if and only if the G -action on A is free , i.e. for all g, h ∈ G , ga = ha for some a ∈ A ⇒ g = h. Remark . We often consider the monoid G itself as a G -act, so it gives rise toa thin category G and the action category G ∫ G . Sometimes, the monoid G isviewed as a category BG with a single object. Then a monoid act could be definedas a functor F : BG → Set . The corresponding action category coincides with thecategory of elements of F (see [24, p. 66, Ex. 2.4.10]). Example 2.2. An abelian group G is called preordered if it is equipped witha preorder ≤ such that g ≤ g ′ implies g + h ≤ g ′ + h for all g, g ′ , h ∈ G . If G + = { g ∈ G | g ≥ } is its positive cone , then g ≤ g ′ is equivalent to g ′ − g ∈ G + .The action of the monoid G + on G is free, and we may identify the action category G + ∫ G with G .A translation on a preordered set P is an order-preserving function F : P → P that satisfies the condition p ≤ F ( p ) for all p ∈ P . The translations of P form amonoid Trans( P ) with composition as the operation.Let A be a G -act. The action by an element g ∈ G now determines a translationon A if and only if a ≤ b implies ga ≤ gb . If this implication holds for all g ∈ G ,then we say that A is an order-preserving G -act . Note that any G -act A is order-preserving if G is commutative. For an order-preserving G -act A , we get a monoidhomomorphism ϕ from G into the monoid of translations Trans( A ). This inducesa monoid embedding ˆ ϕ : G/ Ker ϕ → Trans( A ), where Ker ϕ is the congruencerelation defined by ( g, h ) ∈ Ker ϕ ⇔ ga = ha for all a ∈ A. In particular, ϕ is an embedding if and only if the G -action on A is faithful : for all g, h ∈ G , ga = ha for all a ∈ A implies that g = h .We next give a slight generalization of [11, p. 4, Thm. 2.2]. Proposition 2.3. For any preordered set P , there exists a monoid G and a G -act A such that P and A are isomorphic as thin categories.Proof. We present the proof here for the convenience of the reader. Let G denotethe submonoid of the monoid of all functions P → P consisting of the functions g : P → P for which a ≤ P g ( a ) for all a ∈ P . Define the G -action on A := P bysetting g · a = g ( a ) for all g ∈ G and a ∈ A . Then A is a G -act. It remains to showthat a ≤ P b if and only if a ≤ A b .Assume first that a ≤ A b . By definition, there exists an element g ∈ G such that b = ga . But this means that a ≤ P g ( a ) = ga = b . Conversely, if a ≤ P b , we definea function g : P → P by setting g ( p ) = (cid:26) b, if p = a ; p, otherwise . We then immediately see that g ∈ G and ga = g ( a ) = b , so that a ≤ A b (cid:3) EERO HYRY AND MARKUS KLEMETTI Remark . The monoid G in Proposition 2.3 does not need to be unique: Forexample, the one element set P = {∗} has the trivial monoid action for any monoid G . Note that if G is the monoid of the proof of Proposition 2.3, then the order-preserving elements in G are exactly the translations on A , so that Trans( A ) ⊆ G .2.2. Action categories over a graded ring. Theorem 2.9 will generalize theequivalence of the correspondence theorem of Carlsson and Zomorodian [28, p. 259,Thm. 3.1] mentioned in the Introduction. Moreover, it generalizes the multi-parameter version of the theorem by Carlsson and Zomorodian ([5, p. 78, Thm. 1])as well as the generalization given by Corbet and Kerber ([6, p. 18, Lemma 14]).For a discussion on related finiteness conditions, see [6, p. 3] and Remark 4.16.We begin by defining a certain preadditive category. Definition 2.5. Let A be a G -act, and let S := ⊕ g ∈ G S g be a G -graded ring. The action category over S , denoted G ∫ S A , is the category with the set A as objects,and morphisms ( a, s ) : a → b , where a, b ∈ A and s ∈ M g ∈ G, ga = b S g . Composition for morphisms ( a, s ) : a → ga and ( ga, t ) : ga → hga is defined by( ga, t ) ◦ ( a, s ) = ( a, ts ) . Remark . Keeping a close eye on the domains, we may write s := ( a, s ). Withthis notation, composition is just the multiplication in S . Example 2.7. Let A be a G -act. If R is a commutative ring, then the actioncategory G ∫ R [ G ] A over the monoid ring R [ G ] coincides with the linearized actioncategory R ( G ∫ A ). Indeed, by definition Ob R ( G ∫ A ) = A , andHom R ( G ∫ A ) ( a, b ) = R [ { ( a, g ) | g ∈ G and ga = b } ] . for all a, b ∈ A . Example 2.8. If G is an Abelian group and S := L g ∈ G S g is a G -graded ring, thecategory G ∫ S G is called in [8, p. 358, Def. 2.1] a companion category . In this case,we may identify Hom G ∫ S G ( g, h ) with S h − g .Let A be a G -act. Let S := L g ∈ G S g be a G -graded ring. Recall that a (left) S -module M is A -graded , if1) M = L a ∈ A M a , where M a is an Abelian group for all a ∈ A ;2) S g M a ⊆ M ga for all g ∈ G and a ∈ A .Preparing for Theorem 2.9, we will now define two functors, Φ and Ψ, thatconnect A -graded S -modules to ( G ∫ S A )-modules.Let M be a G ∫ S A -module. By setting sm = M ( s )( m ) for all g ∈ G , s ∈ S g and m ∈ M ( a ), we can define an A -graded S -moduleΦ M := M a ∈ A M ( a ) . ENERALIZED PERSISTENCE AND GRADED STRUCTURES 7 A morphism f : M → N of G ∫ S A -modules consists of homomorphisms of Abeliangroups f a : M ( a ) → N ( a ) with commutative diagrams M ( a ) f a (cid:15) (cid:15) M ( s ) / / M ( ga ) f ga (cid:15) (cid:15) N ( a ) N ( s ) / / N ( ga )for all a ∈ A , g ∈ G and s ∈ S g . These homomorphisms and diagrams obviously giverise to a homomorphism Φ f : Φ M → Φ N of A -graded S -modules with (Φ f ) a = f a for all a ∈ A .Next, let Q be an A -graded S -module. We set (Ψ Q )( a ) = Q a for all a ∈ A . If( a, s ) : a → ga is a morphism, where a ∈ A , g ∈ G and s ∈ S g , we can define ahomomorphism (Ψ Q )(( a, s )) : (Ψ Q )( a ) → (Ψ Q )( ga )by setting (Ψ Q )(( a, s ))( q ) = s · q for all q ∈ Q a . It is clear that Ψ Q is an ad-ditive functor G ∫ S A → Ab , i.e., a G ∫ S A -module. Moreover, if h : Q → P is ahomomorphism of A -graded S -modules, we have a morphism of G ∫ S A -modulesΨ h : Ψ Q → Ψ P given by (Ψ h ) a = h a for all a ∈ A .We are now ready to state Theorem 2.9. Let A be a G -act, and let S := ⊕ g ∈ G S g be a G -graded ring. Theabove functors Φ and Ψ give an isomorphism of categories ( G ∫ S A ) - Mod ∼ = A -gr S - Mod . Proof. It remains to prove that Φ ◦ Ψ = id and Ψ ◦ Φ = id, which is straightforward. (cid:3) Combining this theorem with Example 2.7 gives Corollary 2.10. Let A be a G -act, and let R be a commutative ring. There is anisomorphism of categories R ( G ∫ A ) - Mod ∼ = A -gr R [ G ] - Mod . In particular, if the G -action on A is free, we obtain an isomorphism RA - Mod ∼ = A -gr R [ G ] - Mod . Example 2.11. If A = { e } is a one object set, Theorem 2.9 gives us an isomorphism G ∫ S { e } - Mod ∼ = S - Mod . In the case S = R [ G ], where R is a commutative ring, thismeans that RG - Mod ∼ = R [ G ]- Mod , where RG is the linearization of the 1-objectcategory G . Example 2.12. Let G be a preordered Abelian group with the positive cone G + (see Example 2.2). If R is a commutative ring, then by Corollary 2.10 the categories RG - Mod and G -gr R [ G + ]- Mod are isomorphic. EERO HYRY AND MARKUS KLEMETTI Category algebras and smash products. Let C be a small category, andlet R be a commutative ring. A category algebra R [ C ] is the free R -module withthe basis consisting of the elements e u , where u : c → d is a morphism in C , andwith multiplication defined by e v · e u = ( e vu , if c ′ = d ;0 , otherwisefor morphisms u : c → d and v : c ′ → d ′ in C . Equipped with this product, R [ C ]becomes a ring that has a unit if C is finite.Let A be a G -act and S a G -graded ring. We recall (see [21, p. 390]) that a smash product S A is the free (left) S -module with the basis { p a | a ∈ A } , andwith multiplication defined by the bilinear extension of( s g p a )( t h p b ) = (cid:26) ( s g t h ) p b , if hb = a ;0 , otherwisewhere g, h ∈ G , s g ∈ S g , t h ∈ S h and a, b ∈ A . Equipped with this multiplication, S A is a non-unital ring, i.e. a ring possibly without identity. However, S A has local units . This means that every finite subset of S A is contained in a subringof the form w ( S A ) w , where w is an idempotent of S A . More precisely, let T := { t , . . . , t n } be a finite subset of S A . We may assume that t i = s i p a i , where g i ∈ G , a i ∈ A and s i ∈ S g i for all i ∈ { , . . . , n } . We denote B := { a ∈ A | a = a i or a = g i a i for some i ∈ { , . . . , n }} and w := P a ∈ B p a . It is now straightforward to see that w is idempotent and wt i w = wt i = t i for all i ∈ { , . . . , n } .Let R ′ be a non-unital ring. An R ′ -module M is unital if it satisfies the condition M = R ′ M .The next proposition and its proof are inspired by [2, p. 221, Cor. 2.4]. Proposition 2.13. Let M be an S A -module. Then M is unital if and only if forevery finite subset N ⊆ M there exists a finite subset B ⊆ A such that wn = n forall n ∈ N , where w := P a ∈ B p a .Proof. Assume first that M is unital. Let N := { n , . . . , n p } ⊆ M be a finite set.Now, for all i ∈ { , . . . , p } , the element n i may be written as n i = q X j =1 s i,j n i,j , where s i,j ∈ S A and n i,j ∈ M for all j ∈ { , . . . , q } . This gives us a finite set T = { s i,j | i ∈ { , . . . , p } , j ∈ { , . . . , q }} ⊆ S A. As stated above, we then have a finite subset B ⊆ A such that w = ws for all s ∈ T , where w := P a ∈ B p a . Thus for all i ∈ { , . . . , p } , wn i = w ( q X i =1 s i,j n i,j ) = q X i =1 ( ws i,j ) n i,j = q X i =1 s i,j n i,j = n i . Conversely, suppose that for every finite subset N ⊆ M there exists a finitesubset B ⊆ A such that wn = n for all n ∈ N , where w := P a ∈ B p a . Taking N = { m } for m ∈ M , we get m = wm ∈ S A . (cid:3) ENERALIZED PERSISTENCE AND GRADED STRUCTURES 9 Proposition 2.14. Let R be a commutative ring, G a monoid, and A a G -act.There exists an isomorphism of non-unital rings ϕ : R [ G ∫ A ] → R [ G ] A defined by e ( a,g ) e g p a for all a ∈ A and g ∈ G .Proof. It is easy to see that ϕ is an isomorphism of R -modules. It is also a ringhomomorphism, since for all a, b ∈ A and g, h ∈ G , ϕ ( e ( b,h ) e ( a,g ) ) = (cid:26) ϕ ( e ( a,hg ) ) , if b = ga ;0 , else= (cid:26) e hg p a , if b = ga ;0 , else= ( e h p b )( e g p a )= ϕ ( e ( b,h ) ) ϕ ( e ( a,g ) ) . (cid:3) Proposition 2.15. Let M be an S A -module. Then M = L a ∈ A p a M if and onlyif M is unital.Proof. Assume first that M = L a ∈ A p a M . Let N := { n , . . . , n p } ⊆ M . Since forall i ∈ { , . . . , p } , the element n i may be written as n i = q X j =1 p a i,j n i,j , where a i,j ∈ A and n i,j ∈ M for all j ∈ { , . . . , q } , there exists a finite subset B := { a i,j | i ∈ { , . . . , p } , j ∈ { , . . . , q }} of A . Let w := P a ∈ B p a . Then wn i = w q X j =1 p a i,j n i,j = q X j =1 wp a i,j n i,j = n i , so M is unital by Proposition 2.13.Assume next that M is unital. Let m ∈ M . By Proposition 2.13, we may write m = wm for some w = P a ∈ B p a , where B ⊆ A is finite. Thus m = ( X a ∈ B p a ) m = X a ∈ B p a m, so that M = P a ∈ A p a M . Furthermore, since the elements p a are orthogonal, thesum is direct. (cid:3) Let us denote by S A - Mod the category of unital S A -modules. We willnow define two functors, Γ and Λ, that connect unital ( S A )-modules to ( G ∫ S A )-modules. Let M be a G ∫ S A -module. SetΓ M := M a ∈ A M ( a ) . It is not difficult to check that by setting ( sp a ) m = M (( a, s ))( m a ) for all g ∈ G , s ∈ S g , a ∈ A and m := P b ∈ A m b ∈ Γ M , Γ M becomes an S A -module. Toshow unitality, notice that p a (Γ M ) = M ( a ) for all a ∈ A , which implies that Γ M = L a ∈ A p a (Γ M ). Thus Γ M is unital by Proposition 2.15. If f : M → N isa morphism of G ∫ S A -modules, we can define a homomorphism Γ f : Γ M → Γ N of( S A )-modules by setting (Γ f )( m ) = X a ∈ A f a ( m a )for all m = P a ∈ A m a ∈ Γ M .Next, let Q be a unital S A -module. We define a G ∫ S A -module Λ Q by firstsetting (Λ Q )( a ) = p a Q for all a ∈ A . Let a ∈ A and g ∈ G , s ∈ S g . Given amorphism ( a, s ) : a → ga , we then have a homomorphism of Abelian groups(Λ Q )(( a, s )) : (Λ Q )( a ) → (Λ Q )( ga ) , q ( sp a ) q. Finally, for a homomorphism h : Q → P of S A -modules, there is a morphism of G ∫ S A -modules Λ h : Λ Q → Λ P with (Λ h ) a ( q ) = h ( q ) for all a ∈ A and q ∈ (Λ Q )( a ). Theorem 2.16. Let A be a G -act, and let S := ⊕ g ∈ G S g be a G -graded ring. Thefunctors Γ and Λ give an isomorphism of categories ( G ∫ S A ) - Mod ∼ = S A - Mod . Proof. We need to show that ΓΛ = id and ΛΓ = id.Let Q be a unital S A -module. By Proposition 2.15 we then have(ΓΛ) Q = M a ∈ A (Λ Q )( a ) = M a ∈ A p a Q = Q. Moreover, the S A -module structures of Q and (ΓΛ) Q are the same. Indeed,writing ∗ for the multiplication by S A on (ΓΛ) Q , we get( sp a ) ∗ q = (Λ Q )(( a, s ))( p a q a ) = ( sp a )( p a q a ) = ( sp a ) q for all a ∈ A , g ∈ G , s ∈ S g and q := P a ∈ A p a q a ∈ Q .On the other hand, let M be a G ∫ S A -module. For an object a ∈ A ,((ΛΓ) M )( a ) = p a (Γ M ) = M ( a ) . Furthermore, if ( a, s ) : a → ga is a morphism in G ∫ S A , then((ΛΓ) M )(( a, s ))( m ) = ( sp a ) m = M (( a, s ))( m )for all m ∈ M ( a ), so that ((ΛΓ) M )(( a, s )) = M (( a, s )). (cid:3) Corollary 2.17. Let A be a G -act, and let R be a commutative ring. There existsan isomorphism of categories between the categories of R ( G ∫ A ) -modules and unital R [ G ∫ A ] -modules.Proof. This follows from Proposition 2.14 and Theorem 2.16. (cid:3) Finitely presented R C -modules We will assume in the following that C is a small category and R a commutativering. Recall first that an R C -module M is • finitely generated if there exists an epimorphism M i ∈ I R [Mor C ( c i , − )] → M where I is a finite set, and c i ∈ C for all i ∈ I ; ENERALIZED PERSISTENCE AND GRADED STRUCTURES 11 • finitely presented if there exists an exact sequence M j ∈ J R [Mor C ( d j , − )] → M i ∈ I R [Mor C ( c i , − )] → M → , where I and J are finite sets, and c i , d j ∈ C for all i ∈ I and j ∈ J .For more details on finitely generated and finitely presented objects in an Abeliancategory, we refer the reader to [23, Ch. 3.5].3.1. S -presented and S -generated R C -modules. Let S ⊆ C be a full subcate-gory. The notions of S -generated and S -presented modules will play an importantrole in the rest of this article. Before going into details, we will recall some factsabout the restriction and induction functors along the inclusion i : S ⊆ C .The restriction res S : R C - Mod → RS - Mod is defined by precomposition with i , and the induction ind S : RS - Mod → R C - Mod is its left Kan extension along i . The induction is the left adjoint of the restriction. Note, in particular, thatit thus commutes with colimits. The counit of this adjunction gives us for every R C -module M the canonical morphism µ M : ind S res S M → M, which we will use frequently.More explicitly, for any R C -module M and RS -module N , we have the pointwiseformulas (res S M )( s ) = M ( s ) and (ind S N )( c ) = colim ( t,u ) ∈ ( i/c ) N ( t )for all s ∈ S and c ∈ C . Here ( i/c ) denotes the slice category. Its objects are pairs( s, u ), where s ∈ S and u : s → c is a morphism in C . For ( s, u ) , ( t, v ) ∈ Ob( i/c ), amorphism ( s, u ) → ( t, v ) is a morphism α : s → t in S with vα = u . If C is a poset,the latter formula yields (ind S N )( c ) = colim t ∈ S, t ≤ c N ( t ) . Let A be an R -module and c ∈ C . We define an R C -module A [Mor C ( c, − )] := A ⊗ R R [Mor C ( c, − )]by taking a pointwise tensor product. We note that the functor R - Mod → R C - Mod that sends A to A [Mor C ( c, − )] is right exact for all c ∈ C . Proposition 3.1. Let S ⊆ C be a full subcategory, A an R -module, and s ∈ C .Then ind S res S A [Mor C ( s, − )] ∼ = A [Mor C ( s, − )] . Proof. By Yoneda’s lemma and the aforementioned adjunction, we have the follow-ing isomorphisms:Hom R C ( R [Mor C ( s, − )] , M ) ∼ = M ( s ) ∼ = Hom RS ( R [Mor S ( s, − )] , res S M ) ∼ = Hom R C (ind S R [Mor S ( s, − )] , M ) . This shows us that ind S res S R [Mor C ( s, − )] ∼ = R [Mor C ( s, − )]. In particularcolim ( t,u ) ∈ ( i/d ) R [Mor C ( s, t )] ∼ = R [Mor C ( s, d )] for d ∈ C . Since tensoring commutes with colimits, we see that for all d ∈ C ,(ind S res S A [Mor C ( s, − )])( d ) = colim ( t,u ) ∈ ( i/d ) A [Mor C ( s, t )] ∼ = A ⊗ R colim ( t,u ) ∈ ( i/d ) R [Mor C ( s, t )] ∼ = A ⊗ R R [Mor C ( s, d )] ∼ = A [Mor C ( s, d )] . Therefore ind S res S A [Mor C ( s, − )] ∼ = A [Mor C ( s, − )] as wanted. (cid:3) An R C -module M is said to be S -generated if the natural morphism M s ∈ S M ( s )[Mor C ( s, − )] → M is an epimorphism. Since this morphism factors through the canonical morphism µ M : ind S res S M → M , we see that M is S -generated if and only if µ M is anepimorphism. Proposition 3.2. Let S ⊆ C be a full subcategory. Assume that M is an S -generated R C -module, so that we have an exact sequence of R C -modules → K → M s ∈ S M ( s )[Mor C ( s, − )] → M → . Then the following are equivalent: The canonical morphism µ M : ind S res S M → M is an isomorphism; If there exists an exact sequence of R C -modules → L → N → M → , where N is S -generated, then L is S -generated; K is S -generated; The sequence M s ∈ S K ( s )[Mor C ( s, − )] → M s ∈ S M ( s )[Mor C ( s, − )] → M → is exact; For each s ∈ S , there exist R -modules A s and B s such that the sequence M s ∈ S B s [Mor C ( s, − )] → M s ∈ S A s [Mor C ( s, − )] → M → is exact.When these equivalent conditions hold, we say that M is S -presented .Proof. We will show that 1) ⇒ ⇒ ⇒ ⇒ ⇒ R C -modules0 → L → N → M → . Since the functor res S is exact and the functor ind S right exact, we get a commu-tative diagram with exact rowsind S res S L µ L (cid:15) (cid:15) / / ind S res S N µ N (cid:15) (cid:15) / / ind S res S M µ M (cid:15) (cid:15) / / (cid:15) (cid:15) / / L / / N / / M / / ENERALIZED PERSISTENCE AND GRADED STRUCTURES 13 where µ M is an isomorphism and µ N is an epimorphism. An easy diagram chaseshows us that µ L is an epimorphism, so 2) holds.The implication 2) ⇒ 3) is trivial. Assume next that 3) holds. Now the morphism L s ∈ S K ( s )[Mor C ( s, − )] → K is an epimorphism, so both L s ∈ S K ( s )[Mor C ( s, − )]and K have the same image in L s ∈ S M ( s )[Mor C ( s, − )]. The required exactnessthen follows immediately.Trivially 4) implies 5). Finally, let us assume that 5) holds. By Proposition 3.1,we get a commutative diagram with exact rows L s ∈ S B s [Mor C ( s, − )] ∼ = (cid:15) (cid:15) / / L s ∈ S A s [Mor C ( s, − )] ∼ = (cid:15) (cid:15) / / ind S res S M µ M (cid:15) (cid:15) / / (cid:15) (cid:15) L s ∈ S B s [Mor C ( s, − )] / / L s ∈ S A s [Mor C ( s, − )] / / M / / µ M is an isomorphism by the five lemma. (cid:3) Remark . Proposition 3.2 is due to Djament [9, p. 11, Prop. 2.14]. The readershould be cautious, since we use the term ‘support’ in a different meaning as in [9].The following proposition is a special case of [10, p. 83, Prop.]. For the sake ofclarity, we present a proof using our notation. Proposition 3.4. An R C -module M is finitely presented if and only if there existsa finite full subcategory S ⊆ C such that M ( s ) is finitely presented for all s ∈ S ; M is S -presented.Proof. Assume first that M is finitely presented, so that there exists an exactsequence M j ∈ J R [Mor C ( b j , − )] → M i ∈ I R [Mor C ( a i , − )] → M → , where I and J are finite sets, and a i , b j ∈ C for all i ∈ I and j ∈ J . Evaluating thisat point c ∈ C gives us an exact sequence R m c → R n c → M ( c ) → m c , n c ∈ N , so that 1) holds. For 2), by setting S := { a i | i ∈ I } ∪ { b j | j ∈ J } we immediately see that M is S -presented by Proposition 3.2 5).Assume next that there exists a finite full subcategory S ⊆ C such that 1) and 2)hold. Now M is S -generated, so the natural morphism L s ∈ S M ( s )[Mor C ( s, − )] → M is an epimorphism. Since M ( s ) is finitely generated for all s ∈ S , there existsan epimorphism R n s → M ( s ) for all s ∈ S , where n s ∈ N . Combining theseepimorphisms, we get an epimorphism M t ∈ S R n t [Mor C ( t, − )] → M t ∈ S M ( t )[Mor C ( t, − )] → M and an exact sequence0 → N → M t ∈ S R n t [Mor C ( t, − )] → M → . Because M is S -presented, N must be S -generated by Proposition 3.2 2), so thereexists an epimorphism L t ∈ S N ( t )[Mor S ( t, − )] → N . On the other hand, M ( s ) isfinitely presented, so N ( s ) is finitely generated for all s ∈ S . Thus there exists anepimorphism R m s → N ( s ) for all s ∈ S , where m s ∈ N . Hence we get an exactsequence M t ∈ S R m t [Mor C ( t, − )] → M t ∈ S R n t [Mor C ( t, − )] → M → . (cid:3) From the proof of Proposition 3.4 we immediately get the following corollary: Corollary 3.5. An R C -module M is finitely generated if and only if there exists afinite full subcategory S ⊆ C such that M ( s ) is finitely generated for all s ∈ S ; M is S -generated. Births and deaths relative to S . From now on, we will assume that C is aposet.Let M be an R C -module, S ⊆ C a subset, and c ∈ C . Write S ′ := S \{ c } . Wenote thatcolim d Let C be a poset, M an R C -module, S ⊆ C a subset and c ∈ C .Let λ M,c : colim d Let C be a poset. Let I be an interval of C i.e. a non-empty subsetof C satisfying the condition that if a, b ∈ I , c ∈ C and a ≤ c ≤ b , then c ∈ I . Let R I be the R C -module defined on objects by R I ( c ) = (cid:26) R, when c ∈ I ;0 , otherwise,and with identity morphisms inside the interval. Then the sets of births B C ( R I )and B I ( R I ) both consist of the minimal elements of I . To find the deaths, we notethat R I is ↑ I -presented, so deaths must either be inside I or above it (see Remark4.2).First, let c ∈ S := ( ↑ I ) \ I . Now R I ( c ) = 0. Since S ⊆ ↑ Supp( R I ), we see thatcolim d Let C be a poset, M an R C -module, and S ⊆ C a subset. Then M is S -generated if and only if B S ( M ) ⊆ S ; M is S -presented if and only if B S ( M ) ∪ D S ( M ) ⊆ S .Proof. Both 1) and 2) are proved similarly. We only prove 2) here. Directly fromthe definitions, M is S -presented ⇔ µ M,c : colim d ≤ c, d ∈ S M ( d ) → M ( c ) is an isomorphism for all c ∈ C⇔ λ M,c : colim d Let M be an S -generated R C -module. Then B C ( M ) = B S ( M ) .Proof. By Remark 3.7 it is enough to show that B S ( M ) ⊆ B C ( M ). Let c ∈C\ B C ( M ), so that the natural homomorphism L d Example 3.13. Let k be a field, S ⊆ Z n a subset, and M a k ( N n ∫ Z n )-module.We identify M with the corresponding Z n -graded k [ X , . . . , X n ]-module. Denoteby m := h X , . . . , X n i the maximal homogeneous ideal of k [ X , . . . , X n ]. If N isthe homogeneous submodule of M generated by the union of M s , where s ∈ S , wenotice that ( M/mN ) c = M c / ( mN ) c = M ( c ) / Im( λ M,c ) = S S,c M for all c ∈ Z n . In particular, this yields an isomorphism of k -vector spaces, M/mN ∼ = M c ∈ Z n S S,c M. Remark . Let M be an R C -module and S ⊆ C a subset. Note that for all c ∈ C ,we have c ∈ B S ( M ) if and only if S S,c M = 0. Remark . Let A be an R -module, S ⊆ C a subset, s ∈ S , and c ∈ C . Let S ′ := S \{ c } . If s = c , we see thatcolim d Lemma 3.16. Let M be an R C -module and S ⊆ C a subset. If Supp( M ) ∩ S hasa minimal element c , then S S,c M = 0 .Proof. Assume that c ∈ Supp( M ) ∩ S is minimal. Then M ( d ) = 0 for all d ∈ S with d < c . In particular, colim d Let f : L → M be a morphism of R C -modules, where M is S -generated with an Artinian S ⊆ C . If S S,c f : S S,c L → S S,c M is an epimorphismfor all c ∈ B S ( M ) , then f is an epimorphism.Proof. We first note that Coker f is S -generated, since M is S -generated. Supposethat f is not an epimorphism. Then Coker f = 0, so there exists s ∈ S suchthat (Coker f )( s ) = 0. Hence Supp(Coker f ) ∩ S has a minimal element c by theArtinian property. Now S S,c (Coker f ) = 0 by Lemma 3.16, which implies that c ∈ B S (Coker f ) ⊆ B S ( M ). Since S S,c is right exact, we get Coker S S,c f = 0, so S S,c f is not an epimorphism. (cid:3) Lemma 3.18. Let S ⊆ C be a subset and → L j → N f → M → an exact sequence of R C -modules. The following are equivalent for all c ∈ C : 1) (Ker f )( c ) ⊆ Im λ N,c ; S S,c ( j ) = 0 ; S S,c ( f ) is a monomorphism; ENERALIZED PERSISTENCE AND GRADED STRUCTURES 17 S S,c ( f ) is an isomorphism.Proof. The equivalence of 1) and 3) immediately follows from the fact thatKer S S,c ( f ) = ((Ker f )( c ) + Im λ N,c ) / Im λ N,c . Since S S,c is right exact, we have Ker S S,c ( f ) = Im S S,c ( j ). Therefore 2) implies 3).The equivalence of 3) and 4) holds, because S S,c preserves epimorphisms. (cid:3) We recall that an epimorphism of R C -modules f : N → M is called minimal , if forall morphisms g : L → N , f g is an epimorphism if and only if g is an epimorphism.It is known that an epimorphism f is minimal if and only if for all submodules N ′ ⊆ N N ′ + Ker f = N ⇒ N ′ = N. A minimal epimorphism f : N → M , where N is projective, is called a projectivecover of M (see e.g. [1, p. 28]). Remark . Let A be an R -module and c ∈ C . Then A may be thought of asan R { c } -module, and we note that A [Mor C ( c, − )] ∼ = ind { c } A . In particular, thefunctor A A [Mor C ( c, − )] preserves projectives, since it is the left adjoint of theexact functor res { c } . Proposition 3.20. Let f : N → M be an epimorphism of S -generated R C -modules,where S ⊆ C is Artinian. If (Ker f )( c ) ⊆ Im λ N,c for all c ∈ C , then f is minimal.The converse implication holds if S S,c M is projective for all c ∈ S .Proof. Let (Ker f )( c ) ⊆ Im λ N,c for all c ∈ C . Suppose that N ′ + Ker f = N forsome submodule N ′ ⊆ N . We note that for all c ∈ C , ( N ′ )( c ) + Im λ N,c = N ( c ).This implies that S S,c N ′ = S S,c N for all c ∈ C . Since S is Artinian, we may useProposition 3.17 to conclude that N ′ = N , so f is minimal.Next, let f be minimal, and let S S,c M be projective for all c ∈ S . Thus we canfind sections S S,c M → M ( c ) for all c ∈ S . These induce a morphism h : M s ∈ S S S,s M [Mor C ( s, − )] → M s ∈ S M ( s )[Mor C ( s, − )] → M. Remark 3.15 now implies that S S,c h = id S S,c M for all c ∈ S , so h is an epimorphismby Proposition 3.17.Since S S,c M is projective for all c ∈ S , we see that L s ∈ S S S,s M [Mor C ( s, − )] isalso projective by Remark 3.19 (as a sum of projectives). Thus the morphism h factors through f , and we get a diagram L s ∈ S S S,s M [Mor c ( s, − )] g / / h ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗ N f (cid:15) (cid:15) M that commutes. Now f is minimal, so g is an epimorphism. Applying functor S S,c , where c ∈ S , on the diagram, we see that S S,c f ◦ S S,c g = id, which impliesthat S S,c g is a monomorphism, and therefore an isomorphism. Hence S S,c f is anisomorphism for all c ∈ S . This is equivalent to (Ker f )( c ) ⊆ Im λ N,c for all c ∈ C by Lemma 3.18. (cid:3) Remark . Let M be an S -generated R C -module, where S is Artinian. If S S,c M is projective for all c ∈ S , the morphism h : L s ∈ S S S,s M [Mor c ( s, − )] → M inducedby sections S S,s M → M ( s ) is a projective cover of M .Indeed, as noted earlier, h is an epimorphism with S S,c h = id for all c ∈ S . ThenLemma 3.18 implies that (Ker f )( c ) ⊆ Im λ N,c for all c ∈ C , and the rest followsfrom Proposition 3.20.3.4. Minimality of births and deaths. We will now show how the sets of birthsand deaths relative to a subset S ⊆ C are in a sense minimal if the module is S -generated or S -presented. Proposition 3.22. Let M be an S -generated R C -module, where S ⊆ C is Artinian.Then M is B S ( M ) -generated. Furthermore, B S ( M ) is the minimum element of theset { T ⊆ S | M is T -generated } .Proof. Let ρ be the natural morphism ρ : L s ∈ B S ( M ) M ( s )[Mor C ( s, − )] → M . Re-mark 3.15 shows us that applying the S -splitting functor at c ∈ S yields thecanonical epimorphism S S,c ρ = π : M ( c ) → S S,c M . Thus ρ is an epimorphismby Proposition 3.17.To show the claimed minimality: If M is T -generated for some T ⊆ S , we have B S ( M ) ⊆ B T ( M ) ⊆ T by Proposition 3.9 and Remark 3.7. (cid:3) Next, we introduce a technical lemma. Lemma 3.23. Assume that we have a commutative diagram of R -modules withexact rows L f (cid:15) (cid:15) / / N g (cid:15) (cid:15) / / M h (cid:15) (cid:15) / / / / L ′ / / N ′ / / M ′ where g is a monomorphism. If f is an epimorphism, then h is a monomor-phism. The converse holds if either the natural morphism Coker g → Coker h isa monomorphism or g is an epimorphism.Proof. The snake lemma gives us an exact sequenceKer f → Ker g → Ker h → Coker f → Coker g → Coker h, where Ker g = 0. If Coker f = 0, we get Ker h = 0. If Coker g = 0, we haveKer h ∼ = Coker f , and we are done. If Coker g → Coker h is a monomorphism, wesee that Coker f maps to 0, so Ker h → Coker f is an epimorphism. Since Ker g = 0,the morphism Ker h → Coker f is also a monomorphism. (cid:3) Lemma 3.24. Let M be an S -presented R C -module, where S ⊆ C is Artinian.Assume that we have an exact sequence of R C -modules → L → N f → M → , where N is S -generated and D S ( N ) = ∅ . Then D S ( M ) ⊆ B S ( L ) . Furthermore, if N is B S ( M ) -generated, we have B S ( L ) ⊆ B S ( M ) ∪ D S ( M ) . ENERALIZED PERSISTENCE AND GRADED STRUCTURES 19 Proof. Let c ∈ C . Applying colim d Let M be an S -presented R C -module, where S ⊆ C is Artinian.Then M is B S ( M ) ∪ D S ( M ) -presented. Furthermore, B S ( M ) ∪ D S ( M ) is theminimum element of the set { T ⊆ S | M is T -presented } .Proof. Let us examine an exact sequence0 → L → N → M → , with N of the form N = L s ∈ B S ( M ) A s [Mor C ( s, − )], where A s is an R -module forall s ∈ B S ( M ). Note that such N always exists by Proposition 3.22. Since M is S -presented, Proposition 3.2 2) implies that L is S -generated. Using Proposition 3.25), we notice that if L is T -generated for some T ⊆ S , then M is ( B S ( M ) ∪ T )-presented. Now L is B S ( L )-generated by Proposition 3.22, so we deduce that M is ( B S ( M ) ∪ B S ( L ))-presented. We can now use Lemma 3.24 to see that then M is( B S ( M ) ∪ D S ( M ))-presented.Suppose next that M is also T -presented for some T ⊆ S . As in the proofof Lemma 3.24, we note that B S ( M ) = B S ( N ). The minimality of B S ( M ) inProposition 3.22 implies that B S ( N ) = B S ( M ) ⊆ T , so N is T -generated byProposition 3.22. Thus L is T -generated by Proposition 3.2 2). Therefore we musthave B S ( M ) ⊆ B T ( M ) ⊆ T and B S ( L ) ⊆ B T ( L ) ⊆ T, by Proposition 3.9 1) and Remark 3.7. We use Lemma 3.24 to conclude that B S ( M ) ∪ D S ( M ) = B S ( L ) ∪ B S ( M ) ⊆ T. (cid:3) Remark . Assume that M is an S -presented R C -module, where S ⊆ C is Ar-tinian. Let f : N → M be a projective cover. Then S S,c f is an isomorphism for all c ∈ C if and only if S S,c M is projective for all c ∈ C .To see this, first suppose that S S,c f is an isomorphism for all c ∈ C . Since S S,c preserves projectives for all c ∈ C , we see that S S,c N is projective, and thus S S,c M is projective.Conversely, suppose that S S,c M is projective for all c ∈ C . We may now applyProposition 3.20 and Lemma 3.18 to get isomorphisms S S,c f : S S,c N → S S,c M forall c ∈ C . Remark . In [5], Carlsson and Zomorodian define multiset-valued invariants ξ and ξ for a finitely generated Z n -graded k [ X , . . . , X n ]-module M , where k is afield. The multisets ξ ( M ) and ξ ( M ) indicate the degrees in Z n where the elementsof M are born and where they die, respectively. In more algebraic terms, ξ ( M )and ξ ( M ) consist of the degrees of minimal generators and minimal relations of Mequipped with the multiplicities they occur. Consider an exact sequence0 → L → N f → M → , where N is a free module and f a minimal homomorphism. Since M is S -presentedfor some finite S ⊆ Z n , it is easy to see that ξ ( M ) is a multiset where the underlyingset is B S ( M ) and the multiplicity of c ∈ B S ( M ) is the dimension of M ( c ). Note thatthe choice of S does not matter here, since B S ( M ) = B C ( M ) by Proposition 3.10.We note that L is S -generated by Proposition 3.2 3), so we may apply a similarargument to conclude that ξ ( M ) is a multiset with B S ( L ) as the underlying setand the dimension of L ( c ) as the multiplicity of c ∈ B S ( L ). The next theorem willshow that D S ( M ) is the underlying set of ξ ( M ). Theorem 3.28. Let M be an S -presented R C -module, where S ⊆ C is Artinian.Assume that S S,c M is projective for all c ∈ B S ( M ) . If → L → N f → M → is an exact sequence where f is a projective cover, then B S ( L ) = D S ( M ) .Proof. By Lemma 3.24, it is enough to show that B S ( L ) ⊆ D S ( M ). Let c ∈ B S ( M ).Suppose that we would have c / ∈ D S ( M ). Then λ M,c is a monomorphism. Since f is minimal, by Proposition 3.20 and Lemma 3.18 there exists a natural isomorphism S S,c f : S S,c N = Coker λ N,c → Coker λ M,c = S S,c M. It now follows from Lemma 3.23 that λ L,c is an epimorphism, which is equivalentto c / ∈ B S ( L ). But this is impossible. (cid:3) Presentations with finite support In this section we will prove our main result, Theorem 4.15, which gives a char-acterization for finitely presented modules. We will assume that C is a poset and R a commutative ring.4.1. S -determined R C -modules. Let M be an R C -module. If S ⊆ C is a finiteset such that M is S -presented, we say that S is a finite support of a presentation(FSP) of M . In what follows, we are trying to find a condition equivalent for M having an FSP. Definition 4.1. An R C -module M is S -determined if there exists a subset S ⊆ C such that Supp( M ) ⊆ ↑ S , and for every c ≤ d in C S ∩ ↓ c = S ∩ ↓ d ⇒ M ( c ≤ d ) is an isomorphism. Remark . Let M be an R C -module and S ⊆ C a subset. Denote T := ↑ S .Then the condition Supp( M ) ⊆ T of Definition 4.1 is equivalent to the followingconditions:1) M is T -generated;2) M is T -presented;3) If S ∩ ↓ c = ∅ , then M ( c ) = 0. ENERALIZED PERSISTENCE AND GRADED STRUCTURES 21 To show this, we first note that 1) implies 3), because ↑ S = T . Taking the con-traposition of 3), we get Supp( M ) ⊆ T . Next, note that below every c ∈ D T ( M )there must be some d ∈ Supp( M ) such that d < c . Thus D T ( M ) ⊆ ↑ Supp( M ).Obviously also B T ( M ) ⊆ Supp( M ). We now observe that if Supp( M ) ⊆ T , we get B T ( M ) ∪ D T ( M ) ⊆ ↑ Supp( M ) ⊆ ↑ T = T, This means that Supp( M ) ⊆ T implies 2) by Proposition 3.9. Finally, 3) triviallyfollows from 2). Proposition 4.3. Let M be an S -presented R C -module, where S ⊆ C . Then M is S -determined.Proof. Trivially Supp( M ) ⊆ ↑ S . If c ≤ d in C , we have a commutative diagramcolim e ≤ c, e ∈ S M ( e ) ∼ = (cid:15) (cid:15) / / colim e ≤ d, e ∈ S M ( e ) ∼ = (cid:15) (cid:15) M ( c ) M ( c ≤ d ) / / M ( d )with the vertical isomorphisms being components of the canonical isomorphism ofProposition 3.2, 1). This immediately shows us that M is S -determined. (cid:3) Minimal upper bounds. Let S ⊆ C . We would like to find conditions underwhich S -determined implies S -presented. In general this is false (see Example 4.17),so we first need to apply some technical limitations on the poset C to guaranteethat it is ”small” enough. Notation . Let S ⊆ C be a subset. An element c ∈ C is an upper bound of S , if c ≥ s for all s ∈ S . We denote the set of minimal upper bounds of S by mub C ( S ). Definition 4.5. The poset C is weakly bounded from above if every finite S ⊆ C has a finite number of minimal upper bounds in C . Definition 4.6. The poset C is mub-complete if given a finite non-empty subset S ⊆ C and an upper bound c of S , there exists a minimal upper bound s of S suchthat s ≤ c . Remark . A poset that is weakly bounded from above and mub-complete iscalled a poset with property M in [14]. In contrast to [14], we do not require theempty set to have minimal upper bounds for a poset to be mub-complete. Example 4.8. Let L be a poset where every finite subset K ⊆ L has an infimumand a supremum (i.e. L is a lattice). Then L is weakly bounded from above andmub-complete.A ‘good’ monoid G in [6] is a cancellative monoid that is weakly bounded fromabove as a poset (with the natural order). If G is also commutative, we get thefollowing description of mub-completeness. Proposition 4.9. Let G be a commutative cancellative monoid that is weakly boun-ded from above as a poset (with the natural order). Then G is mub-complete if andonly if there exists a maximal common divisor for each g, h ∈ G . Proof. Assume first that G is mub-complete. Let g, h ∈ G . Since gh is an upperbound of g and h , there exists a minimal upper bound j ∈ G of g and h such that lj = gh for some l ∈ G . We claim that l is a maximal common divisor of g and h .We may write j = ag = bh , where a, b ∈ G . Now gh = lj = lag = lbh, so that g = lb and h = la by cancellativity. Thus l is a common divisor of g and h .For the maximality, let k ∈ G be another common divisor of g and h such that l divides k . We may then write k = k ′ l , where k ′ ∈ G . Furthermore, we have g = ck and h = dk for some c, d ∈ G . Combining these equations, we get lj = gh = ck ′ lh = gdk ′ l. Cancelling l , we see that j = k ′ ch = k ′ dg . Furthermore, cancelling k ′ yields ch = dg ,another upper bound for g and h . Since j is a minimal upper bound of g and h ,we must have k ′ = 1, proving the maximality of l .For the other direction, assume that each pair g, h ∈ G has a maximal commondivisor. Let H := { h , . . . , h n } ⊆ G be a finite non-empty set, and let d be anupper bound of H . We now have d = g h = · · · = g n h n for some g , . . . , g n ∈ G . Let g ′ ∈ G be a maximal common divisor of g , . . . , g n .Hence there exists g ′ i ∈ G such that g i = g ′ i g ′ for all i ∈ { , . . . , n } . Also, d = d ′ g ′ forsome d ′ ∈ G . It is now easy to see that the maximal common divisor of g ′ , . . . , g ′ n is 1, and that d ′ is a minimal upper bound of H . (cid:3) Notation . Let S ⊆ C be a finite subset. We denote the set of minimal upperbounds of non-empty subsets of S byˆ S := [ ∅6 = S ′ ⊆ S mub C ( S ′ ) . We notice that if C is weakly bounded from above, then ˆ S is finite.Using the terminology from [6], a set S ⊆ C is a framing set of M if every c ∈ ↑ Supp( M ) has an element s ∈ S ∩ ↓ c , called a frame of c , such that M ( s ≤ c ′ )is an isomorphism for all s ≤ c ′ ≤ c . Lemma 4.11. If an R C -module M has a framing set S , then M is S -determined.Conversely, if C is weakly bounded from above and mub-complete, and M is S -determined for some finite set S ⊆ C , then ˆ S is a finite framing set of M . Inparticular, if c ∈ C , then every s ∈ mub( S ∩ ↓ c ) ⊆ ˆ S is a frame of c such that S ∩ ↓ c = S ∩ ↓ s .Proof. Assume first that S is a framing set for M . If c ∈ Supp( M ), then thereexists a frame s ∈ S of c , and therefore c ∈ ↑ S . Thus Supp( M ) ⊆ ↑ S . Let c ≤ d in C such that S ∩ ↓ c = S ∩ ↓ d . If d / ∈ ↑ Supp( M ), we have M ( c ) = M ( d ) = 0, and weare done. Otherwise, there exists a frame s ∈ S of d . Since S ∩ ↓ c = S ∩ ↓ d , we seethat s ≤ c ≤ d . Therefore M ( c ≤ d ) is an isomorphism.Assume next that C is weakly bounded from above and mub-complete, and M is S -determined for some finite set S . Since C is weakly bounded from above, ˆ S isfinite. Let c ∈ ↑ Supp( M ). Now there exists an element b ≤ c such that M ( b ) = 0.Since M is S -determined, we see that S ∩ ↓ c ⊇ S ∩ ↓ b = ∅ . Thus c is an upper ENERALIZED PERSISTENCE AND GRADED STRUCTURES 23 bound of the non-empty set S ∩ ↓ c , so by mub-completeness there exists a minimalupper bound s ∈ mub( S ∩ ↓ c ) ⊆ ˆ S such that s ≤ c . It follows that S ∩ ↓ c ⊆ S ∩ ↓ s .Obviously s ≤ c implies S ∩ ↓ s ⊆ S ∩ ↓ c . Hence S ∩ ↓ s = S ∩ ↓ c . If s ≤ c ′ ≤ c , thentrivially ↓ S ∩ s = ↓ S ∩ c ′ , so M ( s ≤ c ′ ) is an isomorphism. (cid:3) Finitely presented R C -modules in mub-complete posets. In the nextproposition we find out how the minimal upper bounds connect to births and deathsrelative to S . This allows us to prove our main result, Theorem 4.15. Proposition 4.12. Let C be weakly bounded from above and mub-complete. Let M be an R C -module that is S -determined for some finite S ⊆ C . If B ˆ S ( M ) ⊆ S , then D ˆ S ( M ) ⊆ ˆ S .Proof. Suppose that B ˆ S ( M ) ⊆ S . This implies that M is ˆ S -generated by Proposi-tion 3.9, so M is B ˆ S ( M )-generated by Proposition 3.22. Let c ∈ D ˆ S ( M ), so that λ M,c is not a monomorphism. This means that there exist d , . . . , d n ∈ ˆ S such that d i < c for all i = { , . . . , n } , and non-zero elements x ∈ M ( d ) , . . . , x n ∈ M ( d n )for which P ni =1 M ( d i ≤ c )( x i ) = 0. If there exists c ′ ∈ ˆ S such that d i ≤ c ′ < c for all i ∈ { , . . . , n } , we may assume that P ni =1 M ( d i ≤ c ′ )( x i ) = 0, since thehomomorphism n M i =1 M ( d i ) → colim d Let C be weakly bounded from above and mub-complete. Let M be an S -determined R C -module, where S ⊆ C is a finite subset. Then ˆˆ S is an FSPof M .Proof. Obviously M is ˆ S -determined, since S ⊆ ˆ S . By Proposition 4.12, it isenough to show that B ˆˆ S ( M ) ⊆ ˆ S , because then D ˆˆ S ( M ) ⊆ ˆˆ S . The rest now followsfrom Proposition 3.9. Since ˆ S ⊆ ˆˆ S , by Remark 3.7 we have B ˆˆ S ( M ) ⊆ B ˆ S ( M ).Let c ∈ C . If c / ∈ ˆ S , then c has a frame s ∈ ˆ S by Lemma 4.11. This means that c / ∈ B ˆ S ( M ), and thus B ˆ S ( M ) ⊆ ˆ S . (cid:3) We sum up Proposition 4.3 and Corollary 4.13 in the next corollary. Corollary 4.14. Let C be weakly bounded from above and mub-complete. An R C -module M has an FSP if and only if M is S -determined for some finite S ⊆ C . Finally, we get our new characterization of finitely presented modules. Theorem 4.15. Let M be an R C -module. If M is finitely presented, then M ( c ) is finitely presented for all c ∈ C M is S -determined for some finite S ⊆ C .Furthermore, if C is weakly bounded from above and mub-complete, and M satisfiesconditions 1) and 2), then M is finitely presented. Proof. Using Proposition 3.4, the first part immediately follows from Proposi-tion 4.3 and the second part from Corollary 4.13. (cid:3) Remark . Let G be a monoid. Theorem 4.15 and Lemma 4.11 show us that the RG -modules of finitely presented type of Corbet and Kerber ([6, p. 19, Def. 15])are the same thing as finitely presented RG -modules.Furthermore, let A be a free G -act that is mub-complete and weakly boundedfrom above as a poset. Starting from the isomorphism RA - Mod ∼ = A -gr R [ G ]- Mod of Corollary 2.10, and using the fact that being finitely presented is a categori-cal property, we get an isomorphism between finitely presented RA -modules andfinitely presented A -graded R [ G ]-modules. Taking A = G now gives the commuta-tive case of [6, p. 25, Thm. 21]. Example 4.17. Let C = { a, b } ∪ Z , where a < n and b < n for all n ∈ Z , and Z hasthe usual ordering. Then the R C -module M := R [Mor C ( a, − )] ⊕ R [Mor C ( b, − )] isobviously finitely presented, but M does not have a finite framing set even thoughit is { a, b } -determined. 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