Families of cellular resolutions, their syzygies, and stability
aa r X i v : . [ m a t h . A C ] M a r Families of cellular resolutions, their syzygies, andstability
Laura JakobssonMarch 19, 2020
Abstract
We study families of cellular resolutions by looking at them as a category andapplying tools from representation stability. We obtain sufficient conditions on thestructure of the family to have a noetherian representation category and apply this toconcrete examples of families. In the study of syzygies we make use of defining thesyzygy module as a representation and find conditions for the finite generation of thisrepresentation. We then show that many families of cellular resolutions coming frompowers of ideals satisfy these conditions and have finitely generated syzygies, includingthe maximal ideals and edge ideals of paths.
Cellular resolutions are a powerful construction for resolving modules given by monomialideals and they contain a lot of structure ([2],[3]). Computational evidence on cellular reso-lutions have long suggested that in some families we have finitely generated syzygies givenby some finite number of resolutions in the family; however, there has not been a satisfac-tory proof of this. In this paper we study families of cellular resolutions from a categoricalperspective and by using representations of categories that is motivated by computationalresults on syzygies. Categorical representation stability, in particular the tools that wereintroduced by Sam and Snowden in [13], has been useful while studying noetherianity andfinite generation of representations. Using tools from representation stability we establishsufficient conditions for families of cellular resolutions to have finitely generated syzygies.The main idea is to define syzygies as a representation of the family and then show finitegeneration for this representation using noetherianity and covering of the cell complexes.This method allows us to use families with non-minimal cellular resolutions to study thesyzygies.The main result of this paper is on the conditions when a family of cellular resolutionshas finitely generated syzygies.
Theorem. If F is a family of cellular resolutions with noetherian representation category Rep S ( F ) such that the cell complex supporting F i is covered by the cell complexes supporting F j , j < i , for all i large enough. Then the syzygy representation σ t is finitely generated forall t . Other than being able to show that certain families have finitely generated syzygies, usingcategorical representations to study cellular resolutions can give new insights and it equipsthem with a structure. In particular it seems to be suited to studying cellular resolutions ofpowers of ideals. We illustrate this by studying the specific examples of families, the powersof maximal ideals and edge ideals of paths are among the examples we prove to have finitelygenerated syzygies.The organisation of the paper is as follows. In Section 2 we cover the needed backgroundmaterial for cellular resolutions, their morphisms and the necessary tools from representationstability. In Section 3 we give the definition of linear families, detailed example of powers of1 = ( x, y, z ), and the results on noetherianity and Gr¨obner properties. Section 4 is devotedto studying the syzygy representation and its relation to the covering of cell complexes andcontains the main theorem. Sections 5 and 6 cover explicit families including edge ideals ofpaths and maximal ideals. Finally in Section 7 we suggest a way of dealing with a family ofcellular resolutions where each resolution is over a different ring. This section addresses someof the limitations the setting of the earlier sections have had and we suggest an alternativeway of approaching the family categorically that allows us to lift the results of the previoussections.
Acknowledgements
I would like to thank Alexander Engstr¨om for many helpful discussions and his guidance.
In this section we define the needed notions of cellular resolutions and their category. Moredetailed information on cellular resolutions can be found in [12] and on the categoricalaspects in [10].
Definition 2.1.
A labelled cell complex X is a regular CW-complex with monomial labelson the faces. The vertices of X have labels x a , x a , . . . , x a r where a , a , . . . , a r ∈ N n .The faces F of X have the least common multiple of the monomial labels of the vertices itcontains, x a F = lcm { x a v : v ∈ F } . The label on the empty face is 1, i.e. x . Definition 2.2.
The degree of a face F is the exponent vector a F of the monomial label. Definition 2.3.
Let S ( − a F ) be the free S -module with a generator F in degree a F . Thenthe cellular complex F X is given by ( F X ) i = L F ∈ X dim F = i − S ( − a F ) with a differential ∂ ( F ) = X G ⊂ F sign( G, F ) x a F − a G G. We call the chain complex F X a cellular resolution if it is acyclic, that is, F X has non-zerohomology only at degree 0. Proposition 2.4 ([12], Def 4.3) . The differentials in the cellular complex can also be de-scribed by monomial matrices, with the columns and rows having the corresponding faces aslabels and the scalar entries coming from the usual differential for reduced chain complex.The free S -modules of F X are then the ones represented by the matrices. Another useful result for cellular resolutions makes use of order of vectors. If a and b are two vectors in N n , we have a (cid:22) b if b − a ∈ N n . Let X be a labelled cell complex, thenwe can define the subcomplex X (cid:22) b to be the complex consisting of all the faces with labels (cid:22) b . Then we have the following. Proposition 2.5 ([12], Prop 4.5) . The cellular free complex F X supported on X is a cellularresolution if and only if X (cid:22) b is acyclic over k for all b ∈ N n . For category-theoretic purposes one needs a morphism between two cellular resolutions.This was defined in [10]. Informally a morphism between two cellular resolutions is a pairof a chain map and a cellular map that do the same thing on the corresponding cells andgenerators of modules. More formally we get the definitions below.
Definition 2.6.
Let g : X → Y be a cellular map between two labelled cell complexes X and Y with label ideals I and J respectively. The set map ϕ g : I → J is the map defined bythe action of g , i.e. label m x ∈ I maps to m y ∈ J if and only if the face x labelled with m x maps to the faces y , . . . , y r labelled by m y , . . . , m y r with m y = lcm( m y , . . . , m y r ) under g ,and m x ∈ I maps to if and only if the face labelled by m x is not mapped to anything in Y . efinition 2.7. We say that a cellular map g : X → Y is compatible with a chain map f : F X → F Y if f ( x ) = ϕ g ( x ) for all x ∈ I , and f i maps the generator e x , associated to face x ∈ X , in F X,i to some linear combination of the generators e y i , i ∈ { , , . . . , r } , associatedto y i ∈ Y with the coefficients in S if and only if g maps x to union of y , y , . . . , y r . This definition of maps between cellular resolutions is very restrictive and balances thetopological and algebraic properties. One thing to note is that the component f in thechain map must be a 1 × S . However, thisdoes leave us with plenty of maps including Morse maps coming from Morse theory and thechange of orientation on the cell complex. Definition 2.8.
Define
CellRes to be the category given by: • A class of objects consisting of cellular resolutions, supported on any regular CW-complex, • A set of morphisms for any pair of objects F X and G Y with individual maps given bythe compatible pairs ( f , f ) . The morphisms between cellular resolutions are an important part in most of the resultsrelating to representation stability. Here we list a few observations and notions for them.The first definition concerns the terminology which we will use later in the paper.
Definition 2.9.
Let ( f , f ) be a morphism of cellular resolutions. It is called a multiplicationby monomial m or a morphism corresponding to a multiplication if ϕ f is a multiplicationby a monomial m . Proposition 2.10.
Embeddings in cell complexes supporting cellular resolutions give cellularmorphisms.Proof.
Let X and Y be two labelled cell complexes supporting cellular resolutions F X and F Y , respectively, and suppose that we have an embedding f : X → Y . Since the embeddingof labelled cell complexes respects the labelling we get that the map ϕ f is an identity map.Then we can find a chain map with f = ϕ f and choose the rest of the matrices with entries1 or -1 based on where f takes the higher dimensional cells. Then these form a compatiblepair and we have a morphism of cellular resolutions corresponding to embeddings. In this section we review the concepts needed from representation stability as defined bySam and Snowden [13]. Let R be a commutative noetherian ring. This assumption is notnecessary but in our setting almost all rings are commutative polynomial rings with finitelymany variables. Hence, they are also noetherian. Let Mod R be the category of R modules.Throughout this section we assume the category C to be essentially small. Recall that thismeans the category C is equivalent to some small category or alternatively it is locally smalland has small number of isomorphism classes as objects (assuming the axiom of choice).For more on the category theory definitions and theorems one can look at [11] for example.We want the category C to be of ”combinatorial nature”, which informally means objectsare finite sets, possibly with some extra structures, and morphisms are functions with extrastructure allowed. Definition 2.11.
Let C be an essentially small category. A representation or a C -moduleover R is a functor C →
Mod R . The representations of C form a category denoted by Rep R ( C ). This is an abelian functorcategory with the morphisms between representations given by natural transformations.Next we want to look at some definitions related to the properties of individual repre-sentations (or modules). Let M be a representation of C . A subrepresentation N of M is asubfunctor of M , that is 3 efinition 2.12. Let M be a representation of C . An element of M is an element of M ( x ) for some x ∈ C . Having defined an element one can then talk about the generating sets for representations.
Definition 2.13.
Let S be any set of elements of M . The smallest subrepresentation of M containing S is said to be generated by S . The representation M is said to be finitelygenerated if it is generated by some finite set of elements. The following representation is one of the main tools used to study noetherianity for therepresentations.
Definition 2.14 ([13]) . The principal projective representation for an element x is thefunctor P x given by P x ( y ) = R [ Hom ( x, y )] .Remark . In the paper of Sam and Snowden [13] they do not explicitly give the morphismpart of the principal projective. The natural choice of maps between the Hom sets in theprincipal projective are post compositions, so this gives then a morphism between P x ( y ) and P x ( z ) if we have a morphism f : y → z .An important fact about the principal projectives is that a representation of C is finitelygenerated if and only if it is a quotient of a finite direct sum of principal projectives. Definition 2.16.
Let M ∈ Rep R ( C ) . We say M is noetherian if every ascending chain ofsubobjects stabilises, or equivalently every subrepresentation is finitely generated.The category Rep R ( C ) is noetherian if every finitely generated representation in it isnoetherian. Next we have the following result.
Proposition 2.17 ([13], Prop 3.1.1.) . The category Rep R ( C ) is noetherian if and only ifevery principal projective is noetherian. One way to study representations is to use pullback functors, and we will use this inSection 6.2. Given a functor Φ :
C → C ′ there is a pullback functor Φ ∗ : Rep R ( C ′ ) → Rep R ( C ). The following finiteness property is particularly useful. Definition 2.18 ([13], Def 3.2.1.) . Let
Φ :
C → C ′ be a functor. Then Φ satisfies property(F) if given any object x ∈ C ′ there exists finitely many y , y , . . . , y n ∈ C and morphisms f i : x → Φ( y i ) such that for any y ∈ C and any morphism f : x → Φ( y ) there exists amorphims g : y i → y such that f = Φ( g ) ◦ f i . Proposition 2.19 ([13],Prop 3.2.3.) . A functor
Φ :
C → C ′ satisfies the property (F) ifand only if Φ ∗ : Rep R ( C ′ ) → Rep R ( C ) maps finitely generated objects to finitely generatedobjects. Finally, we move on to covering the definition of Gr¨obner bases for categories and Gr¨obnercategories from Sam and Snowden [13]. Those will make an appearance in Section 3.3. Let S : C →
Set denote a fixed functor to sets and let S x : C →
Set be the functor givenby S x ( y ) = Hom( x, y ). A principal subfunctor is a subfunctor of S generated by a singleelement. Definition 2.20.
The poset | S | is the set of principal subfunctors of S that is partiallyordered by reverse inclusion. Let P denote the free module R [ S ] and write e f for the element of P ( x ) correspondingto f ∈ S ( x ). An element of P ( x ) is monomial if it is of the form λe f for some λ ∈ R . Asubrepresentation M is monomial if it is spanned by the monomials it contains.To define Gr¨obner basis we need a concept of initial representations and terms. Thefunctor S is orderable if there is a choice of well-order on each S ( x ) such that the inducedmap S ( x ) → S ( y ) is strictly order preserving for every x → y .4uppose S has ordering (cid:22) on it. Then the initial term of an object α ∈ P ( x ) is init( α ) = λ g e g , where g = max (cid:22) { f | λ f = 0 } and α is a direct sum of monomials. Let M be asubfunctor of P . The initial representation of M consists of init( M )( x ) that is the R -spanof init( α ) for α = 0 ∈ M ( x ). Definition 2.21.
Let M be a subrepresentation of P . A set of elements G of M is aGr¨obner basis of M is { init( α ) | α ∈ G } generates init( M ) . Theorem 2.22 ([13], Thm 4.2.4.) . Let S be orderable and | S | be noetherian. Then everysubrepresentation of P has finite Gr¨obner basis. In particular, P is a noetherian object of Rep R ( C ) . Definition 2.23.
Let C be an essentially small category. Then C is called Gr¨obner if forall x ∈ C the functor S x is orderable and the poset | S x | is noetherian.We say that C is quasi-Gr¨obner if there exists some Gr¨obner category C ′ such that thereis a functor Φ : C ′ → C that is essentially surjective and satisfies property (F). Theorem 2.24 ([13], Thm 4.3.2.) . Let C be quasi-Gr¨obner, then Rep R ( C ) is noetherian. In the case the category is directed as well as small we can use the following propositionto determine if it is Gr¨obner. First note that an admissible order is a well-order on a setthat also satisfies if we have any two elements u ≤ v then for any third element t , for which ut and vt make sense, we have ut ≤ vt . Proposition 2.25 ([13], Prop 4.3.4.) . If C is a directed category, then as posets |C x | ∼ = | S x | for all objects x . In particular, C is Gr¨obner if and only if for all x the set |C x | admits anadmissible order and is noetherian as a poset. Cellular resolutions as a whole form a too big class of objects to study via representationstability. Thus we will restrict to the families of cellular resolutions that are essentiallysmall, or small in many cases, and are interesting on their own as a restricted class ofcellular resolutions.
Definition 3.1.
A family of cellular resolutions is an infinite sequence of cellular resolutionssuch that as a subcategory of
CellRes it is essentially small category.Remark . From a representation stability point of view, any essentially small subcategorywould suffice. If we have an indexed sequence then the family will have countably manyobjects, which clearly form a set. So in most cases the only possible cause for the family tonot be essentially small are the morphisms.The above definition of a family of cellular resolutions does not restrict the morphismsin any other way that on the essentially small part. In particular one can choose the cellularresolutions at random without needing any morphisms between them. However these kindof families are not the main interest of our study and we will restrict to ones with morestructure. Note that if we do have any morphisms in the family the Hom sets must be setsand not a class for the essentially small condition to be satisfied. A common way that we usein this paper is to restrict the morphisms to only one compatible pair for each chain map.The following example presents some explicit examples of families of cellular resolutions.
Example 3.3. • The constant family where each of the resolutions is the same. • S/I n where I is any monomial ideal and the consecutive maps are multiplications bygenerators of I . 5 yz x y xyxz yzz x xy x yx z xyzxz z yz y zy x x y x yx z x yzxz xz xyz xy zxy z yz y z y zy Figure 1: The first four cell complexes in the family of cell complexes supporting the non-minimal resolutions. • Cellular resolutions of edge ideals of paths with embeddings as maps between them.
Definition 3.4.
Let F : F → F → . . . → F i → . . . be a family of cellular resolutions.We say that F is linear if there is at least one morphism f i,i +1 : F i → F i +1 betweenconsecutive cellular resolutions, and the other morphisms are compositions of those, i.efor any f i,i + k : F i → F i + k there exists some consecutive morphisms such that f i,i + k = f i + k − ,i + k ◦ f i + k − ,i + k − ◦ . . . ◦ f i +1 ,i +2 ◦ f i,i +1 , except possibly selfmaps F i → F i .Remark . Note that the definition does not require the decomposition of the map f i,i + k to be unique.A common example of linear family is the family of cellular resolution consisting ofpowers of an ideal I . Families consisting of resolutions corresponding to the powers of someideal can be defined to have multiplication maps that correspond to the monomials in theideal, and so the composition part of the linear family is easily satisfied. Of course thefreedom of choice allows one to choose the morphisms such that any power family can benon-linear, so even with families of powers there is need to clarify the morphisms and checklinearity in all cases. We begin with an explicit example of a family of cellular resolutions that has finitely gen-erated syzygies where this can be shown with the help of representation stability. Thepropositions presented in this example are special cases of theorems and propositions in thelater sections.Let S = k [ x, yz ] and let I = ( x, y, z ) be an ideal. We are interested in studying thesyzygies of the modules S/I n for positive integer n .Let X n denote the labelled cell complex with labels from I n . The cell complex X n ismade of a triangle that is subdivided into n smaller triangles. Let the triangle boundedby the vertices x n , y n and z n be called the outer triangle. We refer to the vertices of thecell complex X n by the labels. Consider the family of resolutions supported on the cellcomplexes X n . Figure 1 shows the first few labelled cell complexes in the family. Thisfamily consists of non-minimal cellular resolutions. We assume that the orientation is fixed.Fixing the orientation of the cell complexes means that we only have one cell complex foreach set of labels.Next we want to study the possible morphisms between the cellular resolutions supportedon X n . Recall that in [10] the morphisms are defined as compatible pairs of cellular mapsand chain maps. In the case of the family supported on the cell complexes in Figure 1 wecan approach the morphism by studying the possible cellular maps. Since we are interestedin the syzygies, we do not need the information of all different maps compatible with a6hain map. Thus it suffices to find just one cellular map for each compatible chain map. Inpractice this means we only focus on which cells are mapped to which cells, and we do notcare about how it is mapped as topological map.Recall that in the induce label map ϕ g can be found in the Definition 2.6. Proposition 3.6.
The possible cellular maps g : X n → X n +1 , such that they are a compo-nent of a cellular resolution morphism, are the ones inducing label maps by multiplicationby a variable.Proof. By definition of compatibility the cellular map g must induce a map ϕ g between thelabel ideals. If we have a cellular map where any cell maps into a lower-dimensional cell,the induced label map would not give a map between ideals. Thus the cellular maps haveto map cells to the same dimensional cells.We have three easily found maps for the maps taking cells to same dimensional cells. Thecellular map g taking vertex m to the vertex mx defines a label map ϕ g that is multiplicationby x . A simple computation shows that the cellular map g is compatible with the chain map f between the resolutions supported on X n and X n +1 such that f = ϕ g . This cellular map g gives a morphism of cellular resolutions. Swapping the variable x to either y or z gives asimilar map, the multiplication is just by a different variable and we still get a morphism ofcellular resolutions.In the case n ≥
2, we do not have other maps that do not involve permutations ofvariables. If we are in the case X mapping to X there is the cellular map taking X tothe central triangle of X . This gives us three possible label maps on the vertices which are x xy x yz x xzy yz , y xz , and y xyz xz z yx z yz . One can then use the definition of compatibility to see that we cannot construct a compatiblechain map for any of the three maps above.So we get that the only cellular maps compatible with the cellular resolution morphismsare the multiplications by a variable.Let t , t and t be the maps corresponding to the multiplications by x , y and z , respec-tively.We note that the possible cellular maps are in one to one correspondence to the mor-phisms of the minimal family. Figure 3 shows the corresponding cell complexes of theminimal family to the cell complexes of the non-minimal family of Figure 1.Next we show that for n ≥ t , t and t are ”surjective” together,that is their image cover the cell complex they map to. The condition on n is easily seen byconsidering the maps from X to X where no map maps to the central triangle of X . Proposition 3.7.
For n ≥ , any cell in X n +1 is in the image of t i ( X n ) , i ∈ { , , } , forat least one i .Proof. Let us consider the three maps between X n and X n +1 . The maps t i , i = 1 , ,
3, areembeddings of X n into X n +1 . One can think of these maps as covering a part of X n +1 by X n . If three copies of X n based on t i cover the whole X n +1 we have the desired result.The map t maps X n to the subtriangle of X n +1 bounded by the vertices with labelscontaining the variable x . It leaves a strip of triangles uncovered. This strip is shown inFigure 2. Next consider the map t . We only need to investigate if it covers any of the cellsin the strip, as the other cells are already in the image of t . The map t maps to all labelscontaining y , so in particular it covers all of the triangle strip but the top two ones. Thisis shown in Figure 2. Finally we see that the map t will cover the remaining cells since itwill map to all cells with label z in them, in particular the two top ones.7 n +1 x n − y x n yx n zx n − z y n − x y n +1 xy n y n zy n − z x z n − y z n − xz n yz n z n +1 x n +1 x n − y x n yx n zx n − z y n − x y n +1 xy n y n zy n − z x z n − y z n − xz n yz n z n +1 Figure 2: (a) The image of t is in blue and the uncover triangle strip is in light red in X n +1 .(b) The images of t and t in blue and uncovered top triangles. x yz x y xyxz yzz x xy x yx z xyzxz z yz y zy x x y x yx z x yzxz xz xyz xy zxy z yz y z y zy Figure 3: A minimal family of cell complexes supporting the resolution of powers of I .8he overlap of any two of the maps is X n − with labels multiplied with xy , xz or yz with respect to the maps. The overlap of all three is X n − with labels multiplied by xyz .Here we consider X to be a single point with label 1.Let F be the family given by the cellular resolutions S/I n supported on the subdividedtriangles. We also have the family F that consists of the minimal resolutions of S/I n , whichis also a subcategory of CellRes . The morphisms are pairs of compatible maps, such thatwe only have one pair for each chain map. The first few cell complexes of the resolutionsare shown in Figure 3.We want to show that the category of representations is noetherian. One way for this isto study the principal projectives. In F the Hom sets are finite, so the principal projective P x gives a finitely generated free module for any element. Moreover since any map F n → F m comes from the composition of maps of the form F i → F i +1 for n ≤ i ≤ m −
1, we get thatthe principal projectives are finitely generated by definition.
Proposition 3.8.
Every principal projective representation P x of F is noetherian.Proof. By definition a representation is noetherian if every ascending chain of subobjectsstabilises. In F all the ascending chains are the whole category or some subset of it. Sincethe principal projective is finitely generated the same arguments show the ascending chainsstabilise in the representation stability sense. Corollary 3.9.
Rep S ( F ) is noetherian.Proof. By Proposition 3.8 every principal projective is noetherian. Then applying Proposi-tion 2.17 to F gives the noetherianity of the category.Let us consider the following representation of F . We define a functor s p : F →
Mod S taking a cellular resolution of I n to the p th module of the non-minimal resolution of I n . Thefunctor maps a morphism given by a multiplication of a monomial to the p th componentof chain map between the non-minimal resolutions which is the multiplication by the samemonomial between the non-minimal resolutions. Proposition 3.10.
The representation s p of F defined above is finitely generated.Proof. Since the representation s p takes a morphism of F to the map between the mod-ules given by a corresponding chain map, we have that the morphisms of F give all thepossible maps between the modules that come from a compatible chain map. ThereforeProposition 3.7 shows us that the representation s p is finitely generated by definition.Then we get the following. Proposition 3.11.
The representation σ p : F →
Mod S taking resolution F to its p th syzygy module is finitely generated.Proof. First we note that the p th syzygy module is contained in the p th module of the non-minimal resolution. So we get that σ p is a subrepresentation of s p . Further more we knowthat in a noetherian representation category any subrepresentation of finitely generatedrepresentation is finitely generated. Thus σ p is finitely generated as s p is.9 .2 Noetherianity results for families of cellular resolutions We begin this section by stating a proposition about the noetherianity properties of linearfamilies.
Proposition 3.12.
Let F be a linear family of cellular resolutions with finitely generatedHom sets. Then Rep S ( F ) is noetherian.Proof. We prove the noetherianity by showing that every ascending chain stabilises underall principal projectives. First fix a principal projective P x .By definition of a subobject, the only subobject for some F i in F are the cellular reso-lutions F k with k < i . Then it follows that the possible ascending chains of subobjects areeither the whole family, or some subset of it.Next we want to study the behaviour of the principal projective P x on the differ-ent ascending chains of subobjects. Denote the cellular resolution corresponding to x by F i for some i . Then we have no maps to resolutions F k for k < i , and so P x ( F k ) = S [Hom( F i , F k )] = S for k < i . Thus for the stability point of view it suffices to look at thefamily F from F i onwards. So we have F ≥ i : F i → F i +1 → . . . → F j → . . . Applying the principal projective functor to the family F ≥ i we get the following sequenceof free modules P x ( F ≥ i ) : S [Hom( F i , F i )] → S [Hom( F i , F i +1 )] → . . . → S [Hom( F i , F j )] → . . . The maps between S [Hom( F i , F j )] → S [Hom( F i , F j +1 )] are given by sending the gen-erator e f ∈ S [Hom( F i , F j )], f ∈ Hom( F i , F j ), to e g ◦ f ∈ S [Hom( F i , F j +1 )] where g ∈ Hom( F j , F j +1 ). Each morphism in Hom( F j , F j +1 ) gives a map between the free modules.For rest of the proof we refer to these maps given by the post-composition in Hom-sets aspost composition by a morphism, denoted by p g for g ∈ Hom( F j , F j +1 ).If we have a map from h : S [Hom( F i , F j )] → S [Hom( F i , F k )] for some k > j > i , thenusing the requirement for the family that all morphisms are made of compositions betweenthe consecutive resolutions, we can consider each of the components of h . First we can writethe map as S [Hom( F i , F j )] h ′ −→ S [Hom( F i , F k − )] p g −→ S [Hom( F i , F k )] , where p g is a suitable post-composition. This process can repeated and choosing the suitablecompositions between each consecutive modules until we get h as a composition of p g s. Notethat this is not necessarily unique decomposition. Another important thing to note aboutthe maps p g is that any generator in S [Hom( F i , F j +1 )] is given by the form p g ( e f ) = e g ◦ f where e f ∈ S [Hom( F i , F j )], due to linear structure of the family.Therefore if we have the full family, any generator of S [Hom( F i , F j )] is given by applying j − i p g maps to the generators of S [Hom( F i , F i )]. Hence in this case the principal projectiveis finitely generated or in other words it stabilises.If we are looking at some subfamily as the ascending chain, with the indices of the cellularresolutions denoted by j , j , j , . . . , then the sequence looks the following after applying theprincipal projective S [Hom( F i , F j )] → S [Hom( F i , F j )] → . . . → S [Hom( F i , F j k )] → . . . Again any map S [Hom( F i , F j k )] → S [Hom( F i , F j l )] becomes a composition of p g maps.Note that in this case we might have two p g ◦ p g ′ or more as map between two free modulesbut not the individual components. Since the morphisms have not been restricted all mapsfrom Hom( F i , F j k ) to Hom( F i , F j l ) are given by k − l post-compositions, and any mapin Hom( F i , F j l ) consist of a map from Hom( F i , F j k ) and post-compositions of consecutivemaps. Thus we have that any generator in S [Hom( F i , F j k )] can written as image of some10enerator in S [Hom( F i , F j )] which is finitely generated free module, and so we get that anysubsequence in the family is finitely generated under the principal projective P x .We have shown that any ascending chain of subobjects will be finitely generated underan arbitary principal projective, equivalently all ascending chains of subobjects stabiliseunder the principal projectives. Therefore all the principal projectives are noetherian andby Proposition 2.17 Rep S ( F ) is noetherian. Remark . The noetherian representation depends on the morphisms, so for a generalfamily of cellular resolutions we need at least two conditions to have noetherian representa-tion category: • finitely generated morphisms, i.e. in most cases quotient out the morphisms with thesame chain map. • After finitely many steps the morphism can all be described by just giving the mor-phism between two consecutive cellular resolutions.The above remark gives us the following corollary of Proposition 3.12.
Corollary 3.14. If F is a family of cellular resolutions such that it is linear after somefinite sequence of length i and for all the resolutions F j , F k , j, k ≤ i , Hom( F j , F k ) is finitelygenerated, then Rep S ( F ) is noetherian.Proof. If we look at the family consisting of the part F >i , then by Proposition 3.12 anyprincipal projective is finitely generated on any subsequence. Since the discarded part inthe sequence is finite, we only have finitely many Hom-sets between the cellular resolutions,and each of these sets is finitely generated. So when considering the whole family F or asubsequence that contains cellular resolutions from the first i resolutions, we only need to addfinitely many finite generating sets to the generators of F >i under any principal projective P x . Thus it is still finitely generated, and we have noetherianity for every principal projectiveand by Proposition 2.17 also for Rep S ( F ). A common example of a linear family of cellular resolutions is the family consisting of powerso an ideal with some assumptions on the morphisms. Then one naturally wonders whetherthese very controlled families also satisfy the conditions of being Gr¨obner. For this weconsider the following special type of a family.Let I be a monomial ideal with m generators g , g , . . . , g m . Suppose that for each powerof I , the module S/I k has a cellular resolution. Let F : F → F → . . . → F i → . . . be a family of cellular resolutions where F i = S/I i . Furthermore, let the only maps betweenconsecutive resolutions in the family be multiplications by the generators of I (again onemorphism for each chain map) and the only map from F i to itself is the identity. Moreover,let us suppose all other maps are compositions of the multiplications. This kind of family isnot only a small category but also directed and so we can use the Proposition 2.25 to studyif it is Gr¨obner. Example 3.15.
Let us consider the ideal I = ( x, y, z ) and the cellular resolutions of thepowers again and let F denote the family of non-minimal triangle resolutions. This familysatisfies the conditions described above and in particular it is a directed category. Let F F denote the category of morphisms from F ∈ F where the morphisms are commutingtriangles. We want to study the set |F F | for some F ∈ F .First we want to show that the set |F F | has an admissible order. We label each f ∈ |F F | by the monomial multiplication it is associated to, and then take an ordering on |F F | givenby any monomial order on the monomials. This then gives us the admissible order on |F F | .11he set |F F | is a poset with the order given by x ≤ y if there is a map x → y . Moreoverit is noetherian poset. This can be seen by considering the requirement for descending chaincondition and no anti-chains. Take a descending chain f ≥ f ≥ f ≥ . . . By definition of the order this means we must have a chain of maps F ← F ← F ← . . . in our category where the maps go in the direction of increasing powers. Hence the chainof decreasing cellular resolutions cannot go on forever but will either have to reach theresolution of S/I or stabilise before that. Next we want to look at anti-chains. Observe thatif we have f i : F → F i and f j : F → F j in |F F | then we will have either F i → F j or F j → F i depending on which is the resolution of higher power. Thus we get that f i ≥ f j or f j ≥ f i for i = j . Then if we want an anti-chain we must have that all elements in it correspond tomaps to the same cellular resolution. However, we only have finitely many such maps andso we can only have finite anti-chains.Then by Proposition 2.25 the category F is Gr¨obner.With the above example in mind we formulate the following proposition. Proposition 3.16.
Let I be a monomial ideal. Let F be a family of cellular resolutionswhere F i = S/I i and let the only maps between consecutive resolutions in the family bemultiplications by the generators of I and the only map from F i to itself is the identity.Then F is Gr¨obner.Proof. Let F be a family of cellular resolutions as specified in the proposition and let I bethe defining ideal. Then since all the Hom-sets are finite, and since we have no selfmapsother than the identity, the family F is small and directed as a category. Therefore it sufficesto study the set |F F | for some arbitrary member F of the family. Recall that F F is thecategory of all arrows from F with commuting triangles as morphisms, and that |F F | is theset of isomorphism classes.Each morphisms in F corresponds to multiplication by some monomial that consists ofmultiples of the monomial generators of I . If we consider the arrows in the category F F ,we note that there are no objects that are arrows to powers smaller than F , so we onlyhave objects corresponding to maps to powers higher than F has. In the set of isomorphismclasses |F F | , we note that if two morphisms are associated to different multiplication theyare not isomorphic and if we have two morphisms that correspond to the same monomialmultiplication, then these are isomorphic. Therefore we can label our morphisms uniquelywith the monomials corresponding to multiplication. Then taking any monomial order, sayfor example lexicographic, will give an admissible order on the set |F F | .Next we want to look at |F F | as a poset with the natural order f ≤ g if there existsa morphism f → g . We want to show that this poset is noetherian. First let us considerany anti-chain in it, that is a chain of elements such that any two are not comparable.Let f : F → G and g : F → G ′ be two noncomparable objects. This means there is nomap between G and G ′ that forms a commutative diagram, or in the other direction. Bydefintion the family satisfies that between any consecutive objects we have multiplicationby the monomials in I , and all other maps are compositions of these. Then if G and G ′ are not the same resolutions, we can find a map between them that gives a commutativetriangle with f and g . This tells us that any anti-chain must be made of arrows to thesame resolution. We only have finitely many of such objects, hence there cannot be an infiteanti-chain.Next we want to look at the descending chain condition on |F F | . Take any descendingchain f ≥ f ≥ f ≥ ... and consider the maps that form it as shown in Figure 4.The vertical chain of maps is what gives the order relation. From the definition of thefamily we get that the chain of the vertical maps cannot go on forever as eventually we reach12 G G G G ...... f f f f Figure 4: Commuting triangles in |F F | .the resolution of S/I and there are no resolutions mapping to it or our chain will stabilisebefore it. In either case eventually we must have f i = f i + 1 in the descending chain as weonly have one map that can be repeated. Note that we cannot end in the situation where theresolution that F maps to are the same but we rotate possible maps as these are preciselythe non-comparable cases. Thus the poset |F F | satisfying the descending chain conditionand has no infinite anti-chains, so it is noetherian.Since our choice of F was arbitrary, we can then applying Proposition 2.25 gives us that F is a Gr¨obner category. The main interest for us in the families of cellular resolutions is on their syzygies. The p -th syzygy module can be written as a representation of a family F . First we define arepresentation on the p -th free module. Definition 4.1.
The p-th module representation is a functor s p : F →
Mod R such that s p ( F i ) = p -th free module in the resolution and s p ( F i → F j ) is the restriction ofthe chain map from F i to F j on the p -th component. Next we define the syzygy representation for a family of cellular resolutions.
Definition 4.2.
Let F be a family of cellular resolutions. The p -th syzygy functor σ p : F →
Mod R is defined by taking F ∈ F to its p -th syzygy module and the morphisms are restrictions ofthe chain maps. Proposition 4.3.
The representation σ p is a subrepresentation of s p .Proof. We know that the minimal resolution is contained in a non-minimal one as a directsummand. If the family consist of minimal cellular resolutions, then σ p is a finitely generatedsubmodule of s p by definition of the syzygy module. Otherwise the family F has at least onenon-minimal resolution. Then we can take a natural transformation η from σ p to s p givenby an embedding. Then the natural transformation will have the maps η F : σ p ( F ) → s p ( F )to be monic and hence σ p is a subfunctor of s p .13e are interested in studying the finite generation of the syzygy functor. To do this wewant to make use of the cell complex structure in cellular resolutions and this gives rise tothe following definition of covering. Definition 4.4.
Let F and G be cellular resolutions such that Hom(
F, G ) is not empty. Let X be the cell complex supporting F and let Y be the cell complex supporting G . We say thatwe have covering of Y by X if the images of X under the maps f ∈ Hom(
F, G ) cover Y , ∪ f ∈ Hom(
F,G ) f ( X ) = Y .Let F , F , . . . , F r be cellular resolutions mapping to G . Let X i be the cell complex sup-porting F i and Y be the cell complex supporting G . Then we say that X is a covering of Y if the images of X i under the maps f ∈ Hom( F i , G ) cover Y , ∪ f ∈ Hom( F i ,G ) f ( X i ) = Y . Definition 4.5.
Let F , F , . . . , F r be cellular resolutions mapping to G . Let X i be the cellcomplex supporting F i and Y be the cell complex supporting G . Then we say that we have d-covering of Y by X , X , . . . , X r if the images of d -cells of X i cover d -cells of Y under themaps f ∈ Hom( F i , G ) , ∪ f ∈ Hom( F i ,G ) f ( X i ) = Y .Remark . In general taking the Taylor resolution family will not give a covering. This isdue to not having enough maps between the simplices, see Section 7.1 for more details.
Lemma 4.7.
The p-th module representation is finitely generated if and only if we have ( p − -covering of X i by finitely many X j s with j < i for all i large enough.Proof. First let us fix p . Let F be a family of representations and suppose that the p-th module functor s p is finitely generated. By definition this means we have finitely manygenerators ǫ , ǫ , . . . , ǫ r in s p ( F i ), s p ( F i ) , ldots, s p ( F i ) such that they generate all the modules s p ( F ). Alternatively any generator e of F j , j > i , can be written as an image of one or moreof the ǫ s via maps in the sets Hom( F k , F j ), 1 ≤ k ≤ i .Each of the generators ǫ , ǫ , . . . , ǫ r corresponds to a cell in the first i cell complexes, andmoreover these are all cells of dimension p −
1. Denote these cells by c , c , . . . , c r . From thecompatibility of the cellular resolution maps, if there is a generator mapping to a generatorthen the corresponding cells map to each other. Then the finite generation implies thatchoosing any p − c in some X j , there is a cellular map g belonging to acellular resolution morphism such that g ( c k ) = c for some 1 ≤ k ≤ r . Then any X i has a( p − X ) j s, j < i , for large enough i .Conversely, assume that we have a covering of X i by finitely many X j s with j < i forall i large enough. So given any X j , j > i , there are cell complexes X k , X k , . . . , X k r thatcover X j . We want to show that all X k , X k , . . . , X k r are below i . Suppose that one ofthem is not, call this one X k . Since k > i we have that its ( p − X l , X l , . . . , X l s . Note that composing the cellular map that the X l , X l , . . . , X l s s haveto X k and needed maps from X k to X j , give a map from X l , X l , . . . , X l s to X j . Since X l , X l , . . . , X l s give ( p − X k we can replace X k in the covering set of cellcomplexes by the X l , X l , . . . , X l s . If all of the X l , X l , . . . , X l s are below i , then we have( p − X j by cell complexes below i . Otherwise, we repeat the process forthe cell complexes above i to get ( p − i . The process will always give cell complexes below i , otherwise we would contradict thecomposition properties of the morphisms.Taking the first i cell complexes such that we have ( p − X j , j > i , thentheir ( p − p − s p ( F j ) is reachable via the maps from one ofthe generators of the first i modules for large enough j . we have finitely many ( p − i cell complexes, hence the first i free modules s p ( F i ) have finitely many generatorsall together. Thus we get a finite generating set for the representation s p . Theorem 4.8. If F is a family of cellular resolutions with noetherian representation cat-egory Rep S ( F ) such that the cell complex supporting F i is covered by the cell complexessupporting F j , j < i , for all i large enough. Then the syzygy representation σ p is finitelygenerated for all p . z yzywxw x z xyz xyzwx zw y w y zwx w xyw y w y w x z x yz x yz wx z w xy z xy z w y z y z wx yzw x zw xy zw y zw x yw x w xy w y w x z x yz x yz wx z w x y z x y z wxy z xy z w y z y z wx yz w x z w x y z w xy z w y z w x yzw x zw x y zw xy zw y zw x yw x w x y w xy w X X X X Figure 5: The labelled cell complexes X , X , X , X supporting the resolutions of S/I, S/I , S/I and S/I of Example 4.11. Proof.
Let F i denote the cellular resolutions in the family and let X i be the cell complexsupporting F i .If we have a covering of the cell complexes, then in particular we have a d -covering of F i for all dimensions d by some finite number of lower cellular resolutions for large enough i .Then by Lemma 4.7 we have that s d is finitely generated for all d .Now Proposition 4.3 tells us that σ p is a subrepresenation of s p for all p . Since s p isfinitely generated, then by noetherianity any subrepresentation is also finitely generated.Hence σ p is finitely generated for all p . Remark . If one fixes a family of ideals, then the modules we get from them may havemultiple cellular resolutions, minimal or non-minimal. Then showing that there is no cov-ering for one of the possible cellular resolutions does not imply that the others might nothave it. For example if we look at the Taylor resolution for the ideal I = ( x, y, z ), we willnot get covering of the cell complexes with only three maps. Remark . The condition on having covering for everything above large enough i is aneeded condition. For example consider the following family: Example 4.11.
Let S = k [ x, y, z, w ] be a graded polynomial ring and let I = ( xz, xw, yz, yw )be an ideal. Let us consider the family of cellular resolutions consisting of S/I n . Each ofthese modules has a minimal cellular resolution, and the cell complexes supporting the firstfour modules S/I, S/I , S/I and S/I are shown in Figure 5.These resolutions are minimal and the first two are S h xz xw yz yw i ←−−−−−−−−−−−−−−−−− S − w − y z − y x − w x z ←−−−−−−−−−−−−−−−−−−− y − wz − x ←−−−−−− S ← S d ←−− S d ←−− S d ←−− S ← d = (cid:2) x z x zw x w xyz xyzw xyw y z y zw y w (cid:3) , = − w − y z − w − y z − y x − w − y x z − w − y x z − y x − y
00 0 0 0 0 0 0 0 x z − w x z , and d = − y w − y − z w − z x − y
00 0 w x − y − z w − z x
00 0 0 x . Looking at the first two resolutions we can already see a pattern in them, which appearsto continue if one computes further resolutions. Now we can use the results on the syzygyfunctor to show that the pattern is indeed there and the syzygies are finitely generated. Themaps in this family are again the multiplication maps. On the cellular side they correspondto sending squares to squares. It is then not hard to see that we have covering of the cellcomplex for any of the complexes X i for i ≥
2. Then by Theorem 4.8 the family has finitelygenerated syzygy functors for all p . The finite generation of syzygies holds in general for an n -cube, see Section 5.2. In this section we focus on families coming from powers of ideals and finite generation ofsyzygies in them. S Let S = k [ x , x , . . . , x n ] be a graded polynomial ring and let m be the maximal monomialideal. We know that the minimal resolution of S/ m is supported on the n -simplex. In thissection we want to focus on the families of cellular resolutions given by the powers of m .First we show that the resolutions S/ m k , for k > n -simplex. Definition 5.1.
Let X kn be the labelled cellular complex given by the Newton polytope of m k ,i.e. the vertices are given by the exponent vectors of the monomials in m k , with subdivisionwith the following hyperplanes H i,j = { y ∈ R n | y i = j } for < i ≤ n and j ≥ . The labels are given by the monomials in m k and placed accordingto the Newton polytope.Remark . On the level of vertices and edges, we have that any two vertices that differ bya single variable will be connected by an egde in this subdivision.For an example of the cell complex see Figure 1.16 roposition 5.3.
The module S/ m k has a cellular resolution supported on X kn .Proof. Fix an n and the polynomial ring S . Note that for each n we work over a differentpolynomial ring S n = k [ x , x , . . . , x n ].Let us consider the cellular complex given by X kn . This cell complex has zero reducedhomology.Next consider the cell complex X n k (cid:22) b for any b ∈ N n . Due to the labelling with the leastcommon multiple, any cell in X n k (cid:22) b will contain its boundary and the labeling gives us thatwe cannot create holes into the cell complex with bounding by some b ∈ N n . Then X n k (cid:22) b still has zero reduced homology for any b , and by Proposition 2.5 X kn supports the cellularresolution of m k .Next we want to look at the maps between the powers. From the Example 3.1 we wouldexpect that the maps are gain multiplication by a variable. Proposition 5.4.
Cellular resolution morphism corresponding to multiplication by a vari-able in S give morphism between consecutive powers of m .Proof. Let us fix a variable, say x i , for the morphism. Then we know that the set mapinduced by it is the multiplication by x i , so a vertex with a monomial label l will map to x i l . On the cellular side, this corresponds to the embedding of X k − n to X kn such that X k − n covers the “corner” with x ki . We also know that the embedding has a corresponding chainmap, hence the multiplications by a variable form a map between consecutive powers.The existence of these desired multiplication maps gives us a linear structure on thefamily. Since we cannot have other multiplication maps between distinct cellular resolutionsof the powers of the maximal ideal, we have the following. Proposition 5.5.
The family of cellular resolutions given by S/ m k , for k ∈ N , with multi-plication maps is a linear family of cellular resolutions.Proof. Given any two members of the family, say S/ m i and S/ m j with i < j , the possiblemaps between them must have f a multiplication due to the labels. Moreover, we cansplit any monomial m multiplication to maps given by a variable, the order may vary sothe decomposition is not unique. This gives us the condition for any f i,i + k : F i → F i + k there exists some consecutive morphisms such that f i,i + k = f i + k − ,i + k ◦ f i + k − ,i + k − ◦ . . . ◦ f i +1 ,i +2 ◦ f i,i +1 .Next we want to show that the subdivided simplicial complexes behave as the complexesin Example 3.1. One can use the same method to show the tetrahedron is also coveredafter three subdivisions. However, drawing (or building) the cell complexes gets somewhatcomplicated from dimension four upwards, so we would like to have a more general prooffor these coverings. For this we will make use of the following observations from the low-dimensional cases: the square-free part is not fully covered since nothing can map to thatcell. Once we only have a single square-free vertex, then the cell complex is covered bycopies of one subdivision lower cell complexes.Note that we want to consider embeddings of cell complexes that correspond to themorphisms between S/ m p − and S/ m p , which are in fact the only possible embeddings. Proposition 5.6.
Let n be a positive integer and g : X p − n → X pn be an embedding of cellcomplexes for p > corresponding to the cellular resolution morphisms. Then the embeddingsof g : X n − n → X n − n cover X n − n , and there cannot be a covering in the lower subdivisions.Proof. We want to consider the cell complexes as the labelled complexes to help keep trackof the cells. Here we refer to the vertices by the monomial labels.Firstly showing that n copies of X n − n do not cover X n − n follows from the square-freemonomials and the multiplication maps. Let us consider the cell formed by the square-freevertices. In the ( n −
2) subdivision we have the degree ( n −
1) monomials. In particular this17eans we have an n -simplex formed by these vertices. For it to be covered, we must haveanother cell of the same dimension mapping to it, which implies that at least one copy oflower subdivisions should cover the whole square-free simplex. We know that this cannothappen since we always have a vertex that does not contain any chosen variable. Hence itcannot be covered with a map corresponding to a multiplication by a variable.Next we want to show that the next subdivision is coverable and this property holdsafter it. In the cell complex the parts that are not covered by the maps can also be seen asthe bounded polytope by the hyperplanes H i, = { y ∈ R n | y i = 1 } . In X n − n these planesintersect in a single point, namely the vertex with the label x x . . . x n . So we do not haveany cells that are not covered by the embedded X n − n . Finally one wants to show that thecovering continues in the higher steps. The intersection of the embeddings is formed fromthe vertices that have x x . . . x n in their label. In X in for i ≥ n − Theorem 5.7.
Let m be a maximal ideal of the polynomial ring S = k [ x , x , . . . , x n ] . Thenthe syzygies of the modules S/ m p for p > are finitely generated.Proof. From Proposition 5.5 we have that the family of resolutions S/ m p has linear structure,and we have defined the morphisms to be only three between consecutive powers so the Homsets are finitely generated. Then by Proposition 3.12 we have that the family has noetherianrepresentation category. This together with Proposition 5.6 allows us to apply the Theorem4.8 and we have that the syzygies are finitely generated. In this section let S = k [ x , x , . . . , x n ] be a polynomial ring in 2 n variables. Given an n -dimensional cube, label it with monomials such that along each of the n edge directions weassign a pair of variables x i and x j giving x i to one end of the edges in this direction and x j inthe other. The labels on the vertices are then squarefree monomials of degree n . See Figure5 for an example of 2-dimensional case and Figure 6 for the 3-dimensional labeled cube andthe first subdivision. The labelling of the cube can also be thought of assigning a pair tothe parallel hyperplanes that cut out the cube, with each of the hyperplanes associated toone of the varaibles in the pair. The label on a vertex is then monomial of all the variablesof the hyperplanes it sits in.Let us denote the pairing of variables with P that consists of P , P , . . . , P n where P i isthe set of indices of the pair assigned to the i -the edge direction. Then we can describe theset of labels on the n -cube as the following I P = { x i x i . . . x i n | i j ∈ P j } . Moreover this set is a square-free monomial ideal of S , and the n -cube supports the resolutionof S/I P .By subdividing the n -cube we mean an n -cube that consists of smaller cubes subdividingthe edges into p parts. Now the labeling is done with the same pairs as for the n -cube,but the labels moving along the directions are x pi , x p − i x j , . . . , x i x p − j , x pj . Alternatively,the subdivisions can be thought of cutting by ( p −
1) hyperplanes parallel to the pairs ofhyperplanes bounding to the cube. Then each of these hyperplanes is given a monomial x pi , x p − i x j , . . . , x i x p − j , x pj and the monomial on the vertex is again the monomial given by allthe variables of the hyperplanes where the vertex is. Denote the n cube with subdivision to p edge sections by C pn . Proposition 5.8.
Fix a pairing P on the edge directions. The cell complex C pn supports theresolution of S/I np where I P = { x i x i . . . x i n | i j ∈ P j } .Proof. Let us consider the cell complex ( C pn ) (cid:22) b for any b ∈ N n . With our chosen labelling,bounding a single coordinate in the exponent vector results in cutting the subdivided cube18 zu yzuyzvxzv ywuywvxwvxwu x z u xyz u xyz uvx z uv xyzwu xyzwuvx zwuvx zwu x z v xyz v x zwv xyzwv xyw v x w v x w uv xyw uvxyw v x w u xyw u y z u y z uvy z v y w v y zv y zwu y w u y w uvy zwuv Figure 6: A three dimensional cube and the first subdivision of the three dimensional cubelabeled with the square ideal rules in variables x, y, z, w, u, v .along one of the hyperplanes dividing it, unless all vertices are included or none. If thevector b has entries b i and b j for some pair ij such that b i + b j < p , then ( C pn ) (cid:22) b is an emptycomplex. Now we may assume that in b the entries for each pair are such that we do not getan empty complex. Then ( C pn ) (cid:22) b will still consist of cubes, and will be contractible. Thusit will be acyclic.Next we want to look at the maps between the powers of I P for a fixed P . In particular,we want the maps from (subdivided) cubes to (subdivided) cubes that maps m -cells to m -cells for all m . These maps correspond to embedding the C p − n to C pn , and to make thelabels well behaved for that the algebraic side of the map has to be a multiplication by oneof the monomials of C n . These maps are enough to give a covering of the cell complexes. Proposition 5.9.
Fix a dimension n , then C n covers C pn for p ≥ .Proof. We want to look at the cell maps coming from the multiplication by elements from C n . So we have 2 n multiplication maps between consecutive C p − n and C pn . On the cellcomplex side these correspond to embeddings. The first subdivision to C n consist of 2 n cubes.The first cube C n maps to every n -cube in C pn . This follows from every subdividing cubeis bounded by consecutive hyperplanes, i.e. ones that have varaibles x mi x m ′ j and x m − i x m ′ +1 j associated to them, thus each cube has labels that are multiple of the labels of C n by somemonomial. Hence there is a map taking C n to every n -cube in C pn and we have a covering.Denote the family of cellular resolutions coming from the resolutions supported on thelabeled n -cube by F C . As with the maximal ideals we want to restrict the morphisms suchthat for each chain map there is only one morphism. Proposition 5.10.
For a fixed n the family F C has finitely generated syzygies.Proof. The family F C has 2 n maps between consecutive powers, which are multiplicationsby the monomials in I P . Now all the other maps are compositions of these, so the familysatisfies the condition of being linear. Moreover since the Hom sets are finite, we can apply19he Proposition 3.12 to get that Rep S ( F C ) is noetherian. Now using the Proposition 5.9 wecan apply Theorem 4.8 to get the result. Next we want to look at ideals where the generators have the same degree.
Definition 5.11.
A monomial ideal is said to be equigenerated if all the generators havethe same degree.
The maximal ideals, cube ideals, and their powers are all examples of equigeneratedideals. We know the behaviour of these from the previous sections, so the focus now is onequigenerated ideals that are neither the maximal ideal nor cube ideal or a power of onethem.A useful observation one can make is that given an equigenerated ideal I in n variablesin degree d we have inclusion of I to m dn . If one takes the power of I , this new ideal willhave degree 2 d , and the possible maps between X dn and X dn are given by multiplicationscorresponding to the monomials in m dn . Another point is that we can give a new orientationon the cell complex to emphasize the parts we want. In the equigenerated case a naturalchoice is to consider the subcomplex in X dn coming from deleting all the vertices that arenot in the generators of the equigenrated ideal I . Call this subcomplex X dI . We can writethe resolution of S/ m dn as follows0 ← S d ←− F ⊕ F ′ ← · · · ← F i ⊕ F ′ i d i +1 ←−−− F i +1 ⊕ F ′ i +1 ← . . . where the generators of F i correspond to the cells in X dI and F ′ i has generators correspondingto the other cells in X dn . The map d i can be written as a block matrix (cid:20) A B C (cid:21) where A corresponds to the map of F i +1 to F i , B is the map from F ′ i +1 to F i , and C is the map from F ′ i +1 to F ′ i .First let us consider the equigenerated ideals that given by the bounds on the exponentvector. Each vertex label can be identified with a degree vector a = ( a , a , . . . , a n ) where a n is the degree of x n in the monomial. Let b ∈ N n , then a (cid:22) b if a i ≤ b i . Denote by I (cid:22) b the ideal of monomials bound by b in a fixed m dn . Proposition 5.12.
The cellular resolution of I m (cid:22) b is supported on the cell complex X mdn, (cid:22) mb ,and the cell complexes X kdn, (cid:22) kb have coverings for large enough k .Proof. From the Proposition 2.5 we know that the cell complex given by bounding X dn withthe given vector is an acyclic cell complex. Furthermore, if we bound X mdn, (cid:22) mb with anyvector a (cid:22) b , it is the same as bounding X dn , i.e. giving an acyclic cell complex, and if wechoose some c such that b (cid:22) c , then it gives us the whole complex X dn . Thus bounding X mdn, (cid:22) mb with any vector will give an acyclic cell complex and by Proposition 2.5 it supportsthe cellular resolution of the module given by the monomial labels of X mdn, (cid:22) mb .The monomials generating I m (cid:22) b appear as vertex labels in X mdn, (cid:22) mb , since any monomialin I m (cid:22) b has an exponent vector with entries at most m times the entries of monomials in I (cid:22) b . We want to show that there are no other vertices in the X mdn, (cid:22) mb . To do this weassume there is a vertex that has a monomial label which does not come from generatorsof I m (cid:22) b . The monomial label can be written as a product of m monomials of degree d . Wecan then write the exponent vector of this monomial as ( P mi =1 a i , P mi =1 a i , . . . , P mi =1 a in )where ( a i , a i , . . . , a in ) is the exponent vector of the i -th monomial a i . If this monomialis not a generator in I m (cid:22) b then at least one of the a i has to be outside of I (cid:22) b , that is thereexists at least one a ij > b j for some i and j . We can then focus on those j ’s that containentries > b j , and use k to denote the number of the monomials with j -th entry > b j . Thenwe have mb j ≥ m X i =1 a ij = m X i =0 ,a ij >b a ij + m X i =0 ,a ij ≤ b a ij ,
20e can divide by m since it is a positive integer and get b j ≥ P mi =0 ,a ij >b a ij m + P mi =0 ,a ij ≤ b a ij m . On the first sum we can apply the inequality P mi =1 λ m m ≥ Q mi =1 ( λ i ) /m and use that each ofthe terms is strictly greater than b i to get P mi =0 ,a ij >b a ij m ≥ m Y i =0 ,a ij >b ( a ij ) /k > k Y i =0 b /kj = b j . Then combining this with the earlier inequality we have b j ≥ b j + P mi =0 ,a ij ≤ b a ij m , Which is a contradiction and hence we cannot have any vertices with labels not coming from I m (cid:22) b in X mdn, (cid:22) mb . Moreover, since X mdn, (cid:22) mb satisfies the acyclicity conditions it supports thecellular resolution of I m (cid:22) b .Furthermore, by having all the vertex labels being products of monomials from I (cid:22) b implies that we cannot have cells outside X kdn, (cid:22) kb mapping into X mdn, (cid:22) mb with 1 ≤ k ≤ m − X tdn have covering for large enough t , thus X mdn, (cid:22) mb is alsocovered for large enough m .The cellular resolutions of I m (cid:22) b form a linear family with their structure of maps and bycombining Proposition 5.12 and Theorem 4.8 we get the following: Corollary 5.13.
The family of cellular resolutions of I m (cid:22) b has finitely generated syzygyrepresentation. We want to consider connected equigenerated ideal I such that they have a cellularresolutions supported on the cell complex coming from X dn by removing all vertices with alabel not in I and higher dimensional cells containing those. Denote this cell complex by X dI . Proposition 5.14. If I is an equigenerated ideal in n variables and degree d such that Ihas a cellular resolution supported on X dI and the powers of I are supported on X mdI m , thenthe family of cellular resolutions given by them has finitely generated syzygies.Proof. We will show that the syzygies are finitely generated using subfunctors of s d . Let( F iI ) d denote the d -th free module in the i -th power.Let F m denote the family of cellular resolutions of the powers of the maximal idealand let F i denote the resolution of the i -th power of the maximal ideal. First consider arepresentation of F m defined as followsΦ Id : F m → Mod S sending F i to ( F iI ) d . Now we have that Φ Id ( F i ) is a direct summand of s d ( F i ), so Φ Id is asubfunctor of s p .Moreover we can considered the representation φ Id : F m → Mod S sending F i to the d -thsyzygy module of the resolution of S/I i . Since the syzygy module is a submodule of the( F iI ) d we have that φ Id is a subfunctor of Φ Id , and so it is a subfunctor of s d as well. Thenby noetherianity we know that φ Id is finitely generated for any d as s d is finitely generatedfor any d . Thus we get that the family consisting of powers fo I has finitely generatedsyzygies. 21 234 1 234Figure 7: Graphs G and G of Example 6.2. In this section we focus on the cellular resolutions that are coming from edge ideals. Throughout this section let G denote a simple graph with n vertices and m edges unless otherwisespecified. If v and v are vertices then we denote the edge between them by v v if it exists.We denote the set of edges with E ( G ) and the set of vertices with V ( G ). Definition 6.1.
Let G be a simple graph with vertices numbered by v i , ≤ i ≤ n . The edge ideal associated to G is a monomial ideal in the polynomial ring S = k [ x , x , . . . , x n ] defined as I G = ( x i x j | v i v j ∈ E ( G )) . If we consider edge ideals coming from multiple graphs with numbers of vertices n , n , . . . , n r then we take the polynomial ring with max { n , n , . . . , n r } variables.The possible maps between cellular resolutions coming from edge ideals are particularlycut down by the condition that the first component of the chain map f has to be a 1 × S that has degree greater0. Thus we get that f has to be identity map and on the level of vertices the cellular mapis an embedding. Example 6.2.
Let S = k [ x, y, z, w ] be a graded polynomial ring. Consider the edge ideals I G = ( xy, yz, zw ) and I G = ( xy, xw, yz, zw ) coming from the graphs G and G in theFigure 7.The resolutions of S/I G and S/I G are S h xy yz zw i ←−−−−−−−−−−−−− S − z x − w y ←−−−−−−−−−− S ← S h xy xw yz zw i ←−−−−−−−−−−−−−−−−− S − w − z y − z x − w x y ←−−−−−−−−−−−−−−−−−−− z − wy − x ←−−−−−− S ← X and X respectively. Thesecell complexes are shown in the Figure 8. A possible cellular resolution map between thetwo is given by taking a cellular map g embedding the line into the square with ϕ g = id,and taking a chain map with f = id, f = , f = , and f = 0. This22 x x x x x x x x x x x x x Figure 8: Cell complexes supporting the resolutions of
S/I G and S/I G .then forms a pair of compatible maps that give us the cellular resolution morphism. This isthe only possible map between the two resolution (not counting the possible choices for thecellular map). One of the possible things to look at with edge ideals is of course their powers. In [8]Engstr¨om and Noren showed that the powers of edge ideals of paths have minimal cellularresolutions and gave an explicit description of a cellular resolution that is close to minimal.A n-path is a graph with n vertices and edges 12,23, . . . , ( n − n , and it is denoted with P n . We denote the edge ideal of path P n with I P n . A Newton polytope of a monomial ideal I is a polytope Newt( I ) given by the pan of the exponent vectors of the generators of I . Definition 6.3 ([8], Def 3.1) . Let
Newt( I dP n ) be the Newton polytope of the ideal I dP n , anddefine the hyperplanes H ′ i,j = y ∈ R n | ⌊ ( i − / ⌋ X k =0 y i − k = j and H i,j = { y ∈ R n | y i = j } for < i ≤ n and ≤ j . The the subdivision of Newt( I dP n ) by all H i,j is called Y dn and thesubdivision by all H i,j and H ′ i,j is called Z dn . Proposition 6.4 ([8], Thm 5.6 ) . The cell complexes Y dn and Z dn both support a cellularresolution of I dP n . Now we can study the powers of the edge ideals of chains as a family. One way to definethis family is to take cellular resolutions in our family are taken to be the ones supportedon Y dn and morphism are monomial multiplications, with ones between consecutive powersgiven by I P n . As with the families in earlier sections we aim to show that the syzygy functorfor this family is finitely generated. For this purpose we want to make use of the observationthat Z dn and Y dn are X dn with some cells deleted and cut by few extra hyperplanes. Definethe following cell complex that is a subdivision of Z dn . Definition 6.5.
Let Z dn be subdivision of Z dn given by the hyperplanes H i,j = y ∈ R n | ⌊ ( i − / ⌋ X k =0 y i − k = j , for < i ≤ n and ≤ j . Proposition 6.6.
The cell complex Z dn supports a cellular resolution of I dP n . roof. The subdivision Z dn adds no new vertices to Z dn . Moreover, the labels of the cellcomplex Z dn come from lattice points and bounding by some vector b will give us eitherconvex subcomplex or an empty subcomplex. Thus it will contractible and acyclic andgiving us that it supports the resolution by Proposition 2.5. Proposition 6.7.
For a fixed n the family of cellular resolutions of powers of path ideals P n has finitely generated syzygies.Proof. The cell complexes are refinement of the cell complex we encountered for the maximalideals. We know these are covered eventually, thus we get covering for the complexes sincewe have only four cellular maps that behave the same to the cellular maps with maximalideals.The powers form a linear family which follows from the possible multiplication maps.Then we have noetherianity of the representations and get that the syzygy functor is finitelygenerated as we have covering.
In this section we want to look at the powers of the edge ideals of the complete graph K n on n vertices. We want to look at the resolutions of the powers of edge ideals of completegraphs.The complete graph on n vertices is a graph that contains an edge between any twovertices. This then implies that the edge ideal of the graph K n , denoted by I K n , is given byall pairs x i x j with 1 ≤ i < j ≤ n . These monomials are precisely all the degree 2 monomialsof m n that are bounded by the vector (1 , , . . . , Proposition 6.8.
Let F be family of cellular resolutions coming from the powers of I K n ,then the syzygies of the family are finitely generated.Proof. This follows directly from the Corollary 5.13 by setting b = (1 , , . . . ,
1) and d =2. In this section we want to look at results relating to the edge ideal of the Booth-Luekergraph. In [7] the formulas for the Betti numbers and other invariants of the resolution wereestablished in terms of the graph’s number of edges and vertices. The cellularity of theresolution was not studied in this paper. First we recall the Booth-Lueker edge ideal from[7] and some definitions from [4].For this section let G be a simple graph and let I G be the edge ideal of this graph. Definition 6.9.
For any graph G let BL(G) be the graph with vertex set V ( G ) ∪ E ( G ) andedges uv for every pair of vertices in G and ue for every vertex u incident to an edge e inG. We call BL(G) the BoothLueker graph of G. Let us denote the edge ideal of the Booth-Lueker graph with BL ( I G ). Definition 6.10 ([4], Def 2.1) . Let I be a monomial ideal. The ideal I has linear quo-tients if there is an ordering of the generators ( m , m , . . . , m k ) such that the colon ideal ( m , m , . . . , m j − ) : m j is generated by some subset of variables for each j . Definition 6.11 ([4], Def 2.2) . Let I be a monomial with linear quotients for some order ( m , m , . . . , m k ) . The set of a generator is defined to be set( m j ) = { k ∈ [ n ] | x k ∈
Theorem 6.13 ([4], Thm 3.10) . Suppose that ideal I has linear quotients with respect tosome ordering ( m , . . . , m k ) of the generators, and suppose that I has a regular decompositionfunction. Then the minimal resolution of I obtained as an iterated mapping cone is a cellularand supported on a regular CW-complex. Before applying the above definitions and theorems we make some observations aboutthe Booth-Lueker ideals.
Proposition 6.14. If I is an edge ideal of a simple graph, then the ideal BL ( I ) of theBooth-Lueker graph has linear quotients and a regular decomposition function.Proof. Let I be an edge ideal of a graph with n vertices. The Booth-Lueker ideal of I isgiven by BL ( I ) = ( x x , x x , . . . , x , x n , x x , . . . , x x n , . . . , x n − x n , { x i y k , x j y k | ij is the k-th edge } ) . First we want to show that the ideal has linear quotients for the ordering given above. Notethat we have chosen our ordering such that for a monomial x i x j we have i < j . If weconsider the colon ideal of the form < x x , . . . , x i x j − > : x i x j we can compute it to getthat it has generators x k with k < i or i < k < j . Thus until the y variables the colon idealsatisfies the condition of being generated by variables.Let us look at the colon ideals of the form < x x , . . . , x n − x n , x i y , x j y , . . . , x i t y t > : x j t y t , again one can compute the generators of it and these are x k with k = j t and y i correspondingto an edge with x i t with i < t . Finally the last type of colon ideal we can have is < x x , . . . , x n − x n , x i y , x j y , . . . , x i t y t , x j t y t > : x i t +1 y t +1 and as before computing the generators gives x k with k = i t +1 and y i corresponding to anedge with x i t +1 with i < t + 1. Thus we get that BL ( I ) has linear quotients with respect toto the order given above.Using the order given in the defintion of linearisation, the sets of the generators can beexplicitly computed to obtainset( x i x j ) = { , . . . , i − , i + 1 , . . . , j − } set( x i y k ) = { , . . . , i − , i + 1 , . . . , n } ∪ { n + t | x i y t is a generator of BL ( I ) } Note that if t ∈ set( x i x j ) then t < j . Next we want to check the regularity of the de-composition function. For the generators of the form x i x j we get the following from themultiplication by variables given by the set( x i x j ):set( b ( x t x i x j )) = set( x i x t ) = (cid:26) { , , . . . , t − , t + 1 , . . . , i − } if t < i { , , . . . , i − , i + 1 , . . . , t − } if i > t . It is clear that both of the sets above are contained in set( x i x j ) = { , . . . , i − , i +1 , . . . , j − } .Next let us consider the regularity for the generators of the form x i y k . In this notation x n + j = y j . We can divide the computation of set( b ( x t x i y k )) to two cases, firstly when x i x t ,for any t > n , is not a generator, and secondly when x i x t , for at least one t > n , is agenerator. In the first case the computation gives us the same result as with x i x j . In thesecond case we further subdivide to t < n and t > n .25 < n When t < n we have set( b ( x t x i y k )) = set( x t x i ) which is { , , . . . , t − , t + 1 , . . . , i − } or { , , . . . , i − , i + 1 , . . . , t − } . Both are contained in the set { , . . . , i − , i +1 , . . . , n } ∪ { n + t | x i y t is a generator of BL ( I ) } . t > n When t > n we have set( b ( x t x i y k )) = set( x t x i ) = { , . . . , i − , i + 1 , . . . , n } ∪{ n < r < t | x i x r is a generator of BL ( I ) } . The first part is clearly contained inset( x i y k ), further more we have that { n < r < t | x i x r is a generator of BL ( I ) } ⊆{ n + t | x i y t is a generator of BL ( I ) } since x t must be bofore y k in the ordering of thevariables.Thus we get that the decomposition function satisfies the definition of regularity. Corollary 6.15.
Every ideal of a Booth-Lueker graph has a minimal cellular resolutioncoming from the mapping cone resolution construction.Proof.
By Proposition 6.14 the edge ideal BL ( I G ) of a Booth-Lueker graph has linear quo-tients and a regular decomposition function. Thus we can apply the Theorem 6.13 to BL ( I G )and get that BL ( I G ) has a cellular resolution made with mapping cones.Let us consider the functor from a family of edge ideal resolutions to Booth-Lueker idealresolutions. Given an edge ideal, we can define the Booth-Lueker ideal, and this gives us away to send a cellular resolution of an edge ideal to a cellular resolution to a correspondingBooth-Lueker edge ideal. The morphisms on edge ideals then give well defined morphisms onthe Booth-Lueker ones. For the rest of this section we will denote the variable y k belongingto the k -th edge between x i and x j by y ij Let F I and F J be two cellular resolutions that belong to edge ideals I and J , andsuppose that we have a map between F I and F J . From our earlier observations this maphas to be an embedding. Let us consider the resolutions for the Booth-Lueker ideals BL ( I )and BL ( J ). The generators of I and J are contained in these ideals and moreover we havean embedding of the generators of BL ( I ) to BL ( J ). The cell complexes supporting Booth-Lueker ideal resolutions coming from mapping cones are formed of a piece that correspondsto the complete graph. This can be taken to be minimal or the non-minimal one containedin the cell complex supporting a maximal ideal, and n − y ij . Now we may assume that the ordering on the ideals BL ( I ) and BL ( J ) is such that any monomials from vertices in BL ( J ) that are not in BL ( I )are ordered last. Then we have an embedding of the cell complex supporting the resolutionof BL ( I ) to the cell complex supporting a resolution of BL ( J ) if the corresponding graphshave the same number of vertices. If the underlying graphs have a different number ofvertices, we can still embed the cell complex supporting BL ( I ) to BL ( J ) in the mappingcone construction gives subcomplex at each step and they are glued on in the same wayfor both BL ( I ) and BL ( J ), assuming one is consistent with the choices in building the cellcomplex.Let CellRes E ( n ) denote the category of cellular resolutions coming from edge ideals ofgraphs with at most n vertices, and let CellRes E ( n + n ( n − ) denote the category of cellularresolutions of coming from edge ideals of graphs with at most n + n ( n − vertices. If wetake an edge ideal and go to the Booth-Lueker ideal of it, the resolutions will move fromCellRes E ( n ) to CellRes E ( n + n ( n − ). The resolutions in CellRes E ( n ) are defined over thepolynomial ring with n variables, and we will use the variables x , . . . , x n for this polynomialring. When we use these categories in the context of having a functor the variables in thepolynomial ring for CellRes E ( n + n ( n − ) are denoted by x , . . . , x n , y , y , . . . , y n , y , . . . , y ( n − n . Definition 6.16.
Let
CellRes E ( n ) denote the category of cellular resolutions coming fromedge ideals of graphs with at most n vertices and m edges in k [ x , . . . , x n ] . Then define thefunctor BL : CellRes E ( n ) → CellRes E ( n + n ( n − y sending cellular resolution F G to a minimal resolution of Booth-Lueker edge ideal of GF BL ( G ) . The functor BL takes an embedding of cellular resolutions to an embedding on theBooth-Lueker resolutions. This functor can be restricted to families of cellular resolutions in which case it takesa family over with individual resolutions belonging to CellRes E ( n ) to a family where theresolutions are in CellRes E ( n + n ( n − ).Recall that in Section 2.2 we defined a property for functors called property (F). Nextwe show that in this setting of edge ideals the BL functor satisfies this property. Proposition 6.17.
The restriction of the functor BL between families satisfies the property(F).Proof. Let F denote a family of cellular resolutions coming from edge ideals and let BL( F )denote the family of cellular resolutions obtained by taking each resolution F I in F to aminimal cellular resolution of the associated Booth-Lueker ideal F BL ( I ) .Let F BL ( I ) be an arbitrary cellular resolution in BL( F ). We know that there is an ideal I whose Booth-Lueker ideal BL ( I ) is, and we can choose the resolution F I as our finiteset of elements in F . Then we get a morphism F BL ( I ) → F BL ( I ) that is just the identity,denoted by id. Now pick any F J in F , then the morphism from F BL ( I ) → BL( F J ) is eitheran embedding or there is no morphism. Let us assume we have an embedding and call it g . Then picking the embedding f between F I and F J gives that BL( F ) is an embedding of F BL ( I ) to BL( F J ). Since we only have one possible morphism between two different cellularresolutions coming from different edge ideals, we have that g = BL( F ) = BL( f ) ◦ id.This means we can pullback information on the representation from BL to original graphs.This allows one transfer to finiteness results from the Booth-Lueker ideals to other edgeideals, if there are some. However in this setting of restricting the number of vertices on thegraph one necessarily has only finitely many possible edge ideals that we can get and hencesyzygies by default are finitely generated.We would like to consider sequences of cellular resolutions coming from graphs where wehave not restricted the number of vertices, which means we would have to work over thepolynomial ring with infinitely many variables. In practice we can consider the family whereeach cellular resolution has its own polynomial ring in suitably many variables. In this section we want to consider families of cellular resolutions that do not have restrictionson the number of variables appearing in the polynomial ring. By our previous definitionof family of cellular resolutions this means we would have to work over the polynomialring with infinitely many variables. However, each individual cellular resolution we consideris still assumed to only have finitely many variables in the defining ideal, that is it livesin a polynomial ring with finitely many variables. This observation allows us to considerthe unrestricted family as something consisting of cellular resolutions each over their ownpolynomial ring.For this section we assume all our polynomial rings are over the same base field k . Proposition 7.1. If F is a cellular resolutions over the polynomial ring S = k [ x , . . . , x n ] then the change of variables to the ring S ′ = k [ y , . . . , y m ] , with m ≥ n , gives also a cellularresolution.Proof. Relabelling the variables does not change the relations between the monomials, hencethe cell complex supports a cellular resolution after the change of the ring.
Remark . There is often more than one way of changing the variables to the bigger ring,however not all of the possible changes work with the cellular resolution maps that we wantto have. 27here are multiple options how to change the variables to go from one ring to another.However our aim is to study families of cellular resolutions in this setting so we are onlyinterest in those changes of variable that are compatible with a cellular resolution map. Toillustrate this we have the following example.
Example 7.3.
Let S = k [ x, y, z, w ] and S = k [ x, y, z, w, t ] be two polynomial rings. Letus consider the cellular resolutions for the ideals I P and I P . From Section 6.1.1 we knowthat they both have a minimal cellular resolution. The minimal resolution for I P is theresolution of G from Example 6.2 considered over the ring S . This resolution is supportedon a cell complex formed on three vertices and two edges. We can compute a minimalresolution for I P over the ring S : S h xy yz zw wt i ←−−−−−−−−−−−−−−−−− S − z − wt x − w
00 0 y − t xy z ←−−−−−−−−−−−−−−−−−−− wt − zxxy ←−−−−−− S ← . Changing the ring for the resolution of I P from S to S has multiple ways to do it if weallow all possible changes and permutations. We want to be able to map the resolution tothat of I P , and so we only want those changes that give us a subset of the generators of I P .In practise the possible subsets that can give a map between the resolutions are xy, yz, zw and yz, zw, wt . The changes of variables that give these are x x, y y, z z, w w and x y, y z, z w, w t and x t, y w, z z, w y and x w, y z, z y, w x , and they give maps that are embeddings between the cellular resolutions withpossibly a change of orientation on the cells.The example above then motivates the following definition for a family of cellular reso-lution with no restriction on the polynomial ring. Definition 7.4.
Let F be family of cellular resolutions such that each resolution F i is overa polynomial ring S i . We call such family the unrestricted family of cellular resolutions .The unrestricted family forms a category with the objects being the individual resolu-tions and morphisms are compositions of a change of a ring map and a cellular resolutionmorphism. We can lift many definitions from the earlier sections to the unrestricted family setting.This includes the definition of covering in Definition 4.4. It still holds since the cell complexesdo not change and morphisms behave like the maps between cell complexes in the ordinaryfamily case. Furthermore Definition 3.4 of a linear family can also be applied directly to theunrestricted family.Next we want to consider the representations of the unrestricted family. Let S ∞ denotethe polynomial ring with infinitely many variables. We want to consider representations tothe modules over this infinite ring. Definition 7.5.
A representation of an unrestricted family F is a functor M : F →
Mod S ∞ . Then we can define the following particular representations.
Definition 7.6.
The t -th module representation s ∞ t : F →
Mod S ∞ such that s ∞ t ( F i ) is the free module over S ∞ with generators in the same degrees as the t -thfree module in the resolution, and s ∞ t ( F i → F j ) is the matrix of the chain map from F i to F j on the t -th component. x x x x x x x x Figure 9: Labelled cell complexes for the family in Section 7.1.
Definition 7.7.
Let F be an unrestricted family of cellular resolutions. The t -th syzygyfunctor σ ∞ t : F →
Mod S ∞ is defined by taking F ∈ F to the finitely generated submodule of a free S ∞ -module s ∞ t ( F i ) with the same generators as the t -th syzygy module of F , and the morphisms are restrictionsof the free module maps to the submodule. Note that σ ∞ t is a subfunctor of s ∞ t .Making use of the fact that the unrestricted family behaves similar to the family overa single polynomial ring we get analogues of the results from the previous sections in thesetting of unrestricted families. Proposition 7.8. If F is an unrestricted family of cellular resolutions such that it is linearthen the representations Rep S ∞ ( F ) form a noetherian category.Proof. Suppose that F is an unrestricted family of cellular resolutions such that it is linear.Then the morphisms in this family behave in the same way as for the linear family overa single polynomial ring. Thus we have that the principal projectives are all noetherianfollowing the proof of Proposition 3.12. Then it follows from the principal projectives beingnoetherian that the representation category Rep S ∞ ( F ) is noetherian. Proposition 7.9. If F is an unrestricted family of cellular resolutions such that it is linearand the cell complexes have covering in dimension t for all i large enough, then the syzygyfunctor σ ∞ t is finitely generated for t .Proof. The generators of the modules in s ∞ t ( F ) correspond to the cells in the cell complexsupporting the resolution of F . Therefore we can use the same argument as in the proofof Lemma 4.8. Assume we have t -covering for some large enough i . Then we have thatevery cell complex above i in the family is covered by some finite set of cell complexes.That means there is a finite set of t -cells that cover all other t -cells. This implies that onthe level of free modules, every generator in the t -th modules is reachable from a finite setof generators since the chosen maps correspond to the cellular maps. Then by definitionthe representation s ∞ t is finitely generated for t . Moreover by Proposition 7.8 the linearityimplies that representations are noetherian, in particular this means any subrepresentationof finitely generated representation is finitely generated, so we have that σ ∞ t is finitelygenerated. In this section we want to consider an explicit example of an unrestricted family of cellularresolutions. Let us take the n -simplex with variable labels where we add a new one vari-able at each dimension. Each simplex supports a cellular resolution but over a differentring. This will give us a family of minimal cellular resolutions F and denote the resolutioncorresponding to the n -simplex by F n . This family of cell complexes is shown in Figure 9.29or the morphisms in the family we can take all possible maps, note that between F n and F n +1 there are ( n + 1)! maps. However if we are only interested in which cells map towhich cell, then it is enough to consider only the maps that come from choosing n variablesfrom n + 1 variables. Proposition 7.10.
For a fixed t the representation σ ∞ t of F is finitely generated.Proof. Let F be the family of cellular resolutions supported on the simplices. Then any mapbetween some resolutions F i and F j can be written as composition of consecutive maps, andwe have map between any two of the cellular resolutions. Therefore the family F is a linearfamily and we get that the representation category is noetherian.Next we want to show that for a fixed dimension t we have covering of t dimensional cells.For this we will only consider the maps that give different set of variables after the changeof ring. Then we will have n + 1 maps between the resolutions F n and F n +1 . Let X n denotethe n -simplex. The resolution maps we are considering correspond to the embeddings of X n to X n +1 . There are n + 1 possible embeddigns between X n and X n +1 , which also cover all n -dimensional cells. However, none of the embeddings cover the ( n + 1)-dimensional cell.Thus if we fix a dimension t , taking all cell complexes up X t will then give a covering of the t -dimensional cells with the embeddings. Thus for the family F we have that for a fixed t we have t -covering.Hence by Proposition 7.9 we get that the representation σ ∞ t is finitely generated for afixed t . Finally we list few open questions that arise from the previous sections.In all of our examples the families of cellular resolutions have been linear, so a naturalquestion would be to ask what about non-linear families. Can we find non-linear familiesthat have finitely generated syzygies or satisfy other properties like noetherian representationcategory? Another observation in all our examples is that we used noetherianity to provefinite generation of syzygies, and often the noetherianity of the representation categoryis inherited from the nice structure the family has which in first place suggested finitegeneration of syzygies. One can then ask whether there exists a family of cellular resolutionsthat has finitely generated syzygies but without a noetherian representation category?In this paper we focused on the syzygies of the families of cellular resolutions. Thus onecan ask if the representations can be used to study other properties than syzygies for thefamilies. We also note that the Gr¨obner property was used to study the families and thiscould be an interesting direction to look at. Moreover, the paper of Sam and Snowden [13]contains other structures, like lingual structures, that have not been addressed in this paper.One possible question is do the lingual structures have particular meaning or applicationwith cellular resolutions and can we find families that satisfy the conditions to have thesestructures.As a last open question we pose further work on the unrestricted case of families. Theapproach we have proposed could be considered the naive way to deal with the requirementof having cellular resolutions over different rings, and we have not dwelt very deeply into it.The theory of modules over polynomial ring with infinitely many variables could offer toolsto work further with the proposed setting and also make use of different representations forthese families. Another direction to take with these are the cases where there is polynomialring with a maximal number of variables, in which case the modules can be taken over that.This mixed finite case also allows different permutations of variables within the same ring,which are not morphisms of cellular resolutions in the fixed ring case.30 eferences [1] Thomas Church and Benson Farb. Representation theory and homological stability.
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