The Graph Minor Theorem Meets Algebra - Eric Ramos

The Graph Minor TheoremMeets Algebra

Eric RamosDepartment of Mathematics, Bowdoin College,Brunswick, ME 04011

1 The Graph Minor Theorem

The Graph Minor Theorem of Robertson and Sey-mour is one of the most celebrated results in the his-tory of combinatorics, spanning decades (and hun-dreds of pages) of work. In this note we discuss re-cent work of Miyata, Proudfoot, and the author thatproposes a framework that would allow one to ap-ply the Graph Minor Theorem to algebra and topol-ogy, building on seminal contributions by Sam andSnowden. Though the capstone of this framework isstill only conjectural (See Conjecture 2.3), a weakerversion (See Theorem 2.4) of the capstone has beenproven and already has far ranging consequences intopological combinatorics and algebra. Specifically,we will discuss applications of this framework tothe study of matching complexes and configurationspaces of graphs.

To begin, let’s standardize terminology by defininga graph to be a (non-empty) finite, connected, atmost one-dimensional CW-complex. If one prefers tothink about graphs G in terms of collections of edges,E(G), and vertices, V (G), then our provided defini-tion essentially amounts to saying that our graphswill always be finite, connected, and may have multi-edges and loops. The theory of graphs, having essen-tially begun in work of Euler, has developed into oneof the most foundational subjects in all of mathemat-ics, being pivotal in numerous fields such as combi-natorics, algebra, and many others. For the purposesof this note, we will focus specifically on the theoryof graph minors.

Definition 1.1. Let G be a graph, and let e be anedge that is not a loop. Then the contraction of e isthe (necessarily homotopy equivalent) graph obtainedfrom G by crushing e to a point. If e is an edge (that

may be a loop) whose removal does not disconnectG, then the deletion of e is the graph obtained fromG by removing the edge e without removing its endpoints.

Given two graphs G,G′, we say that G is a mi-nor of G′ if G is isomorphic to a graph that canbe obtained from G′ by a sequence of edge deletionsand contractions. The minor relation imposes a par-tial order on the collection of graphs, and we writeG ≤ G′.

One of the early triumphs in the study of graphminors, which was also one of the great accomplish-ments of early topological graph theory, is the follow-ing theorem, independently discovered by Kuratowskiand Wagner. By definition a graph is planar if it canbe embedded in the plane.

Theorem 1.2 (Kuratowski and Wagner). Let G bea graph. Then G is planar if and only if it admitsneither the complete graph on 5 vertices K5, nor thecomplete bipartite graph K3,3, as a minor.

Important here is that not only does there exist acompletely classifiable collection of so-called “forbid-den” minors, but that this collection is finite. Whileone might expect that this finiteness is a consequenceof the rigidity of the plane, in fact it is the result ofsomething far more general.

Theorem 1.3 (Robertson and Seymour, [RS04]).Let S be any collection of graphs. Then there exists afinite collection of graphs in S that are minimal withrespect to the minor order (restricted to S). Equiva-lently, if S is any collection of graphs which is closedunder taking minors, then there exists some finite col-lection of graphs {Gi} such that for any graph G, Gis in S if and only if G does not admit any of thegraphs Gi as a minor.

The conclusion of the Graph Minor Theorem isoften summarized as saying that the graph minorrelation is a well-quasi-order. The Kuratowski–Wagner theorem tells us that if S is the (minorclosed) class of planar graphs, then the finite col-lection of forbidden minors is precisely {K5,K3,3}.Obviously, however, the graph minor theorem is far

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more powerful than Kuratowski–Wagner, as it guar-antees such a finite forbidden minor classificationmust exist for any minor closed property. It shouldbe noted however, that the neither graph minor the-orem nor its proof provide any way to actually deter-mine the finite collection of forbidden minors for anygiven minor property! While there are some circum-stances where such an explicit characterization hasbeen accomplished, the vast majority remain out ofour reach. Famously, there is a collection of 17,523graphs which are known to be forbidden minors fortoroidal graphs, i.e. graphs that can be embeddedinto the torus, though it is unknown whether this listis exhaustive!

We take the time here to also point out that theGraph Minor Theorem generalizes a long list of wellknown well-quasi-order theorems. These include (inincreasing order of generality) Dickson’s Lemma– that the coordinate-wise poset on Nr is a well-quasi-order – Higman’s Lemma – that the posetof words on a well-quasi-ordered poset is itself well-quasi-ordered – and Kruskal’s Tree Theorem –that the poset of rooted trees is well-quasi-ordered.

2 The Categorical Graph MinorTheorem

Moving on from the classical combinatorics of theGraph Minor Theorem, we would now like to movethe reader in the direction of its categorification. Cat-egorification is not something that has a completelyrigorous definition, but rather something you justkind of “know” when you see it. It can oftentimesbe summarized by the following statement: all non-negative integers, regardless of the counting problemthat spawned them, are secretly the dimensions ofsome vector spaces, whose algebra encodes and ex-pands upon the originating combinatorics. Of course,our ultimate goal is to apply a kind of Graph MinorTheorem to problems arising from algebra and topol-ogy, and so the most natural first step in this processis to take the combinatorial content of the Graph Mi-nor Theorem and expand it into the realms of algebra.

Under this somewhat vague guidance one often

finds that statements like the Graph Minor Theorem– that a given poset is a well-quasi-order – translateto a Noetherianity statement in the algebraic contextof the categorification. Our next major goal will beto make all of this more precise by first introducingthe graph minor category G. Before we get into themost technical details of this construction, we presentan example from topological combinatorics that willhopefully motivate why one would want a kind of“categorical” Graph Minor Theorem.

If G is a graph, we define the matching complexMG to be the simplicial complex whose i-simpliciesare collections of edges of G, {e0, . . . , ei}, with nooverlapping endpoints, i.e. matchings of size i. Thehomology groups of these spaces have been of consid-erable interest in topological combinatorics for manyyears [Wac03, Jon10]. This is especially true of thecases whereG = Kn is a complete graph orG = Kn,m

is a complete bipartite graph.

Now if one knows G ≤ G′, as well as the data ofwhich edges ofG′ were deleted or contracted to obtainG, one can see that the edges of G naturally includeinto those of G′ in such a way that edges which werenon-adjacent inGmust also be non-adjacent inG′. Inparticular, if {e0, . . . , ei} is an i-simplex ofMG, thenone has a naturally associated i-simplex ofMG′ . Thisassociation then induces a map between the abeliangroups

Hi(MG)→ Hi(MG′).

We have therefore now found ourselves in a situationwhere for each graph G we have a finitely generatedabelian group Hi(MG), with the additional structurethat wheneverG is a minor of another graphG′, thereis a natural homomorphism,

Hi(MG)→ Hi(MG′).

In a situation such as this, one would hope that a cat-egorical version of the Graph Minor Theorem wouldimply a kind of finite generation for the entirety ofHi(M•). In other words, it would imply the existenceof a finite collection of graphs {G1, . . . , GN} such thatfor any graph G the group Hi(MG) is spanned by theimages of the groups Hi(MGj ), for all j such thatGj ≤ G. In other words, the algebraic content of

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all possible i-th homology groups Hi(MG) is entirelydetermined by a finite amount of information. Suchextreme finiteness would imply a kind of universalityin the presentations of these groups, which in turnwould imply, as one particular example, a uniformityin the kinds of torsion that can possible appear. Wewill return to the example of the matching complexin later sections.

We are now ready to give a few formal definitions.

Definition 2.1. The Graph Minor Category Gis the category whose objects are graphs, and whosemorphisms are minor morphisms. A minor mor-phism ϕ : G′ → G is a map of sets,

ϕ : V (G′) t E(G′) t {?} → V (G) t E(G) t {?}

satisfying the following conditions:

• ϕ(V (G′)) = V (G) and ϕ(?) = ?;

• if e ∈ E(G′) has endpoints {a, b}, and ϕ(e) 6= ?,then either ϕ(e) = ϕ(a) = ϕ(b) is a vertex of G,or ϕ(e) is an edge of G with endpoints ϕ(a) andϕ(b);

• ϕ maps ϕ−1(E(G)) bijectively onto E(G);

• for any vertex v ∈ V (G), the preimage ϕ−1(v),thought of as a subgraph of G′, is a tree.

The edges of G′ that ϕ maps to the character ? aresaid to be deleted by ϕ, whereas the edges that ϕmaps to a vertex of G are said to be contracted byϕ.

Let us take a moment now to explain more thor-oughly the four conditions of a minor morphism. Thefirst condition states that the morphism only sendsvertices to other vertices (i.e. not to edges), and thatit does so surjectively. This condition also assertsthat the “deletion character” ? must map to itself.The second condition asserts that for any given edgee, with endpoints {a, b}, one of three things musthappen: either the edge is deleted (i.e. mapped tothe deletion character), it is contracted to the vertexϕ(e) in which case a and b must also be mapped toϕ(e), or it is mapped to a new edge whose endpointsmust be the images of the endpoints of a and b. The

third condition states that the edges of G′ that areneither deleted nor contracted can be uniquely identi-fied with edges of G. Note that we will usually thinkabout this condition in the opposite way, that if

ϕ : G′ → G

is a minor morphism, then the edges of G can befound living inside of G′. Finally, the last conditionamounts to saying that minor morphisms are onlyallowed to contract trees within G′, i.e. cycles maynot be contracted. The primary content one shouldtakeaway from this description is that

There exists a morphism ϕ : G′ → G ⇐⇒ G ≤ G′.

One should also note that the category G is not sim-ply the opposite category of the graph minor poset.Indeed, the minor morphisms also encode informationabout which edges are being deleted and contracted,as well as possible movement of the vertices via graphautomorphisms, i.e. permutations of the vertex setthat preserve the adjacency relation.

Definition 2.2. A Gop-module is a covariant func-tor M from Gop to the category of finitely generatedabelian groups. Concretely, a Gop-module is a collec-tion of abelian groups {M(G)}, one for each graphG, such that for every minor morphism ϕ : G′ → G

(equivalently for every realization of G as a minorof G′), there is a homomorphism M(G) → M(G′),defined in such a way so-as to respect compositionof minor morphisms. We say that a Gop-module Mis finitely generated if there is a finite collectionof graphs {Gj} such that for any graph G, M(G) isspanned by the images of the groups M(Gj) inducedby all possible minor morphisms G → Gj . We of-ten refer to the graphs {Gj} as generators of themodule.

Note that we have changed from G to Gop pre-cisely because we want our morphisms to go in thesame direction as the minor relation. Let’s take aquick moment to look at some simple examples ofGop-modules.

• The Trivial Module: For each graph G we setM(G) = Z, whereas for every minor morphism

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we assign the identity map. This module is gen-erated by the graph with a single vertex and noedges.

• The Edge Module: For each graph G we setM(G) = ZE(G), the free abelian group on theedges of G. We have already discussed that anyminor morphism G′ → G induces an inclusionE(G) ↪→ E(G′), and we use this inclusion todefine the map M(G)→M(G′). This module isgenerated by the line segment and the loop.

• The Spanning Tree Module: For each graphG we set M(G) to be the free abelian group onthe spanning trees of G. Any minor morphismG′ → G can be used to map a spanning treeof G to one of G′ by including the edges of thistree into G′ while also adding in the edges of G′

that we contracted by the minor morphism. Thismodule is once again generated by a single point,as a minor morphism to a point is equivalent toa choice of spanning tree.

• The Homology of the Matching Complex:For any fixed i ≥ 0, the collection of groups{Hi(MG)}G form a Gop-module, as outlinedabove. It is conjectured that this module isfinitely generated, though the justification forthis is far from obvious! We will return to thisexample in later sections.

As the usage of “module” suggests in the name Gop-module, virtually any intuition or construction thatone has from the classical theory of rings and moduleswill carry over into this context. In particular, termssuch as kernel, cokernel, and submodule continueto have meaning here using the most natural possibledefinitions. Moreover, the condition of being finitelygenerated has implications that go beyond the mostobvious. For instance,

• [MPR20] Uniform Boundedness of Torsion:If M is finitely generated, then there exists aninteger dM such that for any graph G, the ex-ponent (i.e. largest non-trivial torsion) of thegroup M(G) divides dM .

• [MR20] Uniform Boundedness of Rank: IfM is finitely generated, then there exists a poly-nomial PM (x, y) ∈ Q[x, y] such that, for anygraph G, the rank of the group M(G) is atmost PM (|E(G)|, |V (G)|) · τ(G), where τ(G) isthe number of spanning trees in G.

Note that the Spanning Tree Module illustratesthat the bound given in the second point is actuallysharp.

As suggested earlier, a proper categorification ofthe Graph Minor Theorem should have something tosay about Noetherianity of some algebra. This is in-deed the case.

Conjecture 2.3 (The Categorical Graph Minor The-orem). LetM denote a finitely generated Gop-module.Then all submodules of M are finitely generated.

A proof of Conjecture 2.3 was originally claimed in[MPR20], though a gap was discovered in the proofwhich, as of the writing of the present article, has notyet been filled. To see how this conjecture relates withthe Graph Minor Theorem, let S be any minor-closedset of graphs, and consider the following example. LetM denote the Gop-submodule of the trivial module forwhich

M(G) =

0 if G ∈ S,

Z otherwise.

By definitionM(G) is a submodule of the trivial mod-ule, and therefore M must be finitely generated, pro-vided that Conjecture 2.3 is true. That is, there issome finite collection of graphs {Gj}, for which thegroups M(Gj) contain all algebraic content appear-ing in M . Thus, the containment problem for S (i.e.whether or not M(G) is zero) is determined entirelyby whether or not you have one of the Gj as a minor.This is precisely the Graph Minor Theorem!

Just as the Graph Minor Theorem generalizes awide variety of classical well-quasi-order theorems,one can see that the Categorical Graph Minor Theo-rem generalizes or implies many Noetherianity state-ments. See [SS17] for an overview of such state-ments. Also see [Sno13] for the Noetherianity state-ment associated to Higman’s lemma, and [Bar15] forthe Noetherianity associated to Kruskal’s Tree Theo-

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rem. This last example is especially relevant for whatwe will now call the Weak Categorical Graph MinorTheorem, whose proof follows from the work of [PR],which built on the work of [Bar15].

For a given graph G, its combinatorial genus isdefined as the quantity |E| − |V | + 1. For instance,having combinatorial genus equal to 0 is equivalentto being a tree.

Theorem 2.4 (The Weak Categorical Graph MinorTheorem). Let g ≥ 0 be an integer, and let Gg be thefull subcategory of G whose objects are graphs withcombinatorial genus at most g. If M is a finitelygenerated Gopg -module, all of its submodules are alsofinitely generated.

The theorems of the next section will all be statedin terms of the subcategory Gg, however all of themcould be extended the the whole of G provided thatConjecture 2.3 were verified. One should also notethat this combinatorial genus stratification of thegraph minor category is not the only one that onemight try to apply. It would be interesting to seewhether the subcategories of bounded tree-width orother well known “minor-monotone” graph invari-ants have representations with similar Noetherianitystatements.

3 Applications

In this section we detail some applications of the Cat-egorical Graph Minor Theorem to problems arisingfrom topology. To start, let’s return to the setup fromlast chapter related with the matching complex. Re-call that for a graph G, the matching complexMG isthe simplicial complex whose i-simplices are match-ings of i+ 1 edges of G. We showed last chapter thatfor any fixed i ≥ 0 the assignment G 7→ Hi(MG) isa well-defined Gop-module, and therefore also a Gopg -module for all g. We will now show that this moduleis also finitely generated as a Gopg -module.

To begin, let Ei denote the Gopg -module for whichEi(G) is the free abelian group on collections of i+ 1

edges ofG. For instance, E0 is precisely the restrictionof the Edge Module to Gopg . We see that Ei is finitelygenerated by, for instance, the set of all graphs with

i+ 1 edges. Moreover, the simplicial i-chains ofMG

are easily seen to be a submodule of Ei, and thereforemust also be finitely generated. Taking it one stepfurther the Gopg -module Hi(MG) is a subquotient ofthe simplicial i-chains, and must also be finitely gen-erated. This concludes the proof. Note this exactproof would also prove finite generated for the fullGop-module, provided Conjecture 2.3.

The above proof is an extremely common and ef-fective means for proving that a given Gopg -module isfinitely generated: find some Gopg -module that is eas-ily shown to be finitely generated, and realize yourmodule as an explicit subquotient. In some casesthings don’t work out quite as directly, but often onecan at least find a spectral sequence that converges toyour module, and the ultimate conclusion remains thesame. Once again we note that the feature of finitegeneration has a number of non-trivial consequences.For instance one has the following.

Theorem 3.1 (Miyata and Ramos, [MR20]). Thereexists an integer di,g ≥ 1 such that for any graphG of combinatorial genus at most g, the exponent ofHi(MG) divides di,g.

Torsion in the matching complex is something thathas received a fair amount of attention in recentyears, where it is noted that all torsion thus far dis-covered have orders that are small primes [Jon10].

It should also be noted that one limitation of thisstyle of proof is that it gives you almost no controlover what the generators are. Such control could begiven if one were to develop a robust computationaltheory similar to the classical theory of Gröbnerbases for the representation theories of Sam-SnowdenGröbner Categories, that we touch upon in the fi-nal section. This remains an interesting avenue forfuture research.

The reader may have noticed that the proof of finitegeneration is extremely generalizable. In particular,a large number of graph complexes will have sim-ilar finitely generated homologies, and therefore, forinstance, bounded torsion. Examples of such com-plexes include the complex of bounded degree sub-graphs, complexes of triangle-free subgraphs, com-plexes of t-colorable subgraphs and more. For more

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on these complexes, see the book [Jon07], as well asthe references therein

Another class of interesting examples comes fromthe study of configuration spaces.

Definition 3.2. For any topological space X andany integer n ≥ 1 the configuration space on X

on n-points is the quotient space

Fn(X) = {(x1, . . . , xn) ∈ Xn | xi 6= xj}/Sn,

where the symmetric group Sn acts by permutingcoordinates.

Configuration spaces have been a topic of seriousstudy for decades, although much of what is under-stood relates with cases wherein X is a manifold.More recently, however, there have been consider-able advances made in the theory of configurationspaces of graphs [ADCK19,AK20,Ghr01]. One ques-tion of considerable interest with regards to thesespaces is whether or not their homology ever admitsodd torsion. An affirmative answer to this questionwould have implications as far ranging as physics androbotics [Ghr01].

Let G be a graph and n ≥ 1 a fixed integer. Itis an interesting fact that any minor morphism ϕ :

G′ → G will induce a map Hi(Fn(G))→ Hi(Fn(G′))

[ADCK19]. Using these maps, we now find ourselveswith new Gop-modules G 7→ Hi(Fn(G)). Using theWeak Categorical Graph Minor Theorem, the follow-ing is proven in [PR].

Theorem 3.3 (Proudfoot, and Ramos, [PR]). Forany i, n ≥ 1, the Gopg -module Hi(Fn(G)) is finitelygenerated. In particular, there exists some integerdi,n,g ≥ 1 such that the order of any torsion appearingin Hi(Fn(G)) divides di,n,g.

It was proven by Ko and Park [KP12] that for anygraph the first homology group H1(F2(G)) has tor-sion if and only if G is non-planar, and that any tor-sion which appears must be 2-torsion. The abovetheorem therefore shows that this kind of behavior isone instance of something more general.

As with the matching complex example, it is notcurrently known what the generators are for the mod-ules Hi(Fn(G)) for most choices of i, n, and g. There

are two notable cases where it is known or partiallyknown, however. For i = 1 and any n ≥ 1, the gen-erators are the loop as well as all star graphs (thatis, trees with one vertex of degree ≥ 3 and all othervertices of degree 1) with ≤ n + 1 edges [ADCK19].For i = 2, and n = 3, although the full generating setis not known, we do know what the planar generatorsare. These will be the dumbbell graph of a line seg-ment with a loop on either end, the graph that lookslike the letter Y with a loop attached to one of itsleaves, and the banana graph θ4 of two vertices con-nected by 4 edges [AK20]. As a side note, this thirdgenerator is particularly special, as H2(F3(Θ4)) con-tains a class which is not toric (i.e. a product of twocopies of S1). In fact, it is represented by a surfaceof genus 3 [CL18,WG17]! Further note that all ofthese cases show that the generators do not dependon g when g � 0, a fact which supports the sugges-tion that Theorem 3.3 can be extended to the wholegraph minor category. As with all theorems in thiswork, the proof of Theorem 3.3 for the whole of Gop

would be immediate provided the verification of 2.3.It can be shown that the integer di,n,g of the

previous theorem does not actually depend on n

[MR20]. This strengthening comes from the topologyand combinatorics of the situation, and in particulardepends on more than just the (Weak) CategoricalGraph Minor Theorem. Let’s consider this now.

One of the great early tools used in the study ofconfiguration spaces of manifolds was the idea to “adda point at infinity.” Namely, to introduce a homotopyclass of maps

Fn(X)→ Fn+1(X)

and see the behavior of the induced directed systemon homology. It therefore becomes natural to askwhether one is able to introduce points to Fn(G),whenever G is a graph. The answer to this questionis yes! In fact, for every edge e of G one will have amap,

Fn(G)xe→ Fn+1(G), (1)

which one can think of as introducing a new pointon the edge e. These maps were first constructed,

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Figure 1: An example of the map xe. In this picturethe end points of the edge e are colored in black,while the points coming from the configuration are inwhite.

though only at the level of homology, by the authorin the case of trees [Ram18]. Independently, the fullconstruction at the level of topology described belowwas achieved by An, Drummond-Cole, and Knudsen[ADCK19, ADCK20a], who also expanded it to allgraphs . To illustrate how this map is defined, imag-ine having placed n points on G, and consider thosepoints appearing on an edge e = {a, b}. Fix for nowan identification of e with the interval [0, 1] and list(in order from left to right) all points appearing one as well as the end points a, b. The image of thisconfiguration under xe leaves any points not on e un-changed whereas it replaces the each of the originalpoints on e with the midpoint of itself and the nextmember of the list just constructed. Therefore, forinstance, if e had no points on it originally, the newconfiguration will have precisely one point at the cen-ter of e. On the other hand, if the original configu-ration had one point on the interior of e, then thenew configuration will have two points on e, one atthe mid point of a and the original point and one atthe midpoint of the original point and b. We give anillustration of this map as Figure 1

Using this construction, the following theorem wasproven.

Theorem 3.4 (An, Drummond-Cole, and Knudsen,[ADCK19]). For i ≥ 0 and any graph G, write Hi(G)

for the graded abelian group

Hi(G) :=⊕n

Hi(Fn(G)).

Then the edge stabilization maps (1) induce an actionby the polynomial ring Z[xe]e∈E(G), endowing Hi(G)

with the structure of a finitely generated graded mod-ule over this ring.

The above theorem now allows us to study con-figuration spaces of graphs from the perspective ofcommutative algebra. There are a variety of resultsin this vein, one of which we now spotlight. Recallthat for a finitely generated graded module M over apolynomial ring, the function

n 7→ rank(Mn)

is in eventual agreement with a polynomial – theHilbert polynomial of the module. Let G be agraph not homeomorphic to a loop, and write ∆i

G forthe largest number of connected components that Gcan be broken into by the removal of exactly i verticesof degree at least 3. By convention, ∆i

G = 0 if G hasless than i vertices of degree at least 3. The follow-ing theorem was proven for trees by the author, andfor all graphs by An, Drummond-Cole and Knudsen[ADCK20a].

Theorem 3.5 (An, Drummond-Cole, and Knudsen[ADCK20a]). Let G be a graph that is not homeo-morphic to a loop. Then for all i ≥ 0 the degree ofthe Hilbert polynomial of Hi(G) is precisely ∆i

G − 1.

Note that in followup work An Drummond-Coleand Knudsen also computed the leading coefficient ofthe Hilbert polynomial, once again in terms of invari-ants of G [ADCK20b]. One consequence that imme-diately follows from this theorem is that if G is bi-connected, that is, if removal of any vertex does notdisconnect the graph, then the degree of the Hilbertpolynomial of H1(G) is zero. In particular, the rankofH1(Fn(G)) is constant in n. This fact was observedmuch earlier by Ko and Park [KP12] using totally dif-ferent means. We therefore see that these homologygroups seem to encode an eclectic collection of prop-erties of the graphs including planarity and connec-tivity. It is an active line of research to understandwhat other graph theoretic properties can be foundinside these spaces.

Thinking about the homology groups Hi(Fn(G))

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as a family with two varying parameters, n and G,we have now seen that one obtains finite generationresults by fixing one and varying the other. It is thennatural to ask whether these two orthogonal resultscan be made compatible with one another. This isindeed the case!

Definition 3.6. Let G be a graph. The (free) edgealgebra of G is the polynomial ring on the edges ofG,

AG = Z[E(G)].

We define the universal edge algebra to be thefunctor

A• : Gop → Z−Alg

defined on objects by

G 7→ AG

and on morphisms in the same way as the Edge Mod-ule. An A•-module is a Gop-module M such thatM(G) is an AG-module for every G, and for everyminor morphism ϕ : G′ → G and element a ∈ AG,the following diagram commutes,

M(G)M(ϕ)−−−−→ M(G′)

a·y yA(ϕ)(a)·

M(G)M(ϕ)−−−−→ M(G′)

An A•-moduleM is said to be finitely generated ifthere exists a finite set of graphs {Gj} such that forany graph G, M(G) is spanned (note here that span-ning is in reference to the AG-action) by the imagesof theM(Gj). As with everything else in this section,we may also consider A•,g, defined by restricting A•to Gopg .

We saw above that for any i ≥ 0, the assignment

G 7→ Hi(G)

defines an A•-module. As another example, con-sider the ideal IG of AG generated by products xexe′ ,whenever e, e′ are not adjacent to one another. It isclear that the action of minor morphisms preservesthis condition of being non-adjacent, and thereforeI• is an A•-submodule of A•.

For any fixed graph G, the edge algebra AG clearlysatisfies a Noetherian property by virtue of it being apolynomial ring. What is much less clear is whethereach of these individual Noetherian properties can beglued together, so to speak, to say something aboutmodules over the universal algebra A•. While weonce again must keep the full-strength statement inthe realm of conjecture, one can say the following.

Theorem 3.7 (Miyata, Proudfoot, and Ramos,[MPR20]). If M is a finitely generated A•,g-module,then all A•,g-submodules of M are also finitely gen-erated.

To prove this version of this theorem for Gop-modules, one would need the Categorical graph minortheorem, as well as a kind of universal Gröbner ba-sis approach [MPR20]. By consequence, we see thatfor any finitely generated A•,g-module M , not onlyis M(G) determined by M(Gj) for some finite listof graphs {Gj}, but the syzygies of M(G), in thecommutative algebra sense, are also all determinedby some (possibly different) finite list of graphs. Thisis precisely why the universal exponent of Theorem3.3 does not depend on n.

4 An Outline of the Proof

In this section we provide an outline of the proof ofthe Weak Categorical Graph Minor Theorem 2.4. Wedo this not only to spotlight the beautiful underlyingtheory, due to Sam and Snowden [SS17], but also be-cause we believe it does a good job of illustratinghow the combinatorics of the Graph Minor Theoreminforms the algebra of the Weak Categorical GraphMinor Theorem. The content of this section is a bitmore on the technical side, though we have omittedmany details in an attempt to make it more readable.We also end the work by pointing out what exactlythe difficulty is in lifting the weak result to the fullstrength of Conjecture 2.3, and how one would pre-sumably aim to fix it.

We begin, as Sam and Snowden did in their semi-nal work [SS17], by recalling the Hilbert Basis The-orem. For the purposes of this discussion writeR = k[x1, . . . , xn], where k is a fixed commutative

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Noetherian Ring. The Hilbert Basis Theorem thenstates that all submodules of any finitely generatedmodule must be finitely generated. That is, thatfinitely generated modules over R must be Noethe-rian. The proof of the Hilbert Basis Theorem that wewill concern ourselves with is the standard approachthrough Gröbner bases, and proceeds as follows:To begin, we apply standard reductions to show thatwe only need to prove that submodules (i.e. ideals) ofR itself must be finitely generated. Next, consider thelexicographical order on monomials in R. This ordernot only imposes a well-order on all monomials, butalso has the property of being preserved under multi-plication. In particular, we may therefore define theleading term of any element f ∈ R to be the largestmonomial among all those appearing in f . Now givenany ideal I of R, one defines the initial ideal I0 tobe the ideal generated by the leading terms of all el-ements in I. A standard argument then shows thatI is finitely generated if and only if I0 is, whence itsuffices to prove that all monomial ideals are finitelygenerated. If we encode monomials of R as elementsin Nn, then the standard coordinate-wise partial or-der on Nn is seen to be equivalent to the divisibilitypartial order on monomials. Therefore, that mono-mial ideals are finitely generated is equivalent to thefact that the standard coordinate-wise order on Nn

does not permit infinite anti-chains. This latter factis true according to Dickson’s Lemma, and we aredone.

To summarize, the above proof of the Hilbert BasisTheorem proceeds in three major steps:

1. Reduce the problem from all finitely generatedmodules, to free (finitely generated) modules;

2. Define a well-order on the set of monomialsthat respects the action of the ring. Use thiswell order to reduce the problem to monomial-generated submodules of the free module;

3. Encode divisibility of monomials into some posetthat is known to not have infinite anti-chains.Use this to deduce that all monomial-generatedsubmodules of the free module must be finitelygenerated, concluding the proof.

It is the great innovation of [SS17] that the abovethree steps can be replicated in contexts similar tothe Gopg -modules of the current note. They refer tothis as the theory of Gröbner categories and theirmodules. We consider each of the above three stepsin turn.

To begin, what are the “free” Gopg -modules? Forany fixed graph G of combinatorial genus at most gwe define,

FG(G′) = ZHomGg (G′,G),

the free abelian group on the Hom-set HomGg (G′, G).The maps induced by minor morphisms are then de-fined by precomposition. For instance, if G is thegraph with one vertex and no edges, then,

FG(G′) = ZHomGg (G′,G) = Z{Spanning trees of G′}

is the Spanning Tree module from above. For numer-ous homological reasons related with the vanishingsof certain derived functors, it turns out that the mod-ules FG are each appropriate to be called “free.” Notethat this is analogous to the context of graded mod-ules over the ring R = k[x1, . . . , xn], where there isone free module for each natural number. The samestyle of argument which allowed one to reduce to sub-modules of free modules in the proof of the HilbertBasis Theorem will continue to work here, allowingus to reduce the Weak Categorical Graph Minor The-orem to proving that the submodules of the FG arefinitely generated.

Fixing now a graph G of combinatorial genus atmost g for all time, Sam and Snowden [SS17] definea monomial of FG to be any natural basis elementeϕ ∈ ZHomGg (G

′,G). Our Step 2 insists that we shouldcome up with some well-order on these monomialsthat respects the action of the maps induced by mi-nor morphisms. Unfortunately, this is actually im-possible! Indeed, because minor morphisms includegraph automorphisms, if G has any non-trivial au-tomorphisms then we will not be able to well-orderour monomials in a way consistent with this action.Sam and Snowden come up with a solution to this(fairly common) problem in the following way: In-stead of thinking about our original category Gg, con-

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sider modules over a different category G̃g which ismore rigid than Gg, in the sense that it has no auto-morphisms, while also not being “too different” fromGg. It is proven that in this circumstance a Noethe-rian property for modules over G̃g

opimplies the same

for modules over Gopg .

In the original work [SS17], Sam and Snowdenmake this idea of not being too different precise us-ing what they call Property (F). Property (F) isa feature of a functor Φ : G̃g → Gg that general-izes the property of being a right adjoint. Insteadof getting too deep in the technicalities here, we il-lustrate the spirit of Property (F) with an example.Consider the category FI whose objects are the fi-nite ordinals [n] = {1, . . . , n} and whose morphismsare injections. This category also clearly has auto-morphisms (namely, the permutations), so we insteadconsider the category F̃I of finite ordinals with orderpreserving injections. This category does not havenon-trivial automorphisms, and is also very closelyrelated with FI, in that for any [n] and any injectionof sets [n] ↪→ [m], there is always an ordered injec-tion [n] ↪→ [m] that agrees with the original injectionup to precomposition by some permutation of [n]. Inother words, for any n, there is a finite collection of FI

morphisms (e.g. the permutations of [n]) such thatevery injection originating from [n] agrees with an or-dered injection originating from [n] up to one of thesemorphisms. Sam and Snowden loosely describe thisphenomenon as F̃I having a sort of finite index withinFI [SS17].

Coming back to our Graph Minor Category, thechallenge now becomes to choose the correct cate-gory G̃g. It turns out the category we are looking foris the category whose objects are graphs of genus atmost g, that have been equipped with a choice of arooted and planar spanning tree, as well as a directionand ordering of its extra (outside the given spanningtree) edges. The morphisms of this category will becontractions that preserve all of this structure. Bydemanding all of the extra structure that was addedbe preserved, we have eliminated all automorphisms.Moreover, essentially because any graph can only begiven the extra data of a rooted spanning tree anddirections on its extra edges in finitely many ways,

as well as the fact that the restriction on g disallowsarbitrarily long chains of deletions, the forgetful func-tor G̃g → Gg can be seen to have Property (F). Wecan therefore assume that we have been working withthe category G̃g this entire time, that our fixed graphG has been given the data of both a planar rootedspanning tree, and directions and orderings of its ex-tra edges, and re-examine our Step 2. In this case,the desired well order is presented in [PR].

Step 3 asks us to encode the divisibility relationof monomials into some poset that is known to notadmit infinite anti-chains. So what exactly is the di-visibility relation between monomials in our setting?Well, for traditional monomials over the polynomialring, divisibility meant that there was some some el-ement f of the ring for which one monomial was ftimes the other. Using our definition of monomials infree G̃g-modules we see that this naturally translatesto say that eϕ is divisible by eψ if and only if there issome morphism ζ in G̃g such that ϕ = ψ ◦ ζ. Trans-lating the relationship between minor morphisms andthe minor relation, this tells us that the divisibilityrelationship between monomials is the minor relationlimited to the set of directed and edge-ordered graphscontaining G as a minor. While the Graph MinorTheorem as previously presented does not guaranteethat this poset has no infinite anti-chains (because ofthe extra data we’ve imposed on each graph) there isa stronger labeled version of the Graph Minor The-orem [RS10], as well as an order preserving versionof Kruskal’s Tree Theorem [Bar15], that does give uswhat we want.

Looking closely at everything discussed above, it ishopefully clear that the only thing preventing us fromproving Conjecture 2.3 is choosing the right categoryG̃. This trick of choosing a rooted spanning tree willno longer work here! In fact, to the knowledge ofthe author, there are no currently known “rigidifica-tion” of the Graph Minor Theorem that are imme-diately applicable, just as we relied on the rigidifiedTree Theorem for the bounded genus case. Presum-ably, proving such a rigidification would require oneto have very deep intimate knowledge of how the orig-inal Graph Minor Theorem is proven.

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Acknowledgements

The author would like to send his unending gratitudeto Nicholas Proudfoot. He would also like to send histhanks to the anonymous referees whose suggestionsgreatly improved the exposition of the article. Theauthor was supported by NSF grant DMS-2137628.

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