On a theorem of Lafforgue

We give a new proof, along with some generalizations, of a folklore theorem (attributed to Laurent Lafforgue) that a rigid matroid (i.e., a matroid with indecomposable basis polytope) has only finitely many projective equivalence classes of representations over any given field.


Introduction
A matroid M is called rigid if its base polytope P M has no non-trivial regular matroid polytope subdivisions.Such matroids are interesting for a number of reasons; for example, a theorem of Bollen-Draisma-Pendavingh [8] asserts that for each prime number p, a rigid matroid is algebraically representable in characteristic p if and only if it is linearly representable in characteristic p.A folklore theorem, attributed to L. Lafforgue, asserts that a rigid matroid has at most finitely many representations over any field, up to projective equivalence.This is mentioned without proof in a few places throughout the literature, for example in Alex Fink's Ph.D. thesis [12, p. 10], where he writes: Matroid subdivisions have made prominent appearances in algebraic geometry.[. . .] Lafforgue's work implies, for instance, that a matroid whose polytope has no subdivisions is representable in at most finitely many ways, up to the actions of the obvious groups.
We have been unable to find a proof of this result in the papers of Lafforgue cited by Fink [15,16], though a proof sketch appears in [13,Theorem 7.8].In this paper, we provide a rigorous and efficient proof of Lafforgue's theorem, along with some new generalizations.
What is arguably most interesting about our approach to Lafforgue's theorem is that we deduce it from a purely algebraic statement which has nothing to do with matroids.The only input from matroid theory needed is the fact that the rescaling class functor X M from pastures to sets is representable (see Section 2 below for further details).We believe this to be a nice illustration of the power, and elegance, of the algebraic theory developed by the authors in [3] and [4].

Reformulation and generalizations of Lafforgue's theorem
It is well-known to experts that a matroid M is rigid if and only if every valuated matroid M whose underlying matroid is M is rescaling equivalent to the trivially valuated matroid.Since we could not find a reference for this result, we provide a proof in Appendix B.
Recall from [3] (see also Appendix A) that there is a category of algebraic objects called pastures, which generalize not only fields but also partial fields and hyperfields.According to [1], there is a robust notion of (weak) matroids over a pasture 1 P such that (to mention just a few examples): • Matroids over the Krasner hyperfield K are the same thing as matroids in the usual sense.• Matroids over the tropical hyperfield T are the same thing as valuated matroids.
• Matroids over a field K are the same thing as K-representable matroids, together with a choice of a matrix representation (up to the equivalence relation where two matrices are equivalent if they have the same row space).
For every matroid M there is a functor X M from pastures to sets taking a pasture P to the set of rescaling equivalence classes of (weak) P-representations of M. A matroid M is rigid if and only if X M (T) consists of a single point.For a field K, X M (K) coincides with the set of projective equivalence classes of representations of M over K. Thus Lafforgue's theorem is equivalent to the assertion that if X M (T) is a singleton, then X M (K) is finite for every field K.
Recall from [3] that for every matroid M, the functor X M is representable by a pasture F M canonically associated to M, called the foundation of M. Concretely, this means that Hom(F M , P) = X M (P) for every pasture P, functorially in P.
From this point of view, Lafforgue's theorem is equivalent to the assertion that if Hom(F M , T) = {0}, then Hom(F M , K) is finite for every field K.This is the statement of Lafforgue's theorem that we actually prove in this paper.The advantage of this formulation is that it turns out to be a special case of a result which can be formulated purely in the language of pastures, without any mention of matroids!In fact, the algebraic incarnation of this result holds more generally with pastures (which generalize fields) replaced by bands (which generalize rings).
See Appendix A for an overview of bands, including a definition, some examples, and the key facts needed for the present paper.
2.1.An algebraic generalization of Lafforgue's theorem.In order to state the algebraic result about bands which implies Lafforgue's theorem, we mention (see Proposition 4.4 below) that given a band B and a field K, there is a canonically associated K-algebra ρ K (B) with the universal property that Hom Band (B, S) = Hom K−alg (ρ K (B), S) for every K-algebra S.Moreover, if B is finitely generated (which is the case, for example, when B = F M for some matroid M), then so is ρ K (B).
1 Technically speaking, [1] deals with tracts, not pastures, but the difference between the two is immaterial when considering weak matroids.For the sake of brevity, we do not define tracts in this paper, nor do we consider idylls or ordered blueprints (both of which play a prominent role in [4]).
If B is finitely presented, the set Hom(B, T) has the structure of a finite polyhedral complex Σ B ; cf.Remark 4.5.Moreover, if K is a field, the set Hom Band (B, K) is equal to Hom K−alg (ρ K (B), K), which is in turn equal to the set X B,K (K) of K-points of the finite type affine K-scheme X B,K := Spec(ρ K (B)).(When B = F M for a matroid M, we call X B,K the reduced realization space of M over K.) Our first generalization of Lafforgue's theorem is as follows: Theorem 2.1.For every finitely generated band B and every field K, we have the inequality dim X B,K dim Σ B .In particular, if Hom(B, T) = {0}, then dim Σ B = 0 and thus X B,K (K) = Hom(B, K) is finite for every field K.
Applying Theorem 2.1 to B = F M immediately gives: In the terminology of Remark B.2, Theorem 2.1 in the case B = F M says precisely that for any field K, the dimension of the reduced realization space of M over K is bounded above by the dimension of the reduced Dressian of M.

A relative version of Lafforgue's theorem.
Rudi Pendavingh (private communication) asked if there might be a relative version of Lafforgue's theorem with respect to minors of M.More precisely, Pendavingh asked the following question: Suppose N is an (embedded) minor of M with the property that a valuated matroid structure on M is determined, up to rescaling equivalence, by its restriction to N. Is it then true that, for every field K, there are (up to projective equivalence) at most finitely many extensions of each K-representation of N to a K-representation of M?
We answer Pendavingh's question in the affirmative, proving the following algebraic generalization of Corollary 2.2: Theorem 2.3.Let K be an algebraically closed valued field, and let v : K → T be a non-trivial valuation.If f : B 1 → B 2 is a homomorphism of finitely generated bands, then the fiber dimension of f K : In particular, setting B 1 = F N and B 2 = F M when N is an embedded minor of a matroid M, we find that if the induced map X N (T) → X M (T) has finite fibers (i.e., a valuated matroid structure on N has at most finitely many extensions to M, up to rescaling equivalence) then, for every field k, the natural map X N (k) → X M (k) has finite fibers, i.e., every k-representation of N has at most finitely many extensions to M, up to projective equivalence.
Note that Lafforgue's theorem (Corollary 2.2) follows from the special case of Theorem 2.3 where N is the trivial (empty) matroid and f T : Hom(B 2 , T) → Hom(B 1 , T) has finite fibers.

Some examples
In this section we present examples of both rigid and non-rigid matroids (see Appendix A for some details on our notation).
Example 3.1 (Dress-Wenzel).In [11,Theorem 5.11], Dress and Wenzel showed that if the inner Tutte group F × M of the matroid M is finite, then M is rigid.From our point of view, this is clear, since the inner Tutte group is the multiplicative group of the foundation (cf.[4,Corollary 7.13]) and a non-trivial homomorphism F M → T of pastures would give, in particular, a nonzero group homomorphism F × M → (R, +); however the only torsion element of (R, +) is 0.
For example: (1) The foundation of the Fano matroid F 7 is F 2 , so F 7 is rigid.More generally, any binary matroid has foundation equal to either F ± 1 or F 2 [4, Corollary 7.32] and so it is rigid.
(2) The foundation of the ternary spike T 8 is F 3 (see [5,Proposition 8.9]), so T 8 is also rigid.(3) Dress and Wenzel prove in [11,Corollary 3.8] that the inner Tutte group of any finite projective space of dimension at least 2 is finite, which provides a wealth of additional examples of rigid matroids.(4) Since the automorphism group of the ternary affine plane M = AG(2, 3) acts transitively, all single-element deletions are isomorphic to each other.Let M ′ be any of these deletions.By [5, Proposition 6.2], the foundation of M ′ is equal to the hexagonal (or sixth-root-of-unity) partial field whose multiplicative group is the group of sixth roots of unity in C. Therefore M ′ is rigid.
It is not true that a matroid M is rigid if and only if its inner Tutte group (or, equivalently, its foundation) is finite.For example: Example 3.2 (suggested by Rudi Pendavingh).Let M be the Betsy Ross matroid (cf.[23,Figure 3.3], where M is also called B 11 ).Using the Macaulay2 software described in [10], we have checked that F M is the (infinite) golden ratio partial field One checks easily that Hom(G, T) is trivial, so M is rigid; in particular, the converse of the statement "F M finite implies M rigid" is not true.It is also easy to see directly that G admits only finitely many homomorphisms to any field.
Example 3.3.The matroid U 2,4 is not rigid, since its foundation is the near-regular partial field U = F ± 1 (T 1 , T 2 ) T 1 + T 2 − 1 , which admits infinitely many different homomorphisms to T (map T 1 to 1 and T 2 to any element less than or equal to 1, or vice-versa).And for any field K, the reduced realization space X M (K) is equal to K\{0, 1}, so in particular it is infinite whenever K is.The base polytope of U 2,4 is an octahedron, which admits a regular matroid decomposition into two tetrahedra (see [18, p. 189] for a nice visualization).
Example 3.4.The non-Fano matroid M = F − 7 is not rigid, and it provides an example for which the dimension of the reduced realization spaces X M (K) and X M (T) jumps.The foundation of M is the dyadic partial field D = F ± 1 (T ) T + T − 1 by [5, Prop.8.4], and there is at most one homomorphism F M = D → K into any field K, sending T to the multiplicative inverse of 2 (if it exists, i.e., if char K = 2).In contrast, there are infinitely many homomorphisms D → T (parametrized by the image of f (T ) ∈ T).So dim X M (K) = 0 < 1 = dim X(T).

Proof of the main theorems
The key fact needed for the proof of Theorem 2.1 is the following theorem of Bieri and Groves [7, Theorem A], which is a cornerstone of tropical geometry.For the statement, recall that a semi-valuation from a ring R to R = R ∪ {+∞} is a map v : R → R such that v(0) = +∞, v(xy) = v(x) + v(y), and v(x + y) min{v(x), v(y)} for all x, y ∈ R. (The map v is called a valuation if, in addition, v(x) = +∞ implies that x = 0.) If R is a K-algebra, where K is a valued field (i.e., a field endowed with a valuation v : K → R), a K-semi-valuation is a semi-valuation which restricts to the given valuation on K. Theorem 4.1 (Bieri-Groves).Let K be a field endowed with a real valuation v, and suppose R is a finitely generated K-algebra with Krull dimension equal to n, having generators T 1 , . .., T n .Let X = Spec(R) be the corresponding affine K-scheme.Then the set Remark 4.2.Bieri and Groves assume that X is irreducible and show, more precisely, that Trop(X ) has pure dimension n.Our formulation of the Bieri-Groves theorem (which does not include the purity statement) follows immediately from theirs by decomposing X into irreducible components.
Remark 4.3.More or less by definition, a semi-valuation on a ring R is precisely the same thing as a homomorphism from R to T in the category of bands, and if K is a valued field then a K-semi-valuation on R is the same thing as a homomorphism from R to T which restricts to the given homomorphism v : K → T on K.
Let K be a field, and let Alg K denote the category of K-algebras, i.e. ring extensions R of K together with K-linear ring homomorphisms.We write Hom K (R, S) for the set of K-algebra homomorphisms between two K-algebras R and S. Given a band B, we define the associated K-algebra as where K[B] is the monoid algebra over K and the elements of the nullset N B are interpreted as elements of K[B] (cf.Definition A.1).It comes with a band homomorphism The other main ingredient needed for the proof of Theorem 2.1 is the following technical but important result: Proposition 4.4.Let K a field, B be a band and R = ρ K (B) the associated K-algebra.
(1) The homomorphism α B : B → ρ K (B) is initial for all homomorphisms from B to a K-algebra, i.e., for every K-algebra S the natural map is a bijection.
(2) Assume we are given a valuation v K : K → T, and that B is finitely generated by a 1 , . . ., a n .Let T i = α B (a i ) for i = 1, . .., n, and let X = Spec R. Let exp n : R n → T n be the coordinate-wise exponential map.Then the T i generate R as a K-algebra, and as subsets of T n .
Proof.We begin with (1).The map α * B is injective since R is generated by the subset α B (B), and therefore every homomorphism f : R → S is determined by the composition f • α B : B → S. In order to show that α * B is surjective, consider a band homomorphism f : B → S, which is, in particular a multiplicative map.Therefore it extends (uniquely) to a K-linear homomorphism f : K[B] → S from the monoid algebra K[B] to S. For every ∑ a i ∈ N B , we have ∑ f (a i ) ∈ N S by the definition of a band homomorphism.By the definition of N S , this means that ∑ f (a i ) = 0 in S. Thus f factorizes through f : R = K[B]/ N B → S, and, by construction, we have We continue with (2).Since B is generated by a 1 , . . ., a n as a pointed monoid and α B (B) generates R as a K-algebra, R is generated as a K-algebra by T 1 , . . ., T n .In order to verify that exp n (Trop(X )) ⊂ Hom(B, T), consider a point (v(T 1 ), . . ., v(T n )) ∈ Trop(X ), where v : R → R is a K-semi-valuation.Post-composing v with exp yields a seminorm v ′ : R → T, which is, equivalently, a band homomorphism.Pre-composing v ′ with α B yields a band homomorphism v ′′ : B → T, which is an element of Hom(B, T).By construction, exp n (v(T 1 ), . .., v(T n )) = v ′′ , which establishes the last assertion.Remark 4.5.
(1) Under the assumptions of Proposition 4.4.(2),Hom(B, T) embeds as a subspace of T n , which has a well-defined (Lebesgue) covering dimension in the sense of [21,Chapter 3].As discussed in [17], the subspace topology of Hom(B, T) ⊂ T n is equal to the compact-open topology for Hom(B, T) with respect to the discrete topology for B and the natural order topology for T, which shows that the dimension of Hom(B, T) does not depend on the embedding into T n .
(2) With the topologies just described, exp n defines a continuous injection from Trop(X ) to Hom(B, T) which identifies the former with a closed subspace of the latter.In particular, [21,Prop. 3.1.5]shows that dim Trop(X ) dim Hom(B, T). (3) If in addition to the assumptions of (2), N B is finitely generated as an ideal of B + , then Hom(B, T) is a tropical pre-variety in T n and is therefore the underlying set of a finite polyhedral complex.The dimension of Hom(B, T) as a polyhedral complex is equal to its covering dimension [21, Theorem 2.7 and Section 3.7].
Proof of Theorem 2.1.Let v : K → T be a valuation (which we can take to be the trivial valuation if we like).Let α B : B → R be the canonical homomorphism to the associated K-algebra R = ρ K (B), cf.Proposition 4.4.Let a 1 , . . ., a n ∈ B be a set of generators for B, and for i = 1, . .., n let T i = α B (a i ).By Proposition 4.4, the T i generate R as a K-algebra, i.e., R = K[T 1 , . .., T n ]/I for some ideal I.
Let X = Spec R, so that X (K) = Hom K (R, K).Proposition 4.4 yields a commutative diagram X (K) = Hom K (R, K) Hom(B, K) where the right-hand vertical map is obtained by composing with v : K → T and the lefthand vertical map is induced by composing the embedding of X (K) = Hom K (R, K) into K n via φ → (φ(T i )) n i=1 with the coordinate-wise absolute value v n K : K n → T n .By the Bieri-Groves theorem (Theorem 4.1), the dimension of the affine variety X is equal to the dimension of Trop(X ), as defined in Remark 4. 5 )), and letting Trop(X ) (resp.Trop(Y )) be the tropicalization of X with respect to α with Z an affine subscheme of Y .If we pull back the functions α B 2 (y 1 ), . .., α B 2 (y n ) to a set of generators for the affine coordinate ring of Z, we obtain a commutative diagram Applying the Bieri-Groves theorem to Z, we find that the image of Z(K) under Trop has dimension equal to dim f −1 K (x).In addition, the natural map

Appendix A. Pastures and Bands
More details pertaining to the following overview of bands and pastures can be found in [2].
In this text, a pointed monoid is a (multiplicatively written) commutative semigroup A with identity 1, together with a distinguished element 0 that satisfies 0 • a = 0 for all a ∈ A. The ambient semiring of A is the semiring A + = N[A]/ 0 , which consists of all finite formal sums ∑ a i of nonzero elements a i ∈ A. Note that A is embedded as a submonoid in A + , where 0 is identified with the empty sum.An ideal of A + is a subset I that contains 0 and is closed under both addition and multiplication by elements of A + .Definition A.1.A band is a pointed monoid B together with an ideal N B of B + (called the nullset) such that for every a ∈ A, there is a unique b ∈ A with a + b ∈ N B .We call this b the additive inverse of a, and we denote it by −a.A band homomorphism is a multiplicative map f : B → C preserving 0 and 1 such that ∑ a i ∈ N B implies ∑ f (a i ) ∈ N C .This defines the category Bands.
( For a subset S of B + , we denote by S the smallest ideal of B + that contains S and is closed under the fusion axiom (cf.• The Krasner hyperfield is the pasture K = {0, 1} with nullset

1 =
[6]) (F) if c + ∑ a i and −c + ∑ b j are in S , then ∑ a i + ∑ b j is in S .Definition A.2.A band B is finitely generated if it is finitely generated as a monoid.It is a finitely presented fusion band, which we abbreviate by simply saying that B is finitely presented, if it is finitely generated and N B = S for a finite subset S of N B .The unit group of B is the submonoid B × = {a ∈ B | ab = 1 for some b ∈ B} of B, which is indeed a group.Definition A.3.A pasture is a band P with P × = P − {0} andN P = a + b + c ∈ P + a + b + c ∈ N P .Example A.4.Every ring R is a band, with nullset N R = {∑ a i | ∑ a i = 0 in R}.In fact, this defines a fully faithful embedding Rings → Bands.Every field is a pasture.The following examples of interest are bands which are not rings (we write a − b for a + (−b)):• The regular partial field is the pasture F ± 1 = {0, 1, −1} with nullset N F ± n.1 + n.(−1) n 0 = 1 − 1 .
Proof of Theorem 2.3.Suppose f : B 1 → B 2 is a band homomorphism.Choose generators x 1 , . .., x m for B 1 .Completing f (x 1 ), . . ., f (x m ) to a set of generators for B 2 if necessary, we find a generating set y 1 , . . ., y n for B 2 with m n such that f (