A Commutative Algebra for Oriented Matroids

Discrete Comput Geom 27:73–84 (2002) 

DOI: 10.1007/s00454-001-0053-8 

Discrete & Computational 

Geometry 

© 2002 Springer-Verlag New York Inc. 

A Commutative Algebra for Oriented Matroids ∗ 

R. Cordovil 

Departamento de Matemática, Instituto Superior Técnico, 

Av. Rovisco Pais, 1049-001 Lisboa, Portugal 

cordovil@math.ist.utl.pt 

Abstract. Let V be a vector space of dimension d over a field K and let A be a central 

arrangement of hyperplanes in V. To answer a question posed by K. Aomoto, P. Orlik 

and H. Terao construct a commutative K-algebra U(A) in terms of the equations for the 

hyperplanes of A. In the course of their work the following question naturally occurred: 

◦ Is U(A) determined by the intersection lattice L(A) of the hyperplanes of A 

We give a negative answer to this question. The theory of oriented matroids gives rise to a 

combinatorial analogue of the algebra of Orlik–Terao, which is the main tool of our proofs. 

1. Introduction 

Let M = M([n]) (resp. M = M([n])) denote a matroid (resp. oriented matroid) of 

rank r with ground set [n] :={1, 2,...,n}. Let V be a vector space of dimension d over 

some field K. A (central) arrangement (of hyperplanes) in V, A K ={H 1 ,...,H n }, is 

a finite listed set of codimension one vector subspaces. Given an arrangement A K we 

suppose always chooses a family of linear forms {θ Hi ∈ V ∗ : H i ∈ A K , Ker(θ Hi ) = H i }, 

where V ∗ denotes the dual space of V. The product Q(A) = ∏ H∈A θ H is called the 

defining polynomial of A. There is a matroid M(A K ) on the ground set [n] determined by 

A K : a subset D ⊂ [n] isadependent set of M(A K ) iff there are scalars ζ i ∈ K, i ∈ D, 

not all nulls, such that ∑ i∈D ζ iθ Hi = 0. A circuit is a minimal dependent set with 

respect to inclusion. We denote by L(A K ) the intersection lattice of L(A K ): i.e., the 

set of intersections of hyperplanes in A K , partially ordered by reverse inclusion. Set 

M(A K ) = V \ ⋃ H∈A K 

H. 

∗ This research was supported in part by FCT (Portugal) through program POCTI, the project SAPI- 

ENS/36563/99 and by DONET (European Union Contract ERBFMRX-CT98-0202).

74 R. Cordovil 

Aomoto suggested the study of the (graded) K-vector space AO(A K ), generated by 

the basis {Q(B I ) −1 }, where I is an independent set of M(A K ), B I :={H i ∈ A K : i ∈ I } 

and Q(B I ) = ∏ i∈I θ H i 

denotes the corresponding defining polynomial. In [1] it is 

conjectured that 

dim(AO(A R )) = number of chambers of M(A R ). 

To prove Aomoto’s conjecture, Orlik and Terao have constructed in [8] a commutative 

K-algebra, U(A K ), isomorphic to AO(A K ) as a graded K-vector space in terms of the 

equations {θ H : H ∈ A K }. The authors note that it is not clear whether U(A K ) itself 

depends only on the intersection lattice L(A K ). 

To every oriented matroid M we associate a commutative Z-algebra, denoted by 

A(M). This algebra is the “combinatorial analogue” of the algebra of Orlik–Terao and 

it is the main tool to give a negative answer to the question of Orlik–Terao. 

We use [9] and [10] as a general reference in matroid theory. We refer to [2] and [7] 

for good sources of the theory of oriented matroids and arrangements of hyperplanes, 

respectively. 

2. Two Commutative Algebras 

Let IND l (M) ⊂ ( ) 

[n] 

l be the family of the independent sets of cardinal l of the matroid 

M and set IND(M) = ⋃ l∈N IND l(M). We denote by C = C(M) the set of circuits 

of M. When the smallest element α of a circuit C, |C| > 1, is deleted, the remaining 

set, C\α, is said to be a broken circuit. (Note that our definition is slightly different to 

the standard one. In the standard definition C\α can be empty.) To shorten the notation 

the singleton set {x} is denoted by x. A no broken circuit set of a matroid M is an 

( independent subset of [n] which does not contain any broken circuit. Let NBC l (M) ⊂ 

[n] 

) 

⋃l 

be the set of the no broken circuit sets of cardinal l of M. Set NBC(M) = 

l∈N NBC l(M). We denote by L(M) the lattice of flats of M. (We remark that the 

lattice map ϕ: L(A K ) → L(M(A K )), determined by the one-to-one correspondence 

ϕ ′ : H i ←→ {i}, i = 1,...,n, is a lattice isomorphism.) Consider now an independent 

set X. Let cl M (X) be (or shortly cl(X)) the closure of X in M. Pick an element 

x ∈ cl(X)\X. Let C(X, x) denote the unique circuit of M contained in X ∪ x. For 

every X ∈ IND(M), set 

EA(X) := {x ∈ cl(X)\X: x is the minimum of C(X, x) and C(X, x) ≠ {x}}. 

(The elements of EA(X) are usually called the externally active elements of X.) So, for 

every independent set X of M, X ∈ NBC(M) iff EA(X) =∅. If EA(X) ≠ ∅, let α(X) 

denote the smallest element of EA(X). 

Here, every map X: [n] → {+1, −1, 0} ⊂ Z is called a signed set on [n]. Set 

X + ={l ∈ [n]: X(l) =+1}, X − ={l ∈ [n]: X(l) =−1}. We say that X := X + ∪ X − 

is the support of X, or X is a signed set supporting X. A signed set X conforms to a 

signed set Y if X + ⊂ Y + and X − ⊂ Y − . For every x ∈ [n] and signed set X let X\x 

be the signed set on [n]\x, conforming to X and supporting X\x. We say that a signed 

set X is the union of the signed sets X 1 ,...,X m if X = X 1 ∪···∪X m and every X i 

conforms to X. The reorientation on the subset S ⊂ [n] of the signed set X is the signed

A Commutative Algebra for Oriented Matroids 75 

set, denoted −S X, determined by the equalities 

( −S X) + :={X + \S}∪{S ∩ X − } and ( −S X) − :={X − \S}∪{S ∩ X + }. 

The opposite of a signed set X, denoted −X, is the signed set −X = −[n] X. An oriented 

matroid, denoted M, is a matroid on the ground set [n], denoted M, with an additional 

structure: 

◦ To every circuit C ∈ C(M) is attached two opposite signed sets (signed circuits) 

C and −C supporting C. 

◦ The set of signed circuits of M, denoted C = C(M), verifies a convenient set of 

axioms, see page 103 of [2]. 

The set of all the union of signed circuits of M is called the set of the vectors of the 

oriented matroid. If K is an ordered field the arrangement A K determines an oriented 

matroid M(A K ) on the ground set [n]. Indeed let C ={i 1 ,...,i m }, i 1 < ···

circuit of M(A R ). From the definitions we know that there are well determined scalars 

ζ ij ∈ K ∗ , ζ i1 = 1, such that ∑ m 

j=1 ζ i j 

θ Hij = 0. Set C: [n] →{0, 1, −1} ⊂Z the signed 

set 

⎧ 

⎪⎨ +1 if l ∈ C and λ l > 0, 

C(l) = −1 if l ∈ C and λ l < 0, 

⎪⎩ 

0 if l/∈ C. 

By definition C is one of the two opposite signed circuits of M(A K ) supporting C. 

Note that M(A K ) = M(A K ). The reorientation on the subset S ⊂ [n], of the oriented 

matroid M, is the oriented matroid, denoted −S M, such that C( −S M) := { −S C: C ∈ 

C(M)}. We say also that C( −S M) is the reorientation on the subset S of C(M). (The 

concept of “reorientation” is the combinatorial analogue of the notion of “nonsingular 

projective permissible transformation”.) 

Fix a set E := {e 1 ,...,e n } and let K be a commutative ring with unity 1. Let K[E] 

denote the commutative free K-algebra given by the generators E ∪{1}. For every 

X ⊂ [n], set e X := ∏ i∈X e i, e ∅ := 1. 

Definition 2.1 [8, Definition 2.2 and Proposition 2.3]. 

∂: C(M(A K )) → K[E], C ↦→ 

Consider the map 

m∑ 

ζ ij e C\ij , 

j=1 

where C ={i 1 ,...,i m }, i 1 < ··· 

j=1 ζ i j 

θ Hij 

= 0, ζ i1 = 1. U(A K ) is the 

(commutative) K-algebra given by the generators 1, e 1 ,...,e n , and the relations: 

◦ e i e j = e j e i , ∀i, j = 1,...,n, 

◦ e 2 i 

= 0, ∀i = 1,...,n, 

◦ ∂(C) = 0, ∀C ∈ C(M(A K )). 

We call U(A K ) the Orlik–Terao algebra of A K . 

Now, we introduce the “combinatorial analogue” of the Orlik–Terao algebra.


Definition 2.2. 

Consider the map 

˜∂: C(M) → Z[E], 

C ↦→ 

m∑ 

C(i j )e C\ij , 

j=1 

where C ={i 1 ,...,i m }, i 1 < ···

C, such that C(i 1 ) = 1. A(M) is the (commutative) Z-algebra given by the generators 

1, e 1 ,...,e n , and the relations: 

◦ e i e j = e j e i , ∀i, j = 1,...,n, 

◦ e 2 i 

= 0, ∀i = 1,...,n, 

◦ e i = 0, if i is a loop of M, 

◦ ˜∂(C) = 0, ∀C ∈ C(M), |C| > 1. 

We call A(M) the algebra of the oriented matroid M. 

For every X ⊂ [n], we denote by [X] A (resp. [X] U ), or shortly by [X] orevene X 

when no confusion will result, the residue class in A(M) (resp. U(A K )) determined by 

the element e X . Note that A(M) ∼ = A(M\x) if x is a loop or x is parallel to some other 

element of M. So in what follows we suppose that M is a simple matroid. For every 

circuit C ∈ C(M), we have [C] A = 0. To see this, pick an element x ∈ C if |C| > 1. 

Then 0 = e x · ˜∂(e C ) =±e C . We conclude that if [X] A ≠ 0, then X is an independent 

set of M. 

The “abstract algebra” A(M) has a canonical grading. 

Proposition 2.3. Set A l = A l (M) be the submodule of A(M) generated by the 

elements {[X] A : X ∈ IND l (M)}. The grading A(M) = ⊕ l∈N A l(M) is canonical, 

i.e., it is independent of the knowledge of the oriented matroid M. 

Proof. We know that A l (M) = (0), for all l>r. If A(M) = A 0 = Z (i.e., r = 0) the 

result is clear. Suppose that A(M) ≠ Z. Note that A r ={x ∈ A(M): x · y = 0, ∀y ∈ 

A(M)\Z}. If we know the modules A r ,...,A r−i and A 〈i+1〉 := A r ⊕···⊕A r−i ≠ 

A(M), (i.e., r −i > 1) the module A r−i−1 , i = 0,...,r −2, can be defined recursively 

by 

A r−i−1 ={x ∈ A(M): x · y ∈ A 〈i+1〉 , ∀y ∈ A(M)\Z} / A 〈i+1〉 . 

Proposition 2.4. For every x ∈ [n] there is a unique epimorphism of Z-modules 

p x : A(M) → A(M/x), such that, for every I ∈ IND(M), we have 

p x (e I ) := 

{ 

eI \x if x ∈ I ∈ IND(M), 

0 otherwise. 

(2.1) 

Proof. 

It is enough to prove that 

p x (e X ·˜∂(C)) = 0, ∀X ⊂ [n], ∀C ∈ C(M). (2.2)


We can suppose that X ∩ C =∅and x ∈ X ∪ C. Let Y be a signed set on [n] supporting 

Y ={i 1 ,...,i m }. For convenience of notation set 

˜∂ Y (Y) := 

m∑ 

Y(i j )e Y\ij 

j=1 

∈ Z[E]. 

Let C be one of the opposite signed circuits supporting C. Remember that the signed set 

C\x on [n]\x is a vector (union of signed circuits) of M/x. So we have 

{ 

±eX ·˜∂ C\x (C\x) = 0 if x ∈ C, 

p x (e X ·˜∂(C)) = 

±e X\x ·˜∂ C (C) = 0 if x ∈ X. 

Corollary 2.5. 

are equivalents: 

For every subset X ={i 1 ,...,i m }⊂[n], the following two conditions 

◦ X is an independent set of M, 

◦ [X] A = e i1 e i2 ···e im ≠ 0. 

Proof. It remains to prove that if X is an independent set of M, then [X] A ≠ 0. We 

prove by induction on n. We know that [∅] A = 1. Suppose that the implication is true 

for all the matroids with at most n − 1 elements. Let X, |X| > 0, be an independent set 

of M and pick an element x ∈ X. Suppose for a contradiction that [X] A = 0. X\x is 

an independent set of M/x. From Proposition 2.4 we conclude that 0 = p x ([X] A ) = 

[X\x] A(M/x) 

, a contradiction with the induction hypothesis. 

Proposition 2.6. For every x ∈ [n] there is a unique morphism of Z-modules, i x : A 

(M\x) → A(M), such that, for every I ∈ IND(M\x), we have i x (e I ) = e I . 

Proof. 

The map i x is well determined. Indeed from Corollary 2.5 we know that 

i x (e I ·˜∂(C)) = e I ·˜∂(C) = 0, ∀I ∈ IND(M\x), ∀C ∈ C(M\x). 

Set nbc l :={[I ] A : I ∈ NBC l (M)} and nbc := ⋃ l=0 nbc l. Set M ′ = M\x, 

M ′′ = M/x, A := A(M), A ′ := A(M ′ ) and A ′′ := A(M ′′ ). Consider an independent 

set X ∈ IND l (M), suppose that X /∈ NBC l (M), and set α = α(X). 

Making use of the definition of ˜∂(C(X,α))we can express the element [X] A as a linear 

combination of the elements {[X x ] A : X x = X\x ∪ α, x ∈ C(X,α)\α}. We claim that 

EA(X x ) ⊂ EA(X)\α. Indeed, suppose that β ∈ EA(X x ). If α ∉ C(X x ,β)= C(X,β), 

then we have β ∈ EA(X)\α. If α ∈ C(X x ,β)we have β


Theorem 2.7. 

modules 

For every element x of M, there is a split short exact sequence of 

0 → A(M\x) 

i x 

p x 

−→ A(M) −→ A(M/x) → 0. (2.3) 

We postpone the proof of the theorem. The following corollary is an important direct 

consequence of the above theorem. Corollary 2.8 is similar to a well-known result 

concerning the algebras of Orlik–Solomon, Theorem 3.55 of [7]. 

Corollary 2.8. Suppose that the sequence (2.3) is exact for all the matroids with at 

most n elements. Then nbc(M) is a basis of the module A(M). 

Proof. We prove by induction on n. If n = 0 we know that A(M(∅)) = Z and 

nbc(M(∅)) ={1}. Suppose that n > 0 and that the result is true for all the matroids 

with at most n − 1 elements. By a reordering of the elements of the matroid M we can 

suppose that x = n. It is clear that 

NBC(M ′ ) ={X: X ⊂ [n − 1] and X ∈ NBC(M)}. 

From the induction hypothesis we know that nbc(M ′ ) ={[X] A ′: X ∈ NBC(M ′ )} 

and nbc(M ′′ ) ={[X] A ′′: X ∈ NBC(M ′′ )} are bases of A ′ and A ′′ , respectively. The 

minimal broken circuits of M/n are the minimal sets X such that either X or X ∪{n} 

is a broken circuit of M (see Proposition 3.2.e of [4]). Then 

NBC(M ′′ ) ={X: X ⊂ [n − 1] and X ∪{n} ∈NBC(M)} and (2.4) 

NBC(M) = NBC(M ′ ) ⊎{I ∪ n: I ∈ NBC(M ′′ )}. (2.5) 

We know that nbc(M) ={[X] A : X ∈ NBC(M)} is a generating set of A. So Corollary 

2.8 follows from the exactness of sequence (2.3). 

Theorem 2.7 is a consequence of Lemmas 2.10–2.12 below. 

Lemma 2.9. Suppose that sequence (2.3) is exact for all the matroids with at most 

n − 1 elements. Then for every x ∈ [n], there is an exact sequence of Z-modules 

i x p x 

−→ A −→ A ′′ → 0. 

A ′ 

Proof. From the definitions we know that p x ◦i x , is the null map so Im(i x ) ⊂ Ker(p x ). It 

remains to prove the inclusion Ker(p x ) ⊂ Im(i x ). By a reordering of the elements of [n] 

we can suppose that x = n. Suppose that ∑ m 

i=1 ζ i[I i ] A ∈ Ker(p n ), [I i ] A ∈ nbc(M), ζ i ∈ 

Z, and set 

m∑ 

ζ i [I i ] A = ∑ ζ i ′[I i ′] A + ∑ ζ i ′′[I i ′′] A , i ′ , i ′′ ∈{1,...,m}. 

I i ′ ∌n 

I i ′′ ∋n 

So, 

i=1 

( ) 

m∑ 

p n ζ i [I i ] A = ∑ ζ i ′′[I i ′′\n] A ′′ = 0. (2.6) 

i=1 

I i ′′ ∋n


From (2.4) we conclude that [I i ′′\n] A ′′ ∈ nbc(A ′′ ). From Corollary 2.8 we know that 

nbc(A ′′ ) is a basis of A ′′ . So, (2.6) implies that ζ i ′′ = 0 for every i ′′ . Hence ∑ m 

∑ 

i=1 ζ i[I i ] A = 

I i ′ ∌n ζ i ′[I i ′] A = i n ( ∑ ζ i ′[I i ′] A ′) ∈ Im(i n ). 


n − 1 elements. Let a be an element of the module A(M) = Z ⊕ A 1 ⊕···⊕A r . Then 

a ∈ Z iff p x (a) = 0, for every x ∈ [n]. 

Proof. From (2.1) we see that if a = ζ [∅] A ∈ A 0 (= Z), then p x (a) = 0, for all x ∈ [n]. 

We prove that (p x (a) = 0, ∀x ∈ [n]) ⇒ a ∈ Z. Suppose that p x (a) = 0, ∀x ∈ [n]. 

From Lemma 2.9 we know that 

a ∈ ⋂ 

Ker(p x ) = ⋂ 

Im(i x ) = Z. 

x∈[n] 


n − 1 elements. Then the sequence 0 → A ′ i x 

−→ A is exact. 

Proof. The case n = 1 is trivial. Suppose that n > 1. It is well known that M\x/y = 

M/y\x, for every pair of elements x, y ∈ [n], x ≠ y. Consider the epimorphism 

p ′ y : A(M\x) → A(M\x/y). 

Consider also the monomorphism i ′ x : A(M\x/y) → A(M/y). It is easy to check that 

the following diagram of modules is commutative: 

x∈[n] 

A(M\x) 

⏐ 

↓ p′ y 

i x 

−−−−→ 

A(M) 

⏐ 

↓ p y 

i ′ x 

A(M\x/y) −−−−→ A(M/y). 

We prove the implication i x (a) = i x (b) ⇒ a = b, for every pair a, b ∈ A(M\x). 

We know that 

i x (a) = i x (b) ⇒ (p y ◦ i x (a) = p y ◦ i x (b), ∀y ∈ [n]\x), 

p y ◦ i x (a) = i ′ x ◦ p′ y (a) and p y ◦ i x (b) = i ′ x ◦ p′ y (b), ∀y ∈ [n]\x, 

i ′ x ◦ p′ y (a) = i′ x ◦ p′ y (b) ⇐⇒ p′ y (a) = p′ y (b), ∀y ∈ [n]\x. 

From Lemma 2.10 we know that (p ′ y (a) = p′ y (b), ∀y ∈ [n]\x) ⇔ a − b = ζ ∈ Z. Then 

we have 0 = i x (a) − i x (b) = i x (ζ ) = ζ and so a = b. 

Lemma 2.12. Suppose that sequence (2.3) is exact for all the matroids with at most n 

elements. Then sequence (2.3) for the matroid M([n]) splits, i.e., there is a morphism 

of modules p −1 

x : A′′ → A such that p x ◦ p −1 

x is the identity map and 

A(M) = i x (A(M\x)) ⊕ p −1 

x (A(M/x)).


Proof. We can suppose that x = n. From Corollary 2.8, we know that nbc(M ′ ) 

and nbc(M ′′ ) are bases of A ′ and A ′′ , respectively. There is a morphism of modules 

p −1 

n : A′′ → A well determined by the conditions p −1 

n ([I ] A ′′) := [I ∪ n] A for all I ∈ 

NBC(M ′′ ). It is clear that p n ◦ p −1 

n is the identity map. From (2.5) we conclude that the 

exact sequence (2.3) splits. 

Remark 2.13. With some adjustments, our techniques also give proofs of Proposition 

2.3, Corollary 2.5, Theorem 2.7 and Corollary 2.8 for the algebra of Orlik–Terao. 

3. Applications 

Proposition 3.1. Let V be a vector space of dimension d over an ordered field K, and 

let A be a central arrangement of hyperplanes in V. Consider the algebras U = U(A K ) 

and A = A(M(A K )) and let nbc(U) and nbc(M) be the corresponding no broken 

circuit bases. For a given X ⊂ [n], suppose that 

(1) [X] U = ∑ n 

i=1 ξ i[I i ] U , [I i ] U ∈ nbc(U), ξ i ∈ K and 

(2) [X] A = ∑ n 

i=1˜ξ i [I i ] A , [I i ] A ∈ nbc(M),˜ξ ∈{±1, 0} ⊂Z. 

Then sign(ξ i ) = ˜ξ i , i = 1,...,n. 

We make use of the following lemma: 

Lemma 3.2 [5]. 

Let G = (V, E) be the direct graph defined as follows: 

◦ V(G) = IND(M(A K )). 

◦ −→ II ′ ∈ E(G) is a directed edge of G iff there is a pivotable pair (α, x) such that 

I ′ = I \x ∪ α, where α = α(I ) and x ∈ C(I,α)\α. 

For every pair of vertices X, X ′ of the graph G, there is at most one directed path from 

XtoX ′ . 

Proof. We attach to G two edge-labelling graphs determined respectively by the algebras 

A(M(A K )) and U(A K ). These labelled graphs are denoted G A and G U , respectively. 

Let −→ II 

i ′ be an edge of G where α = α(I ) = I 

i ′ \I, C = C(I,α) = 

{α, x 1 ,...,x i ,...,x m } and I 

i 

′ = I \x i ∪ α. Let C ∈ C(M(A K )) be the signed circuit 

supporting C and such that C(α) = 1. 

(1) Suppose that ∂(C) = e C\α + ∑ m 

i=1 ζ ie C\xi , i.e., [I ] U = ∑ m 

i=1 −ζ i[I 

i ′] 

U. We label 

the edge −→ II 

i ′ of G U with the scalar −ζ i . 

(2) Suppose that ˜∂(C) = e C\α + ∑ m 

i=1 C(x i)e C\xi i.e., [I ] A = ∑ m 

i=1 −C(x i)[I 

i ′] 

A. 

We label the edge −→ II 

i ′ of G A with the scalar −C(x i ). 

From the definitions we know that sign(ζ i ) = C(x i ), for every i = 1,...,m. Let 

P 1 ,...,P s be the list of the maximal length directed paths of G, beginning with the 

vertex I . Let T l denote the last vertex of the path P l , ∀l = 1,...,s. T l is a sink of G,


so [T l ] A ∈ nbc(M) (resp. [T l ] U ∈ nbc(U)). As K is an ordered field, the proposition 

follows. 

Definition 3.3. Let B ={b 1 ,...,b n } be an arbitrary basis of U 1 (resp. A 1 ) such that the 

set IND(M ′ ) := {{j 1 ,..., j s }⊂[n]: b j1 b j2 ···b js ≠ 0} is the family of the independent 

sets of a matroid M ′ ([n]). We say that U fixes M(A K ) (resp. A fixes M) if the following 

condition holds: 

◦ There is a permutation σ ∈ S n and invertible scalars ζ i such that b i = ζ i 

e σ(i) 

, i = 

1,...,n. (Note that σ : M ′ ∼ = M(AK ) (resp. σ : M ′ ∼ = M).) 

Proposition 3.4. In general we cannot reconstruct M from the “abstract algebra” 

A(M). In the case when the algebra A(M) fixes M we can reconstruct the signed set 

of circuits C(M), up to a reorientation and a permutation of the ground set [n]. 

Proof. (The following example is similar to one of [6].) Consider the two direct graphs 

G 1 = (V 1 , E 2 ) and G 2 = (V 2 , E 2 ) (see Figs. 1 and 2): 

◦ V 1 :={v 1 ,...,v 5 }, 

◦ E 1 :={a 1 = −−→ v 1 v 2 , a 2 = −−→ v 2 v 3 , a 3 = v −−→ 

3 v 1 , a 4 = v −−→ 

4 v 1 , a 5 = v −−→ 

1 v 5 , a 6 = v −−→ 

5 v 4 }. 

◦ V 2 :={v 1 ,...,v 5 }, 

◦ E 2 :={b 1 = −−→ v 1 v 2 , b 2 = −−→ v 2 v 3 , b 3 = v −−→ 

4 v 3 , b 4 = v −−→ 

3 v 1 , b 5 = v −−→ 

1 v 5 , b 6 = v −−→ 

5 v 3 }. 

Let M G1 (resp. M G2 ) be the oriented matroid on the ground set {1,...,6} determined 

by the graph G 1 (resp. G 2 ). More precisely: 

◦ M G1 has two pairs of opposite signed circuits C 1 , −C 1 and C 2 , −C 2 where 

C + 1 ={1, 2, 3}, C+ 2 ={4, 5, 6} and C− 1 = C− 2 =∅. 

◦ M G2 has three pairs of opposite signed circuits D 1 , −D 1 , D 2 , −D 2 and D 3 , 

−D 3 where D + 1 ={1, 2, 4}, D+ 2 ={4, 5, 6}, D− 1 = D− 2 =∅, D+ 3 

={1, 2, } and 

={5, 6}. 

D − 3 

Consider the algebras A = A(M G1 ) and B = A(M G2 ). From Definition 2.2 we know 

that A is the commutative Z-algebra generated by the seven elements 1, e 1 ,...,e 6 and 

Fig. 1. Graph G 1 . Fig. 2. Graph G 2 .


the relations 

e i e j = e j e i , e 12 + e 13 + e 23 = 0, e 45 + e 46 + e 56 = 0, e 2 i = 0 (3.1) 

for all i = 1,...,6. Similarly, B is the commutative Z-algebra generated by the seven 

elements 1, e 

1 ′ ,...,e′ 6 

and the relations 

e ′ i e′ j = e ′ j e′ i , e′ 12 + e′ 14 + e′ 24 = 0, e′ 45 + e′ 46 + e′ 56 = 0, e′ i e′ i = 0 (3.2) 

for all i = 1,...,6. (The relation −e 

126 ′ − e′ 125 + e′ 156 + e′ 256 

= 0 is redundant. Indeed, 

from the relations (3.2), we deduce that −e 

126 ′ = e′ 146 + e′ 246 , −e′ 125 = e′ 145 + e′ 245 , 

e 

156 ′ =−e′ 146 − e′ 145 and e′ 256 =−e′ 245 − e′ 246 .) 

Let : A → B be the morphism of Z-algebras determined by the values (1) = 1, 

(e l ) = e 

l ′ + e′ 3 + e′ 4 ,l= 1, 2,(e 3) =−e 

3 ′ and (e l) = e 

l ′ ,l= 4, 5, 6. The map 

is well defined. Indeed 

(e 12 + e 13 + e 23 ) = (e 1 ′ + e′ 3 + e′ 4 )(e′ 2 + e′ 3 + e′ 4 ) − e′ 3 (e′ 1 + e′ 2 + 2e′ 3 + 2e′ 4 ) 

= e 12 ′ + e′ 24 + e′ 14 =0, 

and 

(e 45 + e 46 + e 56 ) = e 45 ′ + e′ 46 + e′ 56 = 0. 

Consider now the morphism of Z-algebras : B → A determined by the values (1) = 

1,(e 

l ′ ) = e l + e 3 − e 4 ,l= 1, 2,(e 

3 ′ ) =−e 3 and (e 

l ′ ) = e l,l= 4, 5, 6. The map 

is well defined. Indeed 

(e 12 ′ + e′ 14 + e′ 24 ) = (e 1 + e 3 − e 4 )(e 2 + e 3 − e 4 ) + e 4 (e 1 + e 2 + e 1 + 2e 3 − 2e 4 ) 

= e 12 + e 13 + e 23 = 0, 

and 

(e 45 ′ + e′ 46 + e′ 56 ) = e 45 + e 46 + e 56 = 0. 

As ◦ = 1 A and ◦ = 1 B , we conclude that the is an isomorphism. 

Suppose now that A = A(M) fixes M and let e 1 ,...,e n be a family of generators of 

A 1 in one-to-one correspondence e i ↔ i with the elements of the ground set [n]ofM. 

Let C be the signed circuit of M supporting C ={i 1 ,...,i k }, k > 1, i 1 < ··· 

and such that C(i 1 ) = 1. By a reordering of the elements of [n] we may suppose 

that i 1 = 1,...,i j = j. Let nbc(M) be the no broken circuit basis of the algebra A 

relative to this new ordering of [n]. It is clear that [1 ··· ĵ ···k] A ∈ nbc(M), for every 

j = 2,...,k, but C\1 ={2,...,k} is a broken circuit. We know that [2 ···k] A = 

− ∑ k 

j=2 C( j)[1 ··· ĵ ···k] A . So, we can recover the signed circuit C. We conclude that 

the base e 1 ,...,e n of A 1 determines C(M) up to a reorientation and a permutation of 

the ground set [n]. 

Theorem 3.5. In general we cannot reconstruct the intersection lattice L(A) from the 

“abstract algebra” U(A).


Proof. Let e 1 ,...,e 6 be the canonical basis of the vector space R 6 . Consider the graph 

G 1 = (V 1 , E 1 ), introduced in Proposition 3.4. To every edge a h = −→ v i v j ∈ E 1 , we attach 

the vector v h = e i − e j . Set 

A R ={H l ⊂ (R 6 ) ⋆ : H l ={x ∈ (R 6 ) ⋆ : 〈x,v l 〉=0},l= 1,...,6}. 

From the definitions, we see that M(A R ) = M G1 . For every one of the four signed 

circuits C ∈ C(M(A R )) we have ∑ 6 

l=1 C(l)v l = 0. So, U(A R ) = A(M(A R )) ⊗ Z R. 

Similarly, we construct an arrangement of hyperplanes B R in (R 6 ) ⋆ , determined by the 

graph G 2 = (V 2 , E 2 ), and we also have U(B R ) = A(M(B R ))⊗ Z R. So U(A R ) ∼ = U(B R ) 

and L(A R ) ≁ = L(BR ). 

We finish with an open question. Orlik and Terao ask in [8]: 

◦ Is U(A C ) the model for any topological invariant of the manifold M(A C ) = 

C d \ ⋃ H∈A 

H 

C 

A partial solution to this problem can be obtained from Proposition 3.4 above and a 

celebrated theorem of Björner and Ziegler [3]: 

Proposition 3.6. Suppose that A C is the complexification of a real arrangement A R 

and the algebra U(A C ) fixes the matroid M(A C ). Then the abstract algebra U(A C ) 

determines the smooth manifold M(A C ) up to homeomorphism. 

Proof. We know that U(A C ) = U(A R ) ⊗ R C and M(A C ) ∼ = M(A R ). From Proposition 

3.4 we conclude that the algebra U(A C ) determines (up to reorientation and a 

permutation of the ground set [n]) the oriented matroid M(A R ). So the smooth manifold 

M(A C ) is well determined up to homeomorphism, see [3]. 

A combinatorial analogue of the question of Orlik–Terao is: 

◦ Which features of the oriented matroid M are reflected in the algebra A(M) 

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2. Björner, A., Las Vergnas, M., Sturmfels, B., White, N., and Ziegler, G. M.: Oriented matroids, second edition. 

Encyclopedia of Mathematics and its Applications, vol. 46. Cambridge University Press, Cambridge, 

1999. 

3. Björner, A., and Ziegler, G. M.: Combinatorial stratification of complex arrangements. J. Amer. Math. Soc. 

5(1) (1992), 105–149. 

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165–170. 

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vol. 300. Springer-Verlag, Berlin, 1992.


8. Orlik, P., and Terao, H.: Commutative algebras for arrangements. Nagoya Math. J. 134 (1994), 65–73. 

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University Press, Cambridge, 1986. 

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Cambridge University Press, Cambridge, 1987. 

Received November 7, 2000, and in revised form May 18, 2001. Online publication November 2, 2001.

A Commutative Algebra for Oriented Matroids

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