Syzygies of Unimodular Lawrence Ideals

Dave Bayer Sorin Popescu Bernd Sturmfels

Abstract: Infinite hyperplane arrangements whose vertices form a lattice are studied from the point of view of commutative algebra. The quotient of such an arrangement modulo the lattice action represents the minimal free resolution of the associated binomial ideal, which defines a toric subvariety in a product of projective lines. Connections to graphic arrangements and to Beilinson’s spectral sequence are explored.

1 Introduction

We are interested in the defining ideals of toric subvarieties in a product of projective lines ℙ^{1} × ℙ^{1} × ⋯ × ℙ^{1}. Writing (x_{i} : y_{i}) for the homogeneous coordinates of the i-th factor ℙ^{1}, these are the binomial ideals in 2n variables of the following form:

J_{L} = ⟨ x^{a}y^{b} − x^{b}y^{a} | a − b ∈ L ⟩ ⊂ S = k[x_{1}, …, x_{n}, y_{1}, …, y_{n}],

where L is a sublattice of ℤ^{n} and k is a field. Here x^{a} = x_{1}^{a_{1}}x_{2}^{a_{2}} ⋯ x_{n}^{a_{n}} for a = (a_{1}, …, a_{n}) ∈ ℕ^{n}. Binomial ideals of the form J_{L} are called Lawrence ideals. They provide the algebraic analogue to the Lawrence construction for convex polytopes [Zi, §6.6]. Lawrence polytopes enjoy remarkable rigidity properties, such as [Zi, Theorem 6.27]. On the algebraic side, rigidity of Lawrence ideals manifests itself in the following result, which appears in [Stu, Theorem 7.1]. Recall for part (d) that the Graver basis consists of all binomials x^{a}y^{b} − x^{b}y^{a} in J_{L} such that the only vector a′ − b′ ∈ L\{0} with 0 ≤ a′ ≤ a and 0 ≤ b′ ≤ b is a − b itself.

Proposition 1.1 The following sets of binomials in a Lawrence ideal J_{L} coincide:

(a) Any minimal set of binomial generators of J_{L}.

(b) Any reduced Gröbner basis for J_{L}.

(c) The universal Gröbner basis for J_{L} (the union of all reduced Gröbner bases).

(d) The Graver basis for J_{L}.

Cellular resolutions, as defined in [BS], provide a natural geometric framework for studying homological, algorithmic and combinatorial properties of monomial and binomial ideals. One instance is the Scarf complex, which gives the minimal resolution for generic monomial ideals [BPS] and generic lattice ideals [PS]. The hull resolution of [BS] generalizes the Scarf complex and provides a cellular resolution for arbitrary co-Artinian monomial modules; however, it need not be minimal. For lattice modules, the hull resolution is compatible with the lattice action and determines a cellular resolution of the corresponding lattice ideal [BS, Theorem 3.9].

In this paper we present a minimal cellular resolution, which happens to also coincide with the hull resolution, for a remarkable class of nongeneric lattice ideals. These are the unimodular Lawrence ideals J_{L}, which are characterized as follows:

Theorem 1.2 For a sublattice L of ℤ^{n} the following conditions are equivalent:

(a) The Lawrence ideal J_{L} possesses an initial monomial ideal which is radical.

(b) Every initial monomial ideal of the Lawrence ideal J_{L} is a radical ideal.

(c) Every minimal generator of J_{L} is a difference of two squarefree monomials.

(d) The lattice L is the image of an integer matrix B with linearly independent columns, such that all maximal minors of B lie in the set {0, 1, −1}.

(e) The lattice L is the kernel of an integer matrix A with linearly independent rows, such that all maximal minors of A lie in {0, m, −m} for some integer m.

(f) The quotient ring S/J_{L} is a normal domain.

Theorem 1.2 is proved in Section 2. If any (and thus all) of these six equivalent conditions for L holds, then we say that the Lawrence ideal J_{L} is unimodular. A first example is the ideal of 2 × 2-minors of a 2 × n-matrix of indeterminates:

J_{L} = I_{2} ( \begin{matrix}x_{1} & x_{2} & x_{3} & ⋯ & x_{n} \\ y_{1} & y_{2} & y_{3} & ⋯ & y_{n} \end{matrix} )  (1.1)

Here L is the kernel of A = ( 1 1 1 ⋯ 1 ), or the image of the matrix B whose rows are e_{i} − e_{i+1}, i ∈ {1, …, n − 1}, the differences of consecutive unit vectors in ℝ^{n}. The minimal resolution of (1.1) is an Eagon-Northcott complex, whose polyhedral model is the hypersimplicial complex of Gel’fand and MacPherson [BS, Ex. 3.15].

In Section 2 we introduce an infinite periodic hyperplane arrangement H_{L} whose vertices are the elements of L. It is shown in Section 3 that this arrangement supports the minimal free resolution of J_{L}, and coincides with the hull complex of J_{L}. In Section 4 we prove that this resolution is universal in the sense that it is stable under all Gröbner deformations. In particular, all initial ideals of J_{L} have the same Betti numbers as J_{L}, and their minimal resolutions are also cellular. We also construct cellular resolutions for the monomial ideals defined by the fibers of L. In Section 5 we discuss Lawrence ideals associated with directed graphs and we present open combinatorial problems. In Section 6 we reinterpret Lawrence ideals in terms of the Audin-Cox homogeneous coordinate ring, and we generalize Beilinson’s spectral sequence from projective space to unimodular toric varieties.

The authors would like to thank the Mathematical Sciences Research Institute in Berkeley for its support while part of this paper was being written. All three authors were partially supported by the NSF during the preparation of this work.

2 Unimodularity and an infinite hyperplane arrangement

In this section we establish some basic facts about unimodular lattices and their Lawrence ideals. We start out by proving the equivalences stated in Section 1. Proof of Theorem 1.2. A monomial ideal is radical if and only if it is generated by squarefree monomials. Clearly, the first term of a binomial x^{a}y^{b} − x^{b}y^{a} is squarefree if and only if the second term is squarefree. The equivalence of (a), (b) and (c) follows directly from Proposition 1.1.

In an exact sequence of free abelian groups,

0 ⟶ ℤ^{n−d} ⟶^{B} ℤ^{n} ⟶^{A} ℤ^{d},  (2.1)

each (n−d) × (n−d)-minor of B is equal to the complementary d × d-minor of A, up to a global constant m. Since the cokernel of B is torsion free, the maximal minors of B are integers with no common factor. This implies m ∈ ℤ and the equivalence of (d) and (e), for L = ker(A) = im(B).

The condition (e) is precisely the defining condition given in [Stu, §8, page 70] for a matrix A to be unimodular. A matrix A is unimodular if and only if its Lawrence lifting Λ(A) is unimodular. Following [Stu, §7, page 55], the Lawrence lifting of the d × n-matrix A is obtained by appending the zero d × n-matrix 0_{d,n} and two copies of the n × n-identity matrix I_{n} as follows:

Λ(A) = ( \begin{matrix}A & 0_{d,n} \\ I_{n} & I_{n} \end{matrix} ) .

Hence the equivalence of (b) and (e) is a reformulation of [Stu, Remark 8.10].

The conjunction of (b) and (e) implies property (f), namely, the lattice L is the kernel of an integer matrix if and only if J_{L} is a prime ideal, and (b) implies normality by [Stu, Proposition 13.15]. To complete the proof, it suffices to show that (f) implies (c). Suppose that (c) is false, i.e., the ideal J_{L} has a minimal generator x^{a}y^{b} −x^{b}y^{a} whose terms are not squarefree. We may assume that this generator is a circuit, which means that it has minimal support. By setting all pairs of variables not appearing in this circuit to zero, we reduce to the case where J_{L} is a principal ideal. But an affine binomial hypersurface is normal if and only if at least one of its monomials is squarefree. This completes our proof of Theorem 1.2.

Here is another, more invariant, formulation of the unimodularity condition.

Proposition 2.1 A lattice L is unimodular if and only if, for every projection π(L) of L to a coordinate sublattice ℤ^{r} ⊂ ℤ^{n}, the quotient ℤ^{r}/π(L) is torsion free.

Proof. We will use the notation of the proof of Theorem 1.2. We first prove the “if” direction. Set m = n − d and consider any nonsingular m × m-submatrix C of B. The image of C in ℤ^{m} equals π(L) for the corresponding coordinate projection π : ℤ^{n} → ℤ^{m}. The finite abelian group ℤ^{m}/π(L) is torsion free if and only if it is zero. Hence π(L) = ℤ^{m} and we conclude that the determinant of the m × msubmatrix of B under consideration is either 1 or −1.

For the “only-if” direction, suppose that L is unimodular. Every coordinate projection π(L) of L is unimodular as well. This can be seen by choosing A (resp. B) to have appropriate unit vectors among its columns (resp. rows). Thus it suffices to show that the group ℤ^{n}/L is torsion free. But this follows from the exact sequence (2.1), which identifies ℤ^{n}/L with the image of the matrix A, showing ℤ^{n}/L ≃ ℤ^{d}.

Let L be any m-dimensional sublattice of ℤ^{n}, and write ℝL for the linear subspace of ℝ^{n} spanned by L. We denote by H_{L} the affine hyperplane arrangement in ℝL obtained by intersecting ℝL with all lattice translates of the coordinate hyperplanes in ℝ^{n}. These are the hyperplanes {x_{i} = j} for 1 ≤ i ≤ n and j ∈ ℤ. Thus H_{L} is an infinite m-dimensional hyperplane arrangement in the vector space ℝL.

It is convenient to embed H_{L} as a hyperplane arrangement in the Euclidean space ℝ^{m} as follows. Let B be an integer n × m-matrix such that L = im(B) as in part (d) of Theorem 1.2. Write b_{i} ∈ ℤ^{m} for the i-th row vector of B. Then H_{L} is isomorphic to the infinite arrangement in ℝ^{m} consisting of the hyperplanes

H_{ij} = { x ∈ ℝ^{m} | b_{i} · x = j } for all i ∈ {1, 2, …, n} and all j ∈ ℤ.

Proposition 2.2 Each lattice point in L is a vertex of the affine hyperplane arrangement H_{L}. There are no additional vertices in H_{L} if and only if L is unimodular.

Proof. We identify L with ℤ^{m} via the matrix B and thus consider H_{L} as the hyperplane arrangement H_{L} = {H_{ij}} in ℝ^{m}. The intersection point of m linearly independent such hyperplanes,

H_{i_{1}j_{1}} ∩ H_{i_{2}j_{2}} ∩ ⋯ ∩ H_{i_{m}j_{m}} = {x},

is defined by the linear equations b_{i_{ν}} · x = j_{ν} for ν = 1, …, m. The point x ∈ ℝ^{m} has integer coordinates for all j_{1}, …, j_{m} ∈ ℤ if and only if det(b_{i_{1}}, …, b_{i_{m}}) = ±1. By part (d) in Theorem 1.2, this means that the lattice L is unimodular. The first assertion holds because each x ∈ ℤ^{m} can be expressed as such an intersection.

From now on we assume that L is a unimodular sublattice of ℤ^{n}. Recall from [Stu, Proposition 8.11] that the Graver basis of the Lawrence ideal J_{L} is given precisely by the circuits of the lattice L. (The circuits of L are the primitive vectors in L whose supports are minimal with respect to inclusion.) We view H_{L} as an infinite regular cell complex, equipped with an action by the abelian group L.

Lemma 2.3 Two vertices a and b of the arrangement H_{L} are connected by an edge if and only if their difference a − b is a circuit of the unimodular lattice L.

Proof. Since L acts transitively on the vertices of H_{L} we may assume that b = 0. Our assertion states that a is a circuit if and only if {0, a} forms an edge in H_{L}. This holds because the circuits in the subspace ℝL of ℝ^{n} are computed by the rule

H_{i_{1}0} ∩ H_{i_{2}0} ∩ ⋯ ∩ H_{i_{m−1}0} = ℝa

for all possible increasing sequences of indices 1 ≤ i_{1} < i_{2} < ⋯ < i_{m−1} ≤ n.

In this section we have introduced a family of hyperplane arrangements H_{L} whose vertices form a lattice L. Such arrangements appear in many parts of the mathematical literature; for example, see [BLSWZ]. The group L acts on the faces of the arrangement H_{L} with finitely many orbits, and we shall be interested in the quotient complex H_{L}/L. Nontrivial examples will be presented in Section 5. First we reinterpret H_{L} and H_{L}/L as minimal free resolutions in the sense of commutative algebra. This will be done in the next two sections by labeling the faces of H_{L} with Laurent monomials, following the general recipes in [BS].

Our warmup example (1.1) is the case where L is spanned by the vectors e_{i} − e_{i+1}, i ∈ {1, …, n − 1}, the differences of the consecutive unit vectors in ℝ^{n}. Thus J_{L} is here the ideal of 2×2-minors, and H_{L} is the hypersimplicial arrangement. The quotient complex H_{L}/L has n−1 distinct maximal faces, called hypersimplices. The hypersimplicial arrangement for n = 3 is depicted in the following figure:

\frac{y_{1}^{2}y_{2}x_{3}^{3}}{x_{1}^{2}x_{2}y_{3}^{3}}  ^{\frac{y_{1}y_{2}x_{3}^{2}}{x_{1}x_{2}y_{3}^{2}}}  ^{\frac{y_{2}x_{3}}{x_{2}y_{3}}}  ^{\frac{x_{1}y_{2}}{y_{1}x_{2}}}  ^{\frac{x_{1}^{2}y_{2}y_{3}}{y_{1}^{2}x_{2}x_{3}}}^{\frac{y_{1}^{2}x_{3}^{2}}{x_{1}^{2}y_{3}^{2}}}  ^{\frac{y_{1}x_{3}}{x_{1}y_{3}}}  1  ^{\frac{x_{1}y_{3}}{y_{1}x_{3}}}^{\frac{y_{1}^{3}x_{2}x_{3}^{2}}{x_{1}^{3}y_{2}y_{3}^{2}}}  ^{\frac{y_{1}^{2}x_{2}x_{3}}{x_{1}^{2}y_{2}y_{3}}}  ^{\frac{y_{1}x_{2}}{x_{1}y_{2}}}  ^{\frac{x_{2}y_{3}}{y_{2}x_{3}}}  ^{\frac{x_{1}x_{2}y_{3}^{2}}{y_{1}y_{2}x_{3}^{2}}}

Figure 1: The hypersimplicial arrangement for n = 3.

Its hypersimplices are up-triangles and down-triangles. Each vertex is labeled by a Laurent monomial. The finite quotient complex H_{L}/L is a torus, subdivided by one vertex, three edges and two 2-cells, one up-triangle and one down-triangle.

3 From hyperplane arrangements to minimal free resolutions

We fix a unimodular sublattice L of dimension m in ℤ^{n}. The Laurent polynomial ring T = k[x_{1}^{±1}, …, x_{n}^{±1}, y_{1}^{±1}, …, y_{n}^{±1}] is a module over the polynomial ring S = k[x_{1}, …, x_{n}, y_{1}, …, y_{n}]. We consider the following monomial S-submodule of T:

M_{L} := S · { x^{a}y^{−a} | a ∈ L } ⊂ T.

Each lattice point a = (a_{1}, …, a_{n}) in L is a vertex of the arrangement H_{L}, and we label that vertex of H_{L} with the corresponding generator of M_{L}, namely,

x^{a}y^{−a} := x_{1}^{a_{1}}x_{2}^{a_{2}} ⋯ x_{n}^{a_{n}}y_{1}^{−a_{1}}y_{2}^{−a_{2}} ⋯ y_{n}^{−a_{n}}.

Each face F of H_{L} is then labeled by the least common multiple m_{F} of the labels of its vertices. (The least common multiple of a set of Laurent monomials is the Laurent monomial whose exponents are the coordinatewise maxima of the given exponents.) The labeled cell complex H_{L} defines a complex of free ℤ^{2n}-graded S-modules

F_{H_{L}} = ⨁_{{F ∈H_{L} \atop F ≠∅}} S(−m_{F}),

where the summand S(−m_{F}) has homological degree dim(F). The differential of the complex F_{H_{L}} is the homogenized differential of the cell complex H_{L}, defined by

∂(F) = ∑_{{F′⊂F \atop cod(F′,F)=1}} ϵ(F, F′) · \frac{m_{F}}{m_{F′}} · F′, for faces F of H_{L}.

Here ϵ(F, F′) is either +1 or −1, indicating the orientation of F′ in the boundary of F. See [BS, §1] for details on this construction. The complex (F_{H_{L}}, ∂) is not S-finite, but has finite length m = rank(L).

As an example consider the complex of free k[x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, y_{3}]-modules defined by the hypersimplicial arrangement in Figure 1. The edge E connecting the module generators m_{1} = \frac{x_{2}y_{3}}{y_{2}x_{3}} and m_{2} = \frac{x_{1}y_{3}}{y_{1}x_{3}} is labeled by their least common multiple, which is the Laurent monomial m_{E} = \frac{x_{1}x_{2}y_{3}}{x_{3}} . This edge E represents the first syzygy (x_{1}y_{2}) · m_{1} − (y_{1}x_{2}) · m_{2} of the monomial submodule M_{L}.

Theorem 3.1 The complex (F_{H_{L}}, ∂) is a minimal ℤ^{2n}-graded free S-resolution of the lattice module M_{L}.

Proof. The complex (F_{H_{L}}, ∂) consists of free S-modules and is clearly ℤ^{2n}-graded. To show that it is a resolution, we apply the exactness criterion in [BS, Proposition 1.2] to the labeled cell complex X = H_{L}. For any (a, b) ∈ ℤ^{2n} we consider the subcomplex X_{≤(a,b)} consisting of all faces F of the arrangement X whose label m_{F} = x^{c}y^{d} satisfies the coordinatewise inequalities c ≤ a and d ≤ b. We shall prove that X_{≤(a,b)} is contractible, by identifying this subcomplex with a convex polytope in ℝL. For instance, the marked pentagon in Figure 1 is X_{≤(2,1,1, 1,1,1)}.

Our labeled hyperplane arrangement can be described as follows. For u ∈ ℝ^{n} we write (⌈u⌉, ⌊u⌋) for the vector in ℤ^{2n} obtained by rounding up and down each coordinate of u. For instance, if u = (−2/5, −1, 7/5) then (⌈u⌉, ⌊u⌋) = (0, −1, 2, −1, −1, 1). Two vectors u and v in the subspace ℝL of ℝ^{n} are called equivalent if (⌈u⌉, ⌊u⌋) = (⌈v⌉, ⌊v⌋). The resulting equivalence classes on ℝL are the (relatively open) faces of X = H_{L}. The face F containing u ∈ ℝL is labeled by the vector (⌈u⌉, −⌊u⌋), or by the corresponding Laurent monomial m_{F} = x^{⌈u⌉}y^{−⌊u⌋}.

For a, b ∈ ℤ^{n} we consider the following subset of the subspace ℝL in ℝ^{n}:

{ u ∈ ℝL | ⌈u⌉ ≤ a and − ⌊u⌋ ≤ b } = { u ∈ ℝL | − b ≤ u ≤ a }. (3.1a)

This is a convex polytope with facet hyperplanes taken from H_{L}. By the construction in the previous paragraph, the complex X_{≤(a,b)} is a polyhedral subdivision of the convex polytope identified in (3.1a). Therefore X_{≤(a,b)} is contractible, and, by [BS, Proposition 1.2], the complex (F_{H_{L}}, ∂) is exact over S.

It follows from the description of the labeled complex X = H_{L} in the secondto-last paragraph that distinct faces F and F′ of X have distinct labels m_{F} ≠ m_{F′}. This shows that the resolution (F_{H_{L}}, ∂) is minimal (compare [BS, Remark 1.4]).

Corollary 3.2 The ℤ^{2n}-graded Betti numbers of M_{L} are 0 or 1.

We retain the following identification for the rest of the paper:

X_{≤(a,b)} = { u ∈ ℝL | − b ≤ u ≤ a }  for a, b ∈ ℤ^{n}.  (3.1b)

In the next section we shall further identify these polytopes with the fibers of the Lawrence ideal J_{L}, i.e., with the congruence classes modulo J_{L} of monomials in S.

Note that the polytope X_{≤(a,b)} is itself a (closed) face of the arrangement X = H_{L} if and only if there exists a vector u ∈ ℝL such that (a, b) = (⌈u⌉, −⌊u⌋). Suppose that this holds. Then each coordinate of a + b is either 0 or 1, and, taking any vertex of X_{≤(a,b)}, we get a lattice point v ∈ L with −b ≤ v ≤ a. By translating our face with the lattice vector v, we obtain now the following conclusion:

Proposition 3.3 Modulo the action by the unimodular lattice L, each face of H_{L} has the form X_{≤(a,b)}, where a ∈ {0, 1}^{n} and b ∈ {0, 1}^{n} have disjoint support.

We next consider the quotient complex H_{L}/L, which is formally defined as the face poset of H_{L} modulo the action by the lattice L. Proposition 3.3 implies:

Corollary 3.4 The number of faces of the quotient complex H_{L}/L is finite.

We apply the algebraic quotient construction in [BS, Section 3] to this quotient complex. The labeling of the ℤ^{2n}-graded cell complex X = H_{L} is consistent with the action by the following lattice which is canonically isomorphic to L,

Λ(L) := { (u, −u) ∈ ℤ^{2n} | u ∈ L }.  (3.2)

Following [BS, Lemma 3.5], the complex (F_{H_{L}}, ∂) has the structure of a complex of ℤ^{2n}-graded free modules over the group algebra S[L]. The rank of F_{H_{L}} over S[L] equals the number of faces of H/L, which is finite by Corollary 3.4. Now the functor in [BS, Theorem 3.2] defines an equivalence of categories between the category of ℤ^{2n}-graded S[L]-modules and the category of ℤ^{2n}/Λ(L)-graded S-modules. Applying this functor to the S[L]-complex (F_{H_{L}}, ∂) we obtain the cellular quotient complex (F_{H_{L}/L}, ∂) which is a complex of ℤ^{2n}/Λ(L)-graded free S-modules. From Theorem 3.1 and [BS, Corollary 3.7] we conclude that (F_{H_{L}/L}, ∂) is the minimal free resolution of S/J_{L} over S. This concludes the proof of the following theorem.

Theorem 3.5 The quotient complex H_{L}/L of the hyperplane arrangement modulo L supports the minimal S-free resolution of the unimodular Lawrence ideal J_{L}.

Corollary 3.6 The minimal free resolution of the unimodular Lawrence ideal J_{L} is independent of the characteristic of the base field k. The number of minimal i-th syzygies of S/J_{L} equals the number of i-dimensional faces of the quotient complex H_{L}/L, and the Betti numbers of S/J_{L} in the ℤ^{2n}/Λ(L)-grading are all 0 or 1.

To write down the matrices in the minimal cellular resolution (F_{H_{L}/L}, ∂) of a unimodular Lawrence ideal J_{L}, one must select a fundamental domain of H_{L} modulo L and identify the cover relation in the poset of faces of H_{L}/L. In higher dimensions it is convenient to use the monomial ideals in Section 4 for that purpose, but in dimensions 2 and 3 we can do the identifications directly on the picture. For instance, if L = ker(1 1 1) and thus J_{L} is the ideal of 2 × 2-minors of a generic 2 × 3-matrix, then the resolution (F_{H_{L}/L}, ∂) is supported by the hypersimplicial arrangement in ℝ^{2} from Figure 1. Here the quotient complex H_{L}/L consists of one vertex, three edges and two triangles, which are glued to form a torus. Hence S/J_{L} has one generator, three first syzygies and two second syzygies:

0 ⟶ S^{2} \underrightarrow{( \begin{matrix}x_{1} & y_{1} \\ x_{2} & y_{2} \\ x_{3} & y_{3} \end{matrix} )} S^{3} \underrightarrow{( x_{2}y_{3}−x_{3}y_{2} x_{3}y_{1}−x_{1}y_{3} x_{1}y_{2}−x_{2}y_{1} )} S ⟶ S/J_{L} ⟶ 0

We now show that the minimal cellular resolution of S/J_{L} equals the hull resolution introduced in [BS, Section 3]. For a, b ∈ ℤ^{n} and t > 0, we write

(t^{a}, t^{b}) = (t^{a_{1}}, …, t^{a_{n}}, t^{b_{1}}, …, t^{b_{n}}) ∈ ℝ^{2n}

Recall that hull(M_{L}) is the complex of bounded faces of the polyhedron P_{t} for large t, where P_{t} is the convex hull of { (t^{a}, t^{b}) | x^{a}y^{b} ∈ M_{L} } ⊂ ℝ^{2n}. The vertices of P_{t} correspond to the minimal generators of M_{L}, and the hull resolution of S/J_{L} is the cellular resolution supported on hull(M_{L})/L. We shall use the following lemma.

Lemma 3.7 Let {a, b} and {c, d} be pairs of integers so a + b = c + d, and let t > 0, t ≠ 1. If |a − b| = |c − d|, then t^{a} + t^{b} = t^{c} + t^{d}. If |a − b| > |c − d|, then t^{a} + t^{b} > t^{c} + t^{d}.

Proof. If |a − b| = |c − d|, then {a, b} = {c, d} as multisets. If |a − b| > |c − d|, suppose to be definite that a > c ≥ d > b. In view of a − c = d − b, we have t^{a} − t^{c} − t^{d} + t^{b} = t^{c}(t^{a−c} − 1) − t^{b}(t^{d−b} − 1) = t^{b}(t^{c−b} − 1)(t^{a−c} − 1) > 0.

Theorem 3.8 The hull resolution of S/J_{L} agrees with the minimal free resolution.

Proof. We show that hull(M_{L}) and H_{L} agree as cell complexes, using the fact that hull(M_{L}) consists of those faces of P_{t} supported by strictly positive inner normals.

Let F be a face of H_{L}. Then F = X_{≤(⌈u⌉,−⌊u⌋)} for some u ∈ ℝL. In other words, the vertices of F are precisely those elements a ∈ L so ⌊u_{i}⌋ ≤ a_{i} ≤ ⌈u_{i}⌉ for each i.

Let v = (t^{−⌈u⌉}, t^{⌊u⌋}). If a is a vertex of F, then v · (t^{a}, t^{−a}) = n + m + (n − m)t^{−1}, where m is the number of coordinates in which u is an integer. Suppose that b ∈ L is not a vertex of F, and let t > 1. Write v_{i} = (t^{−⌈u_{i}⌉}, t^{⌊u_{i}⌋}), so v · (t^{a}, t^{−a}) = ∑_{i=1}^{n} v_{i} · (t^{a_{i}}, t^{−a_{i}}). For each i we have v_{i} · (t^{b_{i}}, t^{−b_{i}}) ≥ v_{i} · (t^{a_{i}}, t^{−a_{i}}), with equality if and only if ⌊u_{i}⌋ ≤ b_{i} ≤ ⌈u_{i}⌉. This follows by applying Lemma 3.7 to the pairs {b_{i} − ⌈u_{i}⌉, ⌊u_{i}⌋ − b_{i}} and {a_{i} − ⌈u_{i}⌉, ⌊u_{i}⌋ − a_{i}}. Thus v · (t^{b}, t^{−b}) > v · (t^{a}, t^{−a}), so v supports F as a face of hull(M_{L}).

Let a and b be two vertices of H_{L} which do not belong to a common face of H_{L}, and let F = X_{≤(⌈u⌉,−⌊u⌋)} be the face of H_{L} determined by u = (a + b)/2. Then |a_{i} − b_{i}| ≥ 2 for some i, for otherwise a and b would both belong to F by Proposition 3.3. Let c be any vertex of F, and let d = a + b − c. Note that for each j, {c_{j}, d_{j}} = {⌊u_{j}⌋, ⌈u_{j}⌉} as multisets, so |c_{j} − d_{j}| ≤ 1. Then a + b = c + d, and |a_{j} − b_{j}| ≥ |c_{j} − d_{j}| for each j, with strict inequality for j = i. Let t > 1. Applying Lemma 3.7 to the pairs {a_{j}, b_{j}} and {c_{j}, d_{j}},

t^{a_{j}} + t^{b_{j}} ≥ t^{c_{j}} + t^{d_{j}} and t^{−a_{j}} + t^{−b_{j}} ≥ t^{−c_{j}} + t^{−d_{j}}

for each j, with strict inequalities for j = i. Let p be the midpoint of the line segment from (t^{a}, t^{−a}) to (t^{b}, t^{−b}), and let q be the midpoint of the line segment from (t^{c}, t^{−c}) to (t^{d}, t^{−d}). We have shown that p − q is a nonzero, nonnegative vector. Therefore, the point p cannot lie on any face of hull(M_{L}), because v·p > v·q for any strictly positive vector v. Thus a and b cannot belong to a common face of hull(M_{L}). We conclude that the cell complexes hull(M_{L}) and H_{L} are equal.

4 Fiber monomial ideals and initial monomial ideals

With any lattice ideal in a polynomial ring one can associate two families of monomial ideals. First, there are the initial monomial ideals, with respect to various term orders. Their minimal free resolutions can always be lifted, by Schreyer’s construction in Gröbner basis theory, to a (possibly nonminimal) resolution of the lattice ideal. Second, we have the fiber monomial ideals, which are generated by the fibers, that is, the equivalence classes of monomials modulo the lattice ideal [PS, Section 2]. It follows from the results in [BS, Section 3] that the resolution of the lattice ideal is always determined by the resolution of a large enough fiber ideal.

In this section we make these results precise for the unimodular Lawrence case. Fix a unimodular sublattice L ⊂ ℤ^{n}. Two monomials m and m′ in S = k[x_{1}, …, x_{n}, y_{1}, …, y_{n}] are considered equivalent if m − m′ lies in the unimodular Lawrence ideal J_{L}. The equivalence classes are finite and called the fibers of J_{L}. For given a, b ∈ ℕ^{n}, let fib(a, b) denote the fiber of the monomial x^{a}y^{b}. We shall identify the fibers with the lattice points in the polytopes of the form (3.1b):

Lemma 4.1 Let X = H_{L} be the labeled cell complex introduced in Section 3. Then the map ϕ : fib(a, b) → X_{≤(a,b)} ∩ L , x^{c}y^{d} ↦ a − c is a bijection.

Proof. A monomial x^{c}y^{d} in S lies in fib(a, b) if and only if the vector (a − c, b − d) ∈ ℤ^{2n} lies in the lattice Λ(L) defined in (3.2). The latter condition means that u = a − c lies in L and b + u = d. Thus u is a vertex of X = H_{L} and its label

x^{u}y^{−u} = x^{a−c}y^{b−d} divides x^{a}y^{b},

which shows that u is actually a vertex of X_{≤(a,b)}. Hence the map ϕ is welldefined. To see that it is a bijection we note that the inverse map is given by ϕ^{−1}(u) = x^{a−u}y^{b+u}. This map is an analogue of the bijection in [PS, (2.1)].

Let ⟨fib(a, b)⟩ denote the ideal in S generated by all monomials in the fiber of x^{a}y^{b}. The vertices of the cell complex X_{≤(a,b)} (considered as a subcomplex of H_{L}) are labeled with certain Laurent monomials of the form x^{u}y^{−u}, with u ∈ L. Consider their preimages under the bijection ϕ, and let Y_{(a,b)} denote the same cell complex as X_{≤(a,b)} but with the vertices labeled by the monomials in fib(a, b). Thus the vertex with label x^{u}y^{−u} in X_{≤(a,b)} becomes the vertex with label x^{a−u}y^{b+u} in Y_{(a,b)}. The labeled cell complex Y_{(a,b)} gives rise to a ℤ^{2n}-graded complex of free S-modules as in [BS, Section 1]. This complex is always exact and minimal:

Theorem 4.2 Let L be a unimodular sublattice of ℤ^{n} and a, b ∈ ℕ^{n}. The labeled cell complex Y_{(a,b)} defines a minimal free resolution of the monomial ideal ⟨fib(a, b)⟩.

Proof. For any c, d ∈ ℕ^{n} we can make the following identification of cell complexes

(Y_{(a,b)})_{≤(c,d)} = X_{≤(min{b,c−a}, min{a,d−b})},  (4.1)

where “min” refers to the coordinatewise minimum, and the labels are shifted appropriately. To see that (4.1) holds, one identifies both sides with the convex polytope

{ u ∈ ℝL | (0, 0) ≤ (a + u, b − u) ≤ (c, d) },

together with its natural subdivision. We conclude that the relevant subcomplexes for the exactness criterion in [BS, Proposition 1.2] are all contractible. As above, in the proof of Theorem 3.1, distinct faces of Y_{(a,b)} have distinct labels. Therefore Y_{(a,b)} supports a minimal cellular resolution of the fiber ideal ⟨fib(a, b)⟩.

Example 4.3 Let L be the corank 1 lattice of all vectors in ℤ^{n} with coordinate sum zero. As discussed at the end of Section 2, X = H_{L} is the hypersimplicial arrangement in ℝ^{n−1}, while J_{L} is the ideal of 2 × 2-minors of the matrix (1.1). Let M be the ideal generated by the monomials x_{1}^{u_{1}}x_{2}^{u_{2}} ⋯ x_{n}^{u_{n}}y_{1}^{v_{1}}y_{2}^{v_{2}} ⋯ y_{n}^{v_{n}} where

( \begin{matrix}u_{1} & u_{2} & ⋯ & u_{n} \\ v_{1} & v_{2} & ⋯ & v_{n} \end{matrix} )  (4.2)

runs over all nonnegative integer matrices having the same row sums and the same column sums. Then the minimal free resolution of M is cellular and supported by a suitably relabeled subcomplex Y_{(a,b)} of the hypersimplicial arrangement X = H_{L}.

For instance, let n = 3 and consider the following monomial ideal

\begin{aligned}M = ⟨ x_{2}^{2}x_{3}^{2}y_{1}^{3}, x_{1}x_{2}x_{3}^{2}y_{1}^{2}y_{2}, & x_{1}^{2}x_{3}^{2}y_{1}y_{2}^{2}, x_{1}x_{2}^{2}x_{3}y_{1}^{2}y_{3}, \\ & x_{1}^{2}x_{2}x_{3}y_{1}y_{2}y_{3}, x_{1}^{3}x_{3}y_{2}^{2}y_{3}, x_{1}^{2}x_{2}^{2}y_{1}y_{3}^{2}, x_{1}^{3}x_{2}y_{2}y_{3}^{2} ⟩ \end{aligned}

Here the matrices (4.2) have row sums 4, 3 and column sums 3, 2, 2. The minimal free resolution of M is supported on the complex Y_{(2,1,1, 1,1,1)} which is a pentagon:

x_{1}^{2}x_{2}^{2}y_{1}y_{3}^{2}  x_{1}x_{2}^{2}x_{3}y_{1}^{2}y_{3}  x_{2}^{2}x_{3}^{2}y_{1}^{3} x_{1}^{3}x_{2}y_{2}y_{3}^{2}  x_{1}^{2}x_{2}x_{3}y_{1}y_{2}y_{3}  x_{1}x_{2}x_{3}^{2}y_{1}^{2}y_{2} x_{1}^{3}x_{3}y_{2}^{2}y_{3}  x_{1}^{2}x_{3}^{2}y_{1}y_{2}^{2}

Figure 2: Minimal free resolution of a fiber in the hypersimplicial arrangement.

Let ≺ be any term order on the polynomial ring S = k[x_{1}, …, x_{n}, y_{1}, …, y_{n}]. Consider the initial monomial ideal in_{≺}(J_{L}) of the unimodular Lawrence ideal J_{L}. We know from Theorem 1.2 that in_{≺}(J_{L}) is a squarefree monomial ideal. We shall describe a minimal cellular free resolution of in_{≺}(J_{L}) and show that it has the same Betti numbers as the resolution (F_{H_{L}/L}, ∂) of the Lawrence ideal J_{L}.

Let H_{L}^{0} be the set of all faces of the infinite hyperplane arrangement H_{L} which contain the origin 0 ∈ L. This is a finite cell complex which we identify with the central hyperplane arrangement in ℝL given by the n coordinate hyperplanes x_{i} = 0. The faces of H_{L}^{0} are cones in ℝL with their apex at the origin. Under the embedding of H_{L} in ℝ^{m} given prior to Proposition 2.2, the complex H_{L}^{0} becomes the central arrangement defined by the hyperplanes H_{10}, H_{20}, …, H_{n0} in ℝ^{m}. For instance, in the example of Figure 1, H_{L}^{0} consists of one 0-face, six 1-faces and six 2-faces.

The term order ≺ extends uniquely to a total order on all Laurent monomials x^{u}y^{−u} and hence on all vertices u of H_{L}. A positive-dimensional cone F in the central arrangement H_{L}^{0} is called ≺-positive if all nonzero vertices u ∈ L of the face F satisfy the inequality 1 ≺ x^{u}y^{−u}. We can represent the term order ≺ by a generic hyperplane H_{≺} not containing the origin in ℝL, such that the ≺-positive cones of H_{L}^{0} are precisely those cones which have bounded, nonempty intersection with H_{≺}. If F is an i-dimensional ≺-positive cone in H_{L}^{0} then F ∩ H_{≺} is an (i−1)-dimensional convex polytope. We write in_{≺}(H_{L}) for the (m−1)-dimensional labeled cell complex consisting of those (bounded) polytopes F ∩ H_{≺}. Here F ∩ H_{≺} inherits the label m_{F} from the face F of H_{L}. Note that m_{F} is a monomial in S since 0 ∈ F.

H_{≺}^{\frac{y_{2}x_{3}}{x_{2}y_{3}}}  ^{\frac{x_{1}y_{2}}{y_{1}x_{2}}}^{\frac{y_{1}x_{3}}{x_{1}y_{3}}}  1  ^{\frac{x_{1}y_{3}}{y_{1}x_{3}}}^{\frac{y_{1}x_{2}}{x_{1}y_{2}}}  ^{\frac{x_{2}y_{3}}{y_{2}x_{3}}}  x_{1}y_{2} x_{1}y_{3} x_{2}y_{3} in_{≺}(H_{L})

Figure 3: Initial monomial ideal of a unimodular Lawrence ideal.

Figure 3 shows such a hyperplane H_{≺} and the resulting complex in_{≺}(H_{L}) arising from the hypersimplicial complex in Figure 1. The 1-dimensional complex in_{≺}(H_{L}) represents the minimal S-free resolution of the monomial ideal in_{≺}(J_{L}):

0 ⟶ S^{2} \underrightarrow{( \begin{matrix}x_{1} & 0 \\ −x_{2} & −y_{2} \\ 0 & y_{3} \end{matrix} )} S^{3} \underrightarrow{( x_{2}y_{3} x_{1}y_{3} x_{1}y_{2} )} S ⟶ S/in_{≺}(J_{L}) ⟶ 0

Theorem 4.4 The minimal free resolution of the initial monomial ideal in_{≺}(J_{L}) of the unimodular Lawrence ideal J_{L} is given by the labeled cell complex in_{≺}(H_{L}).

Proof. The ≺-positive 1-dimensional faces C of H_{L}^{0} are the circuits of L, and their labels m_{C} are the minimal generators in_{≺}(J_{L}). Any higher-dimensional ≺-positive face F of H_{L}^{0} corresponds to a region in the central hyperplane arrangement in ℝL given by the coordinates. Its label m_{F} is a squarefree monomial by Proposition 3.3. We can identify m_{F} with the signed vector in the oriented matroid which represents the region F. Now, every vector in an oriented matroid is a conformal union of the circuits [BLSWZ, Proposition 3.7.2]. This implies that m_{F} is the least common multiple of the labels m_{C} of all circuits C with C ⊆ F. In other words, m_{F} is the least common multiple of those vertices of in_{≺}(J_{L}) which lie on F ∩ H_{≺}.

The labeled cell complex in_{≺}(H_{L}) satisfies the exactness criterion in [BS, Proposition 1.2] because the subcomplex of faces whose label divides x^{a}y^{b} is contractible. This follows from a result of Björner and Ziegler [BLSWZ, Theorem 4.5.7]. Minimality is inherited from H_{L} since different faces in H_{L} have different labels.

Theorem 4.4 tells us that the minimal cellular resolution (F_{H_{L}/L}, ∂) of a unimodular Lawrence ideal J_{L} is a universal resolution in the sense that it is stable with respect to any term order ≺. This generalizes the fact that the minimal generators of J_{L} are a universal Gröbner basis (Proposition 1.1). In other words, the universal Gröbner basis property extends from the generators of J_{L} to all the higher syzygies.

5 Lawrence ideals arising from graphs

Unimodular lattices arise naturally in the study of directed graphs and matroids. See Chapters 1 and 6 in [Whi1] for a first introduction. Detailed information on unimodularity appears in [Whi2, Chapter 3]. For instance, a famous theorem of Seymour [Whi2, Theorem 3.1.1(9)] states that every unimodular lattice can be built up in a simple way (by duality, 1-sums, 2-sums and 3-sums) from graphic lattices and a certain five-dimensional lattice in ℤ^{10}. In this section we discuss the minimal free resolutions of cographic and graphic lattice ideals in combinatorial terms.

Let G = (V, E) be a finite directed graph on d vertices, which we assume are labeled with numbers 1, 2, …, d. Suppose that E has n edges. The edge-node incidence matrix of G is the n×d-matrix whose rows are e_{i} −e_{j} ∈ ℤ^{d} for (i, j) ∈ E. The image of this matrix in ℤ^{n} is denoted by L_{G}, and is called the graphic lattice of G. The orthogonal complement of L_{G} in ℤ^{n} is denoted by L_{G}^{∗}, and is called the cographic lattice of G. In the language of matroid theory, the graphic lattice L_{G} is spanned by the cocircuits of G, and the cographic lattice L_{G}^{∗} is spanned by the circuits of G. The following classical result appears in [Whi2, Theorem 1.5.3].

Proposition 5.1 The graphic and cographic lattices L_{G} and L_{G}^{∗} are unimodular.

We write J_{G} = J_{L_{G}} and J_{G}^{∗} = J_{L_{G}^{∗}} for the two unimodular Lawrence ideals associated with a directed graph G. We call J_{G} the graphic ideal and J_{G}^{∗} the cographic ideal. In fact, these ideals depend only on the undirected graph underlying G, and they can be described as follows. Replace each edge (i, j) of the directed graph G = (V, E) by two directed edges, one from vertex i to vertex j and the other one from vertex j to vertex i. We associate the variables x_{ij} and x_{ji} with these directed edges. This gives a polynomial ring S with 2n variables over k. We interpret the variable x_{ji} as the homogenizing variable for x_{ij}. The pair of variables {x_{ij}, x_{ji}} will play the same role as the pair {x_{i}, y_{i}} in the previous sections.

We first discuss the graphic ideal of G. It can be computed as an ideal quotient:

J_{G} = ⟨ ∏_{j:(i,j)∈E} x_{ij} − ∏_{j:(i,j)∈E} x_{ji} | i = 1, 2, …, d ⟩ : ⟨ ∏_{(r,s)∈E} x_{rs}x_{sr} ⟩^{∞}.

The d binomials listed above correspond to a lattice basis of L_{G}. Hence J_{G} has codimension d. The minimal generators of J_{G} are the cocircuits of the graph G.

Example 5.2 Consider the complete graph on five nodes, G = K_{5}. The graphic ideal J_{K_{5}} is generated by five quartics such as x_{12}x_{13}x_{14}x_{15} −x_{21}x_{31}x_{41}x_{51} and ten sextics such as x_{13}x_{14}x_{15}x_{23}x_{24}x_{25} − x_{31}x_{32}x_{41}x_{42}x_{51}x_{52}. They correspond to the 15 cocircuits of K_{5}. The minimal free resolution of J_{K_{5}} is given by a 4-dimensional simplicial complex with 24 facets, 60 tetrahedra, 50 triangles and 15 edges.

We generalize this example by describing the minimal free resolution of J_{K_{d}}, the graphic ideal of the complete graph on d nodes. Our construction is related to Lie algebra cohomology; a geometric version was found independently by Björner and Wachs [BW]. Let S be the polynomial ring over k in the d(d − 1) variables x_{ij}. Let F_{r−1} denote the free S-module whose basis elements correspond to ordered partitions (A_{1} |A_{2} |…|A_{r}) of the set {1, …, d}, such that 1 ∈ A_{1}, for all 1 ≤ r ≤ d − 1.

Theorem 5.3 The minimal resolution of the graphic ideal J_{K_{d}} is the exact complex

F_{•} : 0 → F_{d−2} ⟶^{∂_{d−2}} F_{d−3} ⟶^{∂_{d−3}} F_{d−4} ⟶ … ⟶ F_{2} ⟶^{∂_{2}} F_{1} ⟶^{∂_{1}} F_{0} → 0, (5.1)

where the differential ∂_{r−1} acts on the basis elements of F_{r−1} by the “cyclic rule”

\begin{aligned}∂_{r−1} & (A_{1} | A_{2} | ⋯ | A_{r}) = (−1)^{r+1} ∏_{i∈A_{r}} ∏_{j∈A_{1}} x_{ij} · (A_{r} ∪ A_{1} | A_{2} | ⋯ | A_{r−1}) \\ & + ∑_{s=2}^{r} (−1)^{s} ∏_{i∈A_{s−1}} ∏_{j∈A_{s}} x_{ij} · (A_{1} | A_{2} | ⋯ | A_{s−2} | A_{s−1} ∪ A_{s} | A_{s+1} | ⋯ | A_{r}). \end{aligned}

Proof. The central hyperplane arrangement H_{L_{K_{d}}}^{0} can be identified with the familiar braid arrangement which consists of the hyperplanes z_{i} = z_{j} in the (d − 1)- dimensional vector space { (z_{1}, z_{2} …, z_{d}) ∈ ℝ^{d} | z_{1} + ⋯ + z_{d} = 0 }. The (r − 2)- faces of H_{L_{K_{d}}}^{0} are naturally labeled by the ordered partition (A_{1} | A_{2} | … | A_{r}) of {1, 2, …, d}. More precisely, the unique face having a given point in its relative interior is determined by sorting the coordinates of that point.

We shall apply the initial ideal construction of Theorem 4.4. Let H_{≺} denote the hyperplane {z_{1} = 1} which represents a lexicographic order with the variables x_{12}, …, x_{1d} being highest. The faces of H_{L_{K_{d}}}^{0} which have bounded intersection with H_{≺} are indexed by those ordered partitions (A_{1} | A_{2} | … | A_{r}) which satisfy 1 ∈ A_{1}. The simplicial complex in_{≺}(H_{L_{K_{d}}}) is the first barycentric subdivision of the (d−2)- simplex, as shown for d = 4 in Figure 4. The vertices of in_{≺}(H_{L_{K_{d}}}) are indexed by the monomials ∏_{i∈A_{1}} ∏_{j∈A_{2}} x_{ij} which represent partitions (A_{1}, A_{2}) with 1 ∈ A_{1}. Higher dimensional faces are labeled by ordered partitions with three or more parts. The cellular resolution given by the labeled simplicial complex in_{≺}(H_{L_{K_{d}}}) has the format (5.1) with the differential mapping (A_{1} | A_{2} | ⋯ | A_{r}) to

∑_{s=2}^{r} (−1)^{s} ∏_{i∈A_{s−1}} ∏_{j∈A_{s}} x_{ij} · (A_{1} | A_{2} | ⋯ | A_{s−2} | A_{s−1} ∪ A_{s} | A_{s+1} | ⋯ | A_{r}).

This complex is the minimal free resolution of the monomial ideal in_{≺}(J_{K_{d}}) as in Theorem 4.4. We lift this complex to the minimal free resolution of J_{K_{d}} by a construction as in [PS, Theorem 5.4]. Because H_{L_{K_{d}}} is simplicial, lifting amounts to adding one more term to each differential, and that term is exactly the remaining cyclic term, which is the first one listed in the statement of Theorem 5.3.

x_{13}x_{23}x_{43} x_{12}x_{13}x_{42}x_{43}  x_{13}x_{14}x_{23}x_{24} x_{12}x_{13}x_{14} x_{12}x_{32}x_{42}  x_{12}x_{14}x_{32}x_{34}  x_{14}x_{24}x_{34}

Figure 4: An initial ideal of the complete graphic Lawrence ideal.

We leave it to a future project to compute the Betti numbers of the graphic ideal J_{G} for arbitrary directed graphs G. We expect that there exist nice formulas in terms of the characteristic polynomials of graphic arrangements following [Za].

Another open combinatorial problem is to find a formula for the Betti numbers of the cographic ideals J_{G}^{∗} which can be described as follows. Let C = (C^{+}, C^{−}) be a signed circuit of the directed graph G; see [BLSWZ, §1.1]. This means that C^{+} and C^{−} are disjoint subsets of the set of edges such that the edges in C^{+} together with the reversals of the edges in C^{−} form a directed cycle which meets each vertex of G at most once. Every signed circuit C = (C^{+}, C^{−}) is coded into a binomial:

∏_{(i,j)∈C^{+}} ∏_{(k,l)∈C^{−}} x_{ij}x_{lk} − ∏_{(i,j)∈C^{+}} ∏_{(k,l)∈C^{−}} x_{ji}x_{kl}.

The cographic ideal J_{G}^{∗} is minimally generated by these binomials where C = (C^{+}, C^{−}) runs over all signed circuits of the directed graph G. For instance, if G = K_{4} is the complete graph on {1, 2, 3, 4} with edges (i, j) for i < j, then there are seven circuits [BLSWZ, bottom of page 3] and our cographic ideal equals

J_{K_{4}}^{∗} = ⟨ x_{12}x_{23}x_{31} − x_{21}x_{32}x_{13} , x_{12}x_{24}x_{41} − x_{21}x_{42}x_{14} , x_{13}x_{34}x_{41} − x_{31}x_{43}x_{14},
x_{23}x_{34}x_{42} − x_{32}x_{43}x_{24} , x_{12}x_{23}x_{34}x_{41} − x_{21}x_{32}x_{43}x_{14} ,
x_{13}x_{32}x_{24}x_{41} − x_{31}x_{23}x_{42}x_{14} , x_{12}x_{24}x_{43}x_{31} − x_{21}x_{42}x_{34}x_{13} ⟩

Since the matroid of K_{4} is self-dual, that is, K_{4} is self-dual under planar duality of graphs, it follows that the cographic ideal J_{K_{4}}^{∗} is isomorphic to the graphic ideal J_{K_{4}} whose minimal resolution is depicted in Figure 4. For d ≥ 5, however, the cographic ideal J_{K_{d}}^{∗} is different from and more complicated than the graphic ideal J_{K_{d}}.

A special case of interest is the complete bipartite graph K_{d,e}. Its cographic ideal J_{K_{d,e}}^{∗} is the Lawrence liftings of the ideal of 2 × 2-minors of a generic matrix. While the minimal free resolution of the ideals of 2 × 2-minors depends on the characteristic of the base field, a result of Hashimoto [RW], their Lawrence liftings have minimal free resolutions which are cellular and characteristic free, by Corollary 3.6.

6 The diagonal embedding of a unimodular toric variety

We next present a different geometric interpretation of the Lawrence ideal J_{L}. It concerns the diagonal embeddings of a toric variety, written in the homogeneous coordinates of Audin-Cox [Au],[Cox]. This connection first appeared in [Oda]. As an application, Beilinson’s spectral sequence is presented for unimodular toric varieties.

Let Σ be a complete fan in the lattice ℤ^{m} and X the associated complete normal toric variety; see e.g. [Fu]. Let b_{1}, …, b_{n} ∈ ℤ^{m} be the primitive generators of the one-dimensional cones of Σ, and let B be the n × m-matrix with row vectors b_{i}. Each b_{i} determines a torus-invariant Weil divisor D_{i} on X, and the group Cl(X) of torus-invariant Weil divisors modulo linear equivalence has the presentation

0 ⟶ ℤ^{m} ⟶^{B} ℤ^{n} ⟶^{π} Cl(X) ⟶ 0,

where π takes the i-th standard basis vector of ℤ^{n} to the linear equivalence class [D_{i}] of the corresponding divisor D_{i}. If the divisor class group is torsion free, so that Cl(X) = ℤ^{n−m}, then we express π by a matrix A, and we are back in (2.1).

Audin [Au] and Cox [Cox] define the homogeneous coordinate ring of X to be the polynomial ring R = k[x_{1}, …, x_{n}], equipped with a grading by the abelian group Cl(X) via the morphism π above. Hence a monomial x^{a} ∈ R has degree [∑_{i=1}^{n} a_{i}D_{i}] ∈ Cl(X). The fan Σ is encoded in the irrelevant ideal B_{Σ} ⊂ R whose monomial generators correspond to complements of facets of Σ. Coherent sheaves on the toric variety X are represented by B_{Σ}-torsion-free Cl(X)-graded modules over R; see [Cox] for the case where Σ is simplicial, and [Mu] for the general case. Closed subschemes of X are defined by B_{Σ}-saturated Cl(X)-graded ideals of R. Furthermore, for any Cl(X)-graded R-module T, and any a ∈ Cl(X), we define its twist T (a) to be the graded R-module with components T (a)_{b} := T_{a+b}. Let O_{X}(a) denote the coherent sheaf on X corresponding to the twisted R-module R(a).

The toric variety X × X has the homogeneous coordinate ring

S = R ⊗_{k} R = k[x_{1}, …, x_{n}, y_{1}, …, y_{n}]

The diagonal embedding X ⊂ X × X defines a closed subscheme, and is represented by a Cl(X) × Cl(X)-graded ideal I_{X} in S. That ideal I_{X} is the kernel of the natural map S = R ⊗ R → k[Cl(X)] ⊗ R , x^{u}y^{v} = x^{u} ⊗ x^{v} ↦ [u] ⊗ x^{u+v}. Explicitly,

I_{X} = ⟨ x^{u}y^{v} − x^{v}y^{u} | π(u) = π(v) in Cl(X) ⟩ ⊂ S.

This formula proves the following observation.

Proposition 6.1 The ideal I_{X} ⊂ S defining the diagonal embedding X ⊂ X × X equals the Lawrence ideal J_{L} for the lattice L = im(B) = ker(π) of principal divisors.

The basic example is projective space X = ℙ^{m}. Here n = m + 1, L = ker( 1 1 1 ⋯ 1 ), Cl(X) = ℤ^{1}, and I_{X} = J_{L} is the ideal (1.1) of 2 × 2-minors of a 2 × n-matrix of indeterminates. The minimal free resolution of I_{X} = J_{L} is an Eagon-Northcott complex. Example 10 in [Oda] demonstrates how Beilinson’s spectral sequence for ℙ^{m} can be derived from this observation.

The results in this paper provide a cellular model for this Eagon-Northcott complex as a hypersimplicial complex (Figure 1). We will show that this derivation of Beilinson’s spectral sequence extends to the general setting of Theorem 3.5. Thus our cellular resolutions provide a partial solution to [Oda, Problem 6].

We say that a toric variety X is unimodular if its lattice L of principal divisors is a unimodular sublattice of ℤ^{n}. This condition is equivalent, by [Stu, Remark 8.10], to saying that the toric variety X is smooth and any other variety obtained by toric flips and flops is smooth as well. Toric varieties with this property appear frequently in representation theory and in integer programming. For example, the toric varieties associated with transportation polytopes and products of minors are unimodular. These toric varieties were studied recently by Babson and Billera [BB].

Suppose now that X is a unimodular toric variety. Then Theorem 3.5 constructs an m-dimensional cellular model for the Cl(X) × Cl(X)-graded minimal resolution for the ideal I_{X} ⊂ S of the diagonal embedding of X in X × X:

L_{•} : 0 ⟶ L_{m} ⟶ … ⟶ L_{2} ⟶ L_{1} ⟶ L_{0} = O_{X×X} ⟶ O_{X×X}/I_{X} ⟶ 0.

Here we identify I_{X} with the associated coherent sheaf I_{X} ⊂ O_{X×X}, which is the ideal sheaf of the diagonal embedding of X. Note that S/I_{X} is Cohen-Macaulay.

Since I_{X} is a Cl(X) × Cl(X)-homogeneous ideal, we can write L_{i} = ⊕_{j=1}^{m_{i}}O(−a_{ij}, −b_{ij}) with a_{ij}, b_{ij} ∈ Cl(X). Here O(−a_{ij}, −b_{ij}) = O(−a_{ij}) ⊠ O(−b_{ij}) = p_{1}^{∗}(O(−a_{ij})) ⊗ p_{2}^{∗}(O(−b_{ij})) denotes the exterior tensor product on X × X of O(−a_{ij}) and O(−b_{ij}), and p_{i} are the projections on the factors of X × X.

Let F be any coherent sheaf on X. By tensoring the above free resolution L_{•} with p_{1}^{∗}(F) we obtain a resolution C_{•} for the restriction of p_{1}^{∗}(F) to the diagonal X ⊂ X × X. As in the case of projective space [Bei], pushing via p_{2}_{∗} a Cartan-Eilenberg injective resolution of C_{•} yields a double complex of O_{X}-modules, whose total chain complex has as cohomology the hyperdirect image Rp_{2}_{∗}(C_{•}). One of the spectral sequences belonging to the double complex Rp_{2}_{∗}(C_{•}) yields the following version of the Beilinson Spectral Sequence. It says that any coherent sheaf F can be reconstructed from the knowledge of the cohomology of a few of its Cl(X)-twists.

Theorem 6.2 Let F be a coherent sheaf on a complete unimodular toric variety X. Then there is a (third quadrant) spectral sequence with E_{1} term

E_{1}^{pq} = ⨁_{j} H^{q}(F ⊗ O_{X}(−a_{pj})) ⊗ O_{X}(−b_{pj})

which converges to the associated graded sheaf of a filtration of F.

In the case of ℙ^{m}, Beilinson’s spectral sequence induces an equivalence between the derived category D^{b}(Coh(ℙ^{m})) of bounded complexes of sheaves with coherent cohomology, and the derived category of modules over the exterior algebra in m + 1 generators. An analogous simple description of D^{b}(Coh(X)) does not hold over an arbitrary unimodular toric variety X, since the bundle E = ⊕_{i,j}O(−a_{ij}) defined as the direct sum (without repetitions) of all “left” summands in the above resolution of the ideal sheaf of the diagonal is in general not exceptional, i.e. Ext_{O_{X}}^{i} (E, E) ≠ 0 for some i > 0. See [AH] for a related approach to describing D^{b}(Coh(X)).

References

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[BB] E. Babson, L.J. Billera: The geometry of products of minors, Discrete Comput. Geometry 20 (1998) 231–249

[BPS] D. Bayer, I. Peeva, B. Sturmfels: Monomial resolutions, Math. Res. Lett. 5 (1998), no. 1-2, 31–46.

[BS] D. Bayer, B. Sturmfels: Cellular resolutions of monomial modules, J. Reine Angew. Math. 502 (1998), 123–140.

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[Zi] G. Ziegler: Lectures on Polytopes, Graduate Text in Mathematics 152, Springer Verlag, 1997.

Dave Bayer, Department of Mathematics, Barnard College, Columbia University,

New York, NY 10027, USA, bayer@math.columbia.edu.

Sorin Popescu, Department of Mathematics, Columbia University, New York, NY 10027, USA, and Department of Mathematics, SUNY at Stony Brook, Stony Brook,

NY 11794, psorin@math.columbia.edu.

Bernd Sturmfels, Department of Mathematics, University of California, Berkeley,

CA 94720, USA, bernd@math.berkeley.edu.