Homology, Homotopy and Applications, vol.4(2), 2002, pp.427–437

RESOLUTIONS AND LATTICES

IRENA PEEVA

(communicated by Clas Löfwall)

Abstract We discuss how lattices and posets can be used as tools to

study minimal free resolutions of monomial or toric ideals.

To Jan–Erik Roos on his sixty–fifth birthday

1. Introduction

In this paper we discuss properties, related to certain lattices and posets, of monomial and toric resolutions.

A lattice is called a finite geometric lattice if it is finite, semimodular, and atomic. The significance of such lattices comes from the fact that a lattice L is a finite geometric lattice if and only if L is the intersection lattice of a central essential hyperplane arrangement, if and only if L is the lattice of flats of a simple matroid. We call a monomial ideal M geometric if its lcm-lattice is geometric. In Section 2, we construct the minimal free resolution F_{M} of S/M as a cellular (simplicial in this case) algebraic complex; this provides a large class of ideals whose minimal free resolutions are simplicial but bigger than the Scarf-complex resolution. We also construct F_{M} as a quotient of Taylor’s resolution. A nice application is Theorem 2.10, which shows that the Poincaré polynomial of the intersection lattice of a central hyperplane arrangement is equal to the Poincaré series of the minimal free resolution of a certain monomial ideal. It leads to a dictionary between some questions/invariants of simple matroids and questions/invariants of minimal free resolutions of geometric ideals. Also, our approach leads to a new (algebraic) proof of Corollary 2.11, which is an important fact in matroid theory and was proved in different ways by Björner, Gelfand–Zelevinsky, Jambu–Terao.

In Section 3 we consider Betti numbers of toric ideals. M. Hochster and R. Stanley have found and proved that such Betti numbers can be computed using simplicial complexes. That formula for the Betti numbers was first published in the first edition of [St] (cf. [St, Theorem 7.9]). For a proof of the formula and applications, see cf. [BH]. We construct a different type of simplicial complexes in Construction 3.1, and we prove in Theorem 3.4(b) that they provide the Betti numbers as well; these simplicial complexes were first introduced in discussions between D. Bayer, B. Sturmfels, and the author. Recently, the computation using lattices of

I thank the referee for suggesting the short proof of Theorem 2.2. Received February 16, 2001, revised January 7, 2002; published on July 12, 2002. 2000 Mathematics Subject Classification: 13D02. Key words and phrases: Syzygies. \begin{matrix}c \\ ◯ \end{matrix} 2002, Irena Peeva. Permission to copy for private use granted.

the Betti numbers of a monomial ideal was developed in [GPW, Theorem 2.1]. We introduce the computation of the Betti numbers of a toric ideal by posets in Construction 3.3 and Theorem 3.4(b). We also discuss the homotopy equivalence between the simplicial complexes and posets that can be used to obtain the Betti numbers. The proof of Theorem 3.4 uses methods from Topological Combinatorics [Bj]. The homotopy equivalence of the complexes X(M) and Γ(M) in Theorem 3.4 can be proved alternatively using Theorem 3.5 on monomial ideals and the methods in [BS]; our proof of this homotopy equivalence does not rely on any results on monomial ideals and is much simpler than the methods developed in [BS].

2. Geometric monomial ideals

Let S = k[x_{1}, …, x_{n}] be the polynomial ring over a field k. In this section M stands for a geometric monomial ideal minimally generated by monomials m_{1}, …, m_{r}. We will construct the minimal free resolution of S/M.

According to [GPW] the lcm-lattice L of M is the lattice with elements labeled by the least common multiples of m_{1}, …, m_{r} ordered by divisibility; in particular the atoms in L are m_{1}, …, m_{r}, the maximal element is lcm(m_{1}, …, m_{r}), and the minimal element is 1 regarded as the lcm of the empty set.

If l_{1}, …, l_{p} are elements, then we denote by l_{1} ∨ ⋯ ∨ l_{p} = lcm(l_{1}, …, l_{p}) the join of these elements. The degree of a set J ⊂ [r] is m_{J} = lcm(m_{j} | j ∈ J). We say that J is independent if m_{J} ≠ m_{J\a} for each a ∈ J. A set J is dependent if it is not independent. The minimal dependent sets are called circuits. If C is a circuit and c is the element in C with the smallest index, then C\c is called a broken circuit. A set J is called a nbc-set if it contains no broken circuit. The nbc-sets form a simplicial complex bc(M) called the broken circuit complex. Note that in the construction of bc(M) we used the order of the minimal monomial generators m_{1}, …, m_{r}; changing the order of m_{1}, …, m_{r} might change the broken circuit complex.

Construction 2.1. We build a cellular (simplicial in this case) resolution of S/M applying Construction 2.1 from [BPS] to bc(M). Let E be the exterior algebra over k on basis elements e_{1}, …, e_{r}. We consider Taylor’s (possibly non-minimal) resolution E, which is the module S ⊗ E equipped with the differential

d(e_{j_{1}} ∧ ⋯ ∧ e_{j_{s}}) = ∑_{1⩽i⩽s}(−1)^{i+1} · \frac{m_{J}}{m_{J\j_{i}}} · e_{j_{1}} ∧ ⋯ ∧ ê_{j_{i}} ∧ ⋯ ∧ e_{j_{s}} ,

where J = {j_{1}, …, j_{s}} and ê_{j_{i}} means that e_{j_{i}} is omitted in the product. Taylor’s resolution is multigraded with deg(e_{j_{1}} ∧ ⋯ ∧ e_{j_{s}}) = lcm(m_{j_{1}}, …, m_{j_{s}}) and homologically graded by deg(e_{j_{1}} ∧ ⋯ ∧ e_{j_{s}}) = s. Set F_{M} to be the subcomplex of E generated as an S-module by the elements {e_{j_{1}} ∧ ⋯ ∧ e_{j_{s}} | {j_{1}, …, j_{s}} is an nbc-set }.

Theorem 2.1. If M is geometric, then F_{M} is the minimal free resolution of S/M.

Proof. The complex F_{M} constructed in 2.1 coincides with Lyubeznik’s resolution [Ly]. Thus, F_{M} is exact. Since the lcm-lattice is geometric, we have that each nbcset is an independent set. Hence, d(F_{M}) ⊂ (x_{1}, …, x_{n})F_{M}, which means that the resolution is minimal.

Next we take a completely different approach and build the minimal free resolution of S/M as a quotient of Taylor’s resolution.

Construction 2.2. We will construct the Orlik–Solomon complex of M as a quotient of Taylor’s resolution by a free submodule; it will be a complex of free modules homologically graded and multigraded.

We consider Taylor’s (possibly non-minimal) resolution E described in Construction 2.1. If J = {j_{1}, …, j_{s}} ⊆ [n] and j_{1} < ⋯ < j_{s}, then we denote by e_{J} the element e_{j_{1}} ∧ ⋯ ∧ e_{j_{s}}.

Denote by L the lcm-lattice of M. We call e_{j_{1}} ∧ ⋯ ∧ e_{j_{s}} a circuit element, dependent element, or independent element if {j_{1}, … j_{s}} is a circuit, dependent set, or independent set, respectively. Set D to be the S-submodule of E generated by all dependent elements. The independent sets form a simplicial complex, hence d(independent element) is an S-combination of independent elements.

If C is a circuit, then

d(e_{j_{1}} ∧ ⋯ ∧ e_{j_{s}}) = ∑_{1⩽i⩽s}(−1)^{i+1}e_{j_{1}} ∧ ⋯ ∧ ê_{j_{i}} ∧ ⋯ ∧ e_{j_{s}} .

Set C to be the S-submodule of E generated by all d(circuit element). Clearly, d(C) = 0.

Note that D⊕C is an ideal in E. Furthermore, we have the inclusion d(D⊕C) ⊆ D ⊕ C because d(C) = 0 and if e_{J} ∈ D then

d(e_{J}) ∈ \begin{cases}C & ifJisacircuit, \\ D & ifJisdependent, butnotacircuit. \end{cases}

We call the complex F_{os(M)} = E/(D⊕C) the Orlik–Solomon complex of M. It is a free S-module. For every circuit C we have that d(e_{C}) is homogeneous with respect to the multigrading and the homological grading; so F_{os(M)} is graded by homological degree and multigraded.

Theorem 2.2. If M is geometric, then F_{os(M)} is the minimal free resolution of S/M. This resolution is a differential graded algebra.

Proof. First, we will prove that F_{os(M)} is exact. For an element α ∈ E we denote by ᾱ its image in F_{os(M)} = E/(D⊕C). Choose a homogeneous element α ∈ E such that d(ᾱ) = 0, that is d(α) ∈ C. Therefore, there exist circuit elements β_{1}, …, β_{q} and coefficients μ_{1}, …, μ_{q} such that d(α) = d(∑_{1⩽i⩽q} μ_{i}β_{i}). Since Taylor’s resolution E is exact, it follows that there exists a γ ∈ E such that

α = ∑_{1⩽i⩽q} μ_{i}β_{i} + d(γ) .

Thus, ᾱ = \overline{d(γ)}. We have ᾱ = d(γ̄) as desired. Hence F_{os(M)} is a free resolution of S/M.

By Construction 2.4, we have d(F_{os(M)}) ⊆ (x_{1}, …, x_{n})F_{os(M)}. Therefore, the resolution F_{os(M)} is minimal.

Taylor’s resolution E is a differential graded algebra. Since, D ⊕ C is an ideal preserved under the differential, it follows that F_{os(M)} is a differential graded algebra as well.

A lattice is a finite geometric lattice if and only if it is the lattice of flats of a simple matroid, if and only if it is the intersection lattice of a central essential hyperplane arrangement. In the latter correspondence, the ground set of the simple matroid is identified with the atoms (the hyperplanes in the arrangement) in the intersection lattice of the arrangement. In this section A stands for a central essential hyperplane arrangement in k^{q} with r hyperplanes. Denote by L the intersection lattice of A.

Construction 2.3. Note that L is atomic and coatomic. We will construct a monomial ideal M, such that L is its lcm-lattice. Let the atoms of L be labeled by ¯1, …, r̄. Label the coatoms of L by 1′, …, n′. Label an atom ī by the monomial

m_{i} = ∏_{ī∨ j′=ˆ1} x_{j} .

Let M be the monomial ideal generated by the monomial labels of the atoms. If there exist two generators m_{p} and m_{q} such that m_{p} divides m_{q}, then {j | q̄ < j′} ⊂ {j | p̄ < j′} which is a contradiction. Hence m_{1}, …, m_{r} are minimal generators of M. Thus, M has r generators and it is an ideal in k[x_{1}, …, x_{n}], where r is the number of atoms of L and n is the number of matroid hyperplanes.

Lemma 2.1. In the notation above, the lcm-lattice of M is L.

For the proof of 2.7 we recall some definitions: Let Δ be a simplicial complex on a vertex set 1, …, n. Its Stanley–Reisner ideal is

I_{Δ} = ( { x_{j_{1}} ⋯ x_{j_{p}} | {j_{1}, …, j_{p}} ∉ Δ} ) .

The Alexander dual complex Δ^{∨} of Δ is

Δ^{∨} = { [n] \ {j_{1}, …, j_{p}} | {j_{1}, …, j_{p}} ∉ Δ } .

The minimal monomial generators of I_{Δ} are

\{ \frac{x_{1} … x_{n}}{x_{j_{1}} ⋯ x_{j_{p}}} | \{j_{1}, …, j_{p}\} is a facet in Δ^{∨}\} .

Proof. Let Δ be the simplicial complex, such that M is its Stanley–Reisner ideal. Let σ_{1}, …, σ_{r} be the facets of the Alexander dual complex Δ^{∨}. The minimal monomial generators of M correspond bijectively to the facets of Δ^{∨} via the correspondence

σ_{i} ∈ Δ^{∨} ⟷ \frac{x_{1} ⋯ x_{n}}{∏_{i∈σ_{i}} x_{i}} ∈ M .

The equality ∏_{ī∨j′=ˆ1} x_{j} = \frac{x_{1} ⋯ x_{n}}{∏_{i∈σ_{i}} x_{i}} yields that σ_{i} = \{j | ī < j′\}. By [GPW, Propo-

sition 2.3(b)] we have that the lcm-lattice of M is isomorphic to the lattice L′ of intersections of the maximal faces of Δ^{∨} enlarged by an additional minimal element ˆ0 and an additional maximal element ˆ1 (the intersections are ordered by reverse inclusion). Note that

σ_{i_{1}} ∩ ⋯ ∩ σ_{i_{s}} = {j | ī_{p} < j′, 1 ⩽ p ⩽ s} = {j | ī_{1} ∨ ⋯ ∨ ī_{s} < j′} .

Now we construct a bijection between L′ and L. Let F ∈ L′, consider the correspondence

⋂_{F ⊆σ_{i}} σ_{i} = F ⟷ ⋀_{j∈F} j′ = ⋁_{i<( {∧ \atop j∈F} j′)} ī = ⋁_{F ⊆σ_{i}} ī.

We conclude that L′ = L.

Example. Let L be the lattice with atoms {¯1}, {¯2}, {¯3}, {¯4}, {¯5} and coatoms (hyperplanes of the matroid) {¯1¯2¯3}, {¯1¯4}, {¯1¯5}, {¯2¯4}, {¯2¯5}, {¯3¯4¯5}; this lattice is taken from [Bj2, 7.6.1]. Label these coatoms as {1′}, {2′}, {3′}, {4′}, {5′}, {6′}. Then

M = (x_{4}x_{5}x_{6}, x_{2}x_{3}x_{6}, x_{2}x_{3}x_{4}x_{5}, x_{1}x_{3}x_{5}, x_{1}x_{2}x_{4}) .

For example, the first generator is x_{4}x_{5}x_{6} because {1′}, {2′}, {3′} are the coatoms bigger than {¯1}.

The unsigned Whitney numbers of the first kind coincide with the face numbers of the broken circuit complex by [Bj2, Theorem 7.4.6]. By Theorem 2.2, the unsigned Whitney numbers of the first kind coincide with the Betti numbers of S/M. Thus, we can translate [Ox, Conjecture 14.2.7] as

Conjecture 2.1. The sequence {b_{i}} of Betti numbers of S/M is log-concave, i.e.

b_{i}^{2} ⩾ b_{i−1}b_{i+1} for 1 ⩽ i.

Conjecture 2.9 raises the following problems in the spirit of Stanley’s Conjecture 4(b) in [St2]:

Problems 2.1. Let T be an arbitrary monomial ideal and {b_{i}} the sequence of Betti numbers of S/T. Find sufficient conditions for {b_{i}} to be log-concave. Find sufficient conditions for {b_{i}} to be unimodal.

One of the most interesting numerical invariants of A is its Poincaré polynomial P_{A}(t). Let μ be the Möbius function of L. The Poincaré polynomial is

P_{A}(t) = ∑_{l∈L} μ(l)(−t)^{rank(l)} .

Theorem 2.2 implies

Theorem 2.3. P_{A}(t) is equal to the Poincaré series P_{S/M}(t) = ∑_{i⩾0} dim Tor_{i}(S/M, k)t^{i} of the minimal free resolution of S/M.

Hilbert’s Syzygy Theorem [Ei, 19.7] together with our results implies the following bound: Let n be the number of matroid hyperplanes (maximal flats) of L. The broken circuit complex is at most n-dimensional.

For m ∈ L we denote by (ˆ0 : m)_{L_{M}} the open interval {l | l ∈ L, ˆ0 < l < m}. The homology H̃_{∗}((ˆ0 : m)_{L}; k) is the homology of the abstract simplicial complex with faces the chains in the poset (ˆ0 : m)_{L} . The homology H̃_{∗}(L; k) = ⊕_{{m∈L \atop m≠ˆ0}} H̃_{i−2}((ˆ0 : m)_{L_{M}}; k) is called the Whitney homology of the lattice L.

The next corollary was proved in different ways independently by Björner, Gelfand–Zelevinsky, Jambu–Terao. Our proof is new and relies on the exactness of Taylor’s resolution (which is easy to prove, see [BPS]), [GPW, Theorem 2.1], and the results in this paper.

Corollary 2.1. Let bc(L) be the broken circuits complex of L, and A the Orlik– Solomon algebra of L. Let f_{i} be the number of i-faces in bc(L). Then

f_{i} = dim A_{i} = ∑_{{m∈L \atop m≠ˆ0}} dim H̃_{i−2}((ˆ0, m)_{L}; k) .

Proof. By construction 2.6 choose a geometric monomial ideal M, such that its lcmlattice is L. The ideal M and the minimal free resolution of S/M are N^{n}-graded. Therefore we have N^{n}-graded Betti numbers b_{i,x^{α}}(S/M) = dim_{k} Tor_{i,α}^{S}(S/M, k) for i ⩾ 0, α = (α_{1}, …, α_{n}) ∈ N^{n} and x^{α} = x_{1}^{α_{1}} ⋯ x_{n}^{α_{n}}. By [GPW, Theorem 2.1], the Whitney homology relates to the Betti numbers of S/M as follows: for i ⩾ 1 and m ∈ L we have

b_{i,m}(S/M) = dim H̃_{i−2}((ˆ0, m)_{L}; k) .

Hence, for i ⩾ 1 we have

b_{i}(S/M) = ∑_{{m∈L \atop m≠ˆ0}} dim H̃_{i−2}((ˆ0, m)_{L}; k) .

Theorem 2.2 implies that b_{i}(S/M) = f_{i}. On the other hand, note that F_{os(M)} ⊗ k = A, so Theorem 2.5 implies that b_{i}(S/M) = dim A_{i}.

Below, we introduce a grading on the Orlik–Solomon algebra. Let the atoms of L be labeled by ¯1, …, r̄ (that is, {¯1, …, r̄} is the ground set of the matroid). Let E be the exterior algebra over C on n generators e_{1}, …, e_{r}; this is a differential algebra with differential d acting as d(e_{i_{1}} ∧⋯∧e_{i_{s}}) = ∑_{j=1}^{s} (−1)^{j+1}e_{i_{1}} ∧⋯∧ê_{i_{j}} ∧⋯∧e_{i_{s}}, (here ê_{i_{j}} means that this variable is not present in the product). Let I be the ideal in E generated by {d(circuit)}. Then A = E/I is called the Orlik–Solomon algebra of L. If A is a complex hyperplane arrangement, then the Orlik–Solomon algebra A is isomorphic to H^{∗}(C^{q} \ A, C).

Consider the polynomial ring S = k[x_{1}, …, x_{n}] over the field k as N^{n}-graded by letting deg(x_{i}) be the i^{th} standard basis vector in N^{n}.

Corollary 2.2. If v = (v_{1}, …, v_{n}), u = (u_{1}, …, u_{n}) ∈ N^{n}, then we set v ∗ u = (max(v_{1}, u_{1}), …, max(v_{n}, u_{n})); this operation makes N^{n} into a semigroup. The Orlik–Solomon algebra is N^{n}-graded by

deg(e_{i}) = deg(m_{i})
deg(e_{i_{1}} ∧ ⋯ ∧ e_{i_{s}}) = deg(lcm(m_{i_{1}}, …, m_{i_{s}})) .

Proof. Clearly, E is N^{n}-graded. Each minimal generator of I has the form d(e_{i_{1}} ∧ ⋯ ∧ e_{i_{s}}) = ∑_{j=1}^{s} (−1)^{j+1}e_{i_{1}} ∧ ⋯ ∧ ê_{i_{j}} ∧ ⋯ ∧ e_{i_{s}} with

ī_{1} ∨ ⋯ ∨ ī_{s} = ī_{1} ∨ ⋯ ∨ ī_{p−1} ∨ ī_{p+1} ∨ ⋯ ∨ ī_{s} for 1 ⩽ p ⩽ s

in the intersection lattice L. Therefore,

lcm(m_{i_{1}}, …, m_{i_{s}}) = lcm(m_{i_{1}}, …, m̂_{p}, …, m_{i_{s}}) for 1 ⩽ p ⩽ s .

This shows that the minimal generators of I are homogeneous. Thus, the quotient A = E/I is N^{n}-graded.

3. Betti numbers of toric ideals

Let A = {a_{1}, …, a_{n}} be a subset of N^{d} \{0}, A be the matrix with columns a_{i}, and suppose that rank(A) = d. Consider the polynomial ring S = k[x_{1}, …, x_{n}] over a field k generated by variables x_{1}, …, x_{n} in N^{d}-degrees a_{1}, …, a_{n} respectively. The prime ideal I_{A}, that is the kernel of the homomorphism k[x_{1}, …, x_{n}] → k[t_{1}, …, t_{d}] mapping x_{i} to t^{a_{i}} = t_{1}^{a_{i1}} ⋯ t_{d}^{a_{id}}, is called a toric ideal; the ring S/I_{A} is called a toric ring. Both the polynomial ring S and I_{A} are N^{d}-graded. If α ∈ N^{d}, then the set of all monomials in S of degree α is called the fiber of α. The minimal free resolution of S/I_{A} over S is N^{d}-graded as well. We are interested in computing the multigraded Betti numbers

b_{i,α}(S/I_{A}) = dim_{k} Tor_{i,α}^{S}(S/I_{A}, k)

for i ⩾ 0 and α ∈ N^{d}.

In this section M stands for a finite set of monomials m_{1}, …, m_{r} in the polynomial ring S. The support supp(m) of a monomial m is the set { i | x_{i} divides m}. The radical rad(m) is the maximal square-free monomial dividing m.

Construction 3.1. Let X(M) be the simplicial complex on vertices the monomials in M and faces { {m_{j}|j ∈ J} | J ⊆ [r], gcd(m_{j}|j ∈ J) ≠ 1 }.

Construction 3.2. Let Γ(M) be the simplicial complex on vertices x_{1}, …, x_{n} and faces the radicals of the monomials in M and all their factors. This complex was introduced in [St].

Furthermore, we introduce the posets P (M), L_{gcd}(M), and L^{√}(M), which can be used to compute the Betti numbers:

Let P̄(M) be the lattice with elements the greatest common divisors of monomials in M ordered by reverse divisibility with an additional minimal element ˆ0. The atoms of P̄(M) are m_{1}, …, m_{r}. The join operation in P̄(M) is taking gcd. If 1 ∈ P̄(M), then it is the maximal element. We call P̄(M) the gcd-lattice of M. Set

L_{gcd}(M) = \begin{cases}P̄(M) \ ˆ0 & if1 ∉ P̄(M) \\ P̄(M) \ \{1, ˆ0\} & if1 ∈ P̄(M). \end{cases}

We denote by Rad(M) the set of the distinct radicals of the monomials in M. Set L^{√}(M) = L_{gcd}(Rad(M)).

Construction 3.3. We denote by MaxRad(M) the set consisting of the distinct maximal radicals of monomials in M. Set P (M) = L_{gcd}(MaxRad(M)). We call P (M) ∪ ˆ0 the radical lattice of M.

The order complex Δ(P) of a poset P is the abstract simplicial complex that is the collection of all chains in the poset. Sometimes we implicitly think of P as a topological space by considering its order complex.

The poset P (M) is the smallest and simplest among L_{gcd}(M), L^{√}(M), P (M). However, none of the simplicial complexes X(M), Γ(M), Δ(P (M)) could be considered as the simplest in general.

Example. Consider the toric surface R = k[uz^{3}, uy^{4}z, uy^{3}, uy^{3}z^{7}, uy^{2}z^{4}] and the toric ideal that is the kernel of k[a, b, c, d, e] → R. Take the fiber M = { a^{5}b^{5}d^{4}, a^{3}d^{5}c^{5}e, a^{2}bd^{3}c^{3}e^{5}, ab^{2}dce^{9}} of α = u^{14}y^{32}z^{48}. The poset P (M) consists of the single element abcde. The poset L^{√}(M) has minimal element abcde, maximal element ad, and two rank 1 elements abd and adce. The poset L_{gcd}(M) is much bigger: it has 11 elements and is not graded. This illustrates that P (M) is much simpler than L_{gcd}(M) and L^{√}(M).

For a simplicial complex Δ on a vertex set 1, …, r let I_{Δ} be the Stanley–Reisner monomial ideal associated to Δ, that is

I_{Δ} = ( { x_{j_{1}} ⋯ x_{j_{p}} | {j_{1}, …, j_{p}} ∉ Δ} ) .

The Alexander dual complex Δ^{∨} of Δ is

Δ^{∨} = { [r] \ {j_{1}, …, j_{p}} | {j_{1}, …, j_{p}} ∉ Δ } .

Theorem 3.1. • (a) Let M = m_{1}, …, m_{r} be a finite set of monomials. The

simplicial complexes X(M), Γ(M), and Δ(P (M)) are homotopy equivalent.

• (b) Let I_{A} be the toric ideal defined by A and α ∈ N^{d}. Denote by M the fiber of α. Then

b_{i,α}(I_{A}) = dim H̃_{i}(X(M); k)
= dim H̃_{i}(Γ(M); k)
= dim H̃_{i}(P (M); k) .

• (c) Under the conditions of (c), let Δ^{∨} be the Alexander dual complex of Γ(M) and I^{∨} be the Stanley–Reisner ideal of Δ^{∨}. For i ⩾ 0 we have that

b_{i,α}(I_{A}) = b_{i,h}(I^{∨}) ,

where h is the square-free product of the variables appearing in the monomials in M. The lcm-lattice of I^{∨} coincides with P (M) enlarged with additional minimal and maximal elements ˆ0, ˆ1.

Example 3.1. Consider the toric surface R = k[uz^{3}, uy^{4}z, uy^{3}, uy^{3}z^{7}, uy^{2}z^{4}] and the toric ideal that is the kernel of the map k[a, b, c, d, e] → R. Take the fiber M = { a^{4}b^{4}d^{3}, ac^{3}d^{2}e^{5}, bce^{9}} of α = u^{11}y^{25}z^{37}. The complex X(M) has the three vertices a^{4}b^{4}d^{3}, ac^{3}d^{2}e^{5}, bce^{9} and the three edges between them. The complex Γ(M) has vertices a, b, c, d, e and facets bce, abd, acde. The poset P (M) has atoms bce, abd, acde and each of its other elements is equal to the gcd of some couple of atoms.

Proof of Theorem 3.4. For simplicity, we set X = X(M), Γ = Γ(M), L_{gcd} = L_{gcd}(M), L^{√} = L^{√}(M), and P = P (M) throughout the proof. We will show in four steps that X ≃ L_{gcd}, L_{gcd} ≃ L^{√}, L^{√} ≃ Γ, and Γ ≃ P.

The set M is a crosscut in the poset L_{gcd}. The corresponding crosscut complex consists of all bounded subsets of M, so it coincides with X. By [Bj, Theorem 10.8] the crosscut complex X is homotopy equivalent to the order complex of L_{gcd}.

Consider the map φ : L_{gcd} → L^{√} that sends a monomial to its radical. This map is order preserving. If p ∈ L^{√}, then the fiber φ^{−1}(p) has maximal element gcd(m_{j}| p divides m_{j}, 1 ⩽ j ⩽ r), hence φ^{−1}(p) is contractible. Quillen’s fiber lemma [Bj, Theorem 10.5] implies that the order complexes of the posets L_{gcd} and L^{√} are homotopy equivalent.

For a monomial m ∈ Rad(M) denote by Γ_{m} the face {x_{j} | x_{j} divides m} of Γ. Then {Γ_{m} | m ∈ Rad(M) } is a cover of the simplicial complex Γ. Moreover, if J ⊆ [r] and Γ_{J} = ⋂_{j∈J} Γ_{m_{j}} is not empty, then Γ_{J} is a full simplex on the vertices {x_{i} | x_{i} divides m_{j} for j ∈ J}, so it is contractible. By the nerve lemma [Bj, Theorem 10.6] we get that Γ is homotopy equivalent to the nerve simplicial complex N on vertex set the monomials in Rad(M) and with faces { J ⊂ [r] | Γ_{J} ≠ ∅ }. Now note that N has faces { J ⊂ [r] | gcd(m_{j}|j ∈ J) ≠ 1 }. Denote by L the lattice of faces of N ordered by inclusion with the minimal element removed. Then the order complex Δ(L) of L is the barycentric subdivision of N. Hence Δ(L) is homotopic to Γ. We will show that L and L^{√} are homotopic. Consider the map ψ : L → L^{√} that sends an element F of L to the monomial rad(gcd(m_{i}|i ∈ F)). This map is order preserving. If p ∈ L^{√}, then the fiber ψ^{−1}(p) has maximal element ⋂_{i∈H} Γ_{m_{i}} where H = { i | p divides m_{i} }, hence ψ^{−1}(p) is contractible. By Quillen’s fiber lemma [Bj, Theorem 10.5] it follows that the order complexes of L and L^{√} are homotopic. Thus, Γ is homotopy equivalent to the order complex of L^{√}.

For a monomial m ∈ MaxRad(M) denote by Γ′_{m} the face {x_{j} | x_{j} divides m} of Γ. Then {Γ′_{m} | m ∈ MaxRad(M) } is a cover of the simplicial complex Γ. The same argument as in the above paragraph can be applied to the set MaxRad(M) instead of to the set Rad(M); using the cover {Γ′_{m} | m ∈ MaxRad(M) } this argument shows that Γ is homotopy equivalent to the order complex of P.

The proof of (a) is completed. In order to prove (b) it suffices to show that

b_{i,α}(I_{A}) = dim H̃_{i}(Γ; k) .

This is stated in [St] and proved in [AH].

Finally, (c) follows from [GPW, Proposition 2.3(b)] because P is the lattice with elements the intersections of the maximal faces of Γ with the minimal element ˆ0 removed.

Remark 3.1. Here we discuss the motivation for Theorem 3.4; it comes from a similar theorem on monomial ideals. Let I be a monomial ideal minimally generated by monomials m_{1}, …, m_{d}. The ideal I and the minimal free resolution of S/I over S are N^{n}-graded. Therefore we have N^{n}-graded Betti numbers

b_{i,x^{α}}(S/I) = dim_{k} Tor_{i,α}^{S}(S/I, k)

for i ⩾ 0, α = (α_{1}, …, α_{n}) ∈ N^{n} and x^{α} = x_{1}^{α_{1}} ⋯ x_{n}^{α_{n}}. These Betti numbers can be computed using various simplicial complexes: Let Γ(m) be the simplicial complex with faces \{J ⊆ [n] | \frac{m}{∏_{i∈J} x_{i}} ∈ I\}. Denote by X_{⪯m} the full simplex with vertices

the minimal monomial generators of I which divide m; let X(m) be the subcomplex of X_{⪯m} obtained by deleting each face whose vertices have least common multiple equal to m. We denote by L_{I} the lattice with elements labeled by the least common multiples of m_{1}, …, m_{d} ordered by divisibility; this is the lcm-lattice of I introduced in [GPW]. Set P (m) to be the open interval (ˆ0, m) in the lattice L_{I}.

The next theorem is the motivation for Theorem 3.4.

Theorem 3.2. Let I be a monomial ideal and m ∈ L_{I}.

(a) The simplicial complexes X(m), Γ(m), and Δ(P (m)) are homotopy equivalent.

(b) We have

b_{i,m}(I) = dim H̃_{i−1}(X(m); k)
= dim H̃_{i−1}(Γ(m); k)
= dim H̃_{i−1}(P (m); k) .

Proof. The proof of [GPW, Theorem 2.1] shows that X(m) and Δ(P (m)) are homotopic; the proof of [BS, Corollary 1.13] shows that X(m) and Γ(m) are homotopic; so (a) holds. The first equality for the Betti number in (b) is proved in [BS, Theorem 1.11], the second equality is proved in [BH, Proposition 1.1], and the third is proved in [GPW, Theorem 2.1].

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Irena Peeva irena@math.cornell.edu

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA