ar X iv : m at h . A C / 9 81 21 26 v 1 2 2 D ec 1 99 8 GENERIC AND COGENERIC MONOMIAL IDEALS

Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions derived from simplicial complexes. For a generic monomial ideal, the associated primes satisfy a saturated chain condition, and the Cohen-Macaulay property implies shellability for both the Scarf complex and the Stanley-Reisner complex. Reverse lexicographic initial ideals of generic lattice ideals are generic. Cohen-Macaulayness for cogeneric ideals is characterized combinatorially; in the cogeneric case the Cohen-Macaulay type is greater than or equal to the number of irreducible components. Methods of proof include Alexander duality and Stanley’s theory of local h-vectors. 1. Genericity of Monomial Ideals Revisited Let M be a monomial ideal minimally generated by monomials m1, . . . , mr in a polynomial ring S = k[x1, . . . , xn] over a field k. For a subset σ ⊆ {1, . . . , r}, we set mσ := lcm(mi | i ∈ σ), and aσ := degmσ ∈ N n the exponent vector of mσ. Here m∅ = 1. For a monomial x a = x1 1 · · ·x an n , we set degxi(x ) := ai, and we call supp(x) := {i | ai 6= 0} ⊆ {1, . . . , n} the support of x . Definition 1.1. A monomial ideal M = 〈m1, . . . , mr〉 is called generic if for any two distinct generators mi, mj of M which have the same positive degree in some variable xs there exists a third monomial generator ml ∈ M which divides m{i,j} = lcm(mi, mj) and satisfies supp(m{i,j}/ml) = supp(m{i,j}). The above definition of genericity is more inclusive than the one given by BayerPeeva-Sturmfels [1], but we will see that this definition permits the same algebraic conclusions as the one in [1]. There are important families of monomial ideals which are generic in the sense of Definition 1.1 but not in the sense of [1]. One such family is the initial ideals of generic lattice ideals as in Theorem 3.1. Here is another one: Example 1.2. The tree ideal M = 〈 ( ∏ s∈I xs n−|I|+1 | ∅ 6 = I ⊆ {1, . . . , n}〉 is generic in the new sense but very far from generic in the old sense. This ideal is Artinian of colength (n+ 1), the number of trees on n+ 1 labelled vertices. Recall that a monomial idealM ⊂ S can be uniquely written as a finite irredundant intersection M = ⋂r i=1 Mi of irreducible monomial ideals (i.e., ideals generated by powers of variables). We say Mi is an irreducible component of M . Definition 1.3. A monomial ideal with irreducible decomposition M = ⋂r i=1 Mi is called cogeneric if the following condition holds: if distinct irreducible components Mi and Mj have a minimal generator in common, there is an irreducible component Ml ⊂ Mi+Mj such thatMl andMi+Mj do not have a minimal generator in common. A monomial ideal M is cogeneric if and only if its Alexander dual M is generic. See [10] or Section 4 for the relevant definitions. Cogeneric monomial ideals will 1 2 EZRA MILLER, BERND STURMFELS, AND KOHJI YANAGAWA be studied in detail in Section 4. The remainder of this section is devoted to basic properties of generic monomial ideals. Let M ⊂ S be a monomial ideal minimally generated by monomials m1, . . . , mr again. The following simplicial complex on r vertices, called the Scarf complex of M , was introduced by Bayer, Peeva and Sturmfels in [1]: ∆M := {σ ⊆ {1, . . . , r} |mτ 6= mσ for all τ 6= σ}. Let S(−aσ) denote the free S-module with one generator eσ in multidegree aσ. The algebraic Scarf complex F∆M is the free S-module ⊕ σ∈∆M S(−aσ) with the differential d(eσ) = ∑ i∈σ sign(i, σ) · mσ mσ\{i} · eσ\{i} where sign(i, σ) is (−1) if i is the j-th element in the ordering of σ. It is known that F∆M is always contained in the minimal free resolution of S/M as a subcomplex [1, §3], although F∆M need not be acyclic in general. However we will see in Theorem 1.5 that it is acyclic if M is generic, as was the case under the old definition. Lemma 1.4. Let M = 〈m1, . . . , mr〉 be a generic monomial ideal. If σ 6∈ ∆M , then there is a monomial m ∈ M such that m divides mσ and supp(mσ/m) = supp(mσ). Proof. Choose σ 6∈ ∆M maximal among subsets of {1, . . . , r} with label aσ. Then mσ = mσ\{i} for some i ∈ σ. If supp(mσ/mi) = supp(mσ), the proof is done. Otherwise, there is σ ∋ j 6= i with degxs mi = degxs mj > 0 for some xs. Since M is generic, there is a monomial m ∈ M which divides m{i,j} and satisfies supp(m{i,j}/m) = supp(m{i,j}). Since m{i,j} divides mσ, the monomial m has the desired property. The following theorem extends results in [1] and is the main result in this section. Theorem 1.5. A monomial ideal M is generic if and only if the following two hold: (a) The algebraic Scarf complex F∆M equals the minimal free resolution of S/M . (b) No variable xs appears with the same non-zero exponent in mi and mj for any edge {i, j} of the Scarf complex ∆M . Proof. Suppose that M is generic. Then (b) is straightforward from the definition, and, using Lemma 1.4, (a) is proved by the same argument as in [1, Theorem 3.2]. Assuming (a) and (b), we show that M is generic. For any generator mi let Ai := {mj | mj 6= mi and degxs mj = degxs mi > 0 for some s}. The set Ai can be partially ordered by letting mj mj′ if m{i,j} divides m{i,j′}. It is enough to produce a monomial ml as in Definition 1.1 whenever mj ∈ Ai is a minimal element for this partial order. Supposing, then, that mj is minimal, use (a) to write m{i,j} mi · ei − m{i,j} mj · ej = ∑ {u,v}∈∆M bu,v · d(e{u,v}) (1) where we may assume (by picking such an expression with a minimal number of nonzero terms) that the monomials bu,v are 0 unless m{u,v} divides m{i,j}. There is at least one monomial ml such that bl,j 6= 0, and we claim ml 6∈ Ai. Indeed, ml divides m{i,j} because m{l,j} does, so if degxt mi < degxt mj (which must occur for some t because mj does not divide mi), then degxt ml ≤ degxt mj . Applying (b) GENERIC AND COGENERIC MONOMIAL IDEALS 3 to m{l,j} we get degxt ml < degxt mj, and furthermore degxt m{i,l} < degxt m{i,j}, whence ml 6∈ Ai by minimality of mj . So if degxs m{i,j} > 0 for some s, then either degxs ml < degxs mj by (b), or degxs ml < degxs mi because ml 6∈ Ai. Remark 1.6. Condition (a) in Theorem 1.5 splits into two parts: minimality and acyclicity. For the Scarf complex of any monomial ideal, minimality is automatic since face labels aσ of ∆M are distinct. It is acyclicity which must be checked. For an arbitrary monomial ideal M , Bayer and Sturmfels [2, §2] constructed a polyhedral complex hull(M) supporting a (not necessarily minimal) free resolution of M . Definition 1.1 suffices to imply that the hull complex equals the Scarf complex: Proposition 1.7. If M is a generic monomial ideal, then the hull complex hull(M) coincides with ∆M , and in this case the hull resolution Fhull(M) = F∆M is minimal. Proof. Essentially unchanged from the proof of [2, Theorem 2.9]. Example 1.2 (continued) The Scarf complex ∆M of M is the first barycentric subdivision of the (n−1)-simplex. By Theorem 1.5, F∆M gives a minimal free resolution of S/M . Miller [10] also constructed a minimal free resolution of S/M as a cohull resolution, derived essentially from the coboundary complex of a permutahedron. 2. Associated Primes and Irreducible Components In this section we study the primary decomposition of a generic monomial ideal M . For a monomial prime P in S, we identify the homogeneous localization (S/M)(P ) with the algebra k[xi | xi ∈ P ]/M(P ), where M(P ) is the monomial ideal of k[xi | xi ∈ P ] gotten from M by setting equal to 1 all the variables not in P . Remark 2.1. If M is a generic monomial ideal then so is M(P ). Let M = ⋂r i=1 Mi be the irreducible decomposition of a monomial ideal M . Then we have {rad(Mi) | 1 ≤ i ≤ r} = Ass(S/M). Note that distinct irreducible components may have the same radical. Bayer, Peeva and Sturmfels [1, §3] give a method for computing the irreducible decomposition of a generic monomial ideal (in the old definition). The generalization of this method by Miller [10, Theorem 5.12] shows that [1, Theorem 3.7] remains valid here, as we will show in Theorem 2.2 below. Recall that codim(I) ≤ codim(P ) ≤ proj-dimS(S/I) ≤ n for any graded ideal I ⊂ S and any associated prime P ∈ Ass(S/I), and codim(I) = proj-dimS(S/I) if and only if S/I is Cohen-Macaulay. There always exists a minimal prime P ∈ Ass(S/I) with codim(P ) = codim(I). But in general there is no P ∈ Ass(S/I) with codim(P ) = proj-dimS(S/I). For example, if I = 〈x1, x2〉 ∩ 〈x3, x4〉, then proj-dimS(S/I) = 3. Theorem 2.2. Let M ⊂ S be a generic monomial ideal. Then (a) For each integer i with codim(M) < i ≤ proj-dimS(S/M), there is an embedded associated prime P ∈ Ass(S/M) with codim(P ) = i. (b) For all P ∈ Ass(S/M) there is a chain of associated primes P = P0 ⊃ P1 ⊃ · · ·⊃ Pt with codim(Pi) = codim(Pi−1)−1 for all i and Pt is a minimal prime of M . 4 EZRA MILLER, BERND STURMFELS, AND KOHJI YANAGAWA Proof. (a) This was proved by Yanagawa [18] under the old definition of genericity. Using Theorem 1.5 and [10, Theorem 5.12], the argument in [18] also works here. (b) It suffices to show that for any embedded prime P of M there is an associated prime P ′ ∈ Ass(S/M) with codim(P ) = codim(P )− 1 and P ′ ⊂ P . The localization P(P ) of P is a maximal ideal of S(P ), and an embedded prime of M(P ), so there is a prime P ′ (P ) ⊂ S(P ) such that P ′ (P ) ∈ Ass(S/M)(P ), codim(P ′ (P )) = codim(P(P )) − 1 and P ′ (P ) ⊂ P(P ) by (a) applied to the generic ideal M(P ). The preimage P ′ ⊂ S of P ′ (P ) ⊂ S(P ) has the expected properties. Remark 2.3. Let M ⊂ S be a generic monomial ideal, and P, P ′ ∈ Ass(S/M) such that P ⊃ P ′ and codimP ≥ codimP ′ + 2. Theorem 2.2 does not state that there is an associated prime between P and P . For example, set M = 〈ac, bd, ab, ab〉. Then 〈a, b〉, 〈a, b, c, d〉 ∈ Ass(S/M), but there is no associated prime between them. Following [1, §3], we next define the extended Scarf complex ∆M∗ of M . Let M := M + 〈x1 , . . . , x D n 〉 (2) with D larger than any exponent on any minimal generator of M . We index the new monomials xs just by their variables xs; so the vertex set of ∆M∗ is a subset of {1, . . . , r} ∪ {x1, . . . , xn}. This subset is proper if M contains a power of a variable. Recall ([1, Corollary 5.5] for the old genericity or [10, Proposition 5.16] for the new) that ∆M∗ is a regular triangulation of an (n − 1)-simplex ∆. The vertex set of ∆ equals {x1, . . . , xn} unless M contains a power of a variable. The restriction of ∆M∗ to {1, . . . , r} equals the Scarf complex ∆M of M . We next determine the restriction of ∆M∗ to {x1, . . . , xn}. The radical rad(M) of M is a square-free monomial ideal. Let V (M) denote the corresponding Stanley-Reisner complex, which consists of all subsets of {x1, . . . , xn} which are not support sets of monomials in M . Then we have the following: Lemma 2.4. For a generic monomial ideal M , the restriction of the extended Scarf complex ∆M∗ to {x1, . . . , xn} coincides with the Stanley-Reisner complex V (M). Proof. Every facet σ of ∆M∗ gives an irreducible component of M ; see [1, Theorem 3.7] and [10, Theorem 5.12]. The radical of that component represents the face σ ∩ {x1, . . . , xn} of V (M). The facets of V (M) arise in this way from the irreducible components whose associated primes are minimal. The following theorem generalizes a result of Yanagawa [18, Corollary 2.4]. For the definition of shellability, see [13, §III.2] or [20, Lecture 8]. Theorem 2.5. Let M be a generic monomial ideal. If M has no embedded associated primes, then M is Cohen-Macaulay. In this case, both ∆M and V (M) are shellable. Proof. The first statement immediately follows from Theorem 2.2. For the second statement we note that all facets σ of ∆M∗ have the following property: |σ ∩ {1, . . . , r}| = codimM and |σ ∩ {x1, . . . , xn}| = dimS/M. (3) In particular, both cardinalities in (3) are independent of the facet σ. On the other hand, ∆M∗ is shellable since it is a regular triangulation of a simplex. A theorem of GENERIC AND COGENERIC MONOMIAL IDEALS 5 Björner [3, Theorem 11.13] implies that the restrictions of ∆M∗ to {1, 2, . . . , r} and to {x1, . . . , xn} are both shellable. We are done in view of Lemma 2.4. Remark 2.6. (a) The shellability of ∆M∗ also implies the following result. If M is generic and P, P ′ ∈ Ass(S/M), then there is a sequence of associated primes P = P0, P1, . . . , Pt = P ′ with codim(Pi + Pi−1) = min{codim(Pi), codim(Pi−1)} + 1 for all 1 ≤ i ≤ t. If M is pure dimensional, this simply says that S/M is connected in codimension 1. (b) A shelling of the boundary complex of a polytope can start from a shelling of the subcomplex consisting of all facets containing a given face; see [20, Theorem 8.12]. The complex V (M) of a generic Cohen-Macaulay monomial ideal M inherits this property, so V (M) has stronger properties than general shellable complexes. Theorem 2.5 and Remark 2.6 suggest the following combinatorial problems: Problem 2.7. (i) Characterize all collections A of monomial primes for which there exists a generic monomial ideal M with A = Ass(S/M). (ii) Characterize the Stanley-Reisner complexes V (M) of Cohen-Macaulay generic monomial ideals M . A necessary condition for (i) is that A satisfy the connectivity in Remark 2.6 (a). But this is not sufficient: for instance, take A to be the minimal primes of a StanleyReisner ring which is Cohen-Macaulay but whose simplicial complex not shellable. For the problem (ii), the Cohen-Macaulayness assumption is essential. Since for all simplicial complex Σ ⊂ 2, there is a (not necessarily Cohen-Macaulay) generic monomial ideal M such that V (M) = Σ. By Theorem 2.5, shellability is a necessary condition for the problem (ii), but it is not sufficient as Remark 2.6 (b) shows. If we put further restrictions on the generators of a generic monomial idealM , then, since the extended Scarf complex ∆M∗ is a triangulation of a simplex, we can apply Stanley’s theory of local h-vectors [13]. The next two results will be reinterpreted in Section 4 in terms of cogeneric ideals using Alexander duality [10]. Again let M be as in (2), and define the excess of a face σ ∈ ∆M∗ to be e(σ) := # supp(mσ)−#σ. This agrees, in our situation, with the definition of excess in [13]. Theorem 2.8. If M is generic and all r generators m1, . . . , mr have support of size c, i.e. #supp(mi) = c for all i, then M has at least (c − 1) · r + 1 irreducible components. Example 2.9. This is false without the assumption that M is generic. For instance, the non-generic monomial idealM = 〈x1, y1〉∩. . .∩〈xn, yn〉 has r = 2 n generators, and each generator has support of size c = n, but M has only n irreducible components. Proof. If c = 1, there is nothing to prove, so we may assume that c ≥ 2. Set Γ = ∆M∗ . The hypothesis on the generators of M means that Γ has n vertices of excess 0 and r vertices of excess c− 1. To prove the assertion, we use the decomposition