MAXIMAL MINIMAL RESOLUTIONS

Srikanth Iyengar and Keith Pardue

September, 1998

Dedicated to Luchezar Avramov on the occasion of his fiftieth birthday

1: Introduction

A free resolution of a module M over a ring R is a complex of free R-modules

⋯ → F_{2} → F_{1} → F_{0} → 0

that is exact everywhere except at F_{0}, where the cokernel is M. Such a resolution always exists. If R is a noetherian ℕ-graded ring, with R_{0} a field, and M is a finitely generated ℤ-graded R-module, then a resolution may be constructed in a minimal way in which each F_{i} is graded and each differential is homogeneous of degree 0. The rank of the ith free module in a minimal free resolution of M is an invariant of M, called the ith Betti number of M and denoted β_{i}^{R}(M). Likewise, β_{ij}^{R}(M), the number of elements of degree j in a minimal set of homogeneous generators of F_{i}, is also an invariant of M, called the (i, j)th graded Betti number of M.

In this paper we study modules with maximal graded Betti numbers. More precisely, consider a set Π consisting of pairs (R, M) where R is a noetherian ℕ-graded ring, with R_{0} a field, and M is a finitely generated graded R-module. Is there some (R′, M′) ∈ Π such that β_{ij}^{R′}(M′) ≥ β_{ij}^{R}(M) for every i and j and every (R, M) ∈ Π? There are trivial examples in which the answer is yes (e.g., Π has only one element) or no (e.g., Π consists of all such pairs). Theorems 1 and 2 below give affirmative answers for certain sets Π defined by conditions on the Hilbert series and the depths of R and M. Theorem 3 allows one to compute the maximal graded Betti numbers for the pairs in the sets considered in Theorems 1 and 2.

Throughout this paper, k denotes a field and Q the polynomial ring k[x_{1}, …, x_{n}] with the usual ℕ-grading given by degx_{i} = 1. The Hilbert series of a finitely generated graded Q-module M is H_{M}(s) = ∑ dim_{k} M_{i}s^{i}. We define d-lexicographic ideals and submodules in Section 2. The Poincaré series of an R-module M is P_{M}^{R}(s; t) = ∑ β_{ij}^{R}(M)s^{j}t^{i}. We write ≼ for coefficient-wise inequality of Laurent series with coefficients in ℤ.

Typeset by AMS-TEX

Theorem 1. Fix power series h_{1}(s), h_{2}(s) ∈ ℤ[[s]], and a non-negative integer d.

Let Π be the set of pairs (Q/𝔞, Q/𝔟) where 𝔞 is a homogeneous ideal with H_{Q/𝔞}(s) = h_{1}(s) and depthQ/𝔞 ≥ d, and 𝔟 ⊇ 𝔞 is a homogeneous ideal with H_{Q/𝔟}(s) = h_{2}(s) and depthQ/𝔟 ≥ d.

If Π is non-empty, then there is a pair (R′, S′) ∈ Π such that P_{S}^{R}(s; t) ≼ P_{S′}^{R′}(s; t) for each (R, S) ∈ Π. We may take R′ = Q/𝔞′ and S′ = Q/𝔟′ where 𝔞′ ⊆ 𝔟′ ⊆ Q are d-lexicographic ideals.

When h_{1}(s) = H_{Q}(s) (so that Q/𝔞 = Q), d = 0, and the characteristic of k is 0, the theorem has been proven independently by Hulett [17] and by Bigatti [8]. When h_{2}(s) = 1 (so that Q/𝔟 = k) and d = 0, the theorem has been proven by Peeva [24]. We were motivated to consider the present form of Theorem 1 by Peeva’s theorem; however, our proof is based on the techniques in [22].

Theorem 2. Fix a power series h(s) ∈ ℤ[[s]], a non-negative integer d, and a finitely generated graded free Q-module F.

Let Π be the set of pairs (Q, F/M), where M ⊆ F is a homogeneous submodule such that H_{F/M}(s) = h(s) and depthF/M ≥ d.

If Π is non-empty, then there exists a pair (Q, F/M′) ∈ Π such that P_{F/M}^{Q}(s; t) ≼

P_{F/M′}^{Q}(s; t) for each (Q, F/M) ∈ Π. We may take M′ to be a d-lexicographic submodule of F.

Note that in Theorem 1, the ring R = Q/𝔞 varies while the ring Q is fixed in Theorem 2.

When d = 0 and the characteristic of k is 0, the theorem has been proven by Hulett [18]. Pardue extended Hulett’s theorem to positive characteristic and also proved a version of the theorem for local rings [22]. Note that the dimension of F/M depends only on h(s). In the case that d = dim F/M, the modules F/M are precisely the Cohen-Macaulay quotient modules of with Hilbert series h(s). In this case, Theorem 2 has been proven by Elias [11]. Our notion of a d-lexicographic submodule is based on a construction in Elias’ paper and our proof of Theorem 2 is based on the proof in his paper.

The first parts of Theorems 1 and 2 are existential in nature: they assure us that there are maximal Poincaré series in the corresponding sets, but say nothing about what these are. In the case of Theorem 2, the maximal Poincaré series may be obtained from the Eliahou-Kervaire resolution of stable ideals [12]:

Theorem (Eliahou, Kervaire). Let 𝔞 be a stable ideal in Q.

The Poincaré series of R = Q/𝔞 as a Q-module is given by

P_{R}^{Q}(s; t) = 1 + ∑_{a∈G(𝔞)} s^{deg a}t(1 + st)^{max(a)−1},

where G(𝔞) is the minimal set of monomial generators of 𝔞.

Stable ideals and “max” of a monomial are defined in Section 4. A d-lexicographic ideal is stable. If M′ is a d-lexicographic submodule in Theorem 2, then F/M′ is isomorphic to a direct sum of cyclic Q-modules, each of which has a d-lexicographic ideal as its annihilator.

Thus, Eliahou and Kervaire’s formula allows one to compute the maximal Poincaré series that arises in Theorem 2. Theorem 3 allows one to compute the maximal Poincaré series that arises in Theorem 1.

Theorem 3. Let 𝔞 and 𝔟 be stable ideals such that 𝔞 ⊆ 𝔟 ∩ (x_{1}, …, x_{n})^{2} and consider S = Q/𝔟 as a module over R = Q/𝔞 via the canonical surjection.

The Poincaré series of S as an R-module is given by

P_{S}^{R}(s; t) = \frac{1 + t(∑_{b∈G(𝔟)} s^{deg b}(1 + st)^{max(b)−1} − ∑_{c∈G(𝔞)∩G(𝔟)} s^{deg c}(1 + st)^{max(c)})}{1 − t^{2} ∑_{a∈G(𝔞)} s^{deg a}(1 + st)^{max(a)−1}},

where G(𝔞) and G(𝔟) are minimal sets of monomial generators of 𝔞 and 𝔟 respectively.

The hypothesis that 𝔞 ⊆ (x_{1}, …, x_{n})^{2} is not stringent. If the degree-one part of 𝔞 is non-trivial then it is spanned by variables x_{1}, …, x_{k} and 𝔟 contains these variables as well. In this case we replace Q by Q/(x_{1}, …, x_{k}) before applying Theorem 3.

In the special case where 𝔟 = (x_{1}, …, x_{n}) (so that S = k), the Poincaré series was obtained by Aramova and Herzog [1] and by Peeva [24], using different techniques. The proof of Theorem 3 builds on the works of Eliahou and Kervaire, of Peeva [24], who endowed their resolution with an algebra structure, and of Avramov [5], who showed how to use multiplicative structures on resolutions to compute Betti numbers.

Section 2 collects notation and definitions needed for the rest of the paper and also states the conditions for the existence of d-lexicographic submodules. In Section 3, we prove Theorems 1 and 2 using deformation techniques. Section 4 reviews some results of Eliahou and Kervaire and of Peeva on stable ideals. These are necessary inputs for the techniques used in Section 5 to prove Theorem 3. In Section 6, we show how to modify our results for local rings. Section 7 discusses related problems for Gorenstein rings.

We are grateful to L. Avramov, A. Geramita, and I. Peeva for helpful discussions related to this paper. We are also grateful to A. Geramita and L. Roberts for their support of our research at Queen’s University.

2: Notation and Definitions

We use the following notations throughout this paper. k is a field and Q = k[x_{1}, …, x_{n}] is a polynomial ring over k. We consider Q as an ℕ-graded ring with deg x_{i} = 1. The ideal 𝔪 = (x_{1}, …, x_{n}) is the graded maximal ideal of Q. F is a free ℤ-graded Q-module with a fixed basis f_{1}, …, f_{r} such that f_{i} is homogeneous of degree d_{i} and d_{1} ≤ ⋯ ≤ d_{r}. Any reference to a submodule of F applies to an ideal as well, since F may be Q.

A monomial of F is an element of the form x^{μ}f_{i} = x_{1}^{μ_{1}} ⋯ x_{n}^{μ_{n}}f_{i}. A monomial submodule M ⊆ F is a submodule generated by monomials. We consider two monomial orders on the monomials of F: lexicographic and reverse lexicographic order.

The lexicographic order on monomials of F is given by x^{μ}f_{i} >_{lex} x^{ν}f_{j} if either

(1) i < j, or

(2) i = j and the first non-zero entry in μ − ν is positive.

The dimension ℓ lexicographic subspace of elements of degree i is the monomial subspace of F_{i} spanned by the first ℓ monomials in the lexicographic order. A submodule L of F is a lexicographic submodule if it is graded and if L_{i} is a lexicographic subspace of F_{i} for every i. For a non-negative integer d, a monomial submodule L of F is a d-lexicographic submodule if x_{n}, …, x_{n−d+1} is a regular sequence on F/L and the image of L in F/(x_{n}, …, x_{n−d+1})F is a lexicographic submodule of the Q/(x_{n}, …, x_{n−d+1})-module F/(x_{n}, …, x_{n−d+1})F. Note that a 0-lexicographic submodule is simply a lexicographic submodule.

An important property of lexicographic order, which is used extensively in this paper, is the existence of d-lexicographic submodules of F under the hypotheses given in the proposition below.

Recall that the Hilbert series of a finitely generated ℤ-graded Q-module M = ⨁_{i} M_{i} is the Laurent series H_{M}(s) = ∑ dim_{k} M_{i}s^{i}.

Proposition 4. Let h(s) ∈ ℤ((s)) be a Laurent series and d a non-negative integer. There is a submodule M ⊆ F such that F/M has Hilbert series h(s) and F/M has depth at least d if and only if there is a d-lexicographic submodule L of F such that F/L has Hilbert series h(s). The d-lexicographic submodule of F with Hilbert series h(s) is unique.

When d = 0 and F = Q, the proposition is a well-known theorem of Macaulay [19].

Proof. Since the depth of F/L is at least d it suffices to prove that if there is such an M, there must be such a d-lexicographic submodule of F. When d = 0 this is proven by Hulett [18].

Assume now that d > 0 and that k is an infinite field. Then there is a sequence of linear forms ℓ_{n}, …, ℓ_{n−d+1} that is regular on F/M. After making a linear change of variables in Q, we may assume that x_{n}, …, x_{n−d+1} is a regular sequence on F/M. Let \overline{Q} = Q/(x_{n}, …, x_{n−d+1}) = k[x_{1}, …, x_{n−d}], and \overline{F} be the free \overline{Q}-module F/(x_{n}, …, x_{n−d+1})F, and \overline{M} be the image of M in \overline{F}. Since x_{n}, …, x_{n−d+1} is a regular sequence on F/M, the Hilbert series of \overline{F} /\overline{M} is (1 − s)^{d}h(s). By Hulett’s theorem there is a lexicographic submodule L̃ ⊆ \overline{F} such that \overline{F} /L̃ has Hilbert series (1 − s)^{d}h(s). Now, consider \overline{Q} as the subring k[x_{1}, …, x_{n−d}] of Q and \overline{F} as a \overline{Q}-submodule of F. Let L be the Q-submodule of F generated by L̃. Then F/L has Hilbert series (1 − s)^{−d}(1 − s)^{d}h(s) = h(s). Since the image of L in \overline{F} is L̃, L is the d-lexicographic submodule that we seek.

Now assume that k is finite. Let k′ be an infinite field containing k. The result in the last paragraph allows us to construct a d-lexicographic submodule L′ of the free Q ⊗_{k} k′- module F ⊗_{k} k′ such that (F ⊗_{k} k′)/L′ has Hilbert series h(s). The generators of L′ are monomials, so they are defined over k. Thus, we may take L to be the submodule of F generated by these monomials. Since F/L has the same Hilbert series and depth as (F ⊗_{k} k′)/L′, the d-lexicographic submodule that we seek is L.

The uniqueness of the d-lexicographic submodule is clear. □

The reverse lexicographic order on monomials of F is given by x^{μ}f_{i} >_{revlex} x^{ν}f_{j} if either

(1) deg x^{μ}f_{i} > deg x^{ν}f_{j}, or

(2) deg x^{μ}f_{i} = deg x^{ν}f_{j}, μ ≠ ν, and the last nonzero entry of μ − ν is negative, or (3) μ = ν, f_{i} and f_{j} have the same degree, and i < j.

Let > be either >_{lex} or >_{revlex}. If f ∈ F is the sum of αx^{μ}f_{i}, with α ∈ k^{∗}, and a k-linear combination of monomials that are smaller than αx^{μ}f_{i} with respect to >, then αx^{μ}f_{i} is called the initial term of f with respect to > and is denoted in(f). The initial term of 0 is 0. If N is a submodule of F, then the submodule of initial terms of N is the submodule of F generated by the initial terms of elements of N; it is denoted in(N). We will also write in_{lex}(N) or in_{revlex}(N) to emphasize which order we are using.

Consider GL(n) as the group of k-linear graded algebra automorphisms of Q, and let GL(F) be the group of Q-linear graded automorphisms of F. Then G = GL(n)⋊GL(F) is an algebraic group that acts on F through k-linear graded automorphisms that take submodules to submodules. Let B be the subgroup of G consisting of all automorphisms taking f_{i} to a Q-linear combination of f_{1}, …, f_{i} and x_{i} to a k-linear combination of x_{1}, …, x_{i}. B is a Borel group of G and is naturally realized as upper triangular matrices. A submodule N of F is Borel-fixed if γ(N) = N for every γ ∈ B. See [21] or [22] for the combinatorial description of a Borel-fixed submodule.

We will use the following two properties of the action of G on F and the reverse lexicographic order on monomials of F.

Proposition 5. Let N be a submodule of F and let G and B be as above. Let > be either lexicographic or reverse lexicographic order. Then there is a Zariski-open set G ⊆ G such that

(1) in(γ_{1}(N)) = in(γ_{2}(N)) for all γ_{1}, γ_{2} ∈ G, and

(2) in(γ(N)) is a Borel-fixed submodule of F for all γ ∈ G.

Proof. See Example I.7 and Proposition VII.1 of [21], or modify the proofs of Theorems 15.18 and 15.20 in [10], which are for the case F = Q. □

A form of Proposition 5 was first proved by Galligo for the action of GL(n) on ℂ{x_{1}, … x_{n}} [14]. Proposition 5 was first proved by Bayer and Stillman in the case F = Q [7].

If γ ∈ G as in Proposition 5, then in(γ(N)) is called the generic initial submodule of N with respect to the chosen order, and is denoted Gin(N) or, to emphasize the order used, Gin_{lex}(N) or Gin_{revlex}(N).

Proposition 6 (Bayer and Stillman). Let N be a submodule of F. Then

depthF/Gin_{revlex}(N) = depthF/N.

Proof. See Theorem 15.13 in [10]. □

3: Maximal Betti Numbers

In this section we prove Theorems 1 and 2, and two corollaries to Theorem 1. We use a modification of a technique developed by Pardue in [22], which is based on deformations used by Hartshorne [16] and Reeves [25] in their studies of Hilbert schemes. The modification used here also has applications to Hilbert schemes, which are given in [23].

Theorem 1. Fix power series h_{1}(s), h_{2}(s) ∈ ℤ[[s]], and a non-negative integer d.

Let Π be the set of pairs (Q/𝔞, Q/𝔟) where 𝔞 is a homogeneous ideal with H_{Q/𝔞}(s) = h_{1}(s) and depthQ/𝔞 ≥ d, and 𝔟 ⊇ 𝔞 is a homogeneous ideal with H_{Q/𝔟}(s) = h_{2}(s) and depthQ/𝔟 ≥ d.

If Π is non-empty, then there is a pair (R′, S′) ∈ Π such that P_{S}^{R}(s; t) ≼ P_{S′}^{R′}(s; t) for each (R, S) ∈ Π. We may take R′ = Q/𝔞′ and S′ = Q/𝔟′ where 𝔞′ ⊆ 𝔟′ ⊆ Q are d-lexicographic ideals.

Proof. Let 𝔞′ and 𝔟′ be d-lexicographic ideals such that (R′, S′) ∈ Π where R′ = Q/𝔞′ and S′ = Q/𝔟′. The existence of these ideals is guaranteed by Proposition 4.

Let 𝔞 ⊆ 𝔟 ⊆ Q be any homogeneous ideals such that (R, S) ∈ Π, where R = Q/𝔞 and S = Q/𝔟. We prove the theorem in three steps.

Step 1: Reduction to the case where 𝔞 and 𝔟 are Borel-fixed ideals.

Extending the ground field does not change depths, Hilbert series, or Poincaré series, so we assume that k is infinite.

Set 𝔞̃ = Gin_{revlex}(𝔞) and 𝔟̃ = Gin_{revlex}(𝔟). Proposition 6 shows that depthQ/𝔞̃ = depthQ/𝔞 and depthQ/𝔟̃ = depthQ/𝔟. Moreover, the Hilbert series of R̃ = Q/𝔞̃ and S̃ = Q/𝔟̃ are h_{1}(s) and h_{2}(s) respectively; hence (R̃, S̃) ∈ Π.

The Gröbner basis deformation (see Proposition 8 of [23] or modify Theorem 15.17 in [10]) gives us a graded k[t]-algebra T and a graded T -module U such that

a) T and U are k[t]-flat;

b) T ⊗_{k[t]} (k[t]/t) ≃ R̃ and U ⊗_{k[t]} (k[t]/t) ≃ S̃; and

c) T ⊗_{k[t]} (k[t]/(t − α)) ≃ R and U ⊗_{k[t]} (k[t]/(t − α)) ≃ S for every α ∈ k^{∗}. Since graded Betti numbers are upper-semicontinuous in flat families, we obtain an inequality of Poincaré series: P_{S}^{R}(s; t) ≼ P R̃ ; S̃(s; t).

Proposition 5 shows that the ideals 𝔞̃ and 𝔟̃ are Borel-fixed. Thus, we replace 𝔞 and 𝔟 with 𝔞̃ and 𝔟̃, and henceforth assume that 𝔞 and 𝔟 are Borel-fixed ideals. Finally, note that 𝔞′ and 𝔟′ are Borel-fixed ideals as well.

Step 2: Reduction to the case where d = 0.

The ideal I is Borel-fixed and depthQ/I ≥ d for each I = 𝔞, 𝔟, 𝔞′ or 𝔟′. By Corollary 15.25 in [10], x = x_{n}, x_{n−1}, …, x_{n−d+1} is a regular sequence on R = Q/𝔞 and S = Q/𝔟, and on R′ = Q/𝔞′ and S′ = Q/𝔟′. This yields equalities of Poincaré series

P_{S}^{R}(s; t) = P_{S/(x)}^{R/(x)}(s; t) and P_{S′}^{R′}(s; t) = P_{S′/(x)}^{R′/(x)}(s; t).

Thus, we may replace Q with Q/(x) and assume that d = 0.

Step 3: Moving to lexicographic ideals.

Note that the ideals 𝔞′ and 𝔟′ are lexicographic, since d = 0.

This step uses some properties of the operation ϕ introduced by Pardue in [22]. For a monomial ideal I, we set ϕ(I) = in_{lex}(σ_{𝕃}(I^{(p)})), where I^{(p)} is the polarization of the monomial ideal in a polynomial ring P = k[z_{ij}]; 𝕃 is a general collection of linear forms ℓ_{ij} ∈ Q; and σ_{𝕃} is the homomorphism from P to Q taking z_{ij} to ℓ_{ij}. For an integer m ≥ 2, the operation ϕ^{m}(I) is defined inductively via ϕ^{m}(I) = ϕ(ϕ^{m−1}(I)). [22] also shows that

(1) ϕ is well defined, in the sense that ϕ(I) is independent of the choice of generic 𝕃; (2) Q/I and Q/ϕ(I) have the same Hilbert series;

(3) if I ⊆ J, then ϕ(I) ⊆ ϕ(J);

(4) if L is a lexicographic ideal, then ϕ(L) = L; and

(5) if Q/I and Q/L have the same Hilbert series and L is lexicographic, then ϕ^{m}(I) = L for sufficiently large m.

In particular, ϕ^{m}(𝔞) = 𝔞′ and ϕ^{m}(𝔟) = 𝔟′ for sufficiently large m. To complete the proof of Theorem 1, we must show that β_{ij}^{Q/𝔞}(Q/𝔟) ≤ β_{ij}^{Q/ϕ(𝔞)}(Q/ϕ(𝔟)) for every i and j.

Let π : P → Q be the homomorphism taking z_{ij} to x_{i}. Then π(𝔞^{(p)}) = 𝔞 and π(𝔟^{(p)}) = 𝔟. Fröberg showed that the kernel of π is generated by a sequence of linear forms that is regular on both P/𝔞^{(p)} and P/𝔟^{(p)} [13]. As in [22], when 𝕃 is generic the kernel of σ_{𝕃} is also generated by a sequence of linear forms regular on both P/𝔞^{(p)} and P/𝔟^{(p)}. This justifies the equalities below:

β_{ij}^{Q/𝔞}(Q/𝔟) = β_{ij}^{P/𝔞^{(p)}}(P/𝔟^{(p)}) = β_{ij}^{Q/σ_{𝕃}(𝔞^{(p)})}(Q/σ_{𝕃}(𝔟^{(p)})) ≤ β_{ij}^{Q/ϕ(𝔞)}(Q/ϕ(𝔟)) ,

and the inequality follows from the fact that graded Betti numbers do not decrease on going to the initial ideal, as was shown in Step 1.

□

Corollary 7. With notation as in Theorem 1, further assume that d is positive and that k is an infinite field. Then we may take the ideals 𝔞′ and 𝔟′ in Theorem 1 to be intersections of prime ideals generated by linear forms.

Proof. By Theorem 1, it suffices to consider R = Q/𝔞 and S = Q/𝔟 where 𝔞 and 𝔟 are d-lexicographic ideals. Let 𝔞′ = σ_{𝕃}(𝔞^{(p)}) and 𝔟′ = σ_{𝕃}(𝔟^{(p)}) where σ_{𝕃} and I^{(p)} are as in Step 3 of the proof of Theorem 1. As in that step, we have for every i and j that β_{ij}^{Q/𝔞}(Q/𝔟) = β_{ij}^{Q/𝔞′}(Q/𝔟′), so that P_{Q/𝔟}^{Q/𝔞}(s; t) = P_{Q/𝔟′}^{Q/𝔞′}(s; t).

Now, we show that 𝔞′ and 𝔟′ are each intersections of prime ideals generated by linear forms. The ideals 𝔞^{(p)} and 𝔟^{(p)} are radical monomial ideals in the polynomial ring P = k[z_{ij}] and are thus the homogeneous ideals of unions of reduced coordinate planes in the projective space ProjP. Since σ_{𝕃} : P → Q gives an inclusion of ProjQ in ProjP as a general n−1 dimensional linear space, the subschemes ProjQ/𝔞′ and ProjQ/𝔟′ of ProjQ are unions of reduced linear spaces. Since the depths of Q/𝔞′ and Q/𝔟′ are positive, we know that 𝔞′ and 𝔟′ are the saturated ideals of these reduced schemes and are thus radical ideals which are intersections of primes generated by linear forms. □

Corollary 8. Fix a power series h(s) ∈ ℤ[[s]], and a non-negative integer d.

Consider the set Π of pairs (Q/𝔞, k), where 𝔞 is a homogeneous ideal with H_{Q/𝔞}(s) = h(s) and depthQ/𝔞 ≥ d, and k = Q/(x_{1}, …, x_{n}).

If Π is non-empty, then there is a pair (R′, k) ∈ Π such that P_{k}^{R}(s; t) ≼ P_{k}^{R′}(s; t) for each (R, k) ∈ Π. We may take R′ = Q/𝔞′ where 𝔞′ is a d-lexicographic ideal. If d > 0 and k is an infinite field, then we may instead take R′ = Q/𝔠 where 𝔠 is an intersection of prime ideals generated by linear forms.

When d = 0, this is proven by Peeva [24] and is also a special case of Theorem 1.

Proof. Let 𝔞′ be the d-lexicographic ideal such that (Q/𝔞′, k) ∈ Π, and let 𝔞 be any ideal such that (Q/𝔞, k) ∈ Π. We must show that P_{k}^{R}(s; t) ≼ P_{k}^{R′}(s; t), where R = Q/𝔞 and R′ = Q/𝔞′.

As in the proof of Theorem 1, we may assume that k is an infinite field and then reduce to the case that 𝔞 is a Borel-fixed ideal such that x_{n−d+1}, …, x_{n} is a regular sequence of linear forms on Q/𝔞. Then both 𝔞 and 𝔞′ are contained in the d-lexicographic ideal 𝔟 = (x_{1}, …, x_{n−d}). So if we let S = Q/𝔟, then Theorem 1 tells us that P_{S}^{R}(s; t) ≼ P_{S}^{R′}(s; t). Now, x_{n−d+1}, …, x_{n} is a regular sequence of linear forms on S while S/(x_{n−d+1}, …, x_{n}) = k. So,

P_{k}^{R}(s; t) = (1 + st)^{n−d}P_{S}^{R}(s; t) ≼ (1 + st)^{n−d}P_{S}^{R′}(s; t) = P_{k}^{R′}(s; t).

Now, we prove the last statement. Let 𝔞′ be as above, and let 𝔠 = σ_{𝕃}(𝔞′^{(p)}) as in the proof of Corollary 7. As in the proof of that corollary, we have that 𝔠 is an intersection of prime ideals generated by linear forms. Since Q/σ_{𝕃}(𝔪^{(p)}) = Q/𝔪 = k, we know that P_{k}^{Q/𝔞′}(s; t) = P_{k}^{Q/𝔠}(s; t). □

Theorem 2. Fix a power series h(s) ∈ ℤ[[s]], a non-negative integer d, and a finitely generated graded free Q-module F.

Let Π be the set of pairs (Q, F/M), where M ⊆ F is a homogeneous submodule such that H_{F/M}(s) = h(s) and depthF/M ≥ d.

If Π is non-empty, then there exists a pair (Q, F/M′) ∈ Π such that P_{F/M}^{Q}(s; t) ≼

P_{F/M′}^{Q}(s; t) for each (Q, F/M) ∈ Π.We may take M′ to be a d-lexicographic submodule of F.

Proof. We may assume that k is an infinite field, as in the proof of Theorem 1. Likewise, we may assume that M is a Borel-fixed submodule of F. Then, as in the proof of Theorem 1, we may reduce to the case in which d = 0. This case is Theorem 31 in [22]. □

Our proof of Theorem 2 is essentially the same as the proof of Elias’ theorem in the Cohen-Macaulay case [11].

4: Multiplicative Structures

This section discusses stable ideals and their minimal free resolutions as given by Eliahou and Kervaire [12], and the algebra structure on these resolutions constructed by Peeva [24].

Since the rings and modules in this section and the next are usually multigraded, we begin with a discussion on multigradings.

Let ℕ^{p} = ⊕_{i=1}^{p}ℕϵ_{i}. We will consider ℕ^{p}-graded rings A such that A_{0} = k, a field. For example, the polynomial ring Q has a natural ℕ^{n}-grading induced by deg x_{i} = ϵ_{i}. If 𝔞 ⊆ Q is a monomial ideal, then Q/𝔞 is also ℕ^{n}-graded.

Let ℕ^{q} = ⊕_{i=1}^{q}ℕϵ_{i} where q ≥ p, so that ℕ^{p} is a summand of ℕ^{q}. For A as above, we consider ℕ^{q}-graded modules M. All ring and module homomorphisms will be homogeneous. If dim_{k} M_{μ} is finite for every μ ∈ ℕ^{q}, then we write H_{M}(s) for the Hilbert series ∑_{μ} dim_{k} M_{μ}s^{μ} ∈ ℤ[[s_{1}, …, s_{q}]]. If W = ⨁_{i∈ℕ} W_{i} where each W_{i} is an ℕ^{q}-graded Amodule, then we consider W as a ℕ^{q+1}-graded A-module by letting an element of degree μ in W_{i} have degree μ + iϵ_{q+1} in W. Thus, W has Hilbert series H_{W} (s; t) = ∑ H_{W_{i}}(s)t^{i}.

As in the usual graded case, a finitely generated ℕ^{q}-graded A-module M has a homogeneous minimal free resolution that is unique up to isomorphism. The number of elements of degree μ in a minimal set of homogeneous generators of the ith free module in this resolution is the (i, μ) Betti number of M, an invariant of M denoted β_{iμ}^{A}(M). The multigraded Poincaré series of M as an A-module is P_{M}^{A}(s; t) = ∑ β_{iμ}^{A}(M)s^{μ}t^{i}. We recover the usual graded Poincaré series by setting s_{1} = ⋯ = s_{q} = s and the ungraded Poincaré series by setting s = 1. Since M is ℕ^{q}-graded so is the vector space Tor_{i}^{A}(M, k) and β_{iμ}^{A}(M) = dim_{k} Tor_{i}^{A}(M, k)_{μ}. Thus, if we view Tor^{A}(M, k) = ⨁_{i} Tor_{i}^{A}(M, k) as an ℕ^{q+1}-graded module, then P_{M}^{A}(s; t) = H_{Tor^{A}(M,k)}(s; t).

Let A be an ℕ^{p}-graded algebra, and let W be a complex of ℕ^{p}-graded A-modules

⋯ → W_{2} → W_{1} → W_{0} → 0.

We write W^{♮} = ⨁ W_{i} for the underlying ℕ^{p+1}-graded A-module. If w ∈ W_{i}, we write |w| = i.

In this paper, a differential graded (DG) algebra over an ℕ^{p}-graded algebra A is a non-negative complex X = (X, ∂) of ℕ^{p}-graded A-modules with X^{♮} a ℕ^{p+1}-graded Aalgebra such that the Leibniz formula ∂(x_{1}x_{2}) = (∂x_{1})x_{2} + (−1)^{|x_{1}|}x_{1}(∂x_{2}) holds for all homogeneous x_{1}, x_{2} ∈ X. A homomorphism φ : (X, ∂) → (X′, ∂′) of DG algebras is a homomorphism of the underlying graded algebras with ∂′φ = φ∂.

A DG module Y over a DG algebra X is a complex (Y, ∂) of ℕ^{p}-graded A-modules with Y^{♮} a ℕ^{p+1}-graded X^{♮}-module such that ∂(xy) = (∂x)y + (−1)^{|x|}x(∂y) for all homogeneous x ∈ X and y ∈ Y.

Let I ⊂ ℕ be a finite set. We denote by max(I) the maximal element of I and by min(I) the minimal element of I. We denote by G(𝔞) the minimal set of monomial generators of a monomial ideal 𝔞. For a monomial a ∈ Q = k[x_{1}, …, x_{n}], set max(a) = max(I), and min(a) = min(I), where I = {i | x_{i} divides a}.

An ideal 𝔞 is said to be stable if it satisfies the following conditions:

(i) 𝔞 is generated by monomials, and (ii) If a ∈ 𝔞, then \frac{x_{i}a}{x_{max(a)}} ∈ 𝔞 for all i ≤ max(a).

Note that d-lexicographic ideals are stable.

For each monomial a ∈ 𝔞 there is a unique generator of 𝔞, which we denote by gen_{𝔞}(a), such that a = gen_{𝔞}(a)a′ with max(gen_{𝔞}(a)) ≤ min(a′). For each monomial a ∈ 𝔞, we define a_{(k)} to be gen_{𝔞}(ax_{k}).

Next we describe the minimal Q-free resolution of a stable ideal, as constructed by Eliahou and Kervaire.

Let 𝔞 be a stable ideal, and let E_{k} be the ℕ^{n+1}-graded Q-module underlying the exterior algebra

\frac{⋀(⨁_{i=1}^{n} Qe_{i})}{(e_{k}, …, e_{n})} ,

with deg e_{i} = ϵ_{i} + ϵ_{n+1}. Consider the graded Q-module

X = Q ⊕ ( ⨁_{a∈G(𝔞)} Qã ⊗ E_{max(a)}),

where ã is a free generator of degree deg(a) + ϵ_{n+1}. For a finite subset I ⊂ ℕ, we denote by [a|I] the element ã ⊗ e_{I} ∈ X. Eliahou and Kervaire show that (X, ∂) with

∂[a|∅] = a,
∂[a|I] = ∑_{k=1}^{card(I)}(−1)^{k}x_{i_{k}}[a|I_{k}] + ∑_{k=1}^{card(I)}(−1)^{k+1} \frac{x_{i_{k}}a}{a_{(i_{k})}} [a_{(i_{k})}|I_{k}] where I_{k} = I ∖ i_{k},

is a minimal ℕ^{n}-graded free resolution of Q/𝔞, of which the ith free module is generated by the elements of degree μ + iϵ_{n+1} where μ ∈ ℕ^{n}. Thus, the multigraded Poincaré series of Q/𝔞 as a Q-module is

P_{Q/𝔞}^{Q}(s; t) = 1 + h_{G(𝔞)}(s; t),

where for any set of monomials C, the polynomial

h_{C}(s; t) = ∑_{c∈C} s^{deg c}t ∏_{i=1}^{max(c)−1}(1 + s_{i}t) ∈ ℤ[s; t] = ℤ[s_{1}, …, s_{n}; t].

Peeva constructed a DG algebra structure on X as follows:

For elements a, a′ ∈ 𝔞, define the meet of a and a′ to be g = gen_{𝔞}(lcm(a, a′)). Suppose that x_{s_{1}} ⋯ x_{s_{f}} = \frac{g}{gcd(g,a)}, with s_{1} ≤ ⋯ ≤ s_{f}. Set a_{0} = a and a_{i+1} = (a_{i})_{(s_{i+1})}, for 0 ≤ i ≤ f − 1. Similarly, if x_{t_{1}} ⋯ x_{t_{f′}} = \frac{g}{gcd(g,a′)} with t_{1} ≤ ⋯ ≤ t_{f′}, then we define a sequence with a′_{0} = a′ and a′_{j+1} = (a′_{i})_{(t_{j+1})}.

The integers s_{1}, …, s_{f} (respectively, t_{1}, …, t_{f′}) are called aa′-transforming (respectively, a′a-transforming) elements. Note that a_{f} = g = b_{f′}.

The product on basis elements [a|I] and [a′|J] in X is defined by the following formulas. In formula (2) we write [a|I, s, J] for the image of ã⊗(e_{I} ∧e_{s} ∧e_{J}) in X. In formulas (3.1-4) the s_{i} are the aa′-transforming elements and the t_{j} are the a′a-transforming elements.

(1)  [a|I] · [a|J]  =0
(2)  [a|I] · [a_{(s)}|J]  =(−1)^{(#I+1)(#J+1)}[a_{(s)}|J] · [a|I]
=\{ \begin{matrix}0 & if s ≤ max(J); \\ \frac{a_{(s)}}{x_{s}} [a|I, s, J] & if s > max(J). \end{matrix}
(3.1)  [a|I] · [a′|J]  =0 if for some i and j, s_{i} ∈ I and t_{j} ∈ J.
(3.2)  [a|I] · [a′|J]  = ∑_{i<z} \frac{aa′}{a_{i}a_{i+1}} [a_{i}|I] · [a_{i+1}|J]

if s_{z+1} ∈ I, s_{i} ∉ I for i ≤ z, and t_{j} ∉ J for all j.

(3.3)  [a|I] · [a′|J]  = ∑_{j<y} \frac{aa′}{a′_{j}a′_{j+1}} [a′_{j+1}|I] · [a′_{j}|J]

if t_{y+1} ∈ J, t_{j} ∉ J for j ≤ y, and s_{i} ∉ I for all i.

(3.4)  [a|I] · [a′|J]  = ∑_{i} \frac{aa′}{a_{i}a_{i+1}} [a_{i}|I] · [a_{i+1}|J] + ∑_{j} \frac{aa′}{a′_{j}a′_{j+1}} [a′_{j+1}|I] · [a′_{j}|J]

otherwise.

MAXIMAL MINIMAL RESOLUTIONS 11

5: Poincaré Series

This section is dedicated to proving the following statement: Theorem 9. Let 𝔞 and 𝔟 be stable ideals such that 𝔞 ⊆ 𝔟 ∩ (x_{1}, …, x_{n})^{2}, and consider S = Q/𝔟 as a module over R = Q/𝔞 via the canonical surjection.

The multigraded Poincaré series of S as an R module is given by

P_{S}^{R}(s; t) = \frac{1 + h_{G(𝔟)}(s; t) − (1 + t)h_{G(𝔞)∩G(𝔟)}(s; t)}{1 − t · h_{G(𝔞)}(s; t)} .

Setting s_{1} = ⋯ = s_{n} = s yields Theorem 3.

The proof is given at the end of this section. This is an outline of the proof: for stable ideals 𝔞 ⊆ 𝔟, we establish that the minimal Q-free resolution of Q/𝔟 is a DG module over the DG algebra resolution of Q/𝔞. This allows us to use the degeneration of the Avramov spectral sequence to compute the Poincaré series of Q/𝔟 over Q/𝔞. Proposition 10. Let 𝔞 and 𝔟 be stable ideals in Q with 𝔞 ⊆ 𝔟. Suppose that X is the minimal free resolution of Q/𝔞 and that Y is the minimal free resolution Q/𝔟, with DG algebra structures as in Section 4.

The k-linear map ϕ: X ⟶ Y defined by

ϕ([a|I]) = \frac{a}{gen_{𝔟}(a)}[gen_{𝔟}(a)|I]

is a homomorphism of DG algebras, extending the canonical surjection Q/𝔞 ↠ Q/𝔟.

The proof of the proposition is based on a series of lemmas. Lemma 11. If a ∈ 𝔟, then gen_{𝔟}(ax_{i}) = gen_{𝔟}(gen_{𝔟}(a)x_{i}) for all i. Proof. This is part of Lemma 1.3 in [12]. □ Lemma 12. If a ∈ 𝔞 and b = gen_{𝔟}(a), then gen_{𝔟}(a_{(i)}) = b_{(i)}, where a_{(i)} = gen_{𝔞}(ax_{i}) and b_{(i)} = gen_{𝔟}(bx_{i}). In particular, ϕ([a_{(i)}|I]) = \frac{a_{(i)}}{b_{(i)}} [b_{(i)}|I] for all i. Proof. Consider the series of equalities:

gen_{𝔟}(a_{(i)}) = gen_{𝔟}(a_{(i)} · \frac{ax_{i}}{a_{(i)}} )  as max(a_{(i)}) ≤ min(\frac{ax_{i}}{a_{(i)}} )
= gen_{𝔟}(ax_{i}) = gen_{𝔟}(gen_{𝔟}(a)x_{i}) by Lemma 11
= gen_{𝔟}(bx_{i}) = b_{(i)}.

This proves the first part of the lemma. Applying the definition of ϕ in Proposition 10 gives the expression for ϕ[a_{(i)}|I]. □

Next, we record some properties of transforming elements.

Lemma 13. Let g be the meet of elements a and a′, and let s_{1}, …, s_{f} be aa′-transforming elements.

(1) x_{s_{i}} divides g, in particular s_{i} ≤ max(g), for all i ≤ f.

(2) Suppose that a <_{lex} a′. Let i be the smallest index such that the power of x_{i} in a differs from the power of x_{i} in a′. Then

a =αβ  with max(α) ≤ i < min(β),
a′ =αx_{i}^{l}β′ for some l ≥ 1,

and i is the first aa′-transforming element.

Proof. (1) is from the definition of the meet. The first part of (2) is from the definition of lexicographic order. The last statement is a straightforward computation using the definition of transforming elements. □

Lemma 14. Let s_{1}, …, s_{f} be aa′-transforming elements. Set b = gen_{𝔟}(a) and b′ = gen_{𝔟}(a′). Set b_{i} = gen_{𝔟}(a_{i}) for 0 ≤ i ≤ f. Similarly, for a′a-transforming elements t_{1}, …, t_{f′}, set b′_{j} = gen_{𝔟}(a′_{j}) for 0 ≤ j ≤ f′.

There exist integers h ≤ f and h′ ≤ f′ with the following properties:

(i) b_{h} = b_{h+1} = ⋯ = b_{f} and b′_{h′} = b′_{h′+1} = ⋯ = b′_{f′}.

(ii) s_{1}, …, s_{h} are bb′-transforming elements, and t_{1}, …, t_{h′} are b′b-transforming ele-

ments. In particular, b_{h} = b′_{h′} is the meet of b and b′, and max(b_{h}) ≤ s_{h+1} and max(b′_{h′}) ≤ t_{h′+1}.

Proof. By symmetry, it is enough to show this for the aa′-transforming elements. To begin with, note that b_{i+1} = (b_{i})_{(s_{i+1})} by Lemma 12.

Let h be the least integer such that b_{h} = b_{h+1}. This is equivalent to max(b_{h}) ≤ s_{h+1}. As max(b_{h+1}) ≤ s_{h+1} ≤ s_{h+2}, we get that b_{h+2} = (b_{h+1})_{(s_{h+2})} = b_{h+1}. An iteration yields b_{h} = b_{h+1} = ⋯ = b_{f}.

Suppose that b_{i−1} ≠ b_{i}; that is, s_{i} < max(b_{i−1}). It is enough to show that s_{i} is the first b_{i−1}b′-transforming element. We prove this when a_{i−1} <_{lex} a′; the proof is similar when a′ <_{lex} a_{i−1}. By Lemma 13, as s_{i} is the first a_{i−1}a′-transforming element, we have

a_{i−1} =αβ  with max(α) ≤ s_{i} < min(β)
a′ =αx_{s_{i}}^{l}β′ for some l ≥ 1.

Thus, s_{i} < max(b_{i−1}) = max(gen_{𝔟}(a_{i−1})) which implies that α ∉ 𝔟. So, b_{i−1} = αγ and b′ = αx_{s_{i}}^{l′}γ′, for some l′ ≥ 1. A further application of Lemma 13 completes the proof. □

Proof of Proposition 10. To prove that ϕ is a homomorphism of DG algebras we need to show that ϕ is a morphism of complexes, and that it is a homomorphism of algebras. Step 1. ϕ is a morphism of complexes.

Let [a|I] be a basis element of X, and let b = gen_{𝔟}(a). The formulas for ∂ and ϕ, along with Lemma 12, yield

ϕ∂([a|I]) = ∑_{k≥1}(−1)^{k} \frac{x_{i_{k}}a}{b}[b|I_{k}] + ∑_{k≥1}(−1)^{k+1} \frac{x_{i_{k}}a}{b_{(i_{k})}}[b_{(i_{k})}|I_{k}] = ∂ϕ([a|I]).

This proves that ϕ is a morphism of complexes. Step 2. ϕ is a homomorphism of algebras.

It is enough to prove that

ϕ([a|I] · [a′|J]) = ϕ([a|I]) · (ϕ[a′|J])

for basis elements [a|I] and [a′|J] of X. There are three cases to consider, depending on which of the multiplication laws at the end of Section 4 is applied to compute the product of [a|I] and [a′|J].

Type 1. If a = a′, then ϕ([a|I] · [a|J]) = ϕ([a|I])ϕ([a|J]) = 0, by multiplication law (1). Type 2. Suppose that a′ = a_{(s)} for some integer s. Set b = ϕ(a). Using Lemma 12, we

get ϕ([a|I]) · ϕ([a_{(s)}|J]) = \frac{aa_{(s)}}{bb_{(s)}} [b|I] · [b_{(s)}|J].

If s ≤ max(J), then ϕ([a|I] · [a|J]) = ϕ([a|I]) · ϕ([a|J]) = 0, by multiplication law (2). If s > max(J), then a direct computation shows that

ϕ([a|I] · [a_{(s)}|J]) = ϕ(\frac{a_{(s)}}{x_{s}} [a|I, s, J]) = \frac{a_{(s)}a}{x_{s}b}[b|I, s, J] = \frac{aa_{(s)}}{bb_{(s)}} [b|I] · [b_{(s)}|J].

This completes the proof for this case.

Type 3. There are four cases to consider. Let the notation be as in Lemma 14.

Case (i). Suppose that s_{z} ∈ I and t_{y} ∈ J. As [a|I] · [a′|J] = 0 by multiplication law (3.1), we must show that [b|I] · [b′|J] = 0.

If z ≤ h and y ≤ h′, then s_{z} is a bb′-transforming element and t_{y} is a b′b-transforming element by Lemma 14. We apply multiplication law (3.1) to see that [b|I] · [b′|J] = 0.

If z > h, then b_{z} = b_{h} is the meet of b and b′, and max(b_{z}) ≤ s_{z} by Lemma 14. As t_{1}, …, t_{h′} are b′b-transforming elements, cf. Lemma 14, it follows from Lemma 13 that t_{j} ≤ max(b_{z}) ≤ s_{z}, for all 1 ≤ j ≤ h′. Note that s_{z} ∈ I; thus, max(I) ≥ t_{j} and so [b′_{j}|I] · [b′_{j−1}|J] = 0 for all 1 ≤ j ≤ h′, by multiplication law (2).

An analogous argument shows that if y > h′ then [b_{i−1}|I] · [b_{i}|J] = 0 for all 0 ≤ i ≤ h. Now, applying the appropriate case of multiplication law (3), we deduce that [b|I] ·

[b′|J] = 0.

Case (iv). Suppose that s_{i} ∉ I and t_{j} ∉ J for all i, j. In the series of equalities below, the first is by multiplication law (3.4), the second is by the result of Type 2 above, and the last equality follows from the fact that b_{i} = b_{i+1} and b′_{j} = b′_{j+1}, for i ≥ h and j ≥ h′, by Lemma 14.

ϕ([a|I] · [a′|J]) = ∑_{0≤i<f} \frac{aa′}{a_{i}a_{i+1}} ϕ([a_{i}|I] · [a_{i+1}|J]) + ∑_{0≤j<f′} \frac{aa′}{a′_{j}a′_{j+1}} ϕ([a′_{j+1}|I] · [a′_{j}|J])

= ∑_{0≤i<f} \frac{aa′}{b_{i}b_{i+1}} [b_{i}|I] · [b_{i+1}|J] + ∑_{0≤j<f′} \frac{aa′}{b′_{j}b′_{j+1}} [b′_{j+1}|I] · [b′_{j}|J]

= ∑_{0≤i<h} \frac{aa′}{b_{i}b_{i+1}} [b_{i}|I] · [b_{i+1}|J] + ∑_{0≤j<h′} \frac{aa′}{b′_{j}b′_{j+1}} [b′_{j+1}|I] · [b′_{j}|J].

Note that s_{1}, …, s_{f} (respectively, t_{1}, …, t_{f′}) are bb′ (respectively b′b) transforming elements, by Lemma 14. As s_{i} ∉ I and t_{j} ∉ J, applying multiplication law (3.4) yields:

ϕ([a|I]) · ϕ([a′|J]) = \frac{aa′}{bb′} [b|I] · [b′|J]

= ∑_{0≤i<h} \frac{aa′}{b_{i}b_{i+1}} [b_{i}|I] · [b_{i+1}|J] + ∑_{0≤j<h′} \frac{aa′}{b′_{j}b′_{j+1}} [b′_{j+1}|I] · [b′_{j}|J]

= ϕ([a|I] · [a′|J]).

The proofs for cases (ii) and (iii) are similar to that for case (iv). □

Now we prove Theorem 9. The techinique is similar to the proof of the theorem in Section 2.1 of [5].

Proof of Theorem 9. Let A = Tor^{Q}(R, k) and B = Tor^{Q}(S, k). Then A is an ℕ^{n+1}-graded ring with A_{0} = k and B is an ℕ^{n+1}-graded A-module by the homology products (cf. Chapter VIII of [20]).

Let X and Y be the Eliahou-Kervaire resolutions of R and S respectively, with DG algebra structures as in Section 5. By Proposition 10, Y is a ℕ^{n+1}-graded DG module over X via the homomorphism ϕ. Notice that as ℕ^{n+1}-graded Q-modules, A = X ⊗_{Q} k and B = Y ⊗_{Q} k. In fact, the homomorphism ϕ ⊗_{Q} k : A → B coincides with the action of A on B given above.

The Avramov spectral sequence (Theorem 3.1 of [4] or Proposition 3.2.4 of [6]),

E_{p,q}^{2} = Tor_{p}^{A}(B, k)_{q} ⟹ Tor_{p+q}^{R}(S, k)

is compatible with the ℕ^{n}-grading. (Here, Tor_{p}^{A}(B, k)_{q} is the ℕ^{n}-graded Q-module generated by elements of degree μ + qϵ_{n+1} in Tor_{p}^{A}(B, k) such that μ ∈ ℕ^{n}.) Since X and Y are minimal and Y is a DG module over X, the sequence collapses with E_{p,q}^{2} = E_{p,q}^{∞} by Proposition 4.1.1 of [4]. This implies an equality of formal power series:

P_{S}^{R}(s; t) = P_{B}^{A}(s; t; t).

Set U = Im(ϕ ⊗ k) ≅ (A/Z), where Z = Ker(ϕ ⊗ k), and let V be an ℕ^{n+1}-graded subspace of A, complementary to U. Peeva’s multiplication laws, given at the end of Section 5, and Proposition 10, together with the assumption that 𝔞 ⊆ (x_{1}, …, x_{n})^{2} ∩ 𝔟, show that

A_{+} · A_{+} = 0,  A_{+} · B_{+} = 0,  A_{+} · B ⊂ U,

where A_{+} and B_{+} are the homogeneous maximal ideals of A and B. Thus, B = U ⊕ V as graded A-modules, with A_{+} · V = 0. This last equality implies that as an A-module, V is a direct sum of copies of k = A/A_{+} with generators in various degrees. In terms of Poincaré series this is tantamount to

P_{B}^{A}(s; t; u) = P_{U}^{A}(s; t; u) + P_{k}^{A}(s; t; u) · H_{V} (s; t).

The short exact sequence 0 ⟶ Z ⟶ A ⟶^{ϕ} U ⟶ 0 yields isomorphisms Tor_{i}^{A}(U, k) = Tor_{i−1}^{A}(Z, k), for i ≥ 1. This, with A_{+} · A_{+} = 0, gives

P_{U}^{A}(s; t; u) = 1 + uP_{Z}^{A}(s; t; u) = 1 + uP_{k}^{A}(s; t; u) · H_{Z}(s; t).

Thus, P_{B}^{A}(s; t; u) = 1 + uP_{k}^{A}(s; t; u)H_{Z}(s; t) + P_{k}^{A}(s; t; u) · H_{V} (s; t).

Using the short exact sequence 0 ⟶ A_{+} ⟶ A ⟶ k ⟶ 0, and A_{+} · A_{+} = 0, we deduce the well known recurrence relation:

P_{k}^{A}(s; t; u) = 1 + uP_{k}^{A}(s; t; u) · H_{A_{+}}(s; t).

This shows that P_{k}^{A}(s; t; u) = (1 − uh_{G(𝔞)}(s; t))^{−1}, since H_{A_{+}} = h_{G(𝔞)}.

It remains to compute H_{V} (s; t) and H_{Z}(s; t). Note that U is the subspace of B spanned by B_{0} = k and basis elements {[c|J]|c ∈ G(𝔞) ∩ G(𝔟)}, and that Z is the subspace of A spanned by elements {[a|I]|a ∈ G(𝔞) ∖ G(𝔟)}. Hence,

H_{U}(s; t) = 1 + h_{G(𝔞)∩G(𝔟)}(s; t),
H_{V} (s; t) = 1 + h_{G(𝔟)}(s; t) − H_{U}(s; t),
H_{Z}(s; t) = 1 + h_{G(𝔞)}(s; t) − H_{U}(s; t).

Putting all this together, and setting u = t, yields the required expression for the Poincaré series. □

6: Local Rings

In this section, we reformulate theorems 1, 2 and 3 and Corollary 8 for local rings. If A is a regular local ring with maximal ideal 𝔪 with a fixed set of minimal generators x_{1}, …, x_{n} of 𝔪, and F is a free A-module (such as A itself) with a fixed basis f_{1}, …, f_{r}, then one can define monomials of F, monomial submodules of F, and lexicographic submodules of F; cf. Section 7 of [22] for these definitions. As in the graded case, we define a dlexicographic submodule to be a monomial submodule L ⊆ F such that x_{n−d+1}, …, x_{n} is a regular sequence on F/L and the image of L in the free A/(x_{n−d+1}, …, x_{n})-module F/(x_{n−d+1}, …, x_{n})F is a lexicographic A/(x_{n−d+1}, …, x_{n})-submodule. If B is a local noetherian ring with maximal ideal 𝔪 and M is a B-module, then the Hilbert series H_{M}(s) is the Hilbert series of the associated graded module gr_{𝔪}M.

Lemma 15. Let A be a regular local ring with maximal ideal 𝔪. If I ⊆ J are monomial ideals in A, B = A/I and C = A/J, then P_{C}^{B}(t) = P_{gr_{𝔪}C}^{gr_{𝔪}B}(t). If F is a free A-module and M ⊆ F is a monomial submodule then P_{F/M}^{A}(t) = P_{gr_{𝔪}(F/M)}^{gr_{𝔪}A}(t).

Proof. The second statement is part of Lemma 32 in [22], and the proof of the first statement is a straightforward generalization of the proof of that lemma. □

Lemma 16. Let h(s) ∈ ℤ((s)) be a Laurent series and d a non-negative integer. There is a submodule M ⊆ F such that F/M has Hilbert series h(s) and F/M has depth at least d if and only if there is a d-lexicographic submodule L of F such that F/L has Hilbert series h(s).

Proof. If h(s) is the Hilbert series of a quotient module of F with depth at least d, then it is also the Hilbert series of a graded quotient module of gr_{𝔪}F with depth at least d. Then, by Proposition 4, there is a d-lexicographic submodule L̃ of gr_{𝔪}F such that (gr_{𝔪}F)/L̃ has Hilbert series h(s). Let L be the submodule of F generated by the monomials in F corresponding to the monomial generators of L̃ in gr_{𝔪}F. This is the d-lexicographic submodule that we seek. □

Remark 17.

We recall the construction of a spectral sequence due to Serre, cf. Chapter II, Section A in [26]. Let (B, 𝔪, k) be a noetherian local ring, and let M and N be finitely generated B-modules. There is a convergent spectral sequence residing in the first quadrant, with

E_{m,q}^{1} = Tor_{m}^{gr_{𝔪}(B)}(gr_{𝔪}(M), gr_{𝔪}(N))_{q}
∂_{m,q}^{r} : E_{m,q}^{r} ⟶ E_{m−1,q+r}^{r} for r ≥ 1

and such that E_{m,∗}^{∞} = ⊕_{q∈ℤ} E_{m,q}^{∞} is the graded object associated to Tor_{m}^{B}(M, N) by an 𝔪-stable filtration. The sequence is constructed as follows: M has a B-free resolution

X : ⋯ ⟶ X_{m+1} ⟶^{∂_{m+1}} X_{m} ⟶^{∂_{m}} X_{m−1} ⟶ ⋯

endowed with a decreasing filtration X = X(0) ⊇ X(1) ⊇ ⋯ with ∂(X(q)) = (∂X)∩X(q), such that the graded object associated to X by this filtration is a homogeneous gr_{𝔪}(B)- free resolution of gr_{𝔪}(M). The complex Y = X ⊗_{B} N has the induced filtration Y (q) = Im(X(q) ⊗_{B} N ⟶ X ⊗_{B} N), and the resulting spectral sequence is the one above. The filtration on H(Y) = Tor^{B}(M, N) is defined by the submodules Im(H(Y (q)) ⟶ H(Y)).

In the special case when N = k, one has E_{m,∗}^{1} = Tor_{m}^{gr_{𝔪}(B)}(gr_{𝔪}(M), k)), and from this it follows that P_{M}^{B}(t) ≼ P_{gr_{𝔪}(M)}^{gr_{𝔪}(B)}(t).

Theorem 18. Fix power series h_{1}(s), h_{2}(s) ∈ ℤ[[s]] and a non-negative integer d.

Let Π be the set of pairs (B, C) where B is a local ring with H_{B}(s) = h_{1}(s) and depthB ≥ d, and C is a homomorphic image of B, with H_{C}(s) = h_{2}(s) and depthC ≥ d.

If Π is non-empty, then there is a pair (B′, C′) ∈ Π such that P_{C}^{B}(t) ≼ P_{C′}^{B′}(t) for each (B, C) ∈ Π.

Proof. Let 𝔪_{B} be the maximal ideal of B. We may replace B and C by their 𝔪_{B}-adic completions without changing their Hilbert series, their depths, or the Poincaré series P_{C}^{B}(t). The structure theory of complete local rings tells us that there is a regular local ring A such that B = A/I and C = A/J with I ⊆ J. Fix a minimal set of generators x_{1}, …, x_{n} for the maximal ideal 𝔪_{A} of A.

Consider the associated graded rings Q = grA, R = grB, and S = grC with respect to the 𝔪_{A}-adic filtration. By Remark 17, P_{C}^{B}(t) ≼ P_{S}^{R}(t). So, with R′ and S′ as in Theorem 1, P_{C}^{B}(t) ≼ P_{S′}^{R′}(t).

By Lemma 16 there are d-lexicographic ideals I′ ⊆ J′ such that B′ = A/I′ and C′ = A/J′ have Hilbert series h_{1}(s) and h_{2}(s) respectively. Then, grB′ = R′ and grC′ = S′ and

P_{C}^{B}(t) ≼ P_{S′}^{R′}(t) = P_{C′}^{B′}(t)

by Lemma 15. □

Theorem 19. Fix a power series h(s) ∈ ℤ[[s]] and a non-negative integer d.

Let Π be the set of pairs (B, k) where B is a local ring such that H_{B}(s) = h(s) and depthB ≥ d.

If Π ≠ ∅, then there is a pair (B′, k) ∈ Π such that P_{k}^{B}(t) ≼ P_{k}^{B′}(t) for each (B, k) ∈ Π.

Proof. The proof is similar to the proof of Theorem 18, except that the problem is reduced to Corollary 8, rather than to Theorem 1. □

Theorem 20. Let A be a regular local ring and F be a finitely generated free A-module. Fix a Laurent series h(s) ∈ ℤ((s)) and a non-negative integer d.

Let Π be the set of pairs (A, F/M) where M ⊆ F is a homogeneous submodule such that H_{F/M}(s) = h and depthF/M ≥ d.

If Π is non-empty, then there exists a pair (A, F/M′) ∈ Π such that P_{F/M}^{A}(t) ≼ P_{F/M′}^{A}(t), for each (A, F/M) ∈ Π. We may take M′ to be a d-lexicographic submodule of F.

Proof. The proof is similar to the proof of Theorem 18, except that it is not necessary to complete A and F, and the problem is reduced to Theorem 2, rather than to Theorem 1.

□

The following statement gives the maximal Poincaré series appearing in Theorem 18. Its proof is immediate from Theorem 3 and Lemma 15. Stable ideals in a regular local ring are defined as in Section 4 for a polynomial ring.

Theorem 21. Let A be a regular local ring and let x_{1}, …, x_{n} be generators of the maximal ideal 𝔪. Let I and J be stable ideals generated by monomials in x_{1}, …, x_{n} such that I ⊆ J ∩ 𝔪^{2}.

The Poincaré series of A/J as an A/I-module is given by

P_{A/J}^{A/I}(t) = \frac{1 + t(∑_{b∈G(J)}(1 + t)^{max(b)−1} − ∑_{c∈G(I)∩G(J)}(1 + t)^{max(c)})}{1 − t^{2} ∑_{a∈G(I)}(1 + t)^{max(a)−1}},

where G(𝔞) and G(𝔟) are minimal sets of monomial generators of 𝔞 and 𝔟 respectively.

7: Gorenstein Rings

A special case of both Theorem 1 and Theorem 2 is that if Π consists of pairs (Q, Q/𝔟) such that Q/𝔟 is Cohen-Macaulay with a given Hilbert series h(s), then there is an element of Π with a maximal Poincaré series. Likewise, a special case of Corollary 7 is that if Π consists of pairs (Q/𝔞, k) such that Q/𝔞 is Cohen-Macaulay with a given Hilbert series h(s), then there is an element of Π with a maximal Poincaré series. These cases suggest two questions.

Question 22. Let h(s) ∈ ℤ[[s]] be the Hilbert series of a graded Gorenstein quotient ring of Q. Is there a graded Gorenstein quotient ring R′ of Q with Hilbert series h(s) such that P_{R}^{Q}(s; t) ≼ P_{R′}^{Q}(s; t) for every graded Gorenstein quotient ring R of Q with Hilbert series h(s)?

If dim R ≥ dim Q − 2, then R = Q/I where I is generated by a regular sequence of degrees that can be determined from h(s). Thus, the Poincaré series of all such R are the same, so that the question becomes trivial. If dim R = dim Q − 3, then Diesel proved that the answer to the question is yes [9]. The question is open when dim R < dim Q−3. When dim R = dim Q − 4, Geramita, Ko and Shin have conjectured that the answer is yes and have given a candidate for R′ [15].

The technique used to prove Theorems 1 and 2 does not help us with this question since it is not always possible to deform a Gorenstein ideal to a monomial Gorenstein ideal. Indeed, in the Artinian case the only monomial Gorenstein ideals are complete intersections, while there are many Hilbert series that arise for Artinian Gorenstein rings that are not the Hilbert series of any complete intersection.

Question 23. Let h(s) ∈ ℤ[[s]] be the Hilbert series of a graded Gorenstein quotient ring of Q. Is there a graded Gorenstein quotient ring R′ of Q with Hilbert series h(s) such that P_{k}^{R}(s; t) ≼ P_{k}^{R′}(s; t) for every graded Gorenstein quotient ring R of Q with Hilbert series h(s)?

If dim R ≥ dim Q − 2, than R is a complete intersection as above. In this case the Poincaré series of all such R are the same as well. See [27] for these series. If dim R < dim Q − 2, then the question is open.

For other theorems and conjectures on maximal Poincaré series, see [2] and [3].

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Purdue University, West Lafayette, Indiana and University of Missouri, Columbia iyengar@ math.missouri.edu

Queen’s University, Kingston, Ontario and Center for Communications Research, Princeton, New Jersey pardue@ ccr-p.ida.org