Homology, Homotopy and Applications, vol.4(2), 2002, pp.17–27

HOMOTOPY LIE ALGEBRAS AND POINCARÉ SERIES OF ALGEBRAS WITH MONOMIAL RELATIONS

LUCHEZAR L. AVRAMOV

(communicated by Clas Löfwall)

Abstract

To every homogeneous ideal of a polynomial ring S over a field K, Macaulay assigned an ideal generated by monomials in the indeterminates and with the same Hilbert function. Thus, from the point of view of Hilbert series residue rings modulo monomial ideals display the most general behavior. The homological perspective reveals a very different picture. Two aspects are particularly relevant to this paper:

If I is generated by monomials, then the Poincaré series of the residue field k of S/I is rational by Backelin [7], and the homotopy Lie algebra of S/I is finitely generated by Backelin and Roos [8]. Constructions of Anick [1] and Roos [15], respectively, show that these properties may fail for general homogeneous ideals.

Recenly, Gasharov, Peeva, and Welker [12] showed that some homological properties of S/I, such as being Golod, depend only on combinatorial data gathered from a minimal set of monomial generators.

Here we prove that these data determine the Poincaré series of k over S/I, along with most of its homotopy Lie algebra. As a consequence, we obtain the surprising result that if the number of generators of the ideal I is fixed, then the number of such Poincaré series is finite, even when K ranges over all fields.

To Jan–Erik Roos on his sixty–fifth birthday

1. Results

Let K be a commutative ring and x a finite set of indeterminates over K. A monomial ideal in the polynomial ring S = K[x] is an ideal I generated by some subset M in x. It is well known, and easy to see, that such an I has a uniquely defined minimal (with respect to inclusion) set of monomial generators M_{I}.

Partly supported by a grant from the National Science Foundation. Received February 15, 2001, revised May 8, 2002; published on July 12, 2002. 2000 Mathematics Subject Classification: 13D. Key words and phrases: monomial ideal, graded Lie algebra, Poincaré series. \begin{matrix}c \\ ◯ \end{matrix} 2002, Luchezar L. Avramov. Permission to copy for private use granted.

For every P ⊆ M_{I} set m_{P} = lcm{p|p ∈ P }; thus, m_{∅} = 1, and m_{{p}} = p for p ∈ M_{I}. The LCM lattice, cf. [12], is the set L_{I} = {m_{P} ∈ S|P ⊆ M_{I}} ordered by divisibility: m_{P} ⩽ m_{Q} if and only if m_{P} |m_{Q}. The GCD graph G_{I} has the elements of L_{I} as vertices, and the pairs {m_{P} , m_{Q}} with gcd{m_{P} , m_{Q}} = 1 as edges.

Set R = S/I and k = R/(x). The K-module ExtR^{•}(k, k), graded by cohomological degree and equipped with Yoneda products, becomes a graded associative algebra. When K is a field it is the universal enveloping algebra of a graded Lie algebra π^{•}(R), called the homotopy Lie algebra of R, cf. [5, §10] for details.

Theorem 1. Let K be a field, let x and x′ be finite sets of indeterminates, let I ⊆ K[x] and I′ ⊆ K[x′] be ideals generated by monomials of degree at least 2, and let R = K[x]/I and R = K[x′]/I′ be the corresponding residue rings.

If λ: L_{I} → L_{I′} is an isomorphism of lattices inducing an isomorphism of graphs G_{I} → G_{I′}, then there is an isomorphism of graded Lie algebras over K,

π^{⩾2}(R) ≅ π^{⩾2}(R′) .

Let n be a natural number. If x ranges over all finite sets of indeterminates and I ranges over all ideals in K[x] generated by n monomials in x, then there exists only a finite number of isomorphism classes of graded Lie algebras π^{⩾2}(K[x]/I).

A proof is presented Section 2. Here we record some corollaries.

Every finite R-module N has a Poincaré series, defined to be the power series

P_{N}^{R}(t) = ∑_{j∈ℕ} rank_{K} Ext_{R}^{j}(N, k)t^{j} ∈ ℤ[[t]] .

The Lie algebra π^{•}(R) always determines the Poincaré series of k over R, namely

P_{k}^{R}(t) = ∏_{i=0}^{∞} \frac{(1 + t^{2i+1})^{ε_{2i+1}(R)}}{(1 − t^{2i+2})^{ε_{2i+2}(R)}}  (1)

where ε_{j}(R) = rank_{K} π^{j}(R) is the jth deviation of R. As ε_{1}(R) = |x|, we have

Corollary 2. There is an equality of of formal power series

\frac{P_{k}^{R}(t)}{(1 + t)^{|x|}} = \frac{P_{k′}^{R′}(t)}{(1 + t)^{|x′|}} .  □

There always is a coefficientwise inequality of formal power series

P_{k}^{R}(t) ≼ \frac{(1 + t)^{|x|}}{1 + t − t P_{R}^{S}(t)} .  (2)

The ring R is called Golod if equality holds. It is easy to see that P_{R}^{S}(t) is determined by L_{I}, cf. Remark 9, so from Corollary 2 we recover [12, (3.5.2)], one of the main results of that paper. It can also be obtained directly from Theorem 1, since by [4] the ring R is Golod if and only if the Lie algebra π^{⩾2}(R) is free.

Corollary 3. The ring R is Golod if and only if the ring R′ is Golod.

The field K is fixed in the results above. They imply that finitely many series

b_{M,K}(t) = \frac{(1 + t)^{|x|}}{P_{k}^{K[x]/(M)}(t)} ∈ ℤ[[t]]  (3)

are obtained by letting x vary over all finite sets of indeterminates and M over all subsets of n monomials. Next we consider what happens when K varies.

Deviations being invariant under field extensions of K, Formula (1) shows that the finiteness property still holds when K is allowed to range over all fields of equal characteristic. However, even when the sets x and M are fixed, the polynomial b_{M,K}(t) may change with the characteristic of K, see Remark 13. To analyze the dependence on the characteristic of K we use an important result of Backelin [7], who proves that b_{M,K}(t) is actually a polynomial in ℤ[t]. In Section 3 we prove

Theorem 4. Let n be a natural number. If K ranges over all fields, x ranges over all finite sets of indeterminates, and M ranges over all subsets of n monomials in K[x], then there exist only finitely many polynomials b_{M,K}(t).

In Section 4 we show that the hypotheses of results above cannot be weakened significantly, nor can their conclusions be strengthened substantially.

2. Graded Lie algebras

The arguments depend on DG (= differential graded) homological algebra. Here DG algebras are non-negatively graded and graded-commutative. Taylor DG algebra resolutions of monomial ideals play a central role. We recall their definition, using notation introduced above and referring for details to [10, §5].

Fixing a linear order ≻ on M_{I}, for every two subsets P, Q of M_{I} we set

inv(P, Q) = |{(p, q) ∈ P × Q | p ≻ q}| .

For each n ∈ ℤ, let T_{n} be the free S-module with basis {e_{P} |P ⊆ M_{I}, |P | = n} and let ∂_{n}: T_{n} → T_{n−1} be the S-linear homomorphism given on e_{P} by

∂(e_{P}) = ∑_{p∈P}(−1)^{inv(\{p\},P ∖\{p\})} \frac{m_{P}}{m_{P ∖\{p\}}} e_{P ∖\{p\}} .  (4)

The pair (T, ∂) is a free resolution of S/I over S. The product of T is defined by

e_{P} · e_{Q} = \begin{cases}(−1)^{inv(P,Q)} gcd(m_{P} , m_{Q}) e_{P ∪Q} & if P ∩ Q = ∅; \\ 0 & otherwise , \end{cases}  (5)

on pairs of basis elements, and extended to all of T by bilinearity. Different choices of the linear order ≻ among the generators of I lead to isomorphic DG algebras.

A DG Γ -algebra is a DG algebra with a system of divided powers operations y ↦ y^{(r)}, defined for every y of even positive degree and all r ∈ ℕ, and compatible with the differential. A complete list of the identities they satisfy can be found in [13, (1.7.1), (1.8.1)]. Morphisms of DG Γ-algebras commute with all their structures.

Remark 5. If U and V are DG Γ-algebras over S, then U ⊗_{S} V has a structure of DG Γ-algebra, functorial in U and V and defined uniquely by the condition that the canonical maps U → U ⊗_{S} V ← V are morphisms, cf. e.g. [13, (1.8.3)]. Every DG algebra over ℚ is a Γ-algebra, with y^{(r)} = y^{r}/(r!), cf. [13, (1.7.2)].

Lemma 6. Every Taylor DG algebra resolution T has a unique structure of DG Γ-algebra, given for each n ∈ 2ℕ ∖ {0} and all r ∈ ℕ by the formula

( ∑_{P ⊆M_{I} ; |P |=n} a_{P} e_{P} )^{(r)} = ∑_{P_{h}⊆M_{I} ; |P_{h}|=n} a_{P_{1}} ⋯ a_{P_{r}} e_{P_{1}} ⋯ e_{P_{r}} .  (6)

Proof. Let γ_{r}( ∑_{P} a_{P} e_{P} ) denote the right hand side of the desired formula. For each element e_{P} the multiplication table (5) gives e_{P}^{q} = 0 for all q ⩾ 2. All elements e_{P} under consideration have even degree, so they commute with each other, hence

( ∑_{P} a_{P} e_{P} )^{r} = ∑_{q_{1}+···+q_{r}=r} \frac{r!}{(q_{1}!) ⋯ (q_{r}!)} a_{P_{1}}^{q_{1}} ⋯ a_{P_{r}}^{q_{r}} e_{P_{1}}^{q_{1}} ⋯ e_{P_{r}}^{q_{r}} = r!γ_{r}( ∑_{P} a_{P} e_{P} )

To prove existence we vary K, so (temporarily) we set S_{K} = K[x_{1}, …, x_{e}] and let T_{K} denote a Taylor DG algebra resolution of S_{K}/M_{I}S_{K} over S_{K}. In T_{ℚ} the equality above yields γ_{r}(y) = y^{r}/(r!) for all y ∈ T_{n}; by Remark 5 this transforms T_{ℚ} into a DG Γ-algebra. The canonical map T_{ℤ} → T_{ℤ} ⊗_{ℤ} ℚ = T_{ℚ} is injective and commutes with the maps y ↦ γ_{r}(y); they form a system of divided powers on T_{ℚ}, hence on T_{ℤ} as well. By Remark 5 the system of divided powers on T_{ℤ} induces one on T_{K} through the isomorphism T_{K} ≅ T_{ℤ} ⊗_{ℤ} K of DG algebras over K.

To prove uniqueness, for each P ⊆ M_{I} we set d_{P} = gcd{p|p ∈ P }. From the formula for divided powers of a product of elements of even degrees, Equation (5), and the formula for divided powers of a product of elements of odd degrees, we get

d_{P}^{r} e_{P}^{(r)} = (d_{P} e_{P} )^{(r)} = ( ± ∏_{p∈P} e_{ℓ})^{(r)} = 0

for all r ⩾ 2. As d_{P} is not a zero-divisor on T, it follows that e_{P}^{(r)} = 0, hence

( ∑_{P} a_{P} e_{P} )^{(r)} = ∑_{q_{1}+···+q_{r}=r} a_{P_{1}}^{q_{1}} ⋯ a_{P_{r}}^{q_{r}} e_{P_{1}}^{(q_{1})} ⋯ e_{P_{r}}^{(q_{r})} = γ_{r}( ∑_{P} a_{P} e_{P} )

due to the formulas for divided powers of sums and of products.

Homotopy Lie algebras for DG Γ-algebras are introduced in [3], cf. also [6].

Remark 7. Let D be a DG Γ-algebra such that the ring D_{0} is noetherian and the D_{0}-module H_{n}(D) is finite for every n ∈ ℤ. If k is a field and D → k is a surjective morphism of DG algebras, then the Eilenberg-Moore extension functors define a graded algebra ExtD^{•}(k, k). It is the universal enveloping algebra of a graded Lie algebra π^{•}(D), and D ↦ π^{•}(D) is a contravariant functor from the category of DG Γ-algebras augmented to k, to the category of graded Lie algebras over k. If δ: D → D′ is a morphism with H_{•}(δ) bijective, then π^{•}(δ) is an isomorphism.

The morphism D → k can be factored as a composition D → U → k of morphisms of DG Γ-algebra, such that the underlying graded module of U is free over the underlying graded algebra of D, and H_{0}(U) = k. Furthermore, the divided powers of the DG Γ-algebra U ⊗_{D} k of Remark 5 are inherited by the homology algebra H_{•}(U ⊗_{D} k). Let Q^{γ} H_{•}(U ⊗_{D} k) denote the residue of H_{•}(U ⊗_{D} k) modulo its subspace spanned by the elements of degree 0, all products of elements of positive degree, and all divided powers of elements of even positive degree. The vector space π^{•}(D) is the graded k-dual of Q^{γ} H_{•}(U ⊗_{D} k).

The next result is part of [2, (5.1)]. The proof there is difficult, because the general properties of homotopy Lie algebras discussed above were not available at the time. We provide a short argument, valid for general graded rings R.

Lemma 8. Let E be the Koszul complex on x, set D = R ⊗_{S} E, and give D the DG Γ -algebra structure described in Remark 5. If ι: R → D is the inclusion, then π^{•}(ι) yields an isomorphism π^{•}(D) ≅ π^{⩾2}(R) of graded Lie algebras.

Proof. The image of x minimally generates R over K, so by a well known theorem of Gulliksen and Schoeller, cf. [13, (1.6.4)] or [5, (6.3.5)], a factorization D → U → k as in Remark 7 can be chosen so that U_{⩽1} = D_{⩽1} and ∂(U) ⊆ (x)U. The DG algebras U ⊗_{R} k and U ⊗_{D} k then have trivial differentials, so the morphism

H_{•}(U ⊗_{ι} k): H_{•}(U ⊗_{R} k) ⟶ H_{•}(U ⊗_{D} k)

of DG Γ-algebras is equal to the surjection U ⊗_{ι} k: U ⊗_{R} k → U ⊗_{D} k whose kernel is generated by U_{1} ⊗_{R} k. It induces an exact sequence of graded vector spaces

0 ⟶ U_{1} ⊗_{R} k ⟶ Q^{γ} (U ⊗_{R} k) ⟶^{Q^{γ} (U⊗_{ι}k)} Q^{γ} (U ⊗_{D} k) ⟶ 0

Since π^{•}(ι) is the k-linear dual of Q^{γ} (U ⊗_{ι} k), we are done.

The following remark is taken from the proofs of [12, (3.1), (3.5.1)].

Remark 9. Let I′ be a monomial ideal in the polynomial ring S′ = K[x′]. Assume there exists an isomorphism λ: L_{I} → L_{I′} of LCM lattices. Their atoms are the monomials in M_{I} and M_{I′}, so λ maps M_{I} bijectively onto M_{I′}; we extend λ to a bijection λ̂ of the Boolean lattice of M_{I} onto that of M_{I′}. Let T′ be the Taylor DG algebra resolution of S′/I′ constructed using the linear order on M_{I′} induced from M_{I} via λ. Formula (4) shows that the map e_{P} ⊗ 1 ↦ e_{λ̂(P)} ⊗ 1 defines an

isomorphism of complexes of vector spaces over K; in particular, P_{R}^{S}(t) = P_{R′}^{S′}(t).

If, in addition, λ is also an isomorphism of GCD graphs, then Formula (5) shows that \overline{λ} is an isomorphism of DG algebras.

Proof of Theorem 1. Let E and T denote Taylor DG algebra resolutions of k = S/(x) and R = S/I, with augmentations ε: E → k and τ: T → R, respectively. By Lemma 6 and Remark 5, the following maps are morphisms of DG Γ-algebas:

R ⊗_{S} E ⟵^{τ⊗_{S}E} T ⊗_{S} E ⟶^{T ⊗_{S}ε} T ⊗_{S} k .

It is well known that H_{•}(τ ⊗_{S} T) and H_{•}(T ⊗_{S} ε) are isomorphisms. This explains the second and third isomorphisms of graded Lie algebras in the sequence

π^{⩾2}(R) ≅ π^{•}(R ⊗_{S} E) ≅ π^{•}(T ⊗_{S} E) ≅ π^{•}(T ⊗_{S} k)

where the first isomorphism is provided by Lemma 8.

If T′ is a Taylor DG algebra resolution of R′ = S′/I′, and E′ the Koszul complex on x′, then by symmetry we obtain isomorphisms of graded Lie algebras

π^{•}(T′ ⊗_{S′} k′) ≅ π^{•}(E′ ⊗_{S′} T′) ≅ π^{•}(R′ ⊗_{S′} E′) ≅ π^{⩾2}(R′) .

By Formula (6), the isomorphism of DG algebras \overline{λ}: T ⊗_{S} k → T′ ⊗_{S′} k′ from Remark 9 commutes with divided powers, so it induces an isomorphism

π^{•}(T ⊗_{S} k) ≅ π^{•}(T′ ⊗_{S′} k′) .

Assembling the sequences of isomorphisms of graded Lie algebras displayed above, we get the desired isomorphism of graded Lie algebras:

π^{⩾2}(R) ≅ π^{⩾2}(R′) .

This completes the proof of the first assertion of Theorem 1. The second assertion follows immediately from the first one and the elementary remark below.

Remark 10. Let I be a monomial ideal in K[x], minimally generated by n monomials. By construction, the number of vertices in the GCD graph G_{I} is equal to the cardinality of the LCM lattice L_{I}, and |L_{I} | ⩽ 2^{n}. Thus, letting K, x, and I vary, while keeping n fixed, one obtains only a finite number of pairs (L_{I}, G_{I}).

3. Poincaré series

In this section it is convenient to refocus from ideals generated by monomial ideals to the minimal sets of generators of such ideals.

We identify the polynomial ring S = K[x] and the semigroup K-algebra of the free commutative monoid [x] generated by x; in particular, we refer to the elements of [x] as monomials in x. An antichain in [x] is a finite set M of monomials in the variables x with the property that no p ∈ M divides any q ∈ M ∖ {p}. The map I ↦ M_{I} is a bijection between the monomial ideals in S and the antichains in [x].

Let M be an antichain in [x]. Fix an order x_{1}, …, x_{e} on the indeterminates x, and for each i = (i_{1}, …, i_{e}) ∈ ℕ^{e} set x^{i} = x_{1}^{i_{1}} ⋯ x_{e}^{i_{e}}. Give S = K[x] the standard ℕ^{e}-grading defined by Deg(x^{i}) = i. This multigrading is inherited by the residue rings R = S/(M) and k = S/(x). For every ℕ^{e}-graded R-module N and each j ∈ ℕ the finite K-vector space Ext_{R}^{j}(N, k) is multigraded. The series

P_{N}^{R}(s, t) = ∑_{i∈ℕ^{e} ; j∈ℕ} rank_{K} Ext_{R}^{j}(N, k)^{i} s^{i} t^{j} ∈ ℤ[s^{±1}][[t]]

is the multigraded Poincaré series of N. Setting 1 = (1, …, 1) ∈ ℕ^{e}, one recovers the Poincaré series in one variable by means of the formula

P_{N}^{R}(t) = P_{N}^{R}(1, t) .

Backelin [7] proves that the multigraded Poincaré series satisfies the condition

b_{M,K}(s, t) · P_{k}^{R}(s, t) = (1 + s_{1}t) ⋯ (1 + s_{e}t)  (7)

for some polynomial b_{M,K}(s, t) ∈ ℤ[s, t] subject to the following restrictions:

deg_{s_{h}}(b_{M,K}(s, t)) ⩽ max{deg_{x_{h}}(p) | p ∈ M} for h = 1, …, e;  (8)
deg_{t}(b_{M,K}(s, t)) ⩽ deg( lcm{p ∈ M}) .  (9)

We now head for a proof of Theorem 4, dealing with Poincaré series P_{k}^{K[x]/(M)}(t) for antichains of monomials M in [x]. The following elementary fact will be needed.

Lemma 11. If M is a fixed antichain M and K ranges over all fields, then there exist only finitely many Poincaré polynomials P_{K[x]/(M)}^{K[x]}(s, t) ∈ ℤ[s, t].

Proof. Let T denote the Taylor resolution of ℤ[x]/(M) over ℤ[x]. Since T ⊗_{ℤ} K is the Taylor resolution of K[x]/(M) over K[x], the isomorphism

Hom_{ℤ[x]}(T, ℤ) ⊗_{ℤ} K ≅ Hom_{K[x]}((T ⊗_{ℤ} K), k)

of complexes of multigraded K-spaces yields for all j ∈ ℤ and i ∈ ℕ^{e} isomorphisms

Ext_{K[x]}^{j}(K[x]/(M), k)^{i} ≅ H_{−j}(Hom_{ℤ[x]}(T, ℤ) ⊗_{ℤ} k)^{i} .

Let P be the set of prime numbers that annihilate some non-zero homology class of Hom_{ℤ[x]}(T, ℤ). For char(K) ∉ P the Künneth formula yields isomorphisms

H_{−j}(Hom_{ℤ[x]}(T, ℤ) ⊗_{ℤ} k)^{i} ≅ (H_{−j}(Hom_{ℤ[x]}(T, ℤ)) ⊗_{ℤ} k)^{i}

for all j and i. Thus, the polynomials P_{K[x]/(M)}^{K[x]}(s, t) are equal for all K with char(K) ∉ P. To finish the proof, it remains to remark that P is finite, because Hom_{ℤ[x]}(T, ℤ) is a finite complex of finitely generated free abelian groups.

Proof of Theorem 4. Let n be a natural number. By Remark 10 we may choose a finite family F_{n} of antichains of n monomials with the following property: All pairs (L_{I′}, G_{I′}) that can be obtained from some ideal I′ minimally generated by n monomials in some polynomial ring over some field K have the form (L_{I}, G_{I}) for some I = (M)K[x] with M ∈ F_{n}. Thus, we obtain

sup_{K} sup_{x} sup_{|M|=n} deg(b_{M,K}(t)) = sup_{K} sup_{M∈F_{n}} deg(b_{M,K}(t)) ⩽ sup_{M∈F_{n}} deg(m_{M}) < ∞

where the equality comes from Formula (3) and Corollary 2, the first inequality from Formula (9), and the second inequality from the finiteness of F_{n}. Choose a natural number d such that deg(b_{M,K}(t)) ⩽ d for all M and all K.

Fix, for the moment, an antichain M and a field K. Since b_{M,K}(0) = 1, there is a unique decomposition b_{M,K}(t) = ∏_{r=0}^{d}(1 − β_{r}t) with β_{r} ∈ ℂ. Let β_{M,K} denote the maximal absolute value of these complex numbers, and let ρ_{M,K} denote the radius of convergence of the rational function (1+t)^{|x|}/(1+t−t P_{K[x]/(M)}^{K[x]}(t)). As Formula (3) shows that 1/β_{M,K} is equal to the radius of convergence of the Poincaré series P_{k}^{K[x]/(M)}(t), we obtain 1/β_{M,K} ⩾ ρ_{M,K} by Formula (2).

On the other hand, Formula (3) and Corollary 2 yield

inf_{K} inf_{x} inf_{|M|=n}(ρ_{M,K}) = inf_{K} inf_{M∈F_{n}}(ρ_{M,K}) .

Since F_{n} is finite, Lemma 11 shows that the second infimum can be computed by using only finitely many fields K, hence it is equal to a real number ρ > 0.

Summing up, we obtain inequalities β_{M,K} ⩽ 1/ρ < ∞ for every choice of M and K. They imply that in every polynomial b_{M,K}(t) the absolute value of the coefficient of t^{r} does not exceed \binom{d}{r}/ρ^{r}. Since each b_{M,K}(t) has integer coefficients, there is only a finite number of distinct polynomials of this form.

4. Discussion

We start with a problem suggested by Theorem 4.

Remark 12. Theorem 4 may be restated as asserting the finiteness of the number

d(n) = sup_{K} sup_{x} sup_{|M|=n} deg (b_{M,K}(t)) .

In view of Formula (3), the ranks of the modules F_{j} for j = 0, …, d(n) in a minimal free resolution F determine the entire Poincaré series of k over K[x]/(M). Thus, knowledge of an effective bound on d(n) would be extremely useful for computational purposes. Using the sets M = {x^{2}|x ∈ x} one gets d(n) ⩾ 2n. Equality is easy to check for n ⩽ 3. For all n, it would be consistent with observations of Charalambous and Reeves [9, p. 2390] based on computer experiments.

It is known that the characteristic of K may affect the Poincaré series of k. To document this fact we present a construction proposed by the referee.

Remark 13. The polynomial b_{M,K}(t) may vary with the characteristic of K. Indeed, let M′ be an antichain of monomials, fix an indeterminate x ∈ x, and consider the antichain M = xM′. For every field K, the ring K[x]/(M) is Golod by a theorem of Shamash [17], so the inequality in Formula (2) becomes an equality. In view of Formula (7), it suffices to show that the Poincaré series P_{K[x]/(M)}^{K[x]}(t) depends on

K. It is equal to P_{K[x]/(M′)}^{K[x]}(t), so examples may be obtained by choosing M′ to be the squarefree monomial ideal associated to a simplicial complex Δ, whose reduced homology groups H̃_{i}(Δ, K) vary with the characteristic of K, cf.[14].

The preceding examples notwithstanding, some terms of b_{M,K}(s, t) are independent of K. This is shown by the remark that follows, which also shows that the degree bounds in Formula (8) are in fact equalities.

Remark 14. Every antichain M in [x] is determined by any one of the polynomials b_{M,K}(s, t) for some K through the congruence

b_{M,K}(s, t) ≡ 1 − ( ∑_{p∈M} s^{Deg(p)})t^{2} mod (t^{3}ℤ[s, t]) .

Indeed, standard equalities, cf. e.g. [5, (7.1.5) and (7.1.1)], yield a congruence

P_{k}^{R}(s, t) ≡ 1 + ( ∑_{h=1}^{e} s_{i})t + ( ∑_{1⩽g<h⩽e} s_{g}s_{h} + ∑_{p∈M} s^{Deg(p)})t^{2} mod (t^{3}ℤ[s][[t]]) .

Writing b_{M,K}(s, t) in the form ∑_{n} p_{n}(s)t^{n} with p_{n}(s) ∈ ℤ[s], we get the desired congruence by comparing the coefficients of 1, t, and t^{2} in Equality (7).

The remark above produces infinitely many polynomials b_{M,K}(s1, t) in two variables. Combining this information with Formulas (7) and (1), we obtain

Remark 15. When I varies over all ideals of S generated by a fixed number of monomials there exist infinitely many power series in two variables of the form P_{k}^{S/I}(s1, t); as a consequence, the isomorphism of Lie algebras in Theorem 1 does not preserve gradings—let alone multigradings—induced by those of S.

Finally, we emphasize that the finiteness conclusions of Theorems 1 and 4 heavily depend on the hypothesis that the ideals involved are generated by monomials.

Remark 16. When J varies over all ideals of S generated by a fixed number of forms of prescribed degrees, the number of power series P_{k}^{S/J}(t) may be infinite, and hence there may exist infinitely many non-isomorphic graded Lie algebras π^{⩾2}(S/J).

Indeed, assume K has characteristic 0. Fröberg, Gulliksen, and Löfwall [11] construct a family of ideals {J_{a} ⊂ K[x_{1}, …, x_{15}] | a ∈ K} generated by 67 quadratic forms, such that the rings R_{a} = S/J_{a} have the same Hilbert series for all a, but there are infinitely many different Poincaré series P_{k}^{R_{a}}(t). Also, Roos [16] exhibits a family of ideals {J_{a} ⊂ K[x_{1}, …, x_{6}] | a ∈ ℕ} generated by 11 quadratic forms, with equal Hilbert series and different Poincaré series for all a ⩾ 2.

Acknowledgements

I should like to thank the referee for two very close readings of the manuscript, and for suggesting—and insisting on—a detailed treatment of the role of the characteristic of the base field. I also want to thank Jörgen Backelin for useful discussions that influenced the final presentation of this material.

References

[1] D. Anick, A counterexample to a conjecture of Serre, Ann. of Math. (2) 115 (1982), 1–33; Comment, ibid. 116 (1982), 661.

[2] L. L. Avramov, The Hopf algebra of a local ring, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 253–277 [Russian; English translation:] Math. USSR-Izv. 38 (1974), 259–284.

[3] L. L. Avramov, Local algebra and rational homotopy, Homotopie Algébrique et Algèbre Locale (Luminy, 1982) Astérisque 113-114, Soc. Math. France, Paris, 1984; pp. 15–43.

[4] L. L. Avramov, Golod homomorphisms, Algebra, algebraic topology, and their interactions (Stockholm, 1983), Lecture Notes Math. 1183, Springer, Berlin-New York, 1986; pp. 58–78.

[5] L. L. Avramov, Infinite free resolutions, in: Six lectures in commutative algebra, Bellaterra, 1996, Progress in Math. 166, Birkhäuser, Boston, 1998; pp. 1–118.

[6] L. L. Avramov, S. Halperin, Through the looking glass: a dictionary between rational homotopy theory and local algebra, Algebra, algebraic topology and their interactions (Stockholm, 1983), Lecture Notes in Math. 1183, Springer, Berlin-New York, 1986; pp. 1–27.

[7] J. Backelin, Les anneaux locaux à relations monomiales ont des séries de Poincaré–Betti rationnelles, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), 607–610.

[8] J. Backelin, J.-E. Roos, When is the double Yoneda Ext-algebra of a local Noetherian ring again Noetherian?, Algebra, algebraic topology and their interactions (Stockholm, 1983), Lecture Notes in Math. 1183, Springer, Berlin-New York, 1986; pp. 101–119.

[9] H. Charalambous, A. Reeves, Poincaré series and resolutions of the residue field over monomial rings, Comm. Algebra 23 (1995), 2389–2399.

[10] R. Fröberg, Some complex constructions with applications to Poincaré series, Séminaire d’Algèbre P. Dubreil 31ème année (Paris, 1977–1978), Lecture Notes in Math. 740, Springer, Berlin-New York, 1979; pp. 272–284.

[11] R. Fröberg, T. H. Gulliksen, C. Löfwall, Flat families of local, artinian algebras with an infinite number of Poincaré series, Algebra, algebraic topology and their interactions (Stockholm, 1983), Lecture Notes in Math. 1183, Springer, Berlin-New York, 1986; pp. 170–191.

[12] V. Gasharov, I. Peeva, V. Welker, The LCM-lattice in monomial resolutions, Math. Res. Letters 6 (1999), 521–532.

[13] T. H. Gulliksen, G. Levin, Homology of local rings, Queen’s Papers Pure Appl. Math. 20, Queen’s Univ., Kingston, ON, 1969.

[14] G. A. Reisner, Cohen-Macaulay quotients of polynomial rings, Adv. in Math. 21 (1976), 30–49.

[15] J.-E. Roos, Relations between Poincaré-Betti series of loop spaces and of local rings, Séminaire d’Algèbre P. Dubreil 31ème année (Paris, 1977–1978), Lecture Notes in Math. 740, Springer, Berlin-New York, 1979; pp. 285–322.

[16] J.-E. Roos, Commutative non Koszul algebras having a linear resolution of arbitrary high order. Applications to torsion in loop space homology, C. R. Acad. Sci. Paris Sér. I 316 (1993), 1123–1128.

[17] J. Shamash, The Poincaré series of a local ring, J. Algebra 12 (1969), 453– 470.

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Luchezar L. Avramov avramov@math.unl.edu

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