HOMOTOPY LIE ALGEBRAS AND POINCARÉ SERIES OF ALGEBRAS WITH MONOMIAL RELATIONS

LUCHEZAR L. AVRAMOV

Dedicated to Jan-Erik Roos

A theorem of Macaulay assigns to every homogeneous ideal of a polynomial ring S one that is generated by monomials in the indeterminates and has the same Hilbert function. Thus, from the point of view of Hilbert series residue rings modulo monomial ideals exhibit the most general behavior. The homological perspective reveals a very different picture, with two aspects 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 [6], and the homotopy Lie algebra of S/I is finitely generated by Backelin and Roos [7]. Constructions due to Anick [1] and Roos [12], respectively, show that these properties may fail for general homogeneous ideals.

For a monomial ideal I, Gasharov, Peeva, and Welker [10] show 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, along with most of its homotopy Lie algebra. As a consequence, we describe new restrictions on the homology of monomial ideals.

Results

Let K be a commutative ring, x a finite set of indeterminates, S = K[x] the polynomial ring, I an ideal of S generated by monomials in x, and R = S/I. Let M_{I} denote the unique minimal set of monomial generators of I, characterized by the property that no element of M_{I} divides any other. For every subset P of M_{I} we set m_{P} = lcm{p | p ∈ P }; in particular, m_{∅} = 1, and m_{{p}} = p for each p ∈ M_{I}.

The LCM lattice L_{I}, defined in [10], is the set {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.

When K is a field we set k = R/Rx and define the Poincaré series of an R-module N to be the power series P_{N}^{R}(t) = ∑_{j∈ℕ} rank_{K} Ext_{R}^{j}(N, k)t^{j} ∈ ℤ[[t]]. The K-vector space ExtR^{•}(k, k), equipped with Yoneda products and graded by cohomological degree, is the universal enveloping algebra of a graded Lie algebra π^{•}(R), called the homotopy Lie algebra of R, cf. [4, §10] for details.

Theorem 1. Let S = K[x] and S′ = K[x′] be polynomial rings over a field K.

If I ⊆ S and I′ ⊆ S′ are ideals generated by monomial of degree at least 2, and λ: L_{I} → L_{I′} is an isomorphism of lattices inducing an isomorphism of graphs G_{I} → G_{I′}, then for the rings R = S/I and R′ = S′/I′ there is an isomorphism

π^{⩾2}(R) ≅ π^{⩾2}(R′)

Date: February 15, 2001.

Partly supported by a grant from the National Science Foundation.

of graded Lie algebras over K, and an equality of formal power series

P_{k}^{R}(t)/(1 + t)^{card x} = P_{k′}^{R′}(t)/(1 + t)^{card^{x′}} .

We present the proof after recording some (possibly, unexpected) corollaries. With card M_{I} fixed there are finitely many choices for L_{I} and G_{I}, so we get:

Corollary 2. There is only a finite number of isomorphism classes of graded Lie algebra π^{•}(S/I), and a finite number of power series P_{k}^{S/I}(t), for I ranging over all the ideals of S generated by a fixed number of monomials.

The situation for general homogeneous ideals is completely different.

Remark 3. When K is infinite, Fröberg, Gulliksen, and Löfwall [9] 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). When K contains ℚ, Roos [13] 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.

Set e = card x, and consider the ℕ^{e}-grading of S defined by setting Deg(x_{h}) = (0, …, 0, 1, 0 …, 0), with 1 in the h’th position. Since I and Sx are monomial ideals, the residue rings R and k inherit multigradings. For each j ∈ ℕ the vector space Ext_{R}^{j}(k, k) then receives a ℕ^{e}-grading. If s = s_{1}, …, s_{e} is a sequence of indeterminates, set s^{i} = s_{1}^{i_{1}} ⋯ s_{e}^{i_{e}} for i = (i_{1}, …, i_{e}). The formal power series

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

is the multigraded Poincaré series of M. Backelin [6] proves it satisfies the condition

\frac{(1 + s_{1}t) ⋯ (1 + s_{e}t)}{P_{k}^{R}(s, t)} = b_{I}(s, t) ∈ ℤ[s, t] (1)

with a polynomial b_{I}(s, t) subject to the following restrictions:

(2) deg_{t} (b_{I}(s, t)) ≤ deg ( lcm{p ∈ M_{I}});

(3) deg_{s_{h}} (b_{I}(s, t)) ≤ max{deg_{x_{h}}(p) | p ∈ M_{I}} for h = 1, …, e .

The next result shows, in particular, that equalities hold in (3).

Corollary 4. There exists a function d: ℕ → ℕ such that an inequality

deg_{t} (b_{I}(s, t)) ≤ d(card M_{I})

holds for all monomial ideals I. For every such ideal there is a congruence

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

Proof. As b_{I}(1, …, 1, t) = (1 + t)^{e}/ P_{k}^{R}(t), Corollary 2 gives the first assertion. Standard isomorphisms, cf. e.g. [11, proof of (3.3.4)], 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_{I}} s^{Deg(p)})t^{2} mod (t^{3}ℤ[s][[t]]) .

Writing b_{I}(s, t) in the form ∑_{n} p_{n}(s)t^{n} with p_{n}(s) ∈ ℤ[s], we obtain the desired congruence by rewriting Equality (1) in the form b_{I}(s, t) P_{k}^{R}(s, t) = ∏_{h=1}^{e}(1 + s_{h}t) and comparing the coefficients of 1, t, and t^{2}.

Remark 5. Considering ideals generated by the squares of the indeterminates one obtains d(n) ≥ 2n, and it can be shown that equality holds for n ≤ 3.

It would be interesting to have an upper bound on d(n) for all n ∈ ℕ.

Remark 6. In view of the congruence in Corollary 4, the coefficient of t^{2} in P_{k}^{R}(s, t) determines the ideal I, so the isomorphism and equality in Theorem 1 do not preserve multidegrees, or even total degrees, coming from gradings of the ring R.

There always is a coefficientwise inequality of formal power series

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

The ring R is Golod if equality holds. When I and I′ have isomorphic LCM lattices an easy observation in [10], reproduced in Remark 12 below, yields P_{R}^{S}(t) = P_{R′}^{S′}(t). Thus, we recover [10, (3.5.2)], one of the main results of that paper:

Corollary 7. The ring R is Golod if and only if the ring R′ is Golod.

Proofs

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 [8, §5].

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

inv(P, Q) = card{(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}, card 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

(5) 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}

on pairs of basis elements, and extend 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 equipped with a system of divided powers operations y ↦ y^{(r)}, defined for all elements of even positive degree and all integers r ≥ 0, and compatible with the differential. A complete list of identities satisfied by divided powers can be found e.g. in [11, (1.7.1), (1.8.1)]. Morphisms of DG Γ-algebras commute with all their structures.

Remark 8. Every DG algebra over ℚ has a unique structure of Γ-algebra, given by y^{(r)} = y^{r}/(r!), cf. [11, (1.7.2)].

If U and V are DG Γ-algebras over S, then U ⊗_{S} V has a functorial in U and V structure of DG Γ-algebra, uniquely defined by the condition that the canonical maps U → U ⊗_{S} V ← V are morphisms, cf. e.g. [11, (1.8.3)].

For instance, the Koszul complex E on x is a tensor product of complexes of length 1; they are trivially Γ-algebras, so induce a DG Γ-algebra structure on E.

We generalize the last observation as follows.

Lemma 9. Every Taylor DG algebra resolution T has a unique structure of DG Γ-algebra; it is given for each n ∈ 2ℕ ∖ {0} and all r ∈ ℕ by the formula

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

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 8 this operation 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}, so they do on T_{ℤ} as well. Finally, by Remark 8 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_{p})^{(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.

The construction of homotopy Lie algebras is extended to DG Γ-algebras in [3], cf. also [5]. We recapitulate some basic facts.

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 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.

We describe a procedure by which π^{•}(D) can be computed.

Remark 10. 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, cf. Remark 8, 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 11. Let E be the Koszul complex on x, set D = R ⊗_{S} E, and give D the DG Γ -algebra structure described in Remark 8. 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. [11, (1.6.4)] or [4, (6.3.5)], a factorization D → U → k as in Remark 10 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 [10, (3.1), (3.5.1)].

Remark 12. 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 isomorphism of graded vector spaces

\overline{λ}: T ⊗_{S} k ⟶ T′ ⊗_{S} k  defined by  \overline{λ}(e_{P} ⊗ 1) = e_{λ̂(P)} ⊗ 1

is an isomorphism of complexes; in particular, P_{R}^{S}(t) = P_{R′}^{S′}(t). If λ is also an isomorphism of GCD graphs, then Formula (5) shows that \overline{λ} is an isomorphism of DG algebras; in particular, TorS_{•}(R, k) ≅ TorS′_{•}(R′, k′) as graded K-algebras.

Proof of Theorem 1. Let E and T denote Taylor DG algebra resolutions of k = S/Sx and R = S/I, with augmentations ε: E → k and τ: T → R, respectively. By Lemma 9 and Remark 8, 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 11.

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 12 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 isomorphism of graded Lie algebras announced in the theorem:

π^{⩾2}(R) ≅ π^{⩾2}(R′) .

The Lie algebra π^{•}(R) determines the Poincaré series of k over R by the formula

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

As rank_{K} π^{1}(R) = card x, the isomorphism of graded Lie algebras yields an equality

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

of formal power series. This completes the proof of the theorem.

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Department of Mathematics, Purdue University, West Lafayette, IN 47907, U.S.A.

E-mail address: avramov@math.purdue.edu