On rational K[π, 1] spaces and Koszul algebras

Stefan Papadima and Sergey Yuzvinsky

February 3, 1997

1 Introduction

Let X be a connected topological space with finite Betti numbers. Then X is said to be a rational K[π, 1] space if its Q-completion Q_{∞}(X) constructed in [8] is aspheric, that is π_{n}Q_{∞}(X) = 0 for all n > 1.

Let A be a connected graded algebra of finite type over a field F. Then A is said to be a Koszul algebra if Ext_{A}^{p,q}(F, F) = 0, for every p ≠ q. (See Definition 2.1 for details.)

Our main result (Theorem 5.1) establishes an equivalence between the topological rational K[π, 1] property of X and the algebraic Koszul property of H^{∗}(X, Q), in the case of a formal space X. The important class of formal spaces was introduced in [30]. One says that X is formal if Q_{∞}(X) depends only on the cohomology algebra H^{∗}(X, Q), see Section 5 for details. We then derive (in Proposition 5.2) that the Koszulness of H^{∗}(X, Q) implies that Q_{∞}(X) is aspheric, for an arbitrary space X. We also derive (in Corollary 5.3) the following lower central series (LCS) formula for a formal rational K[π, 1] space X:

P (X, −t) = ∏_{n≥1}(1 − t^{n})^{ϕ_{n}}.

Here ϕ_{n} = rank(Γ_{n}/Γ_{n+1}), where {Γ_{n}}_{n≥1} is the lower central series of the fundamental group π_{1}(X), and P (X, t) is the Poincaré polynomial of X. The main technical tool in our proofs is Sullivan’s notion of minimal model for commutative differential graded algebras, see Section 3.

The rational K[π, 1] formal spaces, especially those arising in low- dimensional topology, were studied in [24, 5]. More recently the rational K[π, 1] property has come in spotlight in connection with the higher logarithms, see [17]. There is an extensive literature on the Koszul property, see for example [3, 4, 28], and [23] for connections with quantum groups. The rational K[π, 1] spaces and Koszul algebras first appeared together implicitly in [24] and explicitly in [6].

The complements of complex hyperplane arrangements provide a rich supply of formal spaces [9] and have received a lot of attention. In particular the rational K[π, 1] property for these spaces was studied in [13]-[15], [20]- [22]. In this context we are able to prove (in Proposition 5.4) that the above LCS formula implies the rational K[π, 1] property, for arrangements in C^{3}.

The connection with the Koszul property is the following. It is known that the Koszulness of A implies the Koszul duality formula (see 2.3 (iii)), a useful numerical test involving Hilbert series (see Example 5.7). Note however that this formula does not imply in general the Koszul property, see [27, 28]. Finally it is not hard to see that the LCS formula for a formal space X is actually equivalent to the Koszul duality formula for A = H^{∗}(X, Q), if A is quadratic, i.e. generated in degree 1 with relations in degree 2.

It is therefore interesting to know whether the LCS formula implies the rational K[π, 1] property in the particular case of arrangement complements, as asked by Falk and Randell in [15]. In this particular case Corollary 5.3 was proved by Falk [13] and Kohno [22]. It was proved in full generality in [24]; here we give a different simple proof, based on the Koszul duality formula. More results related to Koszulness and LCS-type formulae are known for the so-called fiber-type arrangements introduced in [14], see [29], and for their recent generalization, the hypersolvable arrangements of [19]. (See the remarks in 5.6.)

The paper is organized as follows. In Section 2, we recall the definition and a few basic results about Koszul algebras. In Section 3, we recall the definition of a 1-minimal model and give a concrete realization convenient for our purposes. In Section 4, we prove the main algebraic result (Proposition 4.4). In Section 5, we derive our main results 5.1-5.4 about topological spaces and discuss an example of a hyperplane complement.

2 Koszul algebras: Preliminaries

We collect in this section the definitions and certain basic results about Koszul algebras, see [3, 4, 23]. Let F be a field and let U = ⊕_{n≥0}U_{n} be a positively graded F -algebra. We will always assume that dim U_{n} < ∞ for all n. U is called connected if U_{0} = F.

Considering just graded left U- modules we denote by Ext_{U} the derived functor of the graded homomorphism functor. For two such modules M and N the linear space Ext_{U}(M, N) is bigraded, i.e. Ext_{U}(M, N) = ⊕_{p,q≥0}Ext_{U}^{p,q}(M, N) where p is the homological (resolution) degree and q is the pure (internal) degree coming from the gradings of U, M and N.

Definition 2.1 A positively graded connected F -algebra U is Koszul if

Ext_{U}^{p,q}(F, F) = 0  (2.1)

for every p ≠ q, where F = _{U}F is the trivial graded left U-module (equal to U/U_{>0}).

If V is a finite-dimensional vector space over F let T = T (V) denote the full F -tensor algebra on V. The algebra T is provided with the standard grading ⊕_{n}T_{n} where T_{0} = F and T_{1} = V. If U is a connected graded F - algebra then there exists a canonical graded algebra homomorphism T (U_{1}) → U. The algebra U is called quadratic if this homomorphism is surjective and its kernel I is generated, as an ideal of T, by its degree 2 part I_{2}. We call I the ideal of U.

Definition 2.2 Let U be a quadratic algebra and let I be its ideal. Let I_{2}^{!} be the annihilator in U_{1}^{∗} ⊗ U_{1}^{∗} = (U_{1} ⊗ U_{1})^{∗} of the linear space I_{2} and let I^{!} be the ideal of T (U_{1}^{∗}) generated by I_{2}^{!}. Then the quadratic algebra U^{!} = T (U_{1}^{∗})/I^{!} is called the quadratic dual of U.

Observe that (U^{!})^{!} = U.

For any graded F -linear space V = ⊕_{n}V_{n} with dim V_{n} < ∞ for all n we denote the Hilbert series of V by H(V, t) (= ∑_{n} dim_{F} (V_{n})t^{n}). The following statements will be used often without reference.

Theorem 2.3 (i) If U is Koszul then it is quadratic. For any quadratic algebra the condition (2.1) holds for p < 3 and q ≠ p.

(ii) A quadratic algebra U is Koszul if and only if U^{!} is Koszul.

(iii) If U is Koszul then H(U, t) · H(U^{!}, −t) = 1.

Quite recently, Positselski [27] and Roos [28] have independently constructed examples showing that the converse of 2.3 (iii) is false.

3 1-minimal models of DGA’s

In this section we suppose that F is a field of characteristic 0 and consider algebras over F. Let M = ⊕_{n≥0}M_{n} be a connected DGA. That is M is a connected graded F -algebra which is graded- commutative (i.e. xy = (−1)^{mn}yx, for x ∈ M_{m} and y ∈ M_{n}), equipped with a differential d : M → M, which is homogenous of degree 1, satisfies d^{2} = 0 and acts on M as a graded derivation (i.e. d(x·y) = dx·y +(−1)^{n}x·dy, if x ∈ M_{n}). The DGA homomorphisms are required to be homogenous of degree zero, multiplicative, and to commute with the differentials. Two DGA’s are said to be quasi-isomorphic if they can be connected by a sequence of DGA maps inducing homology isomorphisms.

Assuming that M is generated by M_{1}, the canonical (increasing) filtration of M is defined by M(0) = F and M(n) being the subalgebra of M generated by M(n − 1) and d^{−1}(M(n − 1)) ∩ M_{1}. Recall [7] that M is called minimal if M = ∧M_{1} (i.e. M is freely generated in degree one as a graded commutative algebra) and ⋃_{n} M(n) = M. For every DGA A with H^{0}(A) = F there exists a minimal algebra M as above and a DGA homomorphism f : M → A such that f^{∗} : H^{∗}(M) → H^{∗}(A) is an isomorphism in dimension 1 and a monomorphism in dimension 2. The DGA M is unique up to isomorphism ([16], Theorem 12.1). M is called a 1-minimal model of A.

For the rest of the paper we denote by A = ⊕_{n≥0}A_{n} a graded commutative connected algebra. The 1-minimal model of A (provided with d = 0) can be realized as dual to an important Lie algebra attached to A. Let L = L(A_{1}^{∗}) be the free Lie algebra on A_{1}^{∗}, graded by bracket length. Notice that L_{2} can be identified with A_{1}^{∗} ∧ A_{1}^{∗}. The multiplication of A generates a comultiplication A_{2}^{∗} → A_{1}^{∗} ∧ A_{1}^{∗}. Define the holonomy Lie algebra G = G(A) of A as the factor of L by the Lie ideal generated by the image of the comultiplication. G is a graded Lie algebra and we write G = ⊕_{n≥1}G_{n}. We also consider the filtration of G by the Lie ideals Γ_{n}G where Γ_{n}G is the n-th term of the lower central series of G, i.e. Γ_{1}G = G and Γ_{n+1}G = [Γ_{n}G, G]. Notice that the grading of G is closely related to the filtration, namely Γ_{n}G = ⊕_{k≥n}G_{k}. In particular if we denote by G(n) the nilpotent Lie algebra G/Γ_{n+1}G then G(n) ≈ ⊕_{i=1}^{n}G_{i}, as F -linear spaces.

For every n we define the DGA freely generated in degree one M(n) =: ∧(G(n))^{∗}. The differential d is dual to the Lie bracket G(n) ∧ G(n) → G(n) on (G(n))^{∗} and is extended to M(n) by the graded derivation property. The natural projections G(n + 1) → G(n) generate the embeddings M(n) ⊂ M(n + 1) of DGA’s and we put M = ⋃_{n≥1} M(n). Notice that M(1) ≈ ∧A_{1} as a graded algebra, with differential d = 0, whence there is a natural DGA homomorphism M(1) → A.

In order to analyze M deeper we put V_{n} = G_{n}^{∗}, for every n ≥ 1. Of course V_{1} = A_{1} and d = 0 on V_{1}. More importantly M(n) = ∧(⊕_{i=1}^{n}V_{i}) and

d(V_{n}) ⊂ ∑_{i+j=n} V_{i} ∧ V_{j} ,  (3.1)

for n > 1, where the symbol ∧ applied to different spaces means the tensor product. Now extend f to a DGA homomorphism f : M → A by setting f = 0 on ⊕_{n>1}V_{n}.

Proposition 3.1 ([24] , Lemma 1.8 (i)) Provided with the above homomorphism f, M is a 1-minimal model of A.

Note that there is a shift of the degrees: what we call here V_{n} was called V_{n−1} in [24]. This rescaling has features convenient for our purposes. Put

C^{p}(q) = ⊕_{i_{1}+···+i_{p}=q}V_{i_{1}} ∧ ⋯ ∧ V_{i_{p}} , C(q) = ⊕_{p}C^{p}(q) ,

and notice that M = ⊕_{q}C(q) (as a linear space). The following easily seen properties were noticed in the particular case of arrangements in [13] , for a completely different construction of the 1-minimal model M.

Proposition 3.2 (i) The space C(q) is a subcomplex of M for every q, whence H^{∗}(M) = ⊕_{q}H^{∗}(C(q)).

(ii) The subspace B =: ⊕_{q}H^{q}(C(q)) is the subalgebra of H^{∗}(M) generated by H^{1}(M) = A_{1}. Moreover this algebra is quadratic.

(iii) The homomorphism f^{∗} : H^{∗}(M) → A is 0 on ⊕_{p≠q}H^{p}(C(q)).

(iv) If A is quadratic then B = A and f^{∗} is the projection of H^{∗}(M) to B along ⊕_{p≠q}H^{p}(C(q)).

Proof. (i) The invariance of C(q) with respect to d follows immediately from (3.1) and implies the first statement. (ii) Denote by J the ideal of M(1) = ∧V_{1} = ⊕_{q}C^{q}(q) generated by d(V_{2}). Then d(C^{q−1}(q)) = J_{q} whence B = ⊕_{q}H^{q}(C(q)) = ∧V_{1}/J. (iii) It follows immediately from the construction of f. (iv) If A is quadratic then A = ∧A_{1}/I where I is a homogeneous ideal of ∧A_{1} generated by I_{2}. It follows from the construction of M that V_{1} = A_{1} and d(V_{2}) = I_{2}. Thus B = A. Using (iii) and descending to cohomology we obtain the second statement. □

4 1-minimal models and Koszul algebras

In this section, we prove the main algebraic result. We need to use the universal enveloping algebra U of the Lie algebra G. A more explicit description of U is as follows. Let T be the tensor algebra of the linear space A_{1}^{∗} and τ : A_{1}^{∗} ⊗ A_{1}^{∗} → A_{1}^{∗} ⊗ A_{1}^{∗} be the linear map generated by v ⊗ w ↦ w ⊗ v. Then U is the factor of T by the ideal I generated by (Im(1 − τ)) ∩ ImΔ, where Δ is the comultiplication A_{2}^{∗} → A_{1}^{∗} ⊗ A_{1}^{∗}. This description gives the grading of U (which is induced from the standard grading of T) and shows that U is a quadratic algebra.

Lemma 4.1 If the algebra A is quadratic then it is the quadratic dual of U, i.e. A = U^{!}.

Proof. If A is quadratic it is the factor of ∧A_{1} by the ideal generated by the kernel of the product A_{1} ∧ A_{1} → A_{2}. Equivalently A is the factor of the tensor algebra of A_{1} by the ideal generated by the kernel of the product A_{1} ⊗ A_{1} → A_{2} and the elements a + τ^{∗}a, a ∈ A_{1} ⊗ A_{1}. The result follows since the degree 2 component of this ideal is the annihilator of the degree 2 component of I. □

Remark 4.2 Similarly one can show that in the general case U^{!} is the factor algebra of ∧A_{1} by the ideal generated by the kernel of the multiplication A_{1} ∧ A_{1} → A_{2}.

Now we study the Yoneda algebra Ext = ⊕_{p,q}Ext_{U}^{p,q}(F, F).

Proposition 4.3 The bigraded algebra Ext = ⊕_{p,q}Ext^{p,q} coincides with the algebra H^{∗}(M) = ⊕_{p}H^{p}(M) where the second (pure) grading is induced on H^{p}(M) by the grading of M by the subcomplexes C(q).

Proof. We have Ext^{p} = H^{p}(G, F) = H^{p}(Hom(∧G, F)) ( see e.g. Chapter XIII of [10]). The space of n-cochains of the complex K = Hom(∧G, F) consists of the skew symmetric multilinear functions of n variables from G with the differential δ defined by

(δf)(x_{1}, …, x_{n+1}) = ∑_{1≤i<j≤n+1}(−1)^{i+j}f([x_{i}, x_{j}], x_{1}, …, x̂_{i}, …, x̂_{j}, …, x_{n+1}).

The functions of pure degree q are those ones that are nonzero only if ∑_{i=1}^{n} deg x_{i} = q, where the initial grading on G is used. These functions form a finite dimensional subcomplex K(q) of K and K = ⊕_{q}K(q). Since K(q) is finite dimensional we have

K(q) = (⊕_{i_{1}+i_{2}+···+i_{p}=q} G_{i_{1}} ∧ G_{i_{2}} ∧ ⋯ ∧ G_{i_{p}})^{∗} = C(q)

as graded linear spaces. Thus one can identify K and M as linear spaces. The algebra multiplication on M induces then the standard shuffle multiplication of skew symmetric forms from K. The restrictions to the algebra generators of the differentials δ and d clearly coincide, up to sign. Besides both differentials satisfy the graded Leibnitz condition with respect to the algebra multiplication on M and the shuffle multiplication on K. Thus these differentials coincide up to sign, which implies the statement. □

Proposition 4.4 The graded algebra homomorphism f^{∗} : H^{∗}(M) → A is an isomorphism if and only if A is Koszul.

Proof. Suppose A is Koszul. Then it is quadratic and U is its quadratic dual whence U is also Koszul. Thus

H^{p}(C(q)) = Ext_{U}^{p,q}(F, F) = 0  (4.1)

for q ≠ p. Now Proposition 3.2 (iv) implies that f^{∗} is an isomorphism.

Conversely suppose that f^{∗} is an isomorphism. Proposition 3.2 (iii) implies that A ≈ B and consequently A is quadratic, by Proposition 3.2 (ii). Thus A is the quadratic dual of U and besides (4.1) holds. This implies that U is Koszul whence A is also Koszul. □

Remark 4.5 Proposition 3.2 also implies that both statements of Proposition 4.4 are equivalent to A being quadratic and H^{∗}(M) being generated in degree 1.

5 Formal topological spaces

In this section we interpret the results of the previous section for topological spaces. A space X (connected and with finite Betti numbers) is called formal ([16], p.158) if it has the homotopy type of a simplicial complex K and the DeRham DGA A_{P L}(K) of Q-polynomial forms on the complex K is quasiisomorphic to A = H^{∗}(X, Q) endowed with zero differential. Note that this coincides with the definition given in the Introduction, as follows e.g. from [7] , 12.2. (An enlighting discussion of this definition can be found in [18], pp.235-236.) Among the well-known examples of formal spaces are the compact Kähler manifolds [12] and the complements of complex hyperplane arrangements [9]. If X is formal then a 1-minimal model of A is also a 1minimal model of A_{P L}(K) and it is called a 1-minimal model of X. The holonomy Lie algebra G = G(X) of A is then the Q-version of the R-algebra defined by Chen in [11] and it is called the holonomy Lie algebra of X.

Using Proposition 3.1 the holonomy algebra G of a formal space X has the following interpretation in terms of π_{1}(X). Let Γ_{n} be the n-th term of the lower central series of π = π_{1}(X), i.e. Γ_{1} = π and inductively Γ_{n} = [Γ_{n−1}, π]. The graded abelian group gr_{Γ}^{∗}(π) =: ⊕_{n≥1}(Γ_{n}/Γ_{n+1}) has a natural graded Z-Lie algebra structure induced by the group commutator. Then one has a graded Lie algebra isomorphism G_{∗} ≈ gr_{Γ}^{∗}(π_{1}(X)) ⊗ Q, see e.g. [16] , Theorem 12.2. In particular we have dim G_{n} = ϕ_{n}, where ϕ_{n} = rank(Γ_{n}/Γ_{n+1}). Applying the Poincaré-Birkhoff-Witt theorem we obtain the following expression for the Hilbert series of the universal enveloping algebra U = U(X) of G:

H(U, t) = ∏_{n≥1}(1 − t^{n})^{−ϕ_{n}} .  (5.1)

The space X is a rational K[π, 1], in the sense of the definition given in the Introduction, if and only if its 1-minimal model M is a minimal model of X, i.e. f^{∗} : H^{∗}(M) → H^{∗}(X, Q) is an isomorphism. See [7] , 12.8 (iii) for a proof. Applying Proposition 4.4 we obtain the following.

Theorem 5.1 Suppose X is a formal topological space. Then X is a rational K[π, 1] if and only if H^{∗}(X, Q) is a Koszul algebra.

The above result may be used to get certain information also for nonformal spaces X.

Proposition 5.2 Let X be an arbitrary connected space with finite Betti numbers. If H^{∗}(X, Q) is a Koszul algebra then X is a rational K[π, 1].

Proof. As mentioned before, the K[π, 1] property of X may be checked via the minimal model of X. It is known that one may associate to X a DGA M, called the minimal model of X. As a commutative graded algebra M is freely generated by a graded Q-vector space W = ⊕_{n≥1}W^{n} (see e.g.

[7] , 7.7 and 7.8, for the complete definitions and the precise result). For our purposes it will suffice to recall that X is a rational K[π, 1] if and only if W^{n} = 0 for every n > 1 ([7] , 12.8 (iii)).

Set now A = H^{∗}(X, Q). Our result will follow at once from the fact that W is in general related to A as follows. Firstly A has a so-called bigraded model B, see [18] , Proposition 3.4. It is a DGA freely generated by a graded vector space V = ⊕_{n≥1}V^{n}, with several other defining properties. Secondly one may associate to X a new (perturbed) differential D on B and an induced differential D_{ζ} on V, see [18] , Theorem 4.4 and 4.14. Finally one knows ([18] 4.14) that there is an isomorphism of graded vector spaces W^{∗} ≈ H^{∗}(V, D_{ζ}) (see [30] , Section 8 for a proof).

If A is Koszul we know from Proposition 4.4 that the 1-minimal model M constructed in Section 3 is actually the minimal model of A. It follows then from [24] , Lemma 1.8 (i) that M is in fact the bigraded model of A. Since the free generators V of M are concentrated by construction in degree one, the same property obviously will hold also for W and therefore X is a rational K[π, 1]. □

The next corollary was obtained in [24] , Lemma 4.5, by using the bigraded models of [18]. We present here a new proof, based on the Koszul duality formula.

Corollary 5.3 If a formal space X is a rational K[π, 1] then the LCS formula holds for X (see the Introduction).

Proof. By Theorem 5.1 , A = H^{∗}(X, Q) is Koszul. Since U is the quadratic dual of A, Theorem 2.3 (iii) implies that

P (X, −t) = H(U, t)^{−1}

where P (X, t) is the Poincaré polynomial of X. Then (5.1) completes the proof. □

The most intensively studied class of formal spaces consists of the complements of hyperplane arrangements. Let A be a finite set of linear hyperplanes in C^{ℓ} and X = C^{ℓ} \ ⋃_{H∈A} H. Then A = H^{∗}(X, Q) can be embedded in the DGA of holomorphic differential forms on X (see [9]) whence X is formal. Besides A is determined by combinatorial data, namely by the intersection lattice of A (see [25]). The Hilbert series H(A, t) = P (X, t) is a polynomial of degree ℓ.

The problems related to the topic of this paper were first studied in the arrangement setting in [2, 20, 21, 22, 14, 13]. Falk [13] studied conditions for X to be a rational K[π, 1]. He proved that X is a rational K[π, 1] for every arrangement of so called fiber-type. This is a topologically defined class that coincides with the combinatorially defined class of supersolvable arrangements (see [25] for definitions). Falk [13] and Kohno [22] proved Corollary 5.3 for complements of hyperplane arrangements.

It is still unknown if X can be a rational K[π, 1] for a not supersolvable arrangement or if the LCS formula can hold without X being a rational K[π, 1]. The following result solves the latter problem negatively in a low dimensional case.

Proposition 5.4 If ℓ = 3 and the lower central series formula holds for X then X is a rational K[π, 1].

In order to prove the proposition we need to bring into consideration affine arrangements. Let A be an arrangement in C^{ℓ} and H ∈ A. Pick α ∈ (C^{ℓ})^{∗} with kernel H. Then intersecting every K ∈ A \ {H} with the affine hyperplane H̃ = {v ∈ C^{ℓ}|α(v) = 1} we obtain a set A′ of affine hyperplanes in H̃. Put X′ = H̃ \ ⋃_{K∈A′} K, A′ = H^{∗}(X′, Q) and U′ = U(X′). Set also ϕ′_{n} = rank(Γ_{n}π_{1}(X′)/Γ_{n+1}π_{1}(X′)), n ≥ 1.

One knows that X′ is formal ([25] ,Theorem 5.90). It is also well known that X is homeomorphic to X′ × (C^{∗}), see [25] , Proposition 5.1. This enables us to drop the dimension in the statement of Proposition 5.4 to 2, in particular

H(A′, t) = 1 + mt + nt^{2}.  (5.2)

This also implies that A ≈ A′ ⊗ ∧(e), with deg(e) = 1.

Now we can infer the following two equivalences. On one hand X is a rational K[π, 1] if and only if X′ is a rational K[π, 1]. Indeed, denoting by f′ : (∧V′, d′) → (A′, 0) the 1-minimal model of A′, we may infer from the Künneth theorem that f =: f′ ⊗ id : (∧V′ ⊗ ∧(e), d′ ⊗ id) → (A, 0) will be the 1-minimal model of A. Hence f^{∗} is an isomorphism if and only if f^{′∗} is an isomorphism, as asserted.

On the other hand obviously ϕ_{k} = ϕ′_{k}, if k > 1, and ϕ_{1} = ϕ′_{1} + 1. Since H(A, t) = H(A′, t)(1 + t), it follows that the LCS formula holds for X if and only if it holds for X′. According to (5.1) and (5.2) the latter is equivalent to

H(U′, t) = (1 − mt + nt^{2})^{−1}.

Consequently Proposition 5.4 will follow from the simple lemma below, via Theorem 5.1.

Lemma 5.5 ([26], 3.4, Proposition 2.) Let U′ be a quadratic algebra with

H(U′, t) = (1 − mt + nt^{2})^{−1}.  (5.3)

Then U′ is Koszul.

Proof of the lemma. Let I be the ideal of U′. The condition (5.3) immediately implies that m = dim U′_{1} and n = dim I_{2}. The natural complex

0 → U′ ⊗ I_{2} → U′ ⊗ U′_{1} → U′ → F → 0

is exact except maybe at U′ ⊗ I_{2} for every quadratic algebra. The condition (5.3) implies that it is exact whence it is a graded free resolution of F. Thus Ext_{U′}^{i}(F, F) = 0 for i ≥ 3, which implies the result. □

Proof of the proposition. The Koszul duality formula 2.3 (iii) together with (5.2) and (5.3) imply that A′ and (U′)^{!} have the same Hilbert series. On the other hand one knows that A′ is generated in degree one ([25] , Theorem 5.90) and therefore A′ is a quotient of (U′)^{!}, see Remark 4.2. Consequently A′ ≈ (U′)^{!} and A′ is Koszul. □

Remarks 5.6 The Koszul property for A = H^{∗}(X, Q) was first considered in [29] , where it was proved that A is Koszul for fiber-type arrangements. A substantial generalization of the fiber-type class, namely the hypersolvable class, was recently introduced in [19]. Among other things it was shown there ([19] , Theorem E) that the associated quadratic algebra U^{!} (see 4.2) is always Koszul for a hypersolvable arrangement. Consequently a generalization of the LCS formula ([19] , Theorem C (ii)) holds for this new class. At the same time ([19] , Theorem D (iv),(v)) the implication of Proposition 5.4 holds for hypersolvable arrangements. Moreover one knows that the only rational K[π, 1] hypersolvable complements are the fiber-type ones ([19] , Theorem D (ii),(iv)).

Falk in [13] asked the question if the quadraticity of A is sufficient for X being a rational K[π, 1]. The following example shows that this is not true.

Example 5.7 Consider the hyperplanes given in C^{3} = {(x, y, z)} by the functionals x, y, x+y, z, x−z, y−z, x+y−2z. Then the algebra A is quadratic. (In fact it is Kohno’s parallel arrangement discussed in [15] and [22]. It is mistakenly written in [22] that X is rational K[π, 1] for this arrangement.) One can compute that H(A′, t) = 1+6t+10t^{2}. If X is a rational K[π, 1] then by Corollary 5.3 and (5.1) the series (1 − 6t + 10t^{2})^{−1} equals H(U′, t) whence all its coefficients are nonnegative. This implies that the roots of 1−6t+10t^{2} are real as in the proof of [1] , Lemma 3.4, which is a contradiction.

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Stefan Papadima: Institute of Mathematics of the Academy, P.O. Box 1-764, RO-70700 Bucharest, Romania spapadim@stoilow.imar.ro

Sergey Yuzvinsky: University of Oregon, Eugene, OR 97403, USA yuz@math.uoregon.edu