Homology, Homotopy and Applications, vol.4(2), 2002, pp.227–258

KOSZUL HOMOLOGY AND LIE ALGEBRAS WITH APPLICATION TO GENERIC FORMS AND POINTS

R. FRÖBERG and C. LÖFWAL

(communicated by Larry Lambe)

Abstract We study the Koszul dual for general superalgebras, and apply it to the Koszul homology of a graded algebra. We show that a part of the Koszul homology algebra is related to the homotopy Lie algebra by means of Koszul duality. This is used to study the ”Minimal Resolution Conjecture” and the ”Ideal Generating Conjecture” for sets of generic points in projective space, and for quotients of the polynomial ring (or exterior algebra) modulo generic quadratic forms.

To Jan–Erik Roos on his sixty–fifth birthday

1. Introduction

Consider s generic points in projective space P_{k}^{n}. The Hilbert series of the corresponding coordinate ring A = S/I = k[x_{0}, …, x_{n}]/I is known to be ∑_{i⩾0} min(\binom{n+i}{i}, s)z^{i} [Ge-Or 81]. The Koszul homology of A, H = Tor^{S}(A, k), is known to satisfy H_{i,j} = 0 if j ≠ d + i − 1, d + i for a certain d (depending on n and s), but there are a lot of unanswered questions. The Minimal Resolution Conjecture (MRC) states that for all i and all j, H_{i,j} = 0 or H_{i+1,j} = 0, and the special case that this is true for i = 1 is called the Ideal Generating Conjecture (IGC). The MRC has recently been disproved [Ei-Po 99], while IGC is still open. In this paper we will introduce a new approach to study these problems, namely through the “homotopy Lie algebra”, 𝔤_{A}, of A.

The MRC may be reformulated in terms of 𝔤_{A}. In fact, there are certain numbers m and t (depending on n and s) such that if d > 2 then MRC holds if and only if dim_{k} 𝔤_{A}^{m+1,m+d−1} = t and 𝔤_{A}^{m+2,m+d} = 0. When d = 2 the situation is a bit more complicated. Let η_{A} be the “diagonal” Lie algebra of 𝔤_{A}; i.e., η_{A}^{i} = 𝔤_{A}^{i,i} and let L = ⊕_{i⩾2}η_{A}^{i} and finally Q^{i} = (L/[L, L])^{i}. Then if d = 2, MRC holds if and only if dim_{k} Q^{m+1} = t and Q^{m+2} = 0.

Even if the case d > 2 is easier, we do not know any application of it. However in the case when d = 2, the Lie algebra η_{A} is explicitly given as the free Lie algebra on n + 1 odd variables modulo the ideal generated by the squares of s generic linear forms. We have used a Mathematica program called “liedim” and its C-version

Received June 2, 2000, revised September 12, 2000; published on July 12, 2002. 2000 Mathematics Subject Classification: 13D07, 14M99, 16E45. Key words and phrases: Koszul homology, Lie algebra, generic points, generic forms. \begin{matrix}c \\ ◯ \end{matrix} 2002, R. Fröberg and C. Löfwal. Permission to copy for private use granted.

“cbas” to compute the space Q above for some values of n and s. In this way we have been able to prove MRC and IGC in some new cases.

We have also studied ideals generated by generic quadratic forms in polynomial rings and exterior algebras and ideals generated by generic quadratic Lie elements in free Lie algebras.

The paper is organized as follows. In Section 2 the Koszul dual is defined for general superalgebras, in particular for graded commutative algebras. This will be applied to the Koszul homology algebra, which has generators of both even and odd degrees. In Section 3 the homotopy Lie algebra is introduced for graded commutative algebras through minimal models and in Section 4 a theorem is proved which relates parts of the Koszul homology algebra and the homotopy Lie algebra by means of Koszul duality. In fact, since we are dealing with superalgebras, our theorem may also be applied to quotients of exterior algebras (or to quotients of tensor products of exterior algebras and polynomial algebras). However, in the exterior algebra case, Tor^{S}(A, k) is in general infinite-dimensional and the corresponding Lie algebra is an “ordinary” Lie algebra (non-super).

In Section 5 we introduce the MRC and IGC conjectures and apply the theorem in Section 4 in order to reformulate the conjectures in terms of the Lie algebra L associated to A. We also prove a “monotonicity” property, which implies that if counterexamples to MRC exists for a given n, then there has to be counterexamples for special values of s. For IGC this holds for at most two values of s for each n. In Section 6 we present some theorems about MRC and IGC obtained by computations and by the theory from the previous section. The computations are presented in Section 11.

In Section 7 we study rings of the form k[x_{1}, …, x_{n}]/(f_{1}, …, f_{r}), where f_{1}, …, f_{r} are generic quadratic forms. We prove that the ring is Koszul if and only if r ⩽ n or r ⩾ \binom{n+1}{2} − n^{2}/4. The corresponding result for Lie algebras is also obtained. We study the series of Lie algebras obtained from generic quadratic algebras or defined by generic quadratic Lie relations. We make some conjectures and prove them in some special cases. In Section 8 we state a conjecture about the Lie algebra for the coordinate ring of a set of generic points, and we prove it in some cases. In Section 9 we study the Poincaré series for these coordinate rings. In Section 10 finally, we study the quotient of exterior algebras with generic quadratic forms and their corresponding (ordinary) Lie algebras. In particular we study the exterior algebra in 5 variables modulo 2 or 3 generic quadratic forms. The Hilbert series turned out to be not the ones which were expected.

2. The Koszul dual

We will consider ℤ_{2} × ℕ-graded (or ℤ_{2} × ℕ × ℕ-graded) associative algebras or Lie algebras over a field k of characteristic different from two. The ℤ_{2}-grading (also called the “parity”) of an element x is denoted |x| and we ususally talk about “even” and “odd” elements corresponding to whether |x| = 0 or |x| = 1. This grading makes our algebras “superalgebras”, and it is this grading that determines the sign when two elements in a formula are interchanged; e.g., the graded commutator is defined as [a, b] = ab − (−1)^{|a||b|}ba and the graded Jacobi identity is [[a, b], c] = [a, [b, c]] + (−1)^{|b||c|}[[a, c], b]. The additional ℕ-gradings are called “weights” and they have no effect on signs in formulas.

The Koszul dual A^{!} of a ℤ_{2} ×ℕ-graded connected k-algebra A = k⊕A_{1} ⊕A_{2} ⊕⋯ is again a ℤ_{2} × ℕ-graded connected k-algebra and may be defined as the subalgebra of Ext_{A}(k, k) generated by the elements of homological degree and weight equal to one. The algebra A is called a Koszul algebra if A^{!} = Ext_{A}(k, k). The Koszul dual may be computed using the cobar construction (T((sA_{+})^{∗}), d), cf [Ad 60]. Here A_{+} is the set of elements of positive weight, which is considered to be concentrated in homological degree zero, T stands for “tensor algebra” and s is the suspension functor, which changes parity and raises the homological degree by one, while the weight is left unchanged by s. The differential d is the derivation which extends the map

(sA_{+})^{∗} →^{s} A_{+}^{∗} →^{m^{∗}} (A_{+} ⊗ A_{+})^{∗} →^{(ϕ_{A_{+}} )^{−1}} (sA_{+})^{∗} ⊗ (sA_{+})^{∗} ,

where m: A_{+} ⊗ A_{+} → A_{+} is multiplication and ϕ_{A_{+}} is a natural isomorphism ϕ_{V} applied to V = A_{+}. It is defined as follows. Let V be a ℤ_{2} × ℕ-graded vector space, then ϕ_{V}: (sV)^{∗} ⊗ (sV)^{∗} → (V ⊗ V)^{∗} is defined by ϕ(f ⊗ g) = (f ⊗ g) ∘ (s ⊗ s) for f, g ∈ (sV)^{∗}. Following the strict sign rule, this means that the following holds for a, b ∈ V.

(ϕ(f ⊗ g))(a ⊗ b) = (−1)^{|a|+|g||sa|}f(sa)g(sb) =
(−1)^{|a|+|f||g|}f(sa)g(sb) = −(−1)^{|sa||b|}f(sa)g(sb) .

It is easy to prove (cf [Pr 70], [Lö 86]) that

A^{!} = T((sA_{1})^{∗})/⟨d((sA_{2})^{∗})⟩ .

Here A^{!} only depends on the “1-2-algebra of A”,

T(A_{1})/⟨ker(A_{1} ⊗ A_{1} → A_{2})⟩ ,

which in fact is equal to (A^{!})^{!}.

This gives the following description of the Koszul dual. Let V be a ℤ_{2}-graded finite-dimensional (or locally finite-dimensional ℤ_{2} × ℕ-graded) vector space and R a homogeneous subspace of V ⊗ V. Then

(T(V)/⟨R⟩)^{!} = T((sV)^{∗})/⟨ϕ_{V}^{−1}R^{⊥}⟩ ,

where s is the parity switcher, R^{⊥} = {f ∈ (V ⊗ V)^{∗}; f(R) = 0} and ϕ_{V}: (sV)^{∗} ⊗ (sV)^{∗} → (V ⊗ V)^{∗} is the isomorphism defined above. In coordinates this means the following (we have changed the formulas by a global −1). If {x, y, …} is a basis for V and {X, Y, …} is a basis for (sV)^{∗} dual to {sx, sy, …}, then an element X ⊗ Y is considered to operate on a basis element a ⊗ b by the rule (X ⊗ Y)(a ⊗ b) = 0 if a ≠ x or b ≠ y, (X ⊗ X)(x ⊗ x) = 1 and

(X ⊗ Y)(x ⊗ y) = \{ \begin{matrix}−1 & if x is even and y is odd \\ 1 & otherwise. \end{matrix}

Observe that this somewhat strange rule follows from the fact that the suspension s has to pass x before Y passes sx.

Example 2.1. Let |x| = 0, |y| = 1. Then

(k⟨x, y⟩/⟨xy − yx, y^{2}⟩)^{!} = k⟨X, Y ⟩/⟨XY − Y X, X^{2}⟩ ,

where |X| = 1, |Y | = 0.

This example is in accordance with the known fact that the Ext-algebra of a free graded commutative algebra is free graded commutative with a parity switch (cf. [Ma 88] 3.10) . See Proposition 2.2 below.

For a ℤ_{2}-graded vector space V, let [V, V]_{F} denote the subspace of V ⊗ V generated by the graded commutators. The subscript F refers to the fact that [V, V]_{F} is contained in F(V), the free Lie algebra on V, considered as the Lie subalgebra generated by V in the free associative algebra T(V). Then U(F(V)) = T(V), where U stands for “universal enveloping algebra”. The free graded commutative algebra on V, denoted ⋀(V), is defined as T(V)/⟨[V, V]_{F}⟩. Now V ∧ V = (V ⊗ V)/[V, V]_{F} and hence (V ∧ V)^{∗} may (and will) be identified with ([V, V]_{F})^{⊥} ⊂ (V ⊗ V)^{∗}.

Proposition 2.2. Let V be a ℤ_{2} × ℕ-graded locally finite-dimensional vector space and let s be the parity switcher. Let R be a subspace of V ⊗V which contains [V, V]_{F} and let \overline{R} = R/[V, V]_{F}. Consider the map ϕ_{V} = ϕ: (sV)^{∗}⊗(sV)^{∗} → (V ⊗V)^{∗} defined (as above) by ϕ(f ⊗ g)(a ⊗ b) = (−1)^{|a|+|g||sa|}f(sa)g(sb). Then

(i)ϕ[(sV)^{∗}, (sV)^{∗}]_{F} = ([V, V]_{F})^{⊥}
(ii)(⋀(V)/⟨\overline{R}⟩)^{!} ≅ U(F((sV)^{∗})/⟨ϕ^{−1}R^{⊥}⟩)
(iii)(⋀(V))^{!} ≅ ⋀((sV)^{∗})
(iv)\overline{R}^{∗} ≅ [(sV)^{∗}, (sV)^{∗}]_{F}/ϕ^{−1}R^{⊥} .

Proof. (i). Let f, g ∈ (sV)^{∗} and a, b ∈ V. Then

ϕ(f ⊗ g)(a ⊗ b − (−1)^{|a||b|}b ⊗ a) =
(−1)^{|a||b|+|b|+1}f(sa)g(sb) − (−1)^{|a|+1}f(sb)g(sa) and
ϕ(g ⊗ f)(a ⊗ b − (−1)^{|a||b|}b ⊗ a) =
(−1)^{|f||g|+|a|}g(sa)f(sb) − (−1)^{|f||g|+|a||b|+|b|}g(sb)f(sa) .

Hence ϕ(f ⊗ g) − (−1)^{|f||g|}ϕ(g ⊗ f) is zero on a ⊗ b − (−1)^{|a||b|}b ⊗ a and hence ϕ[(sV)^{∗}, (sV)^{∗}]_{F} ⊂ ([V, V]_{F})^{⊥}. The other inclusion follows by dimension reasons.

(ii). This follows directly from (i) and the description of the Koszul dual above, since ⋀(V)/⟨\overline{R}⟩ ≅ T(V)/⟨R⟩ and

T((sV)^{∗})/⟨ϕ^{−1}R^{⊥}⟩ ≅ U (F((sV)^{∗})/⟨ϕ^{−1}R^{⊥}⟩) ,

which is true since, ϕ^{−1}R^{⊥} ⊂ ϕ^{−1}([V, V]_{F})^{⊥} = [(sV)^{∗}, (sV)^{∗}]_{F}.

(iii). This follows from (i) and (ii) with R = [V, V]_{F}.

(iv). Using (i) we get

\overline{R}^{∗} = (R/[V, V]_{F})^{∗} ≅ ([V, V]_{F})^{⊥}/R^{⊥} ≅
ϕ^{−1}([V, V]_{F})^{⊥}/ϕ^{−1}R^{⊥} = [(sV)^{∗}, (sV)^{∗}]_{F}/ϕ^{−1}R^{⊥} .

In general, for an augmented k-algebra A, the subalgebra of Ext_{A}(k, k) generated by Ext_{A}^{1}(k, k) has relations of degree greater than two. However the relations of degree two may be described explicitly.

Proposition 2.3. Let A be an augmented k-algebra with augmentation ideal I and let gr(A) = k ⊕ I/I^{2} ⊕ I^{2}/I^{3} ⊕ ⋯ be the graded associated k-algebra. Then the 1-2-algebra of Ext_{A}(k, k) is isomorphic to the Koszul dual of gr(A); i.e.,

(Ext_{A}(k, k)^{!})^{!} ≅ (gr(A))^{!} .

Proof. All we need is to consider the beginning of the cobar construction, (sI)^{∗} →^{d} (sI)^{∗} ⊗ (sI)^{∗}, where d is defined as above. We have Ext_{A}^{1}(k, k) = (sI^{2})^{⊥}. Hence

ker (Ext_{A}^{1}(k, k) ⊗ Ext_{A}^{1}(k, k) → Ext_{A}^{2}(k, k)) = im(d) ∩ ((sI^{2})^{⊥} ⊗ (sI^{2})^{⊥}) .

With ϕ and m as above we have ϕ(im(d)) = (ker(m))^{⊥} and

ϕ ((sI^{2})^{⊥} ⊗ (sI^{2})^{⊥}) = (I^{2} ⊗ I + I ⊗ I^{2})^{⊥} .

Hence

ϕ ker (Ext_{A}^{1}(k, k) ⊗ Ext_{A}^{1}(k, k) → Ext_{A}^{2}(k, k)) = (I^{2} ⊗ I + I ⊗ I^{2} + ker(m))^{⊥}

which equals (m^{−1}(I^{3}))^{⊥}, since m(I^{2} ⊗ I) = I^{3}. Thus the 1-2-algebra of Ext_{A}(k, k) is isomorphic to T((sI^{2})^{⊥})/⟨ϕ^{−1}(m^{−1}(I^{3}))^{⊥}⟩.

On the other hand

(gr(A))^{!} = T((sI/I^{2})^{∗})/⟨ϕ_{I/I^{2}}^{−1}(ker(I/I^{2} ⊗ I/I^{2} → I^{2}/I^{3}))^{⊥}⟩

Looking at (I/I^{2} ⊗ I/I^{2})^{∗} as a subspace of (I ⊗ I)^{∗}, the space (ker(I/I^{2} ⊗ I/I^{2} → I^{2}/I^{3}))^{⊥} will be identified with (m^{−1}(I^{3}))^{⊥}. Hence

(gr(A))^{!} ≅ T((sI^{2})^{⊥})/⟨ϕ_{I}^{−1}(m^{−1}(I^{3}))^{⊥}⟩ .

We end this section with a proposition which, combined with Proposition 2.3, will be used in Section 4.

Proposition 2.4. Let L be a ℤ_{2} × ℕ^{+}-graded Lie algebra, locally finite-dimensio-nal over k. Let gr(L) denote the graded associated Lie algebra with respect to the filtration L ⊃ [L, L] ⊃ [L, [L, L] ⊃ ⋯ and let gr(U(L)) denote the graded associated algebra with respect to the filtration obtained from powers of the augmentation ideal. Then gr(U(L)) and U(gr(L)) are naturally isomorphic as graded algebras.

Proof. There is a natural map of Lie algebras from gr(L) to the Lie algebra associated to the algebra gr(U(L)) and hence there is induced a natural map of graded algebras U(gr(L)) → gr(U(L)). This map is surjective, since both algebras are generated by L/[L, L] (depending on the assumption that L has a positive grading). Since, by assumption, both algebras are locally finite-dimensional, it follows that the map is an isomorphism, since the algebras have the same Hilbert series.

3. The homotopy Lie algebra

Let V be a ℤ_{2} × ℕ^{+}-graded vector space over k with basis {x_{1}, …, x_{n}}. We will also denote the free graded commutative algebra on V, ⋀(V), by k[x_{1}, …, x_{n}]. It is the free graded associative algebra k⟨x_{1}, …, x_{n}⟩ modulo the graded commutators x_{i}x_{j} − (−1)^{|x_{i}||x_{j}|}x_{j}x_{i}. Sometimes V will be infinite-dimensional, but always finitedimensional in each weight.

We will consider k-algebras A = S/I, where S = k[x_{1}, …, x_{n}] and I is a homogeneous ideal with respect to both parity and weight. To such a k-algebra A, we will define a ℤ_{2} × ℕ^{+} × ℕ^{+}-graded Lie algebra 𝔤_{A} – the homotopy Lie algebra of A. It may be defined by the property U(𝔤_{A}) = Ext_{A}(k, k), but we will give a more explicit definition (and more suitable for our purposes) using the minimal model of A (cf [Lö 94]).

The minimal model of A is a differential algebra (⋀ V, d), where V is a ℤ_{2} × ℕ × ℕ^{+}-graded vector space and where the first degree is the parity, the second degree is the homological degree and the third degree is the weight. The differential d is a derivation which changes parity, lowers the homological degree by one and preserves weight. Moreover d maps V to ⋀^{⩾2} V = V ∧ V + V ∧ V ∧ V + · · ·. The component of d mapping V to V ∧ V is called d_{2} (its extension as a derivation also satisfies d_{2}^{2} = 0).

The ring A is considered as a differential algebra with zero differential and concentrated in homological degree zero. There is a surjective map ϵ: (⋀ V, d) → A preserving all degrees and inducing an isomorphism in homology. This means that H_{i}(⋀ V, d) = 0 for homological degrees i > 0 and if A = k[x_{1}, …, x_{n}]/I then V in homological degree zero has a basis X_{1}, …, X_{n}, ϵ(X_{i}) = x_{i} and ϵ(im(d)) = I. The construction of (⋀ V, d) is a straightforward procedure of killing cycles in a minimal way (see [Lö 94] for more details).

The Lie algebra 𝔤_{A} is defined by (sV)^{∗} as a vector space, where s changes parity, raises homological degree by one and preserves weight. The Lie product on (sV)^{∗} is in principle defined as the dual of d_{2}. More precisely, it is defined as a map [(sV)^{∗}, (sV)^{∗}]_{F} → (sV)^{∗} in the following way (cf. Proposition 2.2),

[(sV)^{∗}, (sV)^{∗}]_{F} →^{ϕ} ([V, V]_{F})^{⊥} = (V ∧ V)^{∗} →^{d_{2}^{∗}} V^{∗} →^{s^{−1}} (sV)^{∗} .

This map preserves all degrees and it may be checked that it satisfies the graded Jacobi identity. In coordinates the Lie product may be given as follows (cf [Av 84], [Ha 92]).

Suppose {x, y, z, …} is an ordered basis for V and let {X, Y, Z, …} denote the basis for (sV)^{∗} dual to {sx, sy, sz, …}. If d_{2}x = ∑_{y⩽z} c_{xyz}yz then,

[Z, Y] = − ∑_{x}(−1)^{|y|}c_{xyz}X  if y < z
(Y)^{2} = − ∑_{x} c_{xyy}X  if Y is odd.

Inside 𝔤_{A} there is the Lie subalgebra η_{A} generated by 𝔤_{A}^{1,1} (homological degree = weight = 1). It is proved in [Lö 94] that η_{A} coincides with the diagonal subalgebra ⊕_{p⩾1}^{∞}𝔤_{A}^{p,p} and it is proved in [Pr 70], [Lö 86] that U(η_{A}) = A^{!}.

Definition 3.1. Let 𝔤_{A} be the homotopy Lie algebra of A and η_{A} the Lie subalgebra of 𝔤_{A} generated by 𝔤_{A}^{1,1}. The Lie algebra L = (η_{A})^{⩾2} = [η_{A}, η_{A}] is called the “Lie algebra associated to A”. It has two degrees, a parity and a weight (⩾ 2).

Let L be the Lie algebra associated to A and let gr(L) denote the graded associated Lie algebra obtained from the lower central series L ⊃ [L, L] ⊃ [L, [L, L]] ⊃ ⋯; i.e.,

gr(L) = (gr(L))_{1} ⊕ (gr(L))_{2} ⊕ ⋯ = L/[L, L] ⊕ [L, L]/[L, [L, L]] ⊕ · · · .

Each (gr(L))_{i} has an induced grading from L, (gr(L))_{i} = ⊕_{j}(gr(L))_{i}^{j}. The following theorem relates gr(L) to Tor^{S}(A, k), which is the homology of the Koszul complex of A if x_{i} is even for all i. The theorem may be seen as a study of an edge in Avramov’s spectral sequence ([Av 74]), which has as E^{2}-term Tor^{Tor^{S}(A,k)}(k, k) and which converges to U(𝔤_{A}^{⩾2})^{∗}. At this point we want to thank J.-E. Roos, who conjectured part one of the theorem below and encouraged us to supply a proof.

4. The Koszul homology in terms of the homotopy Lie algebra

Theorem 4.1. Let A = S/I = k[x_{1}, …, x_{n}]/I, where I ⊂ (x_{1}, …, x_{n})^{2}, be a ℤ_{2} × ℕ-graded connected k-algebra, commutative in the graded sense by the first degree, which is called the “parity”; the second degree is called the “weight”. Let H = ⊕H_{i,j} = ⊕Tor_{i,j}^{S}(A, k) (here i is the homological degree and j is the weight), which is considered as a graded algebra in the following way; H = k⊕H_{(1)}⊕H_{(2)}⊕⋯, where H_{(i)} = ⊕_{j}H_{j−i,j} . Furthermore parity and weight in A defines a bigrading on each H_{(i)} (which is compatible with the multiplication in H). Let L be the Lie algebra associated to A and gr(L) the graded associated Lie algebra with respect to the lower central series. Then gr(L) has, besides a parity, two gradings, gr(L) = ⊕_{i⩾1,j⩾2}(gr(L))_{i}^{j} (see above). The following natural isomorphisms hold, where in (ii) the operator s just changes the parity.

(i)The 1-2-algebra of H is isomorphic to the Koszul dual of U(gr(L));
i.e., (H^{!})^{!} ≅ (U(gr(L)))^{!}
(ii)H_{(1)} ≅ s(L/[L, L])^{∗}
(iii)ker(H_{(1)} ∧ H_{(1)} → H_{(2)}) ≅ ([L, L]/[L, [L, L]])^{∗}

(iv)If A is Koszul, then H ≅ Ext_{U(L)}(k, k) as algebras, in particular
H_{j−i,j} = H_{(i),j} ≅ (Ext_{U(L)}^{i}(k, k))_{j} .

Corollary 4.2. With notation as in the theorem, suppose x_{i} is even for all i. Then

(i)L^{j} = [L, L]^{j} for j ⩾ n + 2
(ii)If A is Koszul, then gldim(U(L)) < ∞ .

Corollary 4.3. With notation as in the theorem, we have

(i)dim_{k}(H_{1,2})_{even} = dim_{k}(L^{2})_{odd}
(ii)dim_{k}(H_{1,2})_{odd} = dim_{k}(L^{2})_{even}
(iii)dim_{k} H_{2,3} = dim_{k} L^{3}
(iv)dim_{k} H_{3,4} = dim_{k} L^{4}/[L^{2}, L^{2}]
(v)dim_{k}(ker(H_{1,2} ∧ H_{1,2} → H_{2,4})) = dim_{k}[L^{2}, L^{2}]
(vi)dim_{k}(ker(H_{1,2} ⊗ H_{2,3} → H_{3,5}))_{even} = dim_{k}([L^{2}, L^{3}])_{even}
(vii)dim_{k}(ker(H_{1,2} ⊗ H_{2,3} → H_{3,5}))_{odd} = dim_{k}([L^{2}, L^{3}])_{odd}
(viii)dim_{k} ker (H_{2,3} ∧ H_{2,3} ⊕ H_{1,2} ⊗ H_{3,4} → H_{4,6}) =
dim_{k} ([L^{3}, L^{3}] + [L^{2}, L^{4}]) /[L^{2}, [L^{2}, L^{2}]] .

Remark 4.4. It has been known since long that dim_{k} H_{2,3} = dim_{k} L^{3} and this was the starting point for our study of generic points and it is mainly this fact we will use in the applications of the theorem.

Proof of Theorem 4.1. We will give three different proofs. Even if the second one is short, we have included the other two, since we believe the methods used there are interesting in themselves. The first two use the minimal model of A. The third uses the cobar construction of (the dual of) the Koszul complex and the perturbation theory developed by Gugenheim, Stasheff, Lambe and others [Gu-La-St 91], [Hu-Ka 91], [Jo-La 00]. First proof. Let (⋀(V), d) be a minimal model of A as described above. In particular this is a free resolution of A as a module over S ≅ ⋀(V_{0}). Hence Tor^{S}(A, k) may be computed as the homology of ⋀(V) ⊗_{S} k = ⋀(V_{⩾1}) with the induced differential \overline{d}. Due to the minimality of (⋀(V), d) and the fact that the generators in S have positive weight, we may conclude that the weight is always greater than the homological degree. Let W be the subspace of V_{⩾1} defined by elements of minimal possible weight; i.e., W = ⊕_{i⩾1}V_{i,i+1}. Then L = (sW)^{∗}. By degree reasons, the differential \overline{d} maps W to W ∧ W and we will call this map d_{2}. Hence (⋀(W), d_{2}) is a subcomplex of (⋀(V_{⩾1}), \overline{d}) and moreover (\overline{d})^{−1}(W ∧W) = W. Put W^{(1)} = ker(d_{2}) and W^{(2)} = d_{2}^{−1}(W^{(1)} ∧ W^{(1)}). Then W^{(1)} ⊂ W^{(2)} ⊂ W. We have

H_{(1)} = W^{(1)}
ker(H_{(1)} ∧ H_{(1)} → H_{(2)}) = d_{2}(W^{(2)}) .

Claim.

(sW^{(1)})^{⊥} = [L, L]

(sW^{(2)})^{⊥} = [L, [L, L]]

Once this is proved, we get (sH_{(1)})^{∗} = (sW^{(1)})^{∗} = (sW)^{∗}/(sW^{(1)})^{⊥} = L/[L, L] which proves (ii). We also get (sW^{(2)})^{∗} = L/[L, [L, L]]. Let d_{22} denote the restriction of d_{2} to W^{(2)}, then d_{22}: W^{(2)} → W^{(1)}∧W^{(1)}. By the above ker(H_{(1)}∧H_{(1)} → H_{(2)}) = im(d_{22}). To prove (i) we have to prove that ϕ^{−1}(im(d_{22}))^{⊥} is equal to the kernel of

[L/[L, L], L/[L, L]]_{F} → [L, L]/[L, [L, L]] .

But this is given as the kernel of the composition of the following maps.

[(sW^{(1)})^{∗}, (sW^{(1)})^{∗}]_{F} →^{ϕ} (W^{(1)} ∧ W^{(1)})^{∗} →^{d_{22}^{∗}} (W^{(2)})^{∗} → (sW^{(2)})^{∗}

Since the first and the last map are isomorphisms, this kernel is ϕ^{−1} ker(d_{22}^{∗}) = ϕ^{−1}(im(d_{22}))^{⊥} which proves (i).

Now (iii) follows from Proposition 2.2 (iv).

If A is Koszul then W = V_{⩾1} and hence (⋀(V_{⩾1}), \overline{d}) is the same as (⋀(W), d_{2}), which is the (generalized) standard Chevalley-Eilenberg complex (cf [Ch-Ei 48]) to compute the cohomology of the Lie algebra (sW)^{∗} (here the “wedge-degree” is the cohomological degree and the second degree of W is the weight). Hence H = H((⋀(W), d_{2})) = Ext_{U(L)}(k, k) which proves (iv). Proof of Claim.

We have that [L, L] is the image of the map

[(sW)^{∗}, (sW)^{∗}]_{F} →^{ϕ} (W ∧ W)^{∗} →^{d_{2}^{∗}} W^{∗} → (sW)^{∗} .

Hence [L, L] = s^{−1}(im(d_{2}^{∗})) = s^{−1}(ker(d_{2}))^{⊥} = (sW^{(1)})^{⊥}.

To prove the second claim, we have to prove that the map above maps the subspace [(sW)^{∗}, (sW^{(1)})^{⊥}]_{F} onto (sW^{(2)})^{⊥}. We will do this in two steps

1.ϕ([(sW)^{∗}, (sW^{(1)})^{⊥}]_{F}) = (W^{(1)} ∧ W^{(1)})^{⊥}
2.d_{2}^{∗}((W^{(1)} ∧ W^{(1)})^{⊥}) = (W^{(2)})^{⊥} .

To prove 1., we first observe the following general fact. Suppose B → C is a surjective map with kernel D. Then T(B)/⟨D⟩ ≅ T(C) (use right exactness of the tensor product, diagram chasing and induction). Hence also F(B)/⟨D⟩ ≅ F(C). In particular ker([B, B]_{F(B)} → [C, C]_{F(C)}) = [B, D]_{F(B)}. Now since ϕ is a natural isomorphism, ϕ induces an isomorphism

ker {[(sW)^{∗}, (sW)^{∗}]_{F} → [(sW^{(1)})^{∗}, (sW^{(1)})^{∗}]_{F}} →
ker {(W ∧ W)^{∗} → (W^{(1)} ∧ W^{(1)})^{∗}} .

By the general fact above, the first kernel is [(sW)^{∗}, (sW^{(1)})^{⊥}]_{F}. Since the second kernel is (W^{(1)} ∧ W^{(1)})^{⊥}, the first statement is proved. The second statement is a direct consequence of the following general fact (which is easily proven). Suppose f: B → C is a map of vector spaces and let D be a subspace of C. Then f^{∗}(D^{⊥}) = (f^{−1}(D))^{⊥}.

Second proof.

We will use the same constructions and notations as in the first proof. Even if A is not Koszul, we may argue in the same way as in the proof of (iv) above to conclude that H(⋀(W), d_{2}) = Ext_{U(L)}(k, k). Since (⋀(W), d_{2}) is a subcomplex of (⋀(V_{⩾1}), \overline{d}) we get a map of algebras Ext_{U(L)}(k, k) → H. We will study this map in low degrees and prove that the 1-2-algebra of Ext_{U(L)}(k, k) is isomorphic to the 1-2-algebra of H. This, combined with Proposition 2.3 and Proposition 2.4, proves (i). Then (ii) follows from (i) and (iii) and (iv) are proved as before.

Obviously we have Ext_{U(L)}^{1}(k, k) = W^{(1)} = H_{(1)}.

Since (\overline{d})^{−1}(W ∧ W) = W, we have

im(d_{2}) ∩ (W^{(1)} ∧ W^{(1)}) = im(\overline{d}) ∩ (W^{(1)} ∧ W^{(1)}) .

Hence,

(Ext_{U(L)}^{1}(k, k))^{2} = W^{(1)} ∧ W^{(1)}/im(d_{2}) ∩ (W^{(1)} ∧ W^{(1)})
= W^{(1)} ∧ W^{(1)}/im(\overline{d}) ∩ (W^{(1)} ∧ W^{(1)}) = (H_{(1)})^{2} .

It follows that

ker {Ext_{U(L)}^{1}(k, k) ⊗ Ext_{U(L)}^{1}(k, k) → (Ext_{U(L)}^{1}(k, k))^{2}} =
ker {H_{(1)} ⊗ H_{(1)} → (H_{(1)})^{2}}

and hence the 1-2-algebra of Ext_{U(L)}(k, k) is equal to the 1-2-algebra of H. Third proof.

Let (K, d) be the Koszul complex of A with respect to (x_{1}, …, x_{n}); i.e., (K, d) = A[sx_{1}, …, sx_{n}; d(sx_{i}) = x_{i}] (we could also have used (⋀(V_{⩾1}), d) from above). Then (K, d) is a differential graded algebra (a DGA) and H(K, d) = H. Since k is a field, H is a strong deformation retract of (K, d) (an SDR); i.e., there are maps f: (K, d) → (H, 0) and ∇: (H, 0) → (K, d) such that f∇ = id and ∇f is homotopic to id. From this it is possible to define an SDR (T((sK_{+})^{∗}), d_{K}) → (T((sH_{+})^{∗}), 0), where d_{K} is the tensor extension of the differential on K. Now, the multiplication on K_{+} defines a differential d_{2} on T((sK_{+})^{∗}) in the same way as in Section 2; i.e., as the composition

(sK_{+})^{∗} → K_{+}^{∗} → (K_{+} ⊗ K_{+})^{∗} →^{ϕ^{−1}} (sK_{+})^{∗} ⊗ (sK_{+})^{∗}

and (T((sK_{+})^{∗}), d_{K} +d_{2}) is the cobar construction on K^{∗}. Thinking of d_{K} +d_{2} as a perturbation of d_{K}, one may use the perturbation lemma to obtain a differential d on T((sH_{+})^{∗}), such that (T((sH_{+})^{∗}), d) is an SDR of (T((sK_{+})^{∗}), d_{K} + d_{2}). Moreover d is a derivation (see [Jo-La 00] ). But the cohomology of the cobar construction on K^{∗} is Ext_{(K,d)}(k, k) and this is U(𝔤_{A}^{⩾2}) ([Av 74]).

Hence we have a DGA, (T((sH_{+})^{∗}), d), with cohomology U(𝔤_{A}^{⩾2}), which is approximated by (T((sH_{+})^{∗}), d_{2}) = Ext_{H}(k, k). This is the “SDR-version” of Avramov’s spectral sequence. (Compare this with the model (⋀(V_{⩾1}), d) above, which has homology H, approximated by H(⋀(V_{⩾1}), d_{2}) = Ext_{U(𝔤_{A}^{⩾2})}(k, k).)

Put V^{i,j} = (sH_{i−1,j})^{∗} for i > 1 and put V = ⊕V^{i,j}. Then d is determined by the value of d on V, d = d_{2} + d_{3} + ⋯, where d_{n}: V → V^{⊗n}, d_{2} is (in principal) the dual of the multiplication on H and the higher d_{n} are duals of Massey products. We have V^{i,j} = 0 if i > j and hence the elements of V^{i,i} are cycles. The cohomology of (T(V), d) in degree (i, i) is U(L)^{i} and hence

U(L)^{i} = V^{i,i} ⊕ (V ⊗ V)^{i,i}/boundaries ⊕ (V ⊗ V ⊗ V)^{i,i}/boundaries ⊕ ⋯

This decomposition corresponds to the filtration of U(L) with respect to powers of I = U(L)^{+}. This is so, because all elements of (V ⊗ V ⊗ ⋯ ⊗ V)^{i,i}/boundaries are products of elements from V^{j,j}. In particular, I/I^{2} = ⊕_{i}V^{i,i} = (sH_{(1)})^{∗}, I^{2}/I^{3} = ⊕_{i}(V ⊗ V/d_{2}(V))^{i,i} and

ker (I/I^{2} ⊗ I/I^{2} → I^{2}/I^{3}) = im(d_{2}) = ϕ^{−1} (ker(H_{(1)} ⊗ H_{(1)} → H_{(2)}))^{⊥} .

Hence the 1-2-algebra of H is isomorphic to the Koszul dual of gr(U(L)). Thus, using Proposition 2.4, we get (i) and (ii); (iii) follows as before.

Suppose A is Koszul. Then (T(V), d) has cohomology only on the diagonal. Consider j − i as the homological degree of T(V)^{i,j}. We have a map (T(V), d) → (U(L), 0) (where U(L) is considered to be concentrated in homological degree zero), which induces an isomorphism in homology.

Hence (T(V), d) is a minimal (free associative) model of U(L). Such a model may always be constructed for any ℤ_{2} × ℕ-graded connected k-algebra B. This is similar to the construction of models (⋀(V), d) of commutative algebras described above, but now the space (sV)^{∗} is the algebra Ext_{B}^{+}(k, k) and the dual of d_{2} (with the usual sign corrections) is the dual of the Yoneda product. Hence in our case we get Ext_{U(L)}(k, k) ≅ H, which proves (iv).

Problem 4.5. Suppose A is generated by A_{1}. Is the converse of (iv) in Theorem 4.1 true? Or, perhaps weaker, if the double Poincaré series of U(L), P_{U(L)}(x, y), is equal to H(1/x, xy), is it then true that A is Koszul? Observe that the equality P_{U(L)}(x, y) = H(1/x, xy) implies the equality A(z) · A^{!}(−z) = 1.

Added in proof: This problem has now been partially solved by Leonid Positselski (personal communication).

5. The coordinate ring of generic points – IGC and MRC

If X = {P_{1}, …, P_{s}} is a set of points in P_{k}^{n}, we denote the coordinate ring of X, k[X_{0}, …, X_{n}]/I(X), by A_{X}. It is well-known [Ge-Or 81] that the Hilbert series of A_{X} satisfies A_{X}(z) ⩽ ∑_{i⩾0} min(\binom{n+i}{i}, s)z^{i}. There is equality on an open non-empty set in (P_{k}^{n})^{s}, and in case of equality the points are said to be in generic position. A point P = (c_{0} : ⋯: c_{n}) ∈ P_{k}^{n} is called generic if the c_{i}’s are algebraically independent over the prime field of k. A set of mutually generic points is of course in generic position. We will in the sequel always assume that our sets of points are generic. We may also assume that s ⩾ n + 1, since otherwise the points lie in P_{k}^{n−1}.

Definition 5.1. Given n and s ⩾ n + 1, let d ⩾ 2 as a function of n and s be defined by

\binom{n + d − 1}{d − 1} ⩽ s < \binom{n + d}{d} .

It is well-known that

Tor_{i,j}^{S}(A_{X}, k) = 0 if j ≠ i + d − 1, i + d

(here and in the sequel S will always denote k[X_{0}, …, X_{n}]). In particular the minimal generators for the ideal I_{X} are of degrees d and d+1. Also, it is well-known that if s = \binom{n+d−1}{d−1}, then Tor_{i,j}^{S}(A_{X}, k) = 0 if j ≠ i + d − 1 (A_{X} has a linear resolution over S).

It has been conjectured that the map ((I_{X})_{d})^{n+1} → (I_{X})_{d+1}, defined by (f_{0}, …, f_{n}) ↦ ∑_{i=0}^{n} X_{i}f_{i}, is either injective or surjective. This is called the Ideal Generating Conjecture (IGC). It is equivalent to the fact that

Tor_{1,d+1}^{S}(A_{X}, k) = 0 or Tor_{2,d+1}^{S}(A_{X}, k) = 0 .

The generalization that for all i,

Tor_{i,d+i}^{S}(A_{X}, k) = 0 or Tor_{i+1,d+i}^{S}(A_{X}, k) = 0

is called the Minimal Resolution Conjecture (MRC).

The MRC is proved for P^{2} [Ge-Ma 84]; for P^{3} [Ba-Ge 86]; for P^{4} [Wa 95], [La 97]; for P^{n}, n+1 ⩽ s ⩽ n+4 [Ge-Lo 89], [Ca-Ro-Va 91]; for P^{n}, s = \binom{n+2}{2}− n [Lo 93]; for P^{n}, n ⩽ 9, s ⩽ 50 except (n, s) = (6, 11), (7, 12), (8, 13) [Be-Kr 94]; and for P^{n}, s > 6^{n^{3} log n} [Hi-Si 96]. There has some time been computational evidence for some counterexamples by Schreyer, later [Bo 94], and [Be-Kr 94], and now there is shown to exist, for each n ⩾ 6, n ≠ 9, one s which gives a counterexample to MRC [Ei-Po 99]. In [Ge-Gr-Ro 86, Theorems 4.7 and 5.8] IGC is proved in several cases, e.g. for s ⩽ 1 + n + n^{2}/4 and for \binom{n+2}{2} − n ⩽ s < \binom{n+2}{2}. In [Co-Tr-Va 97] it is proved that A_{X} is even Koszul if s ⩽ 1 + n + n^{2}/4.

We will now state some well-known and frequently used facts about the Hilbert series of A_{X}. Let h_{i,j}^{n,s} = dim_{k} Tor_{i,j}^{S}(A_{X}, k), h_{i,j}^{n,s} = h_{i,j} when n and s are considered to be fixed and H(x, y) = ∑ h_{i,j}x^{i}y^{j}.

Proposition 5.2. For any given n and s we have

(i)A_{X}(z) = ∑_{j=0}^{d−1} \binom{n + j}{j}z^{j} + \frac{sz^{d}}{1 − z}

(ii)(1 − z)^{n+1}A_{X}(z) = 1 + ∑_{j=0}^{n}(−1)^{j+1}\binom{n}{j} \{\binom{n + d}{d}\frac{d}{d + j} − s\} z^{d+j}
(iii)(1 − z)^{n+1}A_{X}(z) = H(−1, z)
(iv)h_{j+1,d+j} − h_{j,d+j} = \binom{n}{j} \{\binom{n + d}{d}\frac{d}{d + j} − s\} , 0 ⩽ j ⩽ n .

Proof. We give for convenience a short proof of (ii). It is enough to prove the identity

(1 − z)^{n+1} ∑_{j=0}^{d−1} \binom{n + j}{j}z^{j} = 1 + ∑_{j=0}^{n}(−1)^{j+1}\binom{n}{j}\binom{n + d}{d}\frac{d}{d + j} z^{d+j} .

This can easily be done by proving that both sides have the same derivative. Here is another way to deduce the identity without knowing the right hand side beforehand. Call the left hand side p(z). It is a polynomial of degree n+d, divisible by (1−z)^{n+1}, and it may also be written as

1 − (1 − z)^{n+1} ∑_{j=d}^{∞} \binom{n + j}{j}z^{j} ,

which shows that p(z) − 1 is divisible (as a polynomial) by z^{d}. It follows that p′(z) is divisible by z^{d−1} and by (1 − z)^{n} and hence there is a constant c such that p′(z) = cz^{d−1}(1 − z)^{n} = c ∑_{j=0}^{n}(−1)^{j}\binom{n}{j}z^{j+d−1}. Intergrating this one gets

p(z) = 1 + c ∑_{j=0}^{n}(−1)^{j}\binom{n}{j}\frac{1}{d + j} z^{j+d} .

But the highest coefficient in p(z) is (−1)^{n+1}\binom{n+d−1}{d−1} and hence c(−1)^{n} \frac{1}{d+n} = (−1)^{n+1}\binom{n+d−1}{d−1}, which gives c = −d\binom{n+d}{d}.

Definition 5.3. Let n and s be given and d defined by Definition 5.1. Then m, as a function of n and s, is defined as the least integer such that

\binom{n + d}{d}\frac{d}{d + m} − s ⩽ 0

and t, as a function of n and s, is defined by

t = \binom{n}{m − 1} \{\binom{n + d}{d}\frac{d}{d + m − 1} − s\} .

It is easy to see that if Tor_{i,d+i−1}^{S}(A_{X}, k) = 0, we have Tor_{j,d+j−1}^{S}(A_{X}, k) = 0 for all j ⩾ i, and if Tor_{i,d+i}^{S}(A_{X}, k) = 0, we have Tor_{j,d+j}^{S}(A_{X}, k) = 0 for all j ⩽ i. From this and Proposition 5.2 the following well-known proposition is easily deduced.

Proposition 5.4. For any given n and s the following are equivalent.

(i)MRC holds
(ii)h_{j−1,j+d−1} = h_{j+1,j+d} = 0 for some j
(iii)h_{m−1,m+d−1} = h_{m+1,m+d} = 0
(iv)h_{j,j+d−1} = \binom{n}{j − 1} \{\binom{n + d}{d}\frac{d}{d + j − 1} − s\} and
h_{j+1,j+d} = 0 for some j
(v)h_{m,m+d−1} = t and h_{m+1,m+d} = 0

From Proposition 5.2 the following well-known characterization of IGC immediately follows.

Proposition 5.5. For any given n and s, IGC holds if and only if

h_{2,d+1} = max(n \{\binom{n + d}{d}\frac{d}{d + 1} − s\} , 0) .

It is possible to interprete MRC and IGC in terms of the homotopy Lie algebra of A_{X} (if d > 2) or the Lie algebra associated to A_{X} (if d = 2).

Theorem 5.6. Let X be a set of s generic points in P_{k}^{n} and let A_{X} be the coordinate ring of X. Let 𝔤 be the homotopy Lie algebra of A_{X} and let L be the Lie algebra associated to A_{X} (see Definition 3.1). Let d, m and t be the numbers defined from n and s by Definition 5.1 and 5.3. Then, if d > 2, we have the following

(i)MRC holds ⟺ dim_{k} 𝔤^{m+1,m+d−1} = t and 𝔤^{m+2,m+d} = 0
(ii)IGC holds ⟺ dim_{k} 𝔤^{3,d+1} = max(n \{\binom{n + d}{d}\frac{d}{d + 1} − s\} , 0)

If d = 2, we have the following

(iii)MRC holds ⟺ dim_{k}(L/[L, L])^{m+1} = t and (L/[L, L])^{m+2} = 0
(iv)IGC holds ⟺ dim_{k} L^{3} = max(n \{\frac{(n + 2)(n + 1)}{3} − s\} , 0)

Proof. Suppose first that d > 2. In the third proof of theorem 4.1 it was proved that U(𝔤_{A}^{⩾2}) is the cohomology of (T(V), d), where V^{ij} = (sH_{i−1,j})^{∗} for i > 1 (and zero otherwise). Moreover the image of the differential is contained in ⊕_{n⩾2}V^{⊗n}. In our case, we have that V^{ij} = 0 if j − i ≠ d − 2, d − 1. The differential maps V^{ij} to T(V)^{i+1,j}. Hence, if this is nonzero, d − 2 or d − 3 must be a sum of at least two numbers each of which is d − 1 or d − 2. This is impossible if d > 2. Hence the differential is zero and U(𝔤_{A}^{⩾2}) ≅ T(V). In particular 𝔤_{A}^{i+1,i+d−1} ≅ (H_{i,i+d−1})^{∗} and 𝔤_{A}^{i+1,i+d} ≅ (H_{i,i+d})^{∗}. Now the statements (i) and (ii) follow from Proposition 5.4 (v) and Proposition 5.5.

Suppose that d = 2. The statements (iii) and (iv) follow from the same propositions as above and Theorem 4.1 (ii).

The next theorem shows how the numbers h_{i,j} are changed when the number of points is increased. For a given d, h_{i,i+d} increases with s, while h_{i+1,i+d} decreases. Moreover the rate of change is bounded by \binom{n}{i}.

Theorem 5.7. Let A_{X} be the coordinate ring of a set X of s generic points in P_{k}^{n} and let h_{i,j}^{n,s} = dim_{k} Tor_{i,j}^{S}(A_{X}, k). Then for any i, we have

h_{i,i+d}^{n,s′} ⩽ h_{i,i+d}^{n,s} ⩽ h_{i,i+d}^{n,s′} + \binom{n}{i}(s − s′) and
h_{i+1,i+d}^{n,s} ⩽ h_{i+1,i+d}^{n,s′} ⩽ h_{i+1,i+d}^{n,s} + \binom{n}{i}(s − s′)

whenever s and s′ define the same d; i.e., whenever \binom{n+d−1}{d−1} ⩽ s′ ⩽ s ⩽ \binom{n+d}{d}.

Proof. Let Y = {P_{1}, …, P_{s}} be a set of s generic points and let X = Y \ {P_{s}}. After a coordinate change we may assume that P_{s} = (1 : 0 : ⋯: 0). Let K = I_{X}/I_{Y} be the kernel of the natural map A_{Y} → A_{X}. We get K(z) = A_{Y} (z) − A_{X}(z) = z^{d}/(1 − z), so K_{r} is one-dimensional in each degree r ⩾ d. Let f_{d} + I_{Y} ∈ K, where f_{d} ∉ I_{Y} is homogeneous of degree d; i.e., f_{d}(P_{j}) = 0 for j = 1, …, s − 1 and f_{d}(P_{s}) ≠ 0. Then (X_{i}f_{d})(P_{j}) = 0 for all j and i = 1, …, n, and hence X_{i}f_{d} ∈ I_{Y} for i = 1, …, n. Moreover (X_{0}^{r}f_{d})(P_{j}) = 0 for j = 1, …, s − 1, and (X_{0}^{r}f_{d})(P_{s}) = f_{d}(P_{s}) ≠ 0 for every r ⩾ 0. Thus K_{r+d} is generated by X_{0}^{r}f_{d} + I_{Y}. It follows that K is generated by f_{d} as an S-module and that K ≅ S/(X_{1}, …, X_{n})[−d]. We now consider the long exact Tor_{∗,i+d}^{S}(∗, k)-sequence of 0 ⟶ K ⟶ A_{Y} ⟶ A_{X} ⟶ 0. Since K ≅ S/(X_{1}, …, X_{n})[−d], we have Tor_{j,i+d}^{S}(K, k) = 0 if j ≠ i and

Tor_{i,i+d}^{S}(K, k) = k^{\binom{n}{i}}, and we get an exact sequence

0 ⟶ Tor_{i+1,i+d}^{S}(A_{Y} , k) ⟶ Tor_{i+1,i+d}^{S}(A_{X}, k) ⟶ Tor_{i,i+d}^{S}(K, k) ⟶
Tor_{i,i+d}^{S}(A_{Y} , k) ⟶ Tor_{i,i+d}^{S}(A_{X}, k) ⟶ 0.

The rightmost part of this sequence gives h_{i,i+d}^{n,s−1} ⩽ h_{i,i+d}^{n,s} ⩽ h_{i,i+d}^{n,s−1} + \binom{n}{i}, and the leftmost part gives h_{i+1,i+d}^{n,s} ⩽ h_{i+1,i+d}^{n,s−1} ⩽ h_{i+1,i+d}^{n,s} + \binom{n}{i}. The statement now follows by induction.

Suppose that MRC holds for s′ and s′′ points in P_{k}^{n}. Then, using the previous theorem, we may in some cases conclude that MRC also holds for all s between s′ and s′′.

Theorem 5.8. Suppose MRC holds for s′ and s′′ points in P_{k}^{n}, where for some d ⩾ 2 and some j, 1 ⩽ j ⩽ n,

\binom{n + d}{d}\frac{d}{d + j} ⩽ s′ < s′′ ⩽ \binom{n + d}{d}\frac{d}{d + j − 1} .

Then MRC holds for any s points in P_{k}^{n}, where s′ ⩽ s ⩽ s′′.

Proof. By Theorem 5.4, we have h_{j+1,j+d} = 0 for s′ points (m = j for s′). Hence, by Theorem 5.7, h_{j+1,j+d} = 0 is true also for s points. Also by Theorem 5.4, we have h_{j−1,j+d−1} = 0 for s′′ points. This is true if s′′ < \binom{n+d}{d} \frac{d}{d+j−1}, since then m = j for s′′. If s′′ = \binom{n+d}{d} \frac{d}{d+j−1}, then by Proposition 5.2 for s′′ points, h_{j,d+j−1} − h_{j−1,d+j−1} = 0 and since MRC holds we must have h_{j,d+j−1} = h_{j−1,d+j−1} = 0. Hence, by Theorem 5.7, h_{j−1,j+d−1} = 0 is true also for s points. Hence for s points we have h_{j+1,j+d} = h_{j−1,j+d−1} = 0, and hence MRC holds for s points.

Corollary 5.9. In order to prove MRC for s in the interval \binom{n+d−1}{d−1} ⩽ s < \binom{n+d}{d}, it is enough to prove MRC for s_{j} = ⌈\binom{n+d}{d} \frac{d}{d+j} ⌉, j = 1, …, n − 1, and for s′_{j} = ⌊\binom{n+d}{d} \frac{d}{d+j} ⌋ . In order to prove IGC in the same interval, it is enough to prove IGC for s = s_{1} and s = s′_{1}.

Until now the results in this section hold for any d. However, from now on, we will specialize to the case d = 2. Using Theorem 5.8 and a result by Anna Lorenzini we can prove MRC in the interval [\binom{n+2}{2} − n, \binom{n+2}{2}] for all n. This also follows from the results in [Ge-Gr-Ro 86].

Corollary 5.10. If \binom{n+2}{2} − n ⩽ s ⩽ \binom{n+2}{2} then MRC holds.

Proof. MRC is proved in [Lo 93] for s = \binom{n+2}{2} − n. Since MRC also holds for s = \binom{n+2}{2} and \binom{n+2}{2} \frac{2}{3} ⩽ \binom{n+2}{2} − n ⩽ \binom{n+2}{2} the statement follows from Theorem 5.8.

The following theorem will be useful in the next section, where we present results of calculations in the Lie algebra associated to A_{X}.

Theorem 5.11. Let X be a set of s generic points in P_{k}^{n} and let L be the Lie algebra associated to the coordinate ring of X (see Definition 3.1). Suppose s < \binom{n+2}{2}. We then have

(i) L^{3} = 0 ⇒ MRC (and hence also IGC) holds
for s′ points in P_{k}^{n}, where s ⩽ s′ ⩽ \binom{n+2}{2}
(ii) dim_{k} L^{3} = n \{\frac{(n+2)(n+1)}{3} − s\} ⇒ IGC holds
for s′ points in P_{k}^{n}, where s′ ⩽ s
(iii) dim_{k} L^{3} = n \{\frac{(n+2)(n+1)}{3} − s\} and L^{4} = [L^{2}, L^{2}] ⇒ MRC holds

Proof. Suppose L^{3} = 0. By Corollary 4.3 (iii), we get h_{2,3} = 0. Since also h_{0,2} = 0, MRC holds for s points by Proposition 5.4 (ii). Now MRC also holds for s′′ = \binom{n+2}{2} points. Hence, (i) follows from Theorem 5.8.

Suppose dim_{k} L^{3} = n \{\frac{(n+2)(n+1)}{3} − s\}. Then, by Proposition 5.2 (iv) (d = 2

and j = 1), we get h_{2,3} − h_{1,3} = dim_{k} L^{3}. But dim_{k} L^{3} = h_{2,3} and hence h_{1,3} = 0. By Theorem 5.7, h_{1,3} = 0 is true for any s′ points in P_{k}^{n}, where n + 1 ⩽ s′ ⩽ s and hence (ii) follows.

The last statement follows, since as above h_{1,3} = 0 and, by Corollary 4.3 (iv), h_{3,4} = 0. Now use Proposition 5.4 (ii).

6. Results of calculations

We note that MRC is proved for s < \binom{n+2}{2} if n < 9, except for (n, s) = (6, 11), (7, 12), (8, 13), and also for s ⩽ 50 if n = 9, [Be-Kr 94]. Thus we treat only the case n ⩾ 9.

Theorem 6.1. For n ⩽ 20 and (n+1)(n+2)/3 ⩽ s < (n+1)(n+2)/2 MRC holds.

Proof. Since s < (n + 1)(n + 2)/2 we have d = 2. From the table in Section 11 we get that L^{3} = 0 for n ⩽ 20 and s = ⌈(n + 1)(n + 2)/3⌉. Hence the result follows from Theorem 5.11 (i).

Theorem 6.2. If n ⩽ 20 and s < (n + 1)(n + 2)/2, IGC holds.

Proof. If s ⩾ (n + 2)(n + 1)/3 the conclusion follows from the previous theorem. From the table in Section 11 we get that dim_{k} L^{3} = n \{\frac{(n+2)(n+1)}{3} − s\} for s = ⌊(n + 1)(n + 2)/3⌋. Hence the result follows from Theorem 5.9.

Definition 6.3.

l_{3} = ⌈(n + 1)(n + 2)/3⌉
l_{4} = ⌈n^{2} + n + 1/2 − \frac{1}{2}\sqrt{2n^{4} − 2n^{2} + 1}⌉
l_{5} = ⌈n^{2}/2 + n + 1 − \frac{1}{10}\sqrt{5n^{4} + 20}⌉

We conjecture that if s ⩾ l_{i}, then L^{i} = 0, i = 3, 4, 5, cf. Conjecture 8.1. From the table in Section 11, it follows that the conjecture is true for L^{3} if n ⩽ 20, for L^{4}, if n ⩽ 17, and for L^{5} if n ⩽ 11. We can use the result for L^{4} to extend the interval where we can prove MRC.

Theorem 6.4. If n ⩽ 17 and l_{4} ⩽ s ⩽ \binom{n+2}{2} then MRC holds.

Proof. From the table in Section 11 we get that L^{4} = 0 for s = l_{4}. Hence by Theorem 5.7 it is true also for s ⩾ l_{4}. Corollary 4.3 gives h_{3,4} = 0 for s ⩾ l_{4}. Since we have proved that dim_{k} L^{3} = n \{\frac{(n+2)(n+1)}{3} − s\} for s = ⌊(n+1)(n+2)/3⌋, it follows that h_{1,3} = 0 for that many points and then, by Theorem 5.7, it follows that h_{1,3} = 0 for s ⩽ (n + 1)(n + 2)/3. Hence h_{1,3} = h_{3,4} = 0 for l_{4} ⩽ s ⩽ (n + 1)(n + 2)/3 and hence MRC holds in this interval. But by Theorem 6.1, MRC holds in the interval l_{3} ⩽ s ⩽ \binom{n+2}{2}.

Using Theorem 5.6 (iii), we can show MRC in a few more cases.

Theorem 6.5. MRC is true if

a) n = 9 and d = 2 (i.e., s < 55).

b) (n, s) = (10, 39), (10, 40), (11, 47), and (12, 55).

Proof. For n = 9 and s ⩽ 50, MRC is proved in [Be-Kr 94]. For 50 < s < 55 we have proved that L^{3} = 0 (see Section 11) and hence MRC holds by Theorem 5.11. For the cases in b) we have used the Mathematica program “liedim” to show that [L^{2}, L^{2}] = L^{4}. Then, since dim_{k} L^{3} = t, the statement follows from Theorem 5.6. (In fact, the case (10, 40) may be derived from (10, 39) and (10, 43) using Theorem 5.8.)

It is natural to conjecture that

Tor_{r,s}^{S}(A_{X}, k) ∧ Tor_{t,u}^{S}(A_{X}, k) → Tor_{r+t,s+u}^{S}(A_{X}, k)

is either injective or surjective. We can prove this in some special cases.

Theorem 6.6. Let A_{X} be the coordinate ring of s generic points in P_{k}^{n} and let S = k[x_{0}, …, x_{n}]. If n ⩽ 17 and s ⩾ l_{4} the multiplication

Tor_{1,2}^{S}(A_{X}, k) ∧ Tor_{1,2}^{S}(A_{X}, k) → Tor_{2,4}^{S}(A_{X}, k)

is injective.

Proof. We have shown that L^{4} = 0 in this interval (see Section 11). Thus Corollary 4.3 (v) gives that the multiplication is injective.

Theorem 6.7. Let A_{X} be the coordinate ring of s generic points in P_{k}^{n} and let S = k[x_{0}, …, x_{n}]. For (n, s) =

(3, 6), (5, 13), (6, 17), (6, 18), (7, 21), (7, 22), (8, 26), (8, 27), (9, 32), (9, 33),
(10, 39), (10, 40), (11, 47), (12, 55)

the multiplication

Tor_{1,2}^{S}(A_{X}, k) ∧ Tor_{1,2}^{S}(A_{X}, k) → Tor_{2,4}^{S}(A_{X}, k)

is surjective.

Proof. For these pairs of (n, s), we have that Tor_{i,∗}^{S}(A_{X}, k) for i ⩽ 3 have the expected dimensions for MRC ([Be-Kr 94] and Theorem 6.5). By Corollary 4.3 (v), the kernel of the multiplication map in the theorem has dimension dim_{k}[L^{2}, L^{2}], and by Corollary 4.3 (iv), h_{3,4} = dim_{k} L^{4}/[L^{2}, L^{2}]. Since MRC holds, we know h_{1,2}, h_{2,4}, and h_{3,4}. From Section 11 we know dim_{k} L^{4} for all pairs (n, s) in the theorem. Thus the statement follows by computing dimensions. E.g., if n = 6 and s = 17 we have that h_{1,2} = dim_{k} L^{2} = 11 and hence dim_{k}(Tor_{1,2}^{S}(A_{X}, k)∧Tor_{1,2}^{S}(A_{X}, k)) = 55. Also, h_{3,4} = 0 since MRC holds and m = 2, and, by Proposition 5.2 (iv), we get h_{2,4} = 45. Hence by Corollary 4.3 (iv), L^{4} = [L^{2}, L^{2}], and from the table, dim_{k} L^{4} = 10. Hence the map is surjective, since the kernel is 10-dimensional by Corollary 4.3 (v).

We can check the conjecture about the multiplication in a few more cases. If s = n + 2, it is easy to see that A_{X} is Gorenstein. By reasons of degree, the only possible nonsurjective multiplication is Tor_{i} ∧ Tor_{n−i} → Tor_{n}. This is surjective, since A_{X} is Gorenstein.

In P^{3} the multiplication is injective or surjective for any choice of degrees and number of points. If s ⩾ 7 this follows from Theorem 6.6. If s = 5, A_{X} is Gorenstein. If s = 4 the resolution is linear and the multiplication trivial. For the remaining case s = 6, the multiplication

Tor_{1,2}^{S}(A_{X}, k) ∧ Tor_{1,2}^{S}(A_{X}, k) → Tor_{2,4}^{S}(A_{X}, k)

is surjective by Theorem 6.7. That the multiplication

Tor_{1,2}^{S}(A_{X}, k) ⊗ Tor_{2,3}^{S}(A_{X}, k) → Tor_{3,5}^{S}(A_{X}, k)

is surjective follows in a similar way as in the proof of Theorem 6.7 by determining dim_{k} L^{5} (cf. Section 11).

7. Ideals generated by generic quadratic forms

We call a quadratic form ∑_{1⩽i⩽j⩽n} c_{ij}x_{i}x_{j} in k[x_{1}, …, x_{n}] generic if {c_{ij}} is algebraically independent over the prime field of k (the variables x_{i} are considered to be even). We will now consider ideals (f_{1}, …, f_{t}) in k[x_{1}, …, x_{n}], where f_{k} = ∑_{1⩽i⩽j⩽n} c_{ij}^{(k)}x_{i}x_{j} and {c_{ij}^{(k)}} is algebraically independent over the prime field of k. For simplicity we say that such an ideal is generated by t generic quadratic forms. We will determine for which n and t we have that k[x_{1}, …, x_{n}]/I is a Koszul algebra if I is generated by t quadratic generic forms. We will identify an ideal (g_{1}, …, g_{t}) generated by linearly independent quadratic elements (and thus the quadratic algebra k[x_{1}, …, x_{n}]/(g_{1}, …, g_{t})) with a point in Grass(\binom{n+1}{2}, t). One part of the following theorem is proved in [Co-Tr-Va 97].

Theorem 7.1. Let A = k[x_{1}, …, x_{n}]/(g_{1}, …, g_{t}) be a quadratic algebra, where the variables x_{i} are considered to be even. If t ⩽ n there is an open non-empty subset of Grass(\binom{n+1}{2}, t) on which A is Koszul. If t ⩾ n^{2}/4 + n/2 there is a nonempty countable intersection of open sets in Grass(\binom{n+1}{2}, t) on which A is Koszul. If n < t < n^{2}/4 + n/2, then A is not Koszul on an open non-empty subset of Grass(\binom{n+1}{2}, t). In particular, if A = k[x_{1}, …, x_{n}]/I, where I is generated by t generic quadratic forms, then A is Koszul if and only if t ⩽ n or t ⩾ n^{2}/4 + n/2.

Proof. If t ⩽ n, then almost all A are complete intersections, and complete intersections are Koszul, cf. [Fr-Lö 91].

Now let t ⩾ n^{2}/4 + n/2. Consider the rings A with t fixed as points in G = Grass(\binom{n+1}{2}, t). The subset of G corresponding to rings A with A_{3} = 0 is open. For such A, the Bar complex of k over A is based on the vector space A_{1} ⊕ A_{2}, which has the fixed dimension n + \binom{n+1}{2} − t, and the differential is expressed by means of the multiplication in A. Hence, it is an open condition that the homology is zero in a certain bidegree (i, j), and hence A_{3} = 0 and Tor_{i,j}^{A}(k, k) = 0 is an open condition for fixed (i, j). That A is Koszul, i.e., Tor_{i,j}^{A}(k, k) = 0 if i ≠ j, is therefore true on an intersection of open sets. It suffices to show that this intersection is non-empty. If A = k[x_{1}, …, x_{a}]/(x_{1}, …, x_{a})^{2} ⊗ k[x_{1}, …, x_{b}]/(x_{1}, …, x_{b})^{2}, with a = ⌈n/2⌉ and b = ⌊n/2⌋, we have an example of a Koszul algebra in case t = ⌈n^{2}/4 + n/2⌉, [Fr 75]. If t is larger we could just divide by some more quadratic monomials.

Now let n < t ⩽ \binom{n+2}{3}/n. Then, by [Ho-La 87], there is a non-empty subset of G with {x_{i}g_{j}} linearly independent, where I = (g_{1}, …, g_{t}). By Corollary 4.3 (iii), this gives L^{3} = 0. If A were Koszul, then U(η_{A}) is equal to Ext_{A}(k, k). Then A would be a complete intersection by [Gu 71], a contradiction since t > n.

Finally let \binom{n+2}{3}/n ⩽ t < n^{2}/4 + n/2. Then, by [Ho-La 87], A_{3} = 0 on a nonempty open subset of G. If A is Koszul, then A^{!}(z) = 1/(1 − nz + (\binom{n+1}{2} − t)z^{2}) = 1/((1−nz/2)^{2}+(n^{2}/4+n/2−t)z^{2}). Hence A^{!}(z) has no pole on the real axis. But then the coefficients in A^{!}(z) are not all non-negative, since a series with non-negative coefficients has a singularity on the positive real axis if its radius of convergence is finite.

Remark 7.2. It follows from the theorem above that the “smallest” algebra defined by generic quadratic forms which is not Koszul is obtained by choosing n = 4 and t = 5. This was in fact the first example of a non-Koszul quadratic algebra found by Christer Lech 1975.

To state some conjectures about the coordinate ring of generic quadratic forms, we need a concept from non-commutative graded algebras.

Let B be an ℕ-graded associative algebra and {y_{1}, …, y_{r}} a set of homogeneous elements in B of degrees d_{1}, …, d_{r}, respectively. Then

B/(y_{1}, …, y_{r})(z) ⩾ B(z)/(1 + (z^{d_{1}} + ⋯ + z^{d_{r}})B(z))

and the set is called strongly free in B if there is equality, [An 82]. Any subset of a strongly free set is strongly free. If {y_{1}, …, y_{r}} is a strongly free set of quadratic elements in the free associative algebra k⟨X_{1}, …, X_{n}⟩, we get

k⟨X_{1}, …, X_{n}⟩/(y_{1}, …, y_{r})(z) = 1/(1 − nz + rz^{2}) .

Such an algebra will be called a non-commutative c.i. (complete intersection). They play much the same rôle for non-commutative algebras as complete intersections do for commutative algebras. E.g., the algebra B = k⟨X_{1}, …, X_{n}⟩/(y_{1}, …, y_{r}) is a non-commutative c.i. if and only if k⟨X_{1}, …, X_{n}, Y_{1}, … Y_{r}; dX_{i} = 0, dY_{i} = y_{i}⟩ is a minimal model of B (cf. end of proof of Theorem 4.1). It follows that B is a non-commutative c.i. if and only if gldim(B) ⩽ 2, and hence a non-commutative c.i. is Koszul. Also, for a quadratic algebra A, A^{!} is a non-commutative c.i. if and only if A is Koszul and A_{3} = 0.

We will now consider ideals (h_{1}, …, h_{r}) in k⟨X_{1}, …, X_{n}⟩, where the variables X_{i} are odd and where h_{k} = ∑_{1⩽i⩽j⩽n} d_{ij}^{(k)}[X_{i}, X_{j}] and [X_{i}, X_{j}] = X_{i}X_{j} + X_{j}X_{i}. Such an element is called a Hopf element and the corresponding quotient algebra a quadratic Hopf algebra. We will identify an ideal (h_{1}, …, h_{r}) generated by linearly independent quadratic Hopf elements (and thus also the quadratic Hopf algebra k⟨X_{1}, …, X_{n}⟩/(h_{1}, …, h_{r})) with a point in Grass(\binom{n+1}{2}, r). As before we will say that the ideal (h_{1}, …, h_{r}) is generated by r generic quadratic Hopf elements if the coefficients {d_{ij}^{(k)}} is algebraically independent over the prime field of k.

There is a well-known duality between Grassmanians yielding the following proposition.

Proposition 7.3. The correspondence A ↔ A^{!} defines an isomorphism of projective manifolds between Grass(\binom{n+1}{2}, t) and Grass(\binom{n+1}{2}, \binom{n+1}{2} − t)

Proof. The relations in A^{!} are obtained from those in A by solving a system of linear equations. By Cramer’s rule the solutions may be expressed as rational functions and locally the same determinant may be used in the denominators.

Now the dual version of Theorem 7.1 directly follows.

Theorem 7.4. Let B = k⟨X_{1}, …, X_{n}⟩/(h_{1}, …, h_{r}) be a quadratic Hopf algebra, where the variables X_{i} are considered to be odd. If r ⩽ n^{2}/4 there is a countable intersection of open sets in Grass(\binom{n+1}{2}, r) on which B is Koszul. If r ⩾ \binom{n}{2} there is an open non-empty subset of Grass(\binom{n+1}{2}, r) on which B is Koszul. If n^{2}/4 < r < \binom{n}{2}, then A is not Koszul on an open non-empty subset of Grass(\binom{n+1}{2}, r). In particular, if B = k⟨X_{1}, …, X_{n}⟩/J, where J is generated by r generic quadratic Hopf elements, then B is Koszul if and only if r ⩽ n^{2}/4 or r ⩾ \binom{n}{2} and B is a non-commutative c.i. if and only if r ⩽ n^{2}/4.

Proof. To get the statement about the Koszul property, we just use Theorem 7.1 and Proposition 7.3, observing that \binom{n+1}{2}−n^{2}/4−n/2 = n^{2}/4 and \binom{n+1}{2}−n = \binom{n}{2}. The last statement (proved in [An 82]) follows from the remark above that B is a non-commutative c.i. if and only if B^{!} is Koszul and B_{3}^{!} = 0, which by the proof of Theorem 7.1 is true iff \binom{n+1}{2} − r ⩾ (n + 2)(n + 1)/6 and also \binom{n+1}{2} − r ⩾ n^{2}/4 + n/2 or \binom{n+1}{2} − r ⩽ n. But for n ⩾ 2 we have n ⩽ (n + 2)(n + 1)/6 ⩽ n^{2}/4 + n/2.

Conjecture 7.5. Let A = k[x_{1}, …, x_{n}]/I, where the variables x_{i} are even and where I is generated by quadratic generic elements. Then either A^{!} is a non-commu-tative c.i. or η_{A} is nilpotent, where A^{!} = U(η_{A}).

For a formal power series ∑_{i⩾0} a_{i}z^{i} we let (∑_{i⩾0} a_{i}z^{i})_{+} = ∑_{i⩾0} b_{i}z^{i}, where b_{i} = a_{i} if a_{j} > 0 for all j ⩽ i, and b_{i} = 0 otherwise. A formal power series p(z) with integer coefficients and constant term 1 may be written in the form p(z) = ∏_{k=1}^{∞}(1 + z^{2k−1})^{e_{2k−1}}/(1 − z^{2k})^{e_{2k}}. We denote by Log(p(z)) the series ∑_{i⩾1} e_{i}z^{i}. Recall that, for a graded algebra A, we have U(η_{A}) = A^{!}, so η_{A}(z) = Log(A^{!}(z)).

We now make the conjecture more precise.

Conjecture 7.6. Let A = k[x_{1}, …, x_{n}]/I, where the variables x_{i} are even and where I is generated by \binom{n+1}{2}−r quadratic generic forms. Then η_{A}(z) = (Log(1/(1− nz + rz^{2})))_{+}.

Theorem 7.7. Conjecture 7.6 is true in the following cases

a) If A is Koszul.

b) If n ⩽ 8.

c) If r ⩾ (n^{2} − 1)/3 and n ⩽ 20.

d) If r ⩾ n^{2} − 1/2 − \frac{1}{2} \sqrt{2n^{4} − 2n^{2} + 1} and n ⩽ 17.

e) If r ⩾ n^{2}/2 − \frac{1}{10} \sqrt{5n^{4} + 20} and n ⩽ 10.

f) If (n, r) = (9, 22).

Proof. a) According to Theorem 7.1, A is Koszul if and only if r ⩾ \binom{n}{2} or r ⩽ n^{2}/4. In the first case we know that A is a complete intersection, so η_{A}(z) = nz +(\binom{n+1}{2}− r)z^{2}. A simple calculation shows that (Log(1/(1−nz +rz^{2})))_{+} = nz +(\binom{n+1}{2}−r)z^{2} if and only if r ⩾ (n^{2} − 1)/3. Since \binom{n}{2} ⩾ (n^{2} − 1)/3, the result follows in the first case. In the second case, A^{!} is a non-commutative c.i. (since A is Koszul and A_{3} = 0) and hence A^{!}(z) = 1/(1 − nz + rz^{2}). It follows that η_{A}(z) = Log(A^{!}(z)) = Log(1/(1−nz +rz^{2})). Hence also η_{A}(z) = (Log(1/(1−nz +rz^{2})))_{+}, since (η_{A})_{i} = 0 for some i implies that (η_{A})_{j} = 0 for all j ⩾ i .

b) We show how the result follows for n = 8. Then A is Koszul if \binom{9}{2} − r = t ⩾ 8^{2}/4 + 8/2 = 20; i.e., if A has at least 20 quadratic relations. If A has t quadratic relations, then A^{!} has r = \binom{9}{2} − t = 36 − t quadratic relations, so A^{!}(z) ⩾ 1/(1 − 8z + rz^{2}) and hence η_{A}(z) ⩾ (Log(1/(1 − 8z + rz^{2})))_{+}. To see that there is equality generically when t < 20, it is sufficient to produce one example. Let \overline{A} = A_{X}/y, where A_{X} is the coordinate ring of s points in P^{8}, and y is a linear nonzerodivisor in A_{X}. Then A_{X} and A have \binom{10}{2}−s = 45−s quadratic relations. Thus we can use our calculations for 26 or more points in P^{8}. For 26 to 30 points we verify the conjecture directly from the table in Section 11. Now s ⩾ 30 is equivalent to t ⩽ 15 and to r ⩾ 21. If n = 8 we get (n^{2} −1)/3 = 21, so (Log(1/(1−nz +rz^{2}))_{+} = nz +(\binom{n+1}{2}−r)z^{2}, cf. the proof of a). But from the table we get that (ηĀ)_{3} = 0 for s = 30 and hence this is true also for s > 30. Hence ηĀ = nz+(\binom{n+1}{2}−r)z^{2} = (Log(1/(1−nz+rz^{2}))_{+} and we have proved the conjecture for all t when n = 8. If n < 8 the proof is similar.

c) We shall prove the statement for t ⩽ \binom{n+1}{2} − (n^{2} − 1)/3 = (n^{2} + 3n + 2)/6 relations. As before it suffices to find one example. We get such an example from s points in P^{n} if \binom{n+2}{2} − s ⩽ (n^{2} + 3n + 2)/6; i.e., if s ⩾ (n^{2} + 3n + 2)/3. We refer to Theorem 8.2 a). Observe that r = s − n − 1.

d) We shall prove the statement for t ⩽ \binom{n+1}{2} − (n^{2} − 1/2 − \frac{1}{2} \sqrt{2n^{4} − 2n^{2} + 1}) = (n+1−n^{2})/2+ \frac{1}{2} \sqrt{2n^{4} − 2n^{2} + 1} relations. As before it suffices to find one example. We get such an example from s points in P^{n} if \binom{n+2}{2} − s ⩽ (n + 1 − n^{2})/2 + \sqrt{2n^{4} − 2n^{2} + 1}; i.e., if s ⩾ n^{2} + n + \frac{1}{2} + \frac{1}{2} \sqrt{2n^{4} − 2n^{2} + 1}. We refer to Theorem 8.2 b).

e) As in d) it suffices to consider s ⩾ \frac{n^{2}}{2} + n + 1 − \frac{1}{10} \sqrt{5n^{4} + 20} points in P^{n}. We refer to Theorem 8.2 c).

f) Here we can refer to 8.2 d); i.e., 32 points in P^{9}.

We can also give some partial results.

Theorem 7.8. If A is as in Conjecture 7.6, we have

a) η_{A}(z) ≡ (Log(1/(1 − nz + rz^{2})))_{+} (mod z^{6}) if n ⩽ 10.

b) η_{A}(z) ≡ (Log(1/(1 − nz + rz^{2})))_{+} (mod z^{5}) if n ⩽ 17.

c) η_{A}(z) ≡ (Log(1/(1 − nz + rz^{2})))_{+} (mod z^{4}) if n ⩽ 20.

d) η_{A}(z) ≡ (Log(1/(1 − nz + rz^{2})))_{+} (mod z^{3}).

Proof. We use the same method as in the preceeding proof.

We end this section by reformulating the conjectures above in terms of Hopf algebras.

Conjecture 7.9. Let B = k⟨X_{1}, …, X_{n}⟩/J, where the variables X_{i} are odd and where J is generated by quadratic generic Hopf elements. Then either B is a noncommutative c.i. or η is nilpotent, where B = U(η).

Conjecture 7.10. Let B = k⟨X_{1}, …, X_{n}⟩/J, where the variables X_{i} are odd and where J is generated by r quadratic generic Hopf elements. Then η(z) = (Log(1/(1− nz + rz^{2})))_{+}, where B = U(η).

Using Proposition 7.3 it is easy to see that these new conjectures are equivalent to their commutative versions above. In fact, it is obvious that the old conjectures imply the new ones, since if B is a quadratic Hopf algebra which is a non-commutative c.i., or its underlying Lie algebra is nilpotent, or it has the minimal possible Hilbert series, then the same is true for the generic case.

Suppose that Conjecture 7.9 is true and consider quadratic commutative algebras A defined by t relations. Since A^{!} is a non-commutative c.i. if and only if A is Koszul and A_{3} = 0, it follows from the proof of Theorem 7.1, that the set on which A^{!} is a non-commutative c.i. is an intersection of countably many open sets in Grass(\binom{n+1}{2}, t). Also (using Proposition 7.3), the set on which η_{A} is nilpotent is an open set in Grass(\binom{n+1}{2}, t). One of these sets must be non-empty, since if B is the generic Hopf algebra defined by \binom{n+1}{2} − t quadratic Hopf elements, then B^{!} is a member of one of the sets by Conjecture 7.9. But then also the quadratic commutative algebra defined by t generic relations is a member of the corresponding set.

Suppose now that Conjecture 7.10 is true. Then for each r, there is a quadratic commutative algebra A defined by \binom{n+1}{2}−r relations, such that η_{A}(z) is the correct series. But since this is the minimal possible series, it follows from Proposition 7.3 that the set of A in Grass(\binom{n+1}{2}, \binom{n+1}{2}−r) such that η_{A}(z) has the right value is an intersection of open sets. Since this set is non-empty it contains the generic algebra A.

8. Generic points again

Now let us consider the coordinate ring A of s generic points in P_{k}^{n}. Let \overline{A} = A/y where y is a linear nonzerodivisor. If n + 1 ⩽ s < \binom{n+2}{2} then \overline{A}(z) = 1 + nz + (s − n − 1)z^{2}. This is the series you get if you divide a polynomial ring in n variables with \binom{n+2}{2} − s generic quadratic forms, where s ⩽ (n + 2)(n + 1)/3 (see the proof of

Theorem 7.1). Let A^{!} = U(η_{A}) and \overline{A}^{!} = U(ηĀ). It is natural to conjecture that for any n + 1 ⩽ s < \binom{n+2}{2}, ηĀ behaves as the free Lie algebra on n generators divided by s − n − 1 generic quadratic Lie elements. Hence by Conjecture 7.10, we should have that ηĀ(z) = (Log(1/(1−nz +(s−n−1)z^{2})))_{+}. But since A(z) = (1+z)\overline{A}^{!}(z) we should have η_{A}(z) = (Log((1+z)/(1−nz +(s−n−1)z^{2})))_{+}. Hence it is natural to make the following conjecture.

Conjecture 8.1. Let A be the coordinate ring for s generic points in P_{k}^{n}, where n + 1 ⩽ s < \binom{n+2}{2}. Let A^{!} = U(η_{A}) and L = η_{A}^{⩾2}. Then

η_{A}(z) = (n + 1)z + L(z) = (Log((1 + z)/(1 − nz + (s − n − 1)z^{2})))_{+}.

In particular dim_{k} L^{3} = max((n^{3} − n)/3 − (s − n − 1)n, 0). dim_{k} L^{4} = max((n^{4} − n^{2})/4 − (s − n − 1)n^{2} + \binom{s−n}{2}, 0). dim_{k} L^{5} = max(n^{5}/5 + n^{4} + 2n^{3} + 2n^{2} + 4n/5 − s(n^{3} + 2n^{2} + 2n) + ns^{2}, 0). Hence we conjecture that L^{3} = 0 if and only if s ⩾ l_{3}. L^{4} = 0 if and only if s ⩾ l_{4}. L^{5} = 0 if and only if s ⩾ l_{5}. (For the definition of l_{3}, l_{4} and l_{5}, see Definition 6.3.)

Theorem 8.2. Conjecture 8.1 is true in the following cases:

a) If l_{3} ⩽ s < \binom{n+2}{2} and n ⩽ 20. b) If l_{4} ⩽ s < \binom{n+2}{2} and n ⩽ 17.

c) If l_{5} ⩽ s < \binom{n+2}{2} and n ⩽ 10.

d) If s ⩽ 1 + n + n^{2}/4.

e) If n ⩽ 8.

f) If (n, s) = (9, 32).

Remark. The LHS in b) is approximately 0.293n^{2} and the LHS in c) is approximately 0.276n^{2} for large n.

To prove the theorem we need a lemma.

Lemma 8.3. If L^{i} = 0 for some set of s points in P^{n}, then L^{i} = 0 for t generic points if t ⩾ s.

Proof. If s squares generate L_{i}, then also t generic squares do, if t ⩾ s.

We need a variation of the concept of strongly free sets. We define a set of quadratic elements {y_{1}, …, y_{t}} in k⟨X_{1}, …, X_{n}⟩ to be d-strongly free if

k⟨X_{1}, …, X_{n}⟩/(y_{1}, …, y_{t})(z) ≡ 1/(1 − nz + tz^{2}) (mod z^{d+1}) .

Lemma 8.4. Any subset of a d-strongly free set is d-strongly free.

Proof. This follows from the corresponding proof in [An 82].

Proof of Theorem 8.2. We have shown that L^{3} = 0 for s = l_{3} random points if n ⩽ 20 (see the table in Section 11). Then Lemma 8.3 gives a). For t = ⌊(n+1)(n+2)/3⌋ random points we have shown that dim_{k} L^{3} = −nt + 2\binom{n+2}{3} if n ⩽ 20 (see the table). This gives that the t random squares generating I_{X}^{⊥} is a 3-strongly free set. Then of course t truly generic squares constitute a 3-strongly free set, and Lemma 8.4 shows that s ⩽ t generic squares are 3-strongly free, hence η(z) ≡ (Log((1 + z)/(1 − nz + (s − n − 1)z^{2})))_{+} (mod z^{4}) if n ⩽ 20. We have also shown that L^{4} = 0 if s = ⌈l_{4}⌉ if n ⩽ 17. Hence L^{4} = 0 if s ⩾ l_{4} and n ⩽ 17 by Lemma 8.3. Then L(z) ≡ (Log((1 + z)/(1 − nz + (s − n − 1)z^{2})))_{+} (mod z^{5}) if n ⩽ 17 and b) is proved. c) is proved similarly. In [Co-Tr-Va 97] it is proved that A_{X} is Koszul if s ⩽ 1 + n + n^{2}/4. This gives that (A_{X})^{!}(z) = 1/(1 − nz + (s − n − 1)z^{2}), so d) follows. For e) and f) see the table.

9. Poincaré series of generic points

In this section we study the Poincaré series of the coordinate ring A_{X}; that is, P_{A_{X}} (x, y) = ∑ dim_{k} Tor_{i,j}^{A_{X}} (k, k)x^{i}y^{j}. Here, we allow any s, not only s < \binom{n+2}{2}. In fact, the case s ⩾ \binom{n+2}{2} will be the easy one.

Conjecture 9.1. The Poincaré series P_{A_{X}} (x, y) is rational, where A_{X} is the coordinate ring of a set X of s generic points in P_{k}^{n}.

Theorem 9.2. Conjecture 9.1 is true in the following cases.

a) If s ⩾ \binom{n+2}{2}.

b) If Conjecture 8.1 is true.

Proof. In case a) we may argue as in the proof of Theorem 5.6 to conclude that P_{A_{X}} (x, y) = (1 − x)^{n+1}/(1 − x(H(x, y) − 1)), where H(x, y) is the series of the homology of the Koszul complex (A_{X} is Golod).

In case b) we use the fact that, if A = A_{X}/f where f is some linear nonzerodivisor , then P_{A}(x, y) is rational if and only if U(L)(x, y) is rational, where L is the Lie algebra associated to A, [Lö 86]. If Conjecture 8.1 is true, then either L is nilpotent (and hence U(L)(z) is rational) or U(L)(z) = \frac{1}{(1+z)^{n}(1−nz+(s−n−1)z^{2})}

10. Generic forms in the exterior algebra

If f_{1}, …, f_{t} are generic quadratic forms in k[x_{1}, …, x_{n}], k a field of characteristic 0, there is a well-known conjecture for the Hilbert series of A_{n,t} = k[x_{1}, …, x_{n}]/(f_{1}, …, f_{t}), namely that A_{n,t}(z) = ((1 − z^{2})^{t}/(1 − z)^{n})_{+}. This conjecture is proved in many cases, and no counterexamples are known. When calculating examples of ideals generated by quadratic forms with the expected generic Hilbert series, experience seems to indicate that one does not have to choose all forms at random. In fact, it seems that one can almost always choose the first n forms to be the squares of the variables. This made us look at the problem of Hilbert series of generic forms in the exterior algebra. It turns out that the answer to the question in the exterior algebra, which is analogous to the Hochster-Laksov result for polynomial rings, is not what we expected. Although dim_{k} ∧^{3}(V) = 10 if V is 5-dimensional, two generic forms of degree 2 do not generate ∧^{3}(V).

Theorem 10.1. Consider the exterior algebra E over a vector space with basis {e_{1}, …, e_{5}} (over any field). Let f_{1} = c_{1}e_{1}∧e_{2}+c_{2}e_{1}∧e_{3}+c_{3}e_{1}∧e_{4}+c_{4}e_{1}∧e_{5}+c_{5}e_{2}∧ e_{3}+c_{6}e_{2}∧e_{4}+c_{7}e_{2}∧e_{5}+c_{8}e_{3}∧e_{4}+c_{9}e_{3}∧e_{5}+c_{10}e_{4}∧e_{5} and f_{2} = c_{11}e_{1}∧e_{2}+c_{12}e_{1}∧e_{1}+ c_{13}e_{1}∧e_{4}+c_{14}e_{1}∧e_{5}+c_{15}e_{2}∧e_{3}+c_{16}e_{2}∧e_{4}+c_{17}e_{2}∧e_{5}+c_{18}e_{3}∧e_{4}+c_{19}e_{3}∧e_{5}+c_{20}e_{4}∧e_{5} be two forms in E. Then {e_{1}∧f_{1}, …, e_{5}∧f_{1}, e_{1}∧f_{2}, …, e_{5}∧f_{2}} is linearly dependent.

Proof. It suffices to prove the theorem for generic forms (so we can suppose that the c_{i}’s are algebraically independent over the prime field of k). In fact,

(−c_{3}c_{1}c_{12}c_{20}−c_{3}c_{1}c_{18}c_{14}+c_{3}c_{1}c_{19}c_{13}+c_{16}c_{3}c_{2}c_{14}−c_{16}c_{12}c_{4}+c_{3}c_{2}c_{11}c_{20}+c_{12}c_{3}c_{17}+ c_{18}c_{4}c_{11} − c_{19}c_{3}c_{11} + c_{13}c_{4}c_{15} − c_{13}c_{3}c_{2}c_{17} − c_{3}c_{15}c_{14},

c_{3}c_{1}c_{16}c_{19} − c_{3}c_{1}c_{15}c_{20} − c_{3}c_{1}c_{18}c_{17} − c_{16}c_{7}c_{12} + c_{16}c_{5}c_{14} + c_{5}c_{11}c_{20} + c_{6}c_{17}c_{12} + c_{18}c_{7}c_{11} + c_{13}c_{15}c_{7} − c_{5}c_{13}c_{17} − c_{15}c_{6}c_{14} − c_{6}c_{11}c_{19},

−c_{9}c_{12}c_{16} + c_{3}c_{2}c_{19}c_{16} + c_{9}c_{11}c_{18} + c_{9}c_{13}c_{15} + c_{20}c_{12}c_{5} − c_{20}c_{3}c_{2}c_{15} + c_{12}c_{8}c_{17} − c_{11}c_{8}c_{19} − c_{3}c_{2}c_{18}c_{17} + c_{18}c_{5}c_{14} − c_{5}c_{19}c_{13} − c_{8}c_{14}c_{15},

−c_{16}c_{12}c_{10} + c_{16}c_{3}c_{19} − c_{8}c_{14}c_{16} + c_{12}c_{6}c_{20} + c_{6}c_{14}c_{18} + c_{8}c_{17}c_{13} − c_{3}c_{20}c_{15} + c_{18}c_{11}c_{10} − c_{8}c_{11}c_{20} − c_{13}c_{6}c_{19} + c_{13}c_{15}c_{10} − c_{3}c_{18}c_{17},

−c_{16}c_{14}c_{9} + c_{16}c_{4}c_{19} − c_{9}c_{11}c_{20} + c_{9}c_{17}c_{13} + c_{20}c_{7}c_{12} − c_{20}c_{4}c_{15} − c_{10}c_{17}c_{12} + c_{11}c_{10}c_{19} − c_{7}c_{19}c_{13} − c_{4}c_{18}c_{17} + c_{7}c_{18}c_{14} + c_{14}c_{15}c_{10},

−c_{3}c_{1}c_{13}c_{9} + c_{3}c_{1}c_{12}c_{10} + c_{3}c_{1}c_{8}c_{14} + c_{3}c_{9}c_{11} + c_{6}c_{4}c_{12} − c_{12}c_{3}c_{7} − c_{3}c_{2}c_{10}c_{11} − c_{8}c_{11}c_{4} − c_{5}c_{13}c_{4} + c_{13}c_{3}c_{2}c_{7} − c_{6}c_{3}c_{2}c_{14} + c_{3}c_{5}c_{14},

−c_{3}c_{1}c_{9}c_{16} + c_{3}c_{1}c_{15}c_{10} + c_{3}c_{1}c_{8}c_{17} − c_{16}c_{5}c_{4} + c_{3}c_{2}c_{7}c_{16} + c_{9}c_{6}c_{11} − c_{5}c_{11}c_{10} + c_{6}c_{15}c_{4} − c_{8}c_{7}c_{11} − c_{3}c_{2}c_{6}c_{17} − c_{15}c_{7}c_{3} + c_{3}c_{5}c_{17},

−c_{3}c_{1}c_{18}c_{9} +c_{3}c_{1}c_{8}c_{19} +c_{9}c_{12}c_{6} −c_{3}c_{9}c_{15} −c_{12}c_{10}c_{5} −c_{8}c_{12}c_{7} −c_{5}c_{18}c_{4} +c_{8}c_{15}c_{4} + c_{3}c_{2}c_{18}c_{7} − c_{6}c_{3}c_{2}c_{19} + c_{5}c_{3}c_{19} + c_{3}c_{2}c_{10}c_{15},

c_{3}c_{1}c_{20}c_{8}−c_{3}c_{1}c_{18}c_{10}−c_{3}c_{9}c_{16}+c_{16}c_{8}c_{4}+c_{3}c_{2}c_{10}c_{16}+c_{13}c_{9}c_{6}+c_{5}c_{20}c_{3}−c_{3}c_{2}c_{6}c_{20}− c_{18}c_{6}c_{4} − c_{13}c_{5}c_{10} − c_{8}c_{7}c_{13} + c_{18}c_{7}c_{3},

c_{3}c_{1}c_{9}c_{20}−c_{3}c_{1}c_{10}c_{19}−c_{9}c_{3}c_{17}+c_{9}c_{14}c_{6}−c_{7}c_{3}c_{2}c_{20}+c_{5}c_{4}c_{20}+c_{4}c_{8}c_{17}−c_{5}c_{10}c_{14}− c_{14}c_{8}c_{7} + c_{3}c_{2}c_{17}c_{10} + c_{7}c_{3}c_{19} − c_{4}c_{6}c_{19})

is a relation.

Notice that this also gives a proof of the fact that the ideal (x_{1}^{2}, …, x_{5}^{2}) in S = k[x_{1}, …, x_{5}] can not be extended in any way with two quadratic forms to an ideal I with the expected generic Hilbert series S/I(z) = 1+5z+8z^{2}, if char(k) = 2. It would be natural to make a conjecture about the Hilbert series of the exterior algebra modulo generic forms, similar to the one in the commutative case. The expected series for the exterior algebra in n variables modulo t generic quadratic forms would be A(z) = ((1 + z)^{n}(1 − z^{2})^{t})_{+}. However, the example above shows that this is not true and there are more “probable” counterexamples given in [Gu-Sn 00], where the Hilbert series of exterior algebras modulo generic forms are discussed.

If A is an exterior algebra modulo an ideal generated by quadratic forms, then A^{!} = U(η_{A}), where η_{A} is an ordinary Lie algebra; i.e., all generators are even. If the defining ideal of A is generated by \binom{n}{2} − r quadratic forms, then η_{A} is a free Lie algebra on n generators modulo an ideal generated by r quadratic elements. Recall, that if U(η_{A}) = A^{!}, then η_{A}(z) = log(A^{!}(z)), where log(A^{!}(z)) = ∑_{i⩾1} e_{i}z^{i} if A^{!}(z) = ∏_{i=1}^{∞} 1/((1 − z^{i})^{e_{i}}.

One could ask when a quotient of an exterior algebra on n generators modulo t generic quadratic forms is Koszul. A natural guess, based on the results in Section 7, is that this is true if and only if t ⩾ \binom{n}{2} − n^{2}/4 or t = 0. We can prove this in one direction in the same way as in the proof of Theorem 7.1. The other direction is harder, since the example above shows that the Hochster-Laksov result cannot directly be translated to the exterior algebra case.

Theorem 10.2. Consider an exterior algebra A on n generators modulo t linearly independent quadratic forms as a point in Grass(\binom{n}{2}, t). If t ⩾ \binom{n}{2} − n^{2}/4 there is a non-empty countable intersection of open sets in Grass(\binom{n}{2}, t) on which A is Koszul. Moreover, A is not Koszul if t > 0 and η_{A} is nilpotent, where A^{!} = U(η_{A}).

Proof. The same argument as in the proof of Theorem 7.1 shows that the set of A in Grass(\binom{n}{2}, t), such that A_{3} = 0 and A is Koszul, is a countable intersection of open sets. We may use the same example as before to show that this set is non-empty if t ⩾ \binom{n}{2} − n^{2}/4. Indeed, if a = ⌈n/2⌉ and b = ⌊n/2⌋, a simple computation shows that \binom{a}{2} + \binom{b}{2} = ⌈\binom{n}{2} − n^{2}/4⌉. If η_{A} is nilpotent and (η_{A})_{2} ≠ 0, then 1/A^{!}(z) is a polynomial of degree > n (see above), which is impossible if A is Koszul, since A(z) is a polynomial of degree ⩽ n.

In analogy to the case where the variables are even, it is natural to make the following conjecture.

Conjecture 10.3. If A is a generic quadratic quotient of a polynomial algebra, which is commutative in the graded sense, then A^{!} is either a non-commutative c.i. or η_{A} is nilpotent.

The conjecture is false for 3 generic forms in 5 odd variables, see below. We also make the conjecture more precise in the odd case.

Conjecture 10.4. Let A be an exterior algebra on n variables, modulo \binom{n}{2} − r generic quadratic forms. Then η_{A}(z) = (log(1/(1 − nz + rz^{2})))_{+}.

If (n, r) = (5, 7) or (5, 8), the conjecture is false, see below. We believe, however, that these are the only counterexamples.

Theorem 10.5. Conjecture 10.4 is true for n ⩽ 8 except for (n, r) = (5, 7), (5, 8) when it is false. It is true also for n = 9, r ≠ 21 and in general for r ⩽ n^{2}/4.

Proof. The case r ⩽ n^{2}/4 follows from Theorem 10.2, since then A^{!}(z) = 1/(1 − nz + rz^{2}). For the other cases, it is only necessary to provide an example with the correct series in each case. We refer to the last table in Section 11. Observe that the conjecture is trivially true when (η_{A})^{3} = 0.

As in Section 7 we may reformulate the two conjectures above in terms of Hopf algebras. Consider the free associative algebra k⟨X_{1}, …, X_{n}⟩, where the variables have some parities |X_{i}| ∈ ℤ_{2}. We will call an element f = ∑ c_{ij}[X_{i}, X_{j}] an Hopf element, where [X_{i}, X_{j}] = X_{i}X_{j} − (−1)^{|X_{i}||X_{j}|}X_{j}X_{i}.

Conjecture 10.6. If B = k⟨X_{1}, …, X_{n}⟩/J, where J is generated by r generic Hopf elements, then B is a non-commutative c.i. or η is nilpotent, where B = U(η).

Conjecture 10.7. If B = k⟨X_{1}, …, X_{n}⟩/J, where the variables are even and where J is generated by r generic Hopf elements, then η(z) = (log(1/(1 − nz + rz^{2})))_{+}, where B = U(η).

Finally we discuss the counterexamples to Conjecture 10.4. First we consider (n, r) = (5, 7); i.e., 3 generic quadratic forms in 5 variables. It is shown in [Ei-Ko 94], that the forms x_{1}x_{4} + x_{2}x_{3}, x_{1}x_{5} + x_{2}x_{4}, x_{2}x_{5} + x_{3}x_{4} are generic. The Lie algebra η for this case can be determined in the same way as in [Lö-Ro 97]. The result is that η(z) is periodic, namely η(z) = (5 + 3z) ∑_{i=0}^{∞} z^{2i+1}, while the expected series is 5z + 3z^{2} + 5z^{3}. In fact, η is the periodization of sl(3) equipped with a ℤ_{2}-grading. Secondly, if (n, r) = (5, 8); i.e., 2 generic forms in 5 variables, the expected series is 5z + 2z^{2}. Corollary 4.3 iii) together with Theorem 10.1 shows that L^{3} = η^{3} is 1-dimensional, so we really have a counterexample. We also have L^{4} = 0, so in this case the Lie algebra is nilpotent but its series is not the expected minimal one.

11. Tables

We give the results of our calculations of η_{A}(z) for the coordinate ring of s random points in P^{n}. For the definition of m and t, see Definition 5.3. For the definition of l_{3}, l_{4}, l_{5}, see Definition 6.3.

n s m t η_{A}(z)  l_{3} l_{4} l_{5}
3 7 1 3 4z + 3z^{2}  7 7 7
3 6 2 2 4z + 4z^{2} + 2z^{3} + 3z^{4} + 6z^{5} + ⋯
4 10 1 5 5z + 5z^{2}  10 10 10
5 14 1 7 6z + 7z^{2}  14 14 13
5 13 2 8 6z + 8z^{2} + 5z^{3} + 3z^{4}
6 19 1 9 7z + 9z^{2}  18 18 17
6 18 2 4 7z + 10z^{2} + 4z^{3}
6 17 2 10 7z + 11z^{2} + 10z^{3} + 10z^{4}
7 24 1 12 8z + 12z^{2}  24 23 22
7 23 2 7 8z + 13z^{2} + 7z^{3}
7 22 2 14 8z + 14z^{2} + 14z^{3} + 7z^{4}
7 21 2 21 8z + 15z^{2} + 21z^{3} + 42z^{4} + 84z^{5} + 91z^{6}
8 30 1 15 9z + 15z^{2}  30 28 27
8 29 2 8 9z + 16z^{2} + 8z^{3}
8 28 2 16 9z + 17z^{2} + 16z^{3}
8 27 2 24 9z + 18z^{2} + 24z^{3} + 27z^{4}
8 26 2 32 9z + 19z^{2} + 32z^{3} + 73z^{4} + 160z^{5} + 176z^{6}
9 37 1 18 10z + 18z^{2}  37 34 33
9 36 2 6 10z + 19z^{2} + 6z^{3}
9 34 2 24 10z + 21z^{2} + 24z^{3}
9 33 2 33 10z + 22z^{2} + 33z^{3} + 33z^{4}
9 32 2 42 10z + 23z^{2} + 42z^{3} + 91z^{4} + 126z^{5}
10 44 1 22 11z + 22z^{2}  44 41 39
10 43 2 10 11z + 23z^{2} + 10z^{3}
10 41 2 30 11z + 25z^{2} + 30z^{3}
10 40 2 40 11z + 26z^{2} + 40z^{3} + 10z^{4}
10 39 2 50 11z + 27z^{2} + 50z^{3} + 81z^{4}
10 38 2 60 11z + 28z^{2} + 60z^{3} + 153z^{4} + 288z^{5} + ⋯
11 52 1 26 12z + 26z^{2}  52 48 46
11 48 2 44 12z + 30z^{2} + 44z^{3}
11 47 2 55 12z + 31z^{2} + 55z^{3} + 25z^{4}
11 46 2 66 12z + 32z^{2} + 66z^{3} + 111z^{4}
11 45 2 77 12z + 33z^{2} + 77z^{3} + 198z^{4} + ⋯
12 61 1 30 13z + 30z^{2}  61 56 53
12 60 2 8 13z + 31z^{2} + 8z^{3}
12 56 2 56 13z + 35z^{2} + 56z^{3}
12 55 2 68 13z + 36z^{2} + 68z^{3} + 3z^{4}
12 54 2 80 13z + 37z^{3} + 80z^{3} + 105z^{4}
12 53 2 92 13z + 38z^{2} + 92z^{3} + 208z^{4} + ⋯

n  s m  t η_{A}(z)  l_{3}  l_{4}  l_{5}
13 70 1 35 14z + 35z^{2}  70 64 61
13 64 2 78 14z + 41z^{2} + 78z^{3}
13 63 2 91 14z + 42z^{2} + 91z^{3} + 42z^{4}
13 62 2 104 14z + 43z^{2} + 104z^{3} + 162z^{4}
13 61 2 117 14z + 44z^{2} + 117z^{3} + 283z^{4} + ⋯
14 80 1 40 15z + 40z^{2}  80 73 70
14 73 2 98 15z + 47z^{2} + 98z^{3}
14 72 2 112 15z + 48z^{2} + 112z^{3} + 36z^{4}
14 71 2 126 15z + 49z^{2} + 126z^{3} + 175z^{4} + ⋯
15 91 1 45 16z + 45z^{2}  91 82 79
15 90 2 10 16z + 46z^{2} + 10z^{3}
15 82 2 130 16z + 54z^{2} + 130z^{3}
16 102 1 51 17z + 51z^{2}  102 92 88
16 92 2 160 17z + 61z^{2} + 160z^{3}
17 114 1 57 18z + 57z^{2}  114 103 98
17 103 2 187 18z + 68z^{2} + 187z^{3}
18 127 1 63 19z + 63z^{2}  127 114 109
18 126 2 12 19z + 64z^{2} + 12z^{3} + ⋯
19 140 1 70 20z + 70z^{2}  140 126 120
20 154 1 77 21z + 77z^{2}  154 139 132

Here are the results of our calculations of η(z) for r random quadratic relations in an ordinary Lie algebra on n generators.

\begin{matrix}n & r & η(z) \\ 4 & 5 & 4z + z^{2} \\ 5 & 7 & 5z + 3z^{2} + 5z^{3} + 3z^{4} + 5z^{5} + 3z^{6} ⋯ \\ 5 & 8 & 5z + 2z^{2} + z^{3} \\ 5 & 9 & 5z + z^{2} \\ 6 & 10 & 6z + 5z^{2} + 10z^{3} + 10z^{4} \\ 6 & 11 & 6z + 4z^{2} + 4z^{3} \\ 6 & 12 & 6z + 3z^{2} \\ 7 & 13 & 7z + 8z^{2} + 21z^{3} + 42z^{4} + 84z^{5} + 70z^{6} \\ 7 & 14 & 7z + 7z^{2} + 14z^{3} + 7z^{4} \\ 7 & 15 & 7z + 6z^{2} + 7z^{3} \\ 7 & 16 & 7z + 5z^{2} \\ 8 & 17 & 8z + 11z^{2} + 32z^{3} + 73z^{4} + 160z^{5} + 144z^{6} \\ 8 & 18 & 8z + 10z^{2} + 24z^{3} + 27z^{4} \\ 8 & 19 & 8z + 9z^{2} + 16z^{3} \\ 8 & 20 & 8z + 8z^{2} + 8z^{3} \\ 8 & 21 & 8z + 7z^{2} \end{matrix}

\begin{matrix}n & r & η(z) \\ 9 & 22 & 9z + 14z^{2} + 42z^{3} + 91z^{4} + 126z^{5} \\ 9 & 23 & 9z + 13z^{2} + 33z^{3} \\ 9 & 24 & 9z + 12z^{2} + 24z^{3} \\ 9 & 25 & 9z + 11z^{2} + 15z^{3} \\ 9 & 26 & 9z + 10z^{2} + 6z^{3} \\ 9 & 27 & 9z + 9z^{2} \end{matrix}

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R. Fröberg ralff@matematik.su.se

Department of Mathematics, Stockholm University, S-106 91 Stockholm, Sweden

C. Löfwal clas@matematik.su.se

Department of Mathematics, Stockholm University, Sweden