THE MODULI SPACE OF COMMUTATIVE ALGEBRAS OF FINITE RANK

BJORN POONEN

Abstract. The moduli space of rank-n commutative algebras equipped with an ordered basis is an affine scheme 𝔅_{n} of finite type over ℤ, with geometrically connected fibers. It is smooth if and only if n ≤ 3. It is reducible if n ≥ 8 (and the converse holds, at least if we remove the fibers above 2 and 3). The relative dimension of 𝔅_{n} is \frac{2}{27}n^{3} + O(n^{8/3}). The subscheme parameterizing étale algebras is isomorphic to GL_{n} /S_{n}, which is of dimension only n^{2}. For n ≥ 8, there exist algebras that are not limits of étale algebras. The dimension calculations lead also to new asymptotic formulas for the number of commutative rings of order p^{n} and the dimension of the Hilbert scheme of n points in d-space for d ≥ n/2.

1. The moduli space of based algebras

All rings and algebras are assumed to be commutative, associative, and with 1, unless otherwise specified.

Let n ∈ ℤ_{≥0}. For every ring k, we would like to parameterize the k-algebras that are locally free of rank n as a k-module. But for n ≥ 2 this moduli problem turns out to be not representable by a scheme; the difficulty is related to presence of automorphisms. To rigidify, we equip k-algebras with extra structure. One option would be to consider k-algebras equipped with an ordered d-tuple of generators as a k-algebra: these are parameterized by the Hilbert scheme Hilb^{n}(𝔸^{d}) of n points in 𝔸^{d}. Another option, the one explored in this paper, is to consider k-algebras equipped with an ordered basis.

Define a functor 𝔅_{n}: Schemes^{op} → Sets as follows:

(1) If S is a scheme, an element of 𝔅_{n}(S) is a pair (A, ϕ) (strictly speaking, an isomorphism class of pairs) where A is an O_{S}-algebra and ϕ: A → O_{S}^{⊕n} is an O_{S}-module isomorphism. Two pairs (A, ϕ) and (A′, ϕ′) are considered the same if there is an isomorphism A → A′ that with ϕ and ϕ′ makes a commuting triangle.

(2) If f: T → S is a morphism of schemes, then 𝔅_{n}(f): 𝔅_{n}(S) → 𝔅_{n}(T) maps (A, ϕ) to (f^{∗}A, f^{∗}ϕ).

Proposition 1.1. The functor 𝔅_{n} is representable by an affine scheme of finite type over ℤ.

Proof. Let e_{1}, …, e_{n} be the standard O_{S}-basis of O_{S}^{⊕n}. Given (A, ϕ) ∈ 𝔅_{n}(S), we may use ϕ to transport the multiplication on A to a multiplication on O_{S}^{⊕n}, which is described by giving the n^{3} constants c_{ij}^{ℓ} ∈ H^{0}(S, O_{S}) such that e_{i}e_{j} = ∑_{ℓ=1}^{n} c_{ij}^{ℓ}e_{ℓ}; we also get ϕ(1) = ∑ d_{i}e_{i} for some d_{i} ∈ H^{0}(S, O_{S}). Conversely, given elements c_{ij}^{ℓ}, d_{i} ∈ H^{0}(S, O_{S}) satisfying the polynomial conditions that say that the resulting multiplication law is commutative and associative and has ∑ d_{i}e_{i} as a multiplicative identity, we can recover the algebra uniquely. Thus 𝔅_{n} is representable by the closed subscheme of 𝔸_{ℤ}^{n^{3}+n} cut out by these polynomial conditions on indeterminates c_{ij}^{ℓ} and d_{i}. □

Date: June 4, 2008.

2000 Mathematics Subject Classification. Primary 14D20; Secondary 13E10, 13M99, 14C05.

Key words and phrases. Hilbert scheme of points, finite-dimensional algebras, finite rings.

From now on, 𝔅_{n} denotes the representing scheme. Call 𝔅_{n} the moduli space of based rank-n algebras.

Remark 1.2. Various analogues of 𝔅_{n} have been studied in the literature before, at least over a base field. The paper [Fla68] defines the moduli space for based associative (but not necessarily commutative or unital) algebras, defines a GL_{n}-action as we do below, and studies GL_{n}-orbits in its irreducible components. The moduli space of based associative algebras with 1, along with the GL_{n}-action, is studied in [Gab74,Maz79,Maz82,LBR99]. In particular, [Maz79] contains a detailed study of its number of irreducible components and computes the number explicitly for n = 5 (it is 10). The paper [Maz80] studies the moduli space of based nilpotent commutative associative rank-n algebras over an algebraically closed field of characteristic not 2 or 3, especially for n ≤ 6; it follows from this work that 𝔅_{n} has geometrically irreducible fibers, at least over ℤ[1/6], for all n ≤ 7. For an extensive survey of related work, see [Iar87]: there on page 311 it is written that “There is a virtually complete picture of isomorphism classes of Artin algebras and their deformations for n ≤ 8, but that picture has never been written down in one place”; and presumably it is speaking of the case k = ℂ, but even in that case it seems that the published literature does not contain this. Asymptotic formulas for the dimension of 𝔅_{n} and of the moduli space of based associative algebras with 1 are mentioned in [Ner87], whose primary goal was an asymptotic formula for the dimension of the moduli space of n-dimensional Lie algebras. Finally, the moduli space 𝔅_{n} is implicit in some of the work of Bhargava [Bha04a, Bha04b, Bha04c, Bha05] on parameterizing and enumerating algebras of rank ≤ 5, especially over ℤ.

Remark 1.3. Although we do not do it here, one should eventually try to define moduli spaces of algebras equipped with extra structure such as the resolvents in Bhargava’s work, and also consider their quotients by a GL_{n}-action described below. Some of Bhargava’s work may be interpreted as identifying such moduli spaces for n ≤ 5 as quotients of affine space by a group acting linearly with a dense orbit. For n > 5, the classification of prehomogeneous vector spaces rules out such a simple interpretation, but it may still be possible to identify the moduli spaces as quotients of other simple varieties by group actions. This might yield an approach to counting rank-6 rings over ℤ, for instance. That the situation for n = 6 should not be too bad is hinted at by [Poo07], which proves the finiteness of the set of isomorphism types of rank-6 algebras over any algebraically closed field.

On the other hand, the existence of “pathologies” such as reducibility of 𝔅_{n} for higher n suggests that one should be careful in extrapolating the behavior of orders of number fields from the behavior for low degree. For one arithmetic application, see Proposition 9.7.

Definition 1.4. Given S, define two points of 𝔅(S) as follows:

(1) Ã_{split} is A_{split} := O_{S} × ⋯ × O_{S} equipped with the obvious basis.

(2) • is A_{•} := \frac{O_{S}[x_{1},...,x_{n−1}]}{(x_{1},...,x_{n−1})^{2}} equipped with the basis 1, x_{1}, …, x_{n−1}.

For some calculations it will be simpler to work with based algebras where the first basis vector equals the multiplicative identity:

Definition 1.5. For n ≥ 1, let 𝔅_{n}^{1} be the scheme parameterizing based algebras such that the first basis vector equals 1. It is the closed subscheme of 𝔅_{n} cut out by the equations d_{1} = 1 and d_{i} = 0 for i ≥ 2. For n = 0, we use the convention 𝔅_{0}^{1} := 𝔅_{0}.

2. Action of the general linear group

Let GL_{n} be the general linear group scheme over ℤ. Each M ∈ GL_{n}(S) = Aut_{O_{S}}(O_{S}^{⊕n}) acts on 𝔅_{n}(S) by sending (A, ϕ) to (A, M ∘ ϕ). This defines a left action of GL_{n} on 𝔅_{n}.

Given two elements (A, ϕ) and (A′, ϕ′) of 𝔅_{n}(S), the set Isom(A, A′) of O_{S}-algebra isomorphisms α: A → A′ is in bijection with the set of M ∈ GL_{n}(S) mapping (A, ϕ) to (A′, ϕ′). Namely, α corresponds to the M ∈ GL_{n}(S) making

\xymatrix{A \ar[r]^{ϕ} \ar[d]_{α} & O_{S}^{⊕n} \ar[d]^{M} \\ A′ \ar[r]^{ϕ′} & O_{S}^{⊕n}}

commute.

It follows that the set of isomorphism classes of free rank-n O_{S}-algebras is in bijection with the quotient set GL_{n}(S)\𝔅_{n}(S). Also, given (A, ϕ) ∈ 𝔅_{n}(S), the group of O_{S}-algebra automorphisms Aut(A) is identified with the stabilizer in GL_{n}(S) of (A, ϕ) ∈ 𝔅_{n}(S).

3. Comparison of 𝔅_{n} and 𝔅_{n}^{1}

For n ≥ 1, let H be the subgroup scheme of GL_{n} stabilizing (1, 0, …, 0). It consists of invertible matrices whose first column equals (1, 0, …, 0). If n = 0, we use the convention that H, like GL_{n}, is the trivial group scheme. In any case, the relative dimension of H is n^{2} − n.

The left action of GL_{n} on 𝔅_{n} restricts to a left action of H on 𝔅_{n}^{1}. Also, there is a right action of H on GL_{n} given by multiplication on the right. Combining these gives a left Haction on GL_{n} ×𝔅_{n}^{1} in which N ∈ H(S) maps (M, Ã) ∈ (GL_{n} ×𝔅_{n}^{1})(S) to (MN^{−1}, N · Ã).

The contracted product GL_{n} ∧^{H}𝔅_{n}^{1} is the quotient of GL_{n} ×𝔅_{n}^{1} by this (free) H-action.

Proposition 3.1. The scheme 𝔅_{n} is isomorphic to GL_{n} ∧^{H}𝔅_{n}^{1}.

Proof. Restricting the action GL_{n} ×𝔅_{n} → 𝔅_{n} yields a map

GL_{n} ×𝔅_{n}^{1} → 𝔅_{n}.

which is invariant for the H-action on GL_{n} ×𝔅_{n}^{1}. The fiber above Ã ∈ 𝔅_{n}(S) in GL_{n} ×𝔅_{n}^{1} is a Zariski locally trivial torsor under H, trivialized by any open covering of S that splits the O_{S}-module injection O_{S} → A given by the algebra structure. □

Definition 3.2. Define the moduli stack of locally free rank-n algebras as the Artin stack

𝔄_{n} := [GL_{n} \𝔅_{n}] ≃ [H\𝔅_{n}^{1}].

(Here \ denotes “stack quotient by the group acting on the left”, not set difference!)

4. Comparison with Hilbert schemes

For any n, d ∈ ℤ_{≥0}, let Hilb_{n}(𝔸^{d}) be the Hilbert scheme of n points in 𝔸^{d}: it is the ℤ-scheme whose S-points correspond (functorially in S) to closed subschemes T of 𝔸_{S}^{d} that are flat over S and whose geometric fibers have Hilbert polynomial equal to the constant polynomial n. Let ℌ_{n}(𝔸^{d}) be the ℤ-scheme that parameterizes closed subschemes T as above equipped with an ordered O_{S}-basis for O_{T}. Then ℌ_{n}(𝔸^{d}) is a torsor under GL_{n} over Hilb_{n}(𝔸^{d}). There is a morphism ℌ_{n}(𝔸^{d}) → 𝔅_{n} that maps T ⊆ 𝔸^{d} with an O_{S}-basis to O_{T} with the O_{S}- basis (forgetting the embedding into 𝔸^{d}). This is GL_{n}-equivariant, so we obtain a cartesian diagram

\xymatrix{ℌ_{n}(𝔸^{d}) \ar[r] \ar[d] & 𝔅_{n} \ar[d] \\ Hilb_{n}(𝔸^{d}) \ar[r] & 𝔄_{n}.}

We may view ℌ_{n}(𝔸^{d}) as the moduli space parameterizing rank-n O_{S}-algebras O_{T} equipped with both a basis and a sequence (x_{1}, …, x_{d}) of global sections that generate O_{T} as an O_{S}- algebra (to give these global sections is to give a closed immersion T ↪ 𝔸_{S}^{d}). On the other hand, 𝔅_{n} × 𝔸^{nd} is the moduli space parameterizing rank-n O_{S}-algebras O_{T} equipped with both a basis and an arbitrary sequence (x_{1}, …, x_{d}) of global sections. (The nd coordinates on 𝔸^{nd} specify the coordinates of the x_{i} with respect to the basis.)

Proposition 4.1. The morphism ℌ_{n}(𝔸^{d}) → 𝔅_{n} × 𝔸^{nd} is an open immersion.

Proof. Over a field k, the vector space spanned by the monomials of total degree ≤ e in x_{1}, …, x_{d} increases strictly in dimension with e until it stabilizes, so it must stabilize at or before e = n − 1. Thus x_{1}, …, x_{d} generate the rank-n k-algebra A if and only if the monomials of total degree < n span A. This is a condition on the rank of a matrix whose entries are polynomials in the coordinates of the x_{i} with respect to a fixed basis of A; the condition is expressible as the nonvanishing of certain minors. Those minors, viewed as functions on 𝔅_{n} × 𝔸^{nd}, cut out the complement of ℌ_{n}(𝔸^{d}) in 𝔅_{n} × 𝔸^{nd}. □

Corollary 4.2. The image of ℌ_{n}(𝔸^{d}) → 𝔅_{n} is open in 𝔅_{n}.

Proposition 4.3. If d ≥ n, then ℌ_{n}(𝔸^{d}) → 𝔅_{n} admits a section. The map ℌ_{n}(𝔸^{d}) → 𝔅_{n} is surjective if and only if d ≥ n − 1.

Proof. If d ≥ n, define 𝔅_{n} → ℌ_{n}(𝔸^{d}) by mapping an O_{S}-algebra A with basis e_{1}, …, e_{n}, to the same O_{S}-algebra with the same basis and with algebra generators (x_{1}, …, x_{d}) := (e_{1}, …, e_{n}, 0, 0, …, 0); this is a section.

Given a based O_{S}-algebra, Zariski locally on S we may replace one of the basis elements with 1 and still have a basis, and then the other basis elements are algebra generators. This proves surjectivity for d ≥ n − 1.

We will not have surjectivity for d < n − 1, since A_{•} cannot be generated as an algebra by fewer than n − 1 elements. □

Let k be a field. For any ℤ-scheme V, let V_{k} = V ×_{Spec ℤ} Spec k. For instance, 𝔅_{n,k} denotes

the k-variety (not necessarily irreducible) obtained from 𝔅_{n}.

Corollary 4.4. We have

dim Hilb_{n}(𝔸^{d})_{k} ≤ dim 𝔅_{n,k} − n^{2} + nd.

If d ≥ n − 1, then equality holds.

Proof. By Proposition 4.1,

dim Hilb_{n}(𝔸^{d})_{k} = dim ℌ_{n}(𝔸^{d})_{k} − dim GL_{n,k} ≤ dim 𝔅_{n,k} + dim 𝔸_{k}^{nd} − dim GL_{n,k} .

By 4.3, equality holds if d ≥ n − 1. □

Remark 4.5. If we are interested in only the case n = d, then the connection between Hilbert schemes and 𝔅_{n} is even more direct. There is an open subscheme H_{n} of Hilb_{n}(𝔸^{n}) consisting of the points for which the corresponding quotient of the polynomial algebra is spanned as a vector space by the coordinate functions on 𝔸^{n}. Also, for any rank-n algebra with a fixed basis, the locus of bases is contained in the locus of n-tuples of algebra generators, which is contained in the affine space of all n-tuples of elements, and the first is dense in the third, so the first is dense in the second, which implies that the open subscheme H_{n} of Hilb_{n}(𝔸^{n}) is Zariski dense. Now we have an isomorphism 𝔅_{n} ≃ H_{n} taking a based algebra to the same algebra equipped with the n algebra generators given by the basis. This isomorphism may also be viewed as the composition of the section 𝔅_{n} → ℌ_{n}(𝔸^{n}) of Proposition 4.3 with the morphism ℌ_{n}(𝔸^{n}) → Hilb_{n}(𝔸^{n}). The fact that 𝔅_{n} is isomorphic to a dense open subscheme of Hilb_{n}(𝔸^{n}) explains why many of the properties of 𝔅_{n} we prove in later sections reflect the corresponding properties of Hilbert schemes.

Remark 4.6. By [Fog68] (see especially Corollary 2.10 and the proof of Theorem 2.4 there), Hilb_{n}(𝔸^{2}) is smooth over ℤ with geometrically irreducible fibers. It follows that the same is true of ℌ_{n}(𝔸^{2}), and (by Proposition 4.1) of the image of ℌ_{n}(𝔸^{2}) → 𝔅_{n} viewed as an open subscheme of 𝔅_{n} (the locus parameterizing based algebras that can be locally generated as an algebra by ≤ 2 elements). If n ≤ 3, Proposition 4.3 implies that this locus is the whole scheme 𝔅_{n}, which is then smooth over ℤ with geometrically irreducible fibers.

Remark 4.7. Fix a characteristic p ≥ 0 and a nonnegative integer d. The value of dim Hilb_{n}(X) is the same for every nonempty smooth variety X of dimension d over a field of characteristic p.

5. The moduli spaces for n ≤ 3

Proposition 5.1. For n ≤ 3, the isomorphism type of the ℤ-scheme 𝔅_{n}^{1} is given by the following table:

\begin{matrix}n & 0 & 1 & 2 & 3 \\ 𝔅_{n}^{1} & 𝔸^{0} & 𝔸^{0} & 𝔸^{2} & 𝔸^{6} \end{matrix}

Proof. On 𝔅_{n}^{1} ⊆ 𝔅_{n} ⊆ 𝔸^{n^{3}+n} the values of the coordinates d_{i} are specified, and the values of c_{ij}^{ℓ} when i or j is 1 are forced by the defining polynomials. Thus we may view 𝔅_{n}^{1} as a closed subscheme in the affine space 𝔸^{(n−1)^{2}n} whose coordinates are the indeterminates c_{ij}^{ℓ} for 2 ≤ i, j ≤ n and 1 ≤ ℓ ≤ n.

When n ≤ 2, the commutativity and associativity conditions on these c_{ij}^{ℓ} are vacuous, so 𝔅_{n}^{1} ≃ 𝔸^{(n−1)^{2}n}.

Finally, suppose n = 3. Call a basis for a rank-3 algebra good if it has the form 1, α, β with αβ ∈ O_{S} · 1. We paraphrase [DF40, §15] (see also [GGS02, §4]), where it was shown that rank-3 algebras (over ℤ) equipped with a good basis are in bijection with binary cubic forms. Let 𝔅_{3}^{1,good} be the closed subscheme of 𝔅_{3}^{1} parameterizing rank-3 algebras equipped with a good basis. There is a left action of 𝔾_{a}^{2} on 𝔅_{3}^{1} in which (a, b) ∈ 𝔾_{a}^{2}(S) maps an algebra A equipped with basis 1, α, β to A equipped with basis 1, α + a, β + b. This action restricts to an isomorphism

𝔾_{a}^{2} × 𝔅_{3}^{1,good} → 𝔅_{3}^{1}.

Using the conditions coming from commutativity and associativity, one finds that the multiplication in any based algebra in 𝔅_{3}^{1,good} with basis 1, α, β has the form

α^{2} = −ac + bα − aβ
β^{2} = −bd + dα − cβ
αβ = −ad

for some a, b, c, d, and that conversely any a, b, c, d yield a based algebra in 𝔅_{3}^{1,good}. Thus 𝔅_{3}^{1,good} ≃ 𝔸^{4}, and 𝔅_{3}^{1} ≃ 𝔸^{2} × 𝔸^{4} ≃ 𝔸^{6}. □

Remark 5.2. In Section 8 we will find that for n ≥ 4, the scheme 𝔅_{n}^{1} is not smooth over Spec ℤ, so in particular it is not isomorphic to affine space.

6. The étale locus

Given Ã := (A, ϕ) ∈ 𝔅_{n}(S), we may define an O_{S}-linear trace map Tr: A → O_{S} in the usual way. Let b_{i} = ϕ^{−1}(e_{i}) ∈ H^{0}(S, A); these form an O_{S}-basis of A. Define a regular function Δ: 𝔅_{n} → 𝔸^{1} by setting

Δ(Ã) := det (Tr(b_{i}b_{j})) ∈ H^{0}(S, O_{S}).

Acting on Ã by M ∈ GL_{n}(S) multiplies the matrix in the definition of Δ(Ã) by (M^{−1})^{t} on the left and M^{−1} on the right, so

Δ(M · Ã) = (det M)^{−2}Δ(Ã).

In particular, the zero locus {Δ = 0} and its complement 𝔅_{n}^{et} in 𝔅_{n} are GL_{n}-invariant. Call the open affine subscheme 𝔅_{n}^{et} of 𝔅_{n} the étale locus. The following is standard:

Proposition 6.1. The following are equivalent for Ã:= (A, ϕ) ∈ 𝔅_{n}(S):

(i) Ã ∈ 𝔅_{n}^{et}(S).

(ii) Δ(Ã) ∈ H^{0}(S, O_{S}^{×}).

(iii) The morphism Spec A → S is étale.

(iv) There exists a surjective étale base extension f: T → S and an isomorphism f^{∗}A ≃ f^{∗}A_{split}.

Examples 6.2.

(1) We have Δ(Ã_{split}) = 1, so Ã_{split} ∈ 𝔅_{n}^{et}(S).

(2) If n ≥ 2, Δ(•) = 0, so • ∉ 𝔅_{n}^{et}(S).

Let S_{n} denote the constant group scheme over ℤ corresponding to the symmetric group on n letters. Embed S_{n} in GL_{n} as the subgroup of permutation matrices.

Theorem 6.3. There is a GL_{n}-equivariant isomorphism from the homogeneous space GL_{n} /S_{n} to 𝔅_{n}^{et}.

Proof. We have a morphism GL_{n} → 𝔅_{n}^{et} sending M to M · Ã_{split}. The equivalence of (i) and (iv) in Proposition 6.1 implies that GL_{n} → 𝔅_{n}^{et} is surjective.

It remains to show that the subgroup scheme of GL_{n} stabilizing Ã_{split} equals S_{n}. Equivalently, we must show that for any connected scheme S, the automorphism group of the O_{S}-algebra ∏_{i=1}^{n} O_{S} equals S_{n} (acting by permuting coordinates). This holds, since every automorphism of the S-scheme ∐_{i=1}^{n} S induces a permutation of the connected components, and each map between components is the identity S → S by virtue of being an S-morphism. □

Corollary 6.4. The scheme 𝔅_{n}^{et} is irreducible.

Corollary 6.5. The Zariski closure \overline{𝔅_{n}^{et}} of 𝔅_{n}^{et} in 𝔅_{n} is an irreducible component of 𝔅_{n}.

Corollary 6.6. The affine variety 𝔅_{n,k}^{et} is irreducible and of dimension n^{2}.

Corollary 6.7. The Zariski closure \overline{𝔅_{n,k}^{et}} of 𝔅_{n,k}^{et} in 𝔅_{n,k} is an irreducible component of 𝔅_{n,k} of dimension n^{2}.

Warning 6.8. Conceivably, the base extension (\overline{𝔅_{n}^{et}})_{k} could be strictly larger than \overline{𝔅_{n,k}^{et}}. Hence we cannot conclude that (\overline{𝔅_{n}^{et}})_{k} is irreducible.

Remark 6.9. We could similarly define 𝔅_{n}^{1,et} and prove that 𝔅_{n}^{1,et} ≃ H/S_{n−1}, where S_{n−1} := S_{n} ∩ H ⊆ GL_{n}. The analogues of Corollaries 6.4 to 6.7 for 𝔅_{n}^{1,et} follow. In particular, dim 𝔅_{n,k}^{1,et} = n^{2} − n.

Remark 6.10. The number of isomorphism types of rank-n algebras over an algebraically closed field k is finite if and only if n ≤ 6 [Poo07]. It follows that dim 𝔅_{n,k} = n^{2} and dim 𝔅_{n,k}^{1} = n(n−1) for n ≤ 6, even if k is not algebraically closed. Actually, these dimension formulas hold also for n = 7, at least if k is of characteristic not 2 or 3, because 𝔅_{n,k} is irreducible by [Maz80, Corollary 4], and hence equals \overline{𝔅_{n,k}^{et}}.

Remark 6.11. One might expect at first that all k-algebras are degenerations of étale algebras; i.e., that 𝔅_{n,k}^{et} is Zariski dense in 𝔅_{n,k}. In Section 9 we will disprove this for large n by proving that dim 𝔅_{n,k} ≥ \frac{2}{27}n^{3}. In fact, for every n ≥ 8, the variety 𝔅_{n,k} has irreducible components other than \overline{𝔅_{n}^{et}}: see Proposition 9.6.

7. Connectedness

Proposition 7.1. Suppose n ≥ 1. Then 𝔅_{n}^{1} is the affine cone over a closed subscheme in a weighted projective space. The vertex of the cone is •.

Proof. As in the proof of Proposition 5.1, view 𝔅_{n}^{1} as a closed subscheme of the 𝔸^{(n−1)^{2}n} with coordinates c_{ij}^{ℓ} with 2 ≤ i, j ≤ n and 1 ≤ ℓ ≤ n. Let the weight of c_{ij}^{ℓ} be 2 if ℓ = 1, and 1 otherwise. Then the equations expressing commutativity and associativity are homogeneous. The origin in 𝔸^{(n−1)^{2}n} corresponds to the multiplication table for •. □

Corollary 7.2. The point • ∈ 𝔅_{n}^{1}(k) belongs to every irreducible component of 𝔅_{n,k}^{1}.

Remark 7.3. The observation that every algebra in 𝔅_{n,k}^{1} can be connected to • is due to Manjul Bhargava.

Corollary 7.4. The point • belongs to every irreducible component of 𝔅_{n,k}.

Proof. The inclusion 𝔅_{n}^{1} → 𝔅_{n} factors as

(7.5)  𝔅_{n}^{1} ⟶ GL_{n} ×𝔅_{n}^{1} ⟶ 𝔅_{n}

in which the first map sends Ã to (1_{n}, Ã), and the second map is a torsor under H, by Proposition 3.1. Since GL_{n} and H are irreducible, the base extension of (7.5) to k induces bijections on irreducible components. □

As a corollary, we obtain

Theorem 7.6. The varieties 𝔅_{n,k}^{1} and 𝔅_{n,k} are connected.

8. Smoothness

Proposition 8.1. For n ≠ 2, the tangent space T_{•}(𝔅_{n,k}^{1}) has dimension n(n − 1)^{2}/2.

Proof. By Proposition 5.1, we may assume n ≥ 3. A deformation of • to a based algebra in 𝔅_{n,k}^{1}(k[ϵ]/(ϵ^{2})) is given in terms of a basis e_{1}, …, e_{n} (with e_{1} = 1) by the \frac{n(n−1)}{2} · n multiplication constants c_{ij}^{ℓ} for 2 ≤ i ≤ j ≤ n and 1 ≤ ℓ ≤ n, in which c_{ij}^{ℓ} = γ_{ij}^{ℓ}ϵ with γ_{ij}^{ℓ} ∈ k. In assuming i ≤ j, we have already imposed commutativity, so it remains to examine the restrictions on the γ_{ij}^{ℓ} imposed by associativity.

The condition (e_{2}e_{2})e_{3} = e_{2}(e_{2}e_{3}) implies γ_{22}^{1} = 0 and γ_{23}^{1} = 0. Similarly γ_{ij}^{1} = 0 for all i, j. No other conditions are imposed, so the dimension equals the number of (i, j, ℓ) satisfying 2 ≤ i ≤ j ≤ n and 2 ≤ ℓ ≤ n. This is \frac{n(n−1)}{2} · (n − 1). □

Corollary 8.2. For n ≠ 2, the tangent space T_{•}(𝔅_{n,k}) has dimension n(n − 1)^{2}/2 + n.

Proof. Using (7.5), we find dim T_{•}(𝔅_{n,k}) = dim T_{•}(𝔅_{n,k}^{1}) + dim GL_{n} − dim H. □

Corollary 8.3. For n ≥ 4, the point • is singular on both 𝔅_{n,k}^{1} and 𝔅_{n,k}.

Proof. The irreducible component \overline{𝔅_{n,k}^{1,et}} of 𝔅_{n,k}^{1} contains • and its dimension n(n − 1) is less than dim T_{•}(𝔅_{n,k}^{1}) = n(n − 1)^{2}/2 if n ≥ 4. □

Proposition 8.4. For n ∈ ℤ_{≥0}, the following are equivalent:

(1) n ≤ 3.

(2) 𝔅_{n,k}^{1} is smooth.

(3) 𝔅_{n,k} is smooth.

Proof. By Proposition 3.1, 𝔅_{n,k}^{1} is smooth if and only if 𝔅_{n,k} is. For n ≤ 3 use Proposition 5.1 or Remark 4.6. For n ≥ 4 use Corollary 8.3. □

Example 8.5. Let R be a discrete valuation ring with uniformizer π. Let A be the based R/π^{2}R-algebra with basis 1, x, y, z satisfying x^{2} = πx, y^{2} = πy, z^{2} = πz, xy = πz, yz = πx, zx = πy. Then A does not lift to a free R/π^{3}R-algebra of rank 4, let alone a free R-algebra of rank 4: the associative relation x^{2}y = x(xy) fails for any lift to R/π^{3}R. This violation of the infinitesimal lifting criterion for smoothness shows again that 𝔅_{4} is not smooth.

9. Dimension: lower bound

Lemma 9.1. Suppose that n, d, r ∈ ℤ_{≥0} satisfy r ≤ d(d + 1)/2 and n = 1 + d + r. Then

dim 𝔅_{n,k} ≥ r (\frac{d(d + 1)}{2} − r) + n^{2} − (d^{2} + dr).

Proof. Let 𝔪 = (x_{1}, …, x_{d}) ⊆ k[x_{1}, …, x_{d}]. Then dim 𝔪^{2}/𝔪^{3} = d(d + 1)/2. Let Z_{d,r} be the variety parameterizing (V, ϕ) where V is a codimension-r subspace of 𝔪^{2}/𝔪^{3}, and ϕ is a k-linear isomorphism \frac{k[x_{1},...,x_{d}]}{𝔪^{3}}/V → k^{n}. The forgetful map (V, ϕ) ↦ V exhibits Z_{d,r} as a GL_{n}-torsor over the Grassmannian Gr(r, d(d + 1)/2), so

dim Z_{d,r} = dim Gr(r, d(d + 1)/2) + dim GL_{n} = r (\frac{d(d + 1)}{2} − r) + n^{2}.

We next compute the dimension of the fibers of the morphism π: Z_{d,r} → 𝔅_{n,k} defined by (V, ϕ) ↦ (A_{V} , ϕ) where A_{V} := \frac{k[x_{1},...,x_{d}]}{𝔪^{3}}/V. For fixed V, to give a k-algebra isomorphism from A_{V} to some A_{V′} is to give (ℓ, η_{1}, …, η_{d}) where ℓ is a k-linear isomorphism 𝔪/𝔪^{2} → 𝔪/𝔪^{2} such that Sym^{2} ℓ: 𝔪^{2}/𝔪^{3} → 𝔪^{2}/𝔪^{3} maps V to V′ and η_{1}, …, η_{d} ∈ \frac{𝔪^{2}}{𝔪^{3}}/V′. (The isomorphism A_{V} → A_{V′} attached to such data is given by x_{i} ↦ ℓ(x_{i})+η_{i}.) It follows that given (V, ϕ), the dimension of the fiber of Z_{d,r} → 𝔅_{n,k} containing it equals the number of parameters needed to specify ℓ and η_{1}, …, η_{d}. Thus the fibers have dimension d^{2} + dr.

Hence

dim 𝔅_{n,k} ≥ dim π(Z_{d,r}) = r (\frac{d(d + 1)}{2} − r) + n^{2} − (d^{2} + dr).

□

Theorem 9.2. For n ≥ 1,

dim 𝔅_{n,k} ≥ \begin{cases}\frac{2}{27}n^{3} + \frac{1}{9}n^{2} + \frac{5}{3}n − 1, & if n ≡ 0 (mod 3) \\ \frac{2}{27}n^{3} + \frac{1}{9}n^{2} + \frac{14}{9}n − \frac{20}{27}, & if n ≡ 1 (mod 3) \\ \frac{2}{27}n^{3} + \frac{1}{9}n^{2} + \frac{5}{3}n − \frac{37}{27}, & if n ≡ 2 (mod 3). \end{cases}

Proof. Calculus shows that for fixed n ≥ 1, the bound in Lemma 9.1 (with r = n − 1 − d) as a function of d increases up to a point around 2n/3 − 7/6 + o(1) and decreases thereafter. Evaluating the bound on the two integers on each side shows that the maximum at nonnegative integers occurs at d = ⌊\frac{2n−2}{3}⌋, and the maximum value is as shown. □

Remark 9.3. The idea to use local algebras with 𝔪^{3} = 0 to obtain a lower bound of the form (\frac{2}{27} + o(1))n^{3} is old, but Theorem 9.2 seems to be more precise than other results of its type.

Corollary 9.4. If n ≥ 11, then dim 𝔅_{n,k} > n^{2}.

Remark 9.5. It is natural to guess that the true dimension of 𝔅_{n,k} equals n^{2} for n ≤ 10, and equals the lower bound of Theorem 9.2 for n ≥ 11.

Proposition 9.6. If n ≥ 8, then 𝔅_{n,k} is reducible.

Proof. It suffices to give a rank-8 algebra that cannot be deformed to an étale algebra, since then rank-n algebras with the same property can be obtained by taking a product with an étale algebra of rank n − 8. An explicit rank-8 algebra with this property was first given in [IE78]. An example somewhat simpler than the one given there is

\frac{k[a, b, c, d]}{(a^{2}, ab, b^{2}, c^{2}, cd, d^{2}, ad − bc)},

whose construction grew out of discussions of a working group at Berkeley (Jonah Blasiak, Dustin Cartwright, David Eisenbud, Daniel Erman, Mark Haiman, the present author, Bernd Sturmfels, Mauricio Velasco, and Bianca Viray). The method for showing that a point P of 𝔅_{n,k} corresponding to this algebra is not in the component \overline{𝔅_{n,k}^{et}} is the same as in [IE78]: one calculates that dim T_{P} (𝔅_{n,k}) < dim \overline{𝔅_{n,k}^{et}} = n^{2} (or rather, one does an equivalent calculation in the Hilbert scheme). □

Proposition 9.7. For each n ≥ 8, there exists a prime p and an 𝔽_{p}-algebra that is not isomorphic to A/pA for any order A in an étale ℚ-algebra.

Proof. We will show that any algebra coming from a point of 𝔅_{n}(𝔽_{p}) − \overline{𝔅_{n}^{et}}(𝔽_{p}) has the property; the algebra

A_{0} := \frac{𝔽_{p}[a, b, c, d]}{(a^{2}, ab, b^{2}, c^{2}, cd, d^{2}, ad − bc)} × 𝔽_{p} × ⋯ × 𝔽_{p}

from the proof of Proposition 9.6, with n − 8 factors of 𝔽_{p}, is an example.

If A is an order in an étale ℚ-algebra, and we choose a ℤ-basis, then the corresponding section s: Spec ℤ → 𝔅_{n} is such that s(Spec ℚ) ∈ 𝔅_{n}^{et}, so s(Spec ℤ) ⊆ \overline{𝔅_{n}^{et}}, and s(Spec 𝔽_{p}) ∈ \overline{𝔅_{n}^{et}}(𝔽_{p}). In other words, the point of 𝔅_{n}(𝔽_{p}) corresponding to A/pA with its basis is in \overline{𝔅_{n}^{et}}(𝔽_{p}), so A/pA ≄ A_{0}. □

Question 9.8. Can one characterize the k-algebras that correspond to points of \overline{𝔅_{n,k}^{et}}?

Closely related is the following:

Question 9.9. What functor does the ℤ-scheme \overline{𝔅_{n}^{et}} represent?

10. Dimension: upper bound

We now work toward an asymptotically matching upper bound on dim 𝔅_{n,k}, namely dim 𝔅_{n,k} ≤ \frac{2}{27}n^{3} + O(n^{8/3}). Such a result is announced in [Ner87], who writes that the proof is nearly identical to the proof he gives for the moduli space of n-dimensional Lie algebras. We will give details of a proof for 𝔅_{n,k}.

The approach towards both those results is to adapt the proof (begun in [Hig60] and completed in [Sim65]) that the number of p-groups of order p^{n} is p^{\frac{2}{27} n^{3}+O(n^{8/3})}. As suggested to us by Hendrik Lenstra, there is an analogy between the powers of the maximal ideal of a local finite-rank k-algebra and the descending p-central series of a p-group. Although there seems to be no direct connection between finite-rank k-algebras and finite p-groups, the combinatorial structure in the two enumeration proofs are nearly identical.

10.1. Symmetric bilinear maps. This subsection is inspired by [Sim65, §2], which studied alternating bilinear maps.

Throughout this subsection, we fix the following notation. Let V and W be vector spaces over a field k. Let m = dim V and n = dim W. For Proposition 10.1 and its two corollaries, we fix a symmetric bilinear map ( , ): V × V → W. For any subspace U ≤ V, let (U, U) be the k-span of the set {(u_{1}, u_{2}) : u_{1}, u_{2} ∈ U}.

Proposition 10.1. Suppose that (V, V) = W but for no proper subspace U ≤ V is (U, U) = W. Then there exists a basis x_{1}, …, x_{m} of V such that for 1 ≤ i ≤ m − 1, if V_{i} is the span ⟨x_{1}, …, x_{i}⟩ then (x_{i}, x_{i+1}) ∉ (V_{i}, V_{i}).

Proof. We show by induction that for r = 0, 1, …, m that there exist x_{1}, …, x_{r} such that (x_{i}, x_{i+1}) ∉ (V_{i}, V_{i}) for 1 ≤ i ≤ r − 1. If r ≤ 1, this is trivial. The proof of the inductive step is as in the proof of [Sim65, Proposition 2.1]. □

Corollary 10.2. Under the hypothesis of Proposition 10.1, m ≤ n + 1.

Corollary 10.3. There exists a subspace U ≤ V such that dim U ≤ n + 1 and (U, U) = (V, V).

Proof. Apply Corollary 10.2 to a minimal subspace U satisfying (U, U) = (V, V). □

Proposition 10.4. Suppose that k is a finite field 𝔽_{q} of order q. Let V, W, m, n be as before, and let s ∈ ℤ_{>0}. The number of symmetric bilinear maps ( , ): V × V → W such that for some s-dimensional subspace U ≤ V we have (U, U) = W, but for no (s − 1)-dimensional subspace U′ ≤ V is (U′, U′) = W is less than or equal to

q^{\frac{m^{2}}{2} (n−s)+O((m+n)^{8/3})}.

Proof. See the proof of [Sim65, 2.4–2.6]. The only differences are (1) We have q instead of a prime p; this is of no consequence.

(2) Since we deal with symmetric forms in place of his alternating forms, the \binom{m}{2} that appears in the proof of Sims’ Proposition 2.6 should be replaced by \binom{m+1}{2}. Both of these are close enough to m^{2}/2 that the difference in the resulting exponent is dominated by the O((m + n)^{8/3}) term.

□

10.2. Finite local 𝔽_{q}-algebras. Our next step is to count rank-n local 𝔽_{q}-algebras A with residue field 𝔽_{q}. Given such an A, we will define a collection of data, and then prove that the data uniquely determine A up to isomorphism.

Let 𝔪 be the maximal ideal of A. For i ≥ 0, let m_{i} = dim_{𝔽_{q}} 𝔪^{i}/𝔪^{i+1}. Let g_{01} = 1. Let V := 𝔪/𝔪^{2} and W := 𝔪^{2}/𝔪^{3}. The multiplication in A induces a symmetric bilinear map ( , ): V × V → W such that (V, V) = W. Let V̄ ≤ V be a subspace of minimum dimension such that (V̄ , V̄) = W. Let s := dim V̄. By Corollary 10.3, s ≤ m_{2} + 1. By Proposition 10.1, we may choose a basis x_{1}, …, x_{s} of V̄ such that for i ∈ [1, s − 1] we have (x_{i}, x_{i+1}) ∉ (V_{i}, V_{i}), where V_{i} := ⟨x_{1}, …, x_{i}⟩. For i ∈ [0, s], let W_{i} = (V_{i}, V_{i}) and d_{i} = dim W_{i}. So 0 = d_{0} ≤ d_{1} < d_{2} < ⋯ < d_{s} = m_{2}. Choose a basis y_{1}, …, y_{m_{2}} of W such that W_{i} = ⟨y_{1}, …, y_{d_{i}}⟩. Given i and j ∈ (d_{i−1}, d_{i}] we may assume y_{j} = (x_{b}, x_{i}) for some b ∈ [1, i] depending on j. Extend x_{1}, …, x_{s} to a basis x_{1}, …, x_{m_{1}} of V. Choose a representative g_{1i} ∈ A of each x_{i}. For each i and each j ∈ (d_{i−1}, d_{i}], take g_{2j} = g_{1b}g_{1i} as the representative of y_{j} for the same b as above. By induction on i, the map

\frac{𝔪^{i−1}}{𝔪^{i}} ⊗ V̄ → \frac{𝔪^{i}}{𝔪^{i+1}}

induced by multiplication in A is surjective for each i ≥ 2. Thus in particular, by induction on i, for i ≥ 3, we can choose g_{i1}, …, g_{i,m_{i}} ∈ A representing a basis of 𝔪^{i}/𝔪^{i+1} such that each g_{ij} equals g_{i−1,r}g_{1ℓ} for some r ∈ [1, m_{i−1}] and ℓ ∈ [1, s]. For i = 3, if X_{h} denotes the image of

\frac{𝔪^{2}}{𝔪^{3}} ⊗ V_{h} → \frac{𝔪^{3}}{𝔪^{4}}

we may assume moreover that the g_{3j} = g_{2r}g_{1ℓ} are chosen so that the first few have ℓ = 1 and span X_{1}, and the next few have ℓ = 2 and with the previous ones span X_{2}, and so on up to ℓ = s. Then in each product g_{2r}g_{1ℓ} arising at the ℓ^{th} stage (i.e., mapping to X_{ℓ} − X_{ℓ−1}), we have r > d_{ℓ−1}, since otherwise g_{2r} = g_{1b}g_{1c} for some b ≤ c ≤ ℓ − 1 and the class of g_{2r}g_{1ℓ} = g_{1b}g_{1ℓ}g_{1c} in 𝔪^{3}/𝔪^{4} lies in X_{ℓ−1}, and hence is not a new basis element of the ℓ^{th} stage.

The g_{ij} for i ≥ 0 and j ∈ [1, m_{i}] form a basis for A. Define c_{ijℓuv} ∈ 𝔽_{q} by

g_{ij}g_{1ℓ} = ∑ c_{ijℓuv}g_{uv}.

Proposition 10.5. The isomorphism type of a local 𝔽_{q}-algebra A with residue field 𝔽_{q} is determined by the sequence (m_{i})_{i≥0} and the c_{ijℓuv} for (i, j, ℓ, u, v) satisfying the following conditions:

i = 1, 1 ≤ j ≤ ℓ ≤ m_{1}, u ≥ 2, 1 ≤ v ≤ m_{u}; or
i = 2, 1 ≤ ℓ ≤ s, d_{ℓ−1} < j ≤ m_{2}, u ≥ 3, 1 ≤ v ≤ m_{u}; or
i ≥ 3, 1 ≤ j ≤ m_{i}, 1 ≤ ℓ ≤ s, u > i, 1 ≤ v ≤ m_{u};

where s and the d_{i} are determined by the c_{1jℓuv}.

Proof. Given the data above, we construct the vector space underlying A by taking the 𝔽_{q}- vector space with basis g_{ij}. Since A is generated as a 𝔽_{q}-algebra by the g_{1ℓ} for 1 ≤ ℓ ≤ m_{1}, it suffices to show that the given data determine g_{ij}g_{1ℓ} for i ≥ 1 as a linear combination of the g_{uv}.

If i = 1, then by commutativity we may assume j ≤ ℓ, in which case the value of g_{1j}g_{1ℓ} is already given by the c_{1jℓuv}.

We next prove by strong induction on ℓ that g_{2j}g_{1ℓ} is determined for all j and all ℓ ≤ s. Suppose that the values g_{2j}g_{1ℓ′} have been determined for all j whenever ℓ′ < ℓ, and we want to determine g_{2j}g_{1ℓ}. If j > d_{ℓ−1}, the relevant c_{2jℓuv} have already been given. So assume j ≤ d_{ℓ−1}. Then g_{2j} = g_{1b}g_{1c} for some b ≤ c ≤ ℓ − 1, so g_{2j}g_{1ℓ} = (g_{1b}g_{1ℓ})g_{1c}. Here g_{1b}g_{1ℓ} is a known combination of the g_{uv} for u ≥ 2 and all v. But g_{2v}g_{1c} has been determined already by the inductive hypothesis, and g_{uv}g_{1c} for u ≥ 3 is already given (since c ≤ ℓ − 1 ≤ s), so the product g_{2j}g_{1ℓ} = (g_{1b}g_{1ℓ})g_{1c} is determined.

We now know that multiplication by g_{1ℓ} on all the g_{ij} with i ≥ 1 is determined provided that ℓ ≤ s. To extend this to all ℓ ≤ m_{1}, we use induction on i. The case i = 1 is already given by the known c_{1jℓuv}. So assume i ≥ 2. Then g_{ij} = g_{i−1,r}g_{1b} for some r, b with b ≤ s, so g_{ij}g_{1ℓ} = (g_{i−1,r}g_{1ℓ})g_{1b}. By the inductive hypothesis, g_{i−1,r}g_{1ℓ} is a known combination of g_{uv} with u ≥ 2, so its product with g_{1b} is determined (since b ≤ s). □

Proposition 10.6. The number of rank-n local 𝔽_{q}-algebras with residue field 𝔽_{q} is q^{\frac{2}{27} n^{3}+O(n^{8/3})} as n → ∞. The implied constant can be chosen independent of q.

Proof. The lower bound comes from Lemma 9.1 and Theorem 9.2: using standard formulas for # Gr(r, d(d + 1)/2)(𝔽_{q}) and # GL_{n}(𝔽_{q}), one obtains a lower bound that is at least a constant times q raised to the right hand side in Theorem 9.2.

Now for the upper bound. To each local 𝔽_{q}-algebra A with residue field 𝔽_{q} we associate the finite list of positive integers m_{0}, m_{1}, …, m_{t} summing to n, and the c_{ijℓuv} ∈ 𝔽_{q} as in Proposition 10.5. The number of possibilities for m_{0}, m_{1}, …, m_{t}, including the choice of t, is at most 2^{n−1} (given n stars, place or do not place a bar between each consecutive pair of stars); this is O(q^{n^{8/3}}), and hence may be ignored.

Below, “log” means logarithm to the base q. The number of possibilities for s is at most n + 1, so we may assume s is fixed. For each s, Proposition 10.4 shows that the log of the number of possibilities for the c_{1jℓ2v} giving that value of s is at most

\frac{m_{1}^{2}}{2} (m_{2} − s) + O((m_{1} + m_{2})^{8/3}) = \frac{m_{1}^{2}}{2} (m_{2} − s) + O(n^{8/3}).

The log of the number of possibilities for the c_{1jℓuv} with u ≥ 3 is \frac{m_{1}(m_{1}+1)}{2}(n − 1 − m_{1} − m_{2}). Thus the log of the number of possibilities for all the c_{1jℓuv} is at most

\frac{m_{1}^{2}}{2} (n − 1 − m_{1} − s) + O(n^{8/3}).

The log of the number of possibilities for the c_{2jℓuv} is

(∑_{ℓ=1}^{s}(m_{2} − d_{ℓ−1})) (n − 1 − m_{1} − m_{2}).

Since 0 ≤ d_{0} ≤ d_{1} < d_{2} < … < d_{s}, we have d_{ℓ−1} ≥ ℓ − 2, so the value above is at most

(m_{2}s − \frac{s^{2}}{2} ) (n − 1 − m_{1} − m_{2}) + O(n^{2}).

The log of the number of possibilities for the c_{ijℓuv} for a fixed i ≥ 3 is

m_{i}s(n − 1 − m_{1} − ⋯ − m_{i}).

Thus the log of the number of possibilities for all the c_{ijℓuv} is at most

\frac{m_{1}^{2}}{2} (n−1−m_{1}−s)+(m_{2}s − \frac{s^{2}}{2} ) (n−1−m_{1}−m_{2})+∑_{i≥3} m_{i}s(n−1−m_{1}−⋯−m_{i})+O(n^{8/3}),

which up to O(n^{2}) is the same as the expression M in [Sim65, p. 165]. Thus by the analysis there it is at most \frac{2}{27}n^{3} + O(n^{8/3}). □

Proposition 10.7. The number of rank-n local 𝔽_{q}-algebras is q^{\frac{2}{27} n^{3}+O(n^{8/3})} as n → ∞. The implied constant can be chosen independent of q.

Proof. The lower bound follows from Proposition 10.6. Each local 𝔽_{q}-algebra has residue field 𝔽_{q^{d}} for some d | n, so by Proposition 10.6 their total number is

∑_{d|n}(q^{d})^{\frac{2}{27} (n/d)^{3}+O((n/d)^{8/3})}.

This sum has at most n terms, and each is at most q^{\frac{2}{27} n^{3}+O(n^{8/3})}, so the result follows. □

10.3. Algebras of finite rank.

Lemma 10.8. Suppose that f: ℝ_{≥0} → ℝ is convex and f(0) = 0. Then f(x_{1} + ⋯ + x_{n}) ≥ f(x_{1}) + ⋯ + f(x_{n}) for any x_{1}, …, x_{n} ∈ ℝ_{≥0}.

Proof. This is a special case of the Hardy-Littlewood-Pólya majorization inequality [HLP88, Theorem 108], for instance. □

Theorem 10.9. The number of rank-n 𝔽_{q}-algebras is q^{\frac{2}{27} n^{3}+O(n^{8/3})} as n → ∞. The implied constant can be chosen independent of q.

Proof. The lower bound follows from Proposition 10.7. A general 𝔽_{q}-algebra is a product of local ones. Thus we may specify a rank-n 𝔽_{q}-algebra by giving a partition n = λ_{1} + ⋯ + λ_{m} and a local 𝔽_{q}-algebra A_{i} of rank λ_{i} for each i.

The number of partitions (even if we did not impose an ordering on the λ_{i}) is ≤ 2^{n−1} < q^{O(n^{8/3})}, so it will suffice to bound the number of algebras for a fixed partition. By Proposition 10.7, this number is

≤ q^{f(λ_{1})+···+f(λ_{m})}

where f(x) := \frac{2}{27}x^{3} + cx^{8/3} for some universal constant c > 0. Lemma 10.8 gives the desired upper bound. □

Theorem 10.10. We have dim 𝔅_{n,k} = \frac{2}{27}n^{3} + O(n^{8/3}) uniformly in k.

Proof. We may replace k by its minimal subfield. For fixed n, dim 𝔅_{n,ℚ} = dim 𝔅_{n,𝔽_{p}} for all but finitely many primes p, so it suffices to obtain a bound for dim 𝔅_{n,𝔽_{p}} that is uniform in p. Theorem 10.9 estimates #𝔅_{n,𝔽_{p}}(𝔽_{p^{e}}) for every e (the latter count includes the choice of a basis, but the number of choices is only O(p^{n^{2}})). Taking e → ∞ and applying the Lang-Weil bounds [LW54], we obtain the desired result. □

11. Commutative rings of finite order

As a bonus, Theorem 10.9 leads to an asymptotic formula for the number of (commutative) rings of order p^{n}, namely p^{\frac{2}{27} n^{3}+O(n^{8/3})}. To prove this, we follow [KP70], which proved the analogous formula p^{\frac{4}{27} n^{3}+O(n^{8/3})} for the number of associative rings of order p^{n} that are not necessarily commutative or unital. We begin with the commutative analogue of [KP70, Theorem 3.1].

Lemma 11.1. The number of (commutative) rings of order p^{n} up to isomorphism is at most p^{n^{2}+n} times the number of rank-n 𝔽_{p}-algebras up to isomorphism.

Proof. ([KP70]) For each ring R of order p^{n}, choose generators x_{1}, …, x_{m} of the additive group of R such that their orders p^{a_{1}}, …, p^{a_{m}} multiply to p^{n}. For i ≤ m and 0 ≤ j < a_{i}, let y_{ij} = p^{j}x_{i}. Rename all the y_{ij} in any order as z_{1}, …, z_{n}. Then z_{i}z_{j} = ∑ c_{ijk}z_{k} for some c_{ijk} ∈ {0, 1, …, p − 1}. Construct the rank-n 𝔽_{p}-algebra A having the same structure constants c_{ijk} considered in 𝔽_{p}. Associativity, commutativity, and existence of 1 for R imply the corresponding properties for A.

The construction above defines a map from the set of isomorphism classes of rings of order p^{n} to the set of pairs (a⃗, A) where a⃗ = (a_{1}, …, a_{m}) is a sequence of positive integers summing to n, and A is a based rank-n 𝔽_{p}-algebra. Reversing the construction shows that this map is injective.

The number of sequences a⃗ is 2^{n−1} ≤ p^{n}, and the number of choices of basis for a rank-n 𝔽_{p}-algebra is # GL_{n}(𝔽_{p}) ≤ p^{n^{2}}. □

Theorem 11.2. The number of commutative rings of order p^{n} up to isomorphism is p^{\frac{2}{27} n^{3}+O(n^{8/3})} as n → ∞. The implied constant can be chosen independent of p.

Proof. Combine Theorem 10.9 and Lemma 11.1. □

Theorem 11.3. The number of commutative rings of order ≤ N up to isomorphism is exp(\frac{2}{27}(log N)^{3}/(log 2)^{2} + O((log N)^{8/3})) as N → ∞.

Proof. The lower bound is attained already by considering the commutative rings of order 2^{⌊log_{2} N⌋}, by Theorem 11.2.

For the upper bound, it suffices to prove the same bound for the number r_{N} of commutative rings of exact order N, since r_{0}+⋯+r_{N} ≤ (N +1) max_{i∈[0,N]} r_{i}, and N +1 < exp((log N)^{8/3}) for large N.

By Theorem 11.2, there exists c ≥ 0 such that the number of rings of order p^{n} is ≤ p^{(2/27)n^{3}+cn^{8/3}}. Write N = ∏_{p} N_{p}, where N_{p} is a power of the prime p. Then

log r_{N} ≤ ∑_{p|N} (\frac{2}{27} (\frac{log N_{p}}{log p} )^{3} + c (\frac{log N_{p}}{log p} )^{8/3}) log p

≤ ∑_{p|N} (\frac{2}{27} \frac{(log N_{p})^{3}}{(log 2)^{2}} + c\frac{(log N_{p})^{8/3}}{(log 2)^{5/3}} )

≤ \frac{2}{27} \frac{(log N)^{3}}{(log 2)^{2}} + c\frac{(log N)^{8/3}}{(log 2)^{5/3}} ,

by Lemma 10.8. □

The main contribution to the number of finite rings may come from those of the form (\frac{ℤ_{2}[[x_{1},...,x_{d}]]}{𝔪^{3}})/V where ℤ_{2} is the ring of 2-adic integers, 𝔪 = (2, x_{1}, …, x_{d}), and V is an 𝔽_{2}- subspace of 𝔪^{2}/𝔪^{3}. If this is so, then we would obtain the following information about the “typical” finite ring:

Conjecture 11.4. As N → ∞, the fraction of (isomorphism classes of) commutative rings A of order ≤ N satisfying the following conditions tends to 1:

(1) The size of A is 2^{n}, where 2^{n} is the largest power of 2 less than or equal to N.

(2) The ring A is local.

(3) The residue field of A is 𝔽_{2}.

(4) If 𝔪 is the maximal ideal of A, then 𝔪^{3} = 0 but 𝔪^{2} ≠ 0.

(5) The 𝔽_{2}-dimension of 𝔪/𝔪^{2} is 2n/3 + O(1).

(6) The characteristic of A is 8 (i.e., we have 8 = 0 but 4 ≠ 0 in A).

Similar questions have been raised for groups: see [Man99], for instance.

12. Hilbert schemes revisited

Finally, we use the results and techniques in Sections 9 and 10 to estimate the dimension of Hilb_{n}(𝔸^{d}) when d is at least about as large as n.

Corollary 12.1. If d ≥ n − 1,

dim Hilb_{n}(𝔸^{d})_{k} = \frac{2}{27}n^{3} + nd + O(n^{8/3}).

Proof. Substitute Theorem 10.10 in Corollary 4.4. □

We may generalize Corollary 12.1 to cover a larger range of d.

Theorem 12.2. Suppose that d = αn for some nonnegative α ∈ ℚ. Set

c_{α} := \begin{cases}2/27, & if α ≥ 2/3 \\ \frac{α^{2}}{2} (1 − α) & if 0 ≤ α ≤ 2/3. \end{cases}

Then for α ≥ 1/2,

dim Hilb_{n}(𝔸^{d})_{k} = c_{α}n^{3} + nd + O(n^{8/3}).

For α < 1/2, we have only the inequality

dim Hilb_{n}(𝔸^{d})_{k} ≥ c_{α}n^{3} + O(n^{2}).

Proof. Let 𝔅_{n,k,d} be the image of ℌ_{n}(𝔸^{d})_{k} → 𝔅_{n,k}, so 𝔅_{n,k,d} parameterizes rank-n algebras that can be generated by d elements. By Corollary 4.2, 𝔅_{n,k,d} is open in 𝔅_{n,k}. The proof of Corollary 4.4 generalizes to show that

dim Hilb_{n}(𝔸^{d})_{k} = dim 𝔅_{n,k,d} − n^{2} + nd,

so it suffices to estimate dim 𝔅_{n,k,d}.

For d ≥ 2n/3, we still have

dim 𝔅_{n,k,d} = \frac{2}{27}n^{3} + O(n^{8/3}),

since the algebras constructed in Lemma 9.1 for the lower bound on dim 𝔅_{n,k} in Theorem 9.2 could be generated by about 2n/3 elements.

For d = αn with α < 2/3, we can use Lemma 9.1 to obtain

dim 𝔅_{n,k,d} ≥ c_{α}n^{3} − O(n^{2}).

If 1/2 ≤ α ≤ 2/3, we obtain a matching upper bound (but with error term O(n^{8/3})) by generalizing the proof of Theorem 10.10 as follows. Proceed as in the proof of Proposition 10.6, but at the end impose the additional condition that m_{1} ≤ αn. This translates, in the notation of [Sim65, p. 165], into the constraint x ≤ α in addition to the constraints x + y ≤ 1, 0 ≤ y ≤ x. The constant in front of the n^{3} in the lower bound obtained is the maximum value of

B(x, y) := x^{2}(1 − x − y)/2 + y^{2}(1 − x − y)/2 + y(1 − x − y)^{2}/2

subject to these constraints. For fixed x ≥ 1/2, calculus shows that the maximum value of B(x, y) for y ∈ [0, 1 − x] occurs at y = 0, and the value there is \frac{x^{2}}{2} (1 − x) = c_{x}. Calculus shows also that the maximum value of B(x, y) on the triangle with vertices (0, 0), (1/2, 0), (1/2, 1/2) is attained at (1/2, 0). By the previous two sentences, for 1/2 ≤ α ≤ 2/3, the maximum value of B(x, y) on the region defined by x ≤ α in addition to the constraints x + y ≤ 1, 0 ≤ y ≤ x is \frac{α^{2}}{2} (1 − α), attained at (α, 0). □

Remark 12.3. Before the present work, it seems that asymptotic bounds for dim Hilb_{n}(𝔸^{d})_{k} were known only as n → ∞ for fixed d: it is shown in [BI78] that this dimension is bounded above and below by universal positive constants times n^{2−2/d}.

Acknowledgements

I thank Hendrik Lenstra for the insight that enumeration of finite-rank algebras might be related to enumeration of p-groups. I thank also Manjul Bhargava, Bas Edixhoven, Mark Haiman, and David Helm for discussions, and I thank the referee for several helpful suggestions. This work was begun at the trimester on “Explicit methods in number theory” at the Institut Henri Poincaré in 2004, and continued at U. C. Berkeley and at the workshop on “Rings of low rank” at the Lorentz Center in 2006. The author’s research is supported partially by NSF grant DMS-0301280.

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Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

E-mail address: poonen@math.berkeley.edu
URL: http://math.berkeley.edu/~poonen/