ESSENTIAL DIMENSION OF FINITE GROUPS

NIKITA A. KARPENKO AND ALEXANDER S. MERKURJEV

Abstract. We prove that the essential dimension of a p-group G over a field F containing a primitive p-th root of unity is equal to the least dimension of a faithful representation of G over F.

The notion of the essential dimension ed_{F} (G) of a finite group G over a field F was introduced in [5]. The integer ed_{F} (G) is equal to the smallest number of algebraically independent parameters needed to classify all Galois G-algebras over any field extension of F. If V is a faithful linear representation of G over F, then ed_{F} (G) ≤ dim(V) (cf. [2, Prop. 4.11]). The essential dimension of G can be smaller than dim(V) for every faithful representation V of G over F. For example, we have ed_{F} (ℤ/3ℤ) = 1 over F = ℚ or any field F of characteristic 3 (cf. [12], [2, Example 2.3]) and ed_{ℂ}(S_{5}) = 2 (cf. [5, Th. 6.5]).

In this paper we prove that if G is a p-group and F is a field of characteristic different from p containing a primitive p-th root of unity, then ed_{F} (G) coincides with the least dimension of a faithful representation of G over F.

In the paper the word “scheme” means a separated scheme of finite type over a field and “variety” an integral scheme.

Acknowledgment: We are grateful to Z. Reichstein for useful conversations.

1. Preliminaries

1.1. Severi-Brauer varieties. (cf. [1]) Let A be a central simple algebra of degree n over a field F. The Severi-Brauer variety P = SB(A) of A is the variety of right ideals in A of dimension n. For a field extension L/F, the algebra A is split over L if and only if P (L) ≠ ∅ if and only if P_{L} ≃ ℙ_{L}^{n−1}.

The change of field map deg : Pic(P) → Pic(P_{L}) = ℤ for a splitting field extension L/F identifies Pic(P) with eℤ, where e is the exponent (period) of A. In particular, P has divisors of degree e. The algebra A is split over L if and only if P_{L} has a divisor of degree 1 (a hyperplane).

Date: June, 2007.

Key words and phrases. Algebraic groups, finite p-groups, essential dimension, representations, K-theory. 2000 Mathematical Subject Classifications: 20G15; 14C35.

The work of the first author has been partially supported by the Collaborative Research Centre 701 “Spectral Structures and Topological Methods in Mathematics” of the Bielefeld University.

The work of the second author has been supported by the NSF grant DMS #0652316.

1.2. Groupoids and gerbes. Let X be a groupoid over F in the sense of [17]. We assume that for any field extension L/F, isomorphism classes of objects in the category X (L) form a set which we denote by X̂(L). We can view X̂ as a functor from the category Fields/F of field extensions of F to Sets.

Example 1.2.1. If G is an algebraic group over F, then the groupoid BG is defined as the category of G-torsors over some base over F. Hence the functor B̂G takes a field extension L/F to the set of isomorphism classes of G-torsors over L.

Special examples of groupoids are gerbes banded by a commutative group scheme C over F. There is a bijection between the set of isomorphism classes of gerbes banded by C and the Galois cohomology group H^{2}(F, C) (cf. [7, Ch. 4] and [14, Ch. 4, §2]). The gerbe BC corresponds to the trivial element of H^{2}(F, C).

Example 1.2.2. (Gerbes banded by μ_{n}) Let A be a central simple F -algebra and n an integer with [A] ∈ Br_{n}(F) = H^{2}(F, μ_{n}). Let P be the Severi-Brauer variety of A and S a divisor on P of degree n. Denote by X_{A} the gerbe banded by μ_{n} corresponding to [A]. For a field extension L/F, the set X̂_{A}(L) has the following direct description (cf. [4]): X̂_{A}(L) is nonempty if and only if P is split over L. In this case X̂_{A}(L) is the set of equivalence classes of the set

{f ∈ L(P)^{×} : div(f) = nH − S_{L}, where H is a hyperplane in P_{L}},

and two functions f and f′ are equivalent if f′ = fh^{n} for some h ∈ L(P)^{×}.

1.3. Essential dimension. Let T : Fields/F → Sets be a functor. For a field extension L/F and an element t ∈ T (L), the essential dimension of t, denoted ed(t), is the least tr. deg_{F} (L′) over all subfields L′ ⊂ L over F such that t belongs to the image of the map T (L′) → T (L). The essential dimension ed(T) of the functor T is the supremum of ed(t) over all t ∈ T (L) and field extensions L/F.

Let G be an algebraic group over F. The essential dimension ed(G) of G is the essential dimension of the functor taking a field extension L/F to the set of isomorphism classes of G-torsors over Spec L. If G is a finite group, we view G as a constant group over a field F and write ed_{F} (G) for the essential dimension of G over F.

Example 1.3.1. Let X be a groupoid over F. The essential dimension of X, denoted by ed(X), is the essential dimension ed(X̂) of the functor X̂ defined in §1.2. In particular, ed(BG) = ed(G) for an algebraic group G over F.

1.4. Canonical dimension. (cf. [3], [11]) Let F be a field and C a class of field extensions of F. A field E ∈ C is called generic if for any L ∈ C there is an F -place E ⇝ L.

The canonical dimension cdim(C) of the class C is the minimum of the tr. deg_{F} E over all generic fields E ∈ C.

Let T : Fields/F → Sets be a functor. Denote by C_{T} the class of splitting fields of T, i.e., the class of field extensions L/F such that T (L) ≠ ∅. The canonical dimension of T, denoted cdim(T), is the canonical dimension of the class C_{T}.

If X is a scheme over F, we write cdim(X) for the canonical dimension of X viewed as a functor L ↦ X(L) = Mor_{F} (Spec L, X).

Example 1.4.1. Let X be a groupoid over F. We define the canonical dimension cdim(X) of X as the canonical dimension of the functor X̂. In particular, if G is an algebraic group over F, then cdim(BG) coincides with cdim(G) as defined in [3].

Example 1.4.2. (A relation between essential and canonical dimension) Let T : Fields/F → Sets be a functor. We define the “contraction” functor T^{c} : Fields/F → Sets as follows. For a field extension L/F, we have T^{c}(L) = ∅ if T (L) is empty and T^{c}(L) is a one element set otherwise. If X is a regular and complete variety over F viewed as a functor then ed(X^{c}) = cdim(X) and this integer is equal to the smallest dimension of a closed subvariety Z ⊂ X such that there is a rational morphism X −−→ Z. (cf. [11, Cor. 4.6]).

1.5. Valuations. Let K/F be a regular field extension, i.e., for any field extension L/F, the ring K ⊗_{F} L is a domain. We write KL for the quotient field of K ⊗_{F} L.

Let v be a valuation on L over F with residue field R. Let O be the valuation ring and M its maximal ideal. As K ⊗_{F} R is a domain, the ideal M̃:= K ⊗_{F} M in the ring Õ := K ⊗_{F} O is prime. The localization ring ÕM̃ is a valuation ring in KL with residue field KR. The corresponding valuation ṽ on KL is called the canonical extension of v on KL. Note that the groups of values of v and ṽ coincide.

2. Essential and canonical dimension of gerbes banded by (μ_{n})^{s}

In this section we relate the essential and canonical dimension of gerbes banded by (μ_{n})^{s}. The following statement is a generalization of [4, Th. 7.1].

Theorem 2.1. Let X be a gerbe banded by (μ_{n})^{s} over an arbitrary field F with n > 1 and s ≥ 0. Then ed(X) = cdim(X) + s.

Proof. The gerbe X is given by an element in H^{2}(F, μ_{n})^{s} = Br_{n}(F)^{s}, i.e., by an s-tuple of central simple algebras A_{1}, A_{2}, …, A_{s} with [A_{i}] ∈ Br_{n}(F). Let P be the product of the Severi-Brauer varieties P_{i} := SB(A_{i}), i = 1, …, s. As the classes of splitting fields for X and P coincide, we have cdim(X) = cdim(P). We shall prove that ed(X) = cdim(P) + s.

Let S_{i} be a divisor on P_{i} of degree n. Let L/F be a field extension and f_{i} ∈ L(P_{i})^{×} with div(f_{i}) = nH_{i} − (S_{i})_{L} for i = 1, …, s. Write ⟨f_{i}⟩ for the corresponding element in X̂(L) (cf. §1.2).

By Example 1.4.2, there is a closed subvariety Z ⊂ P and a rational dominant morphism P −−→ Z with dim(Z) = cdim(P). We view F (Z) as a subfield of F (P). As P (L) ≠ ∅ and P is regular, there is an F -place γ : F (P) ⇝ L (cf. [11, §4.1]). Since Z is complete, the valuation ring of the restriction γ|_{F (Z)} : F (Z) ⇝ L dominates a point in Z. It follows that Z(L) ≠ ∅. Choose a point y ∈ Z such that F′ := F (y) ⊂ L.

Since P (F′) ≠ ∅, there are g_{i} ∈ F′(P_{i})^{×} with div(g_{i}) = nH′_{i} − (S_{i})_{F′} for i = 1, …, s. As Pic(P_{i})_{L} = ℤ, there are functions h_{i} ∈ L(P_{i})^{×} with div(h_{i}) = (H′_{i})_{L} − H_{i}. We have

div(g_{i})_{L} = div(f_{i}) + div(h_{i}^{n}),

hence

a_{i}g_{i} = f_{i}h_{i}^{n}

for some a_{i} ∈ L^{×}. Hence ⟨f_{i}⟩ = ⟨a_{i}g_{i}⟩ in X (L), therefore ⟨f_{i}⟩ is defined over the field F′(a_{1}, a_{2}, …, a_{s}). It follows that

ed⟨f_{i}⟩ ≤ tr. deg_{F} (F′) + s ≤ dim(Z) + s = cdim(P) + s,

hence ed(X) ≤ cdim(P) + s.

We shall prove the opposite inequality. As P (F (Z)) ≠ ∅, there are f_{i} ∈ F (Z)(P_{i})^{×} with div(f_{i}) = nH_{i}−(S_{i})_{F (Z)}. Consider the field L := F (Z)(t_{1}, t_{2}, …, t_{s}), where t_{i} are variables, and the point ⟨t_{i}f_{i}⟩ ∈ X̂(L).

We claim that ed⟨t_{i}f_{i}⟩ ≥ cdim(P) + s. Suppose ⟨t_{i}f_{i}⟩ is defined over a subfield L′ ⊂ L, i.e., there are g_{i} ∈ L′(P_{i})^{×} and h_{i} ∈ L(P_{i})^{×} with t_{i}f_{i} = g_{i}h_{i}^{n}. We shall show that tr. deg_{F} (L′) ≥ cdim(P) + s.

The sequence of variables t_{1}, t_{2}, …, t_{s} yields a valuation v on L over F with residue field F (Z) and the group of values ℤ^{s} such that v(t_{i}) = e_{i}, where the e_{i} denote the standard basis elements of ℤ^{s}. Write w for the restriction of v on L′.

Let K = F (P). We extend canonically the valuations v and w to valuations ṽ and w̃ on KL and KL′ respectively (cf. §1.5). Note that f_{i} ∈ K(Z)^{×}, g_{i} ∈ (KL′)^{×} and h_{i} ∈ (KL)^{×}. We have

e_{i} = ṽ(t_{i}f_{i}) ≡ w̃(g_{i}) (mod n).

Since n > 1, the elements w(g_{i}) generate a subgroup of ℤ^{s} of finite index. It follows that the value group of w̃ is of rank s, hence rank(w) = rank(w̃) = s.

Let R be the residue field of w. By [18, Ch. VI, Th. 3, Cor. 1], we have

(1)  tr. deg_{F} (L′) ≥ tr. deg_{F} (R) + rank(w) = tr. deg_{F} (R) + s.

As P (L′) ≠ ∅, there is an F -place F (P) ⇝ L′. Composing it with the place L′ ⇝ R given by w, we get an F -place F (P) ⇝ R. As P is complete, we have P (R) ≠ ∅, i.e., R is a splitting field of P. On the other hand, R is a subfield of F (Z) that is the residue field of v and a generic splitting field of P. Hence R is also a generic splitting field of P. By definition of the canonical dimension,

cdim(P) = tr. deg_{F} F (Z) ≥ tr. deg_{F} (R) ≥ cdim(P).

It follows that tr. deg_{F} (R) = cdim(P) and therefore, we have tr. deg_{F} (L′) ≥ cdim(P) + s by (1). The claim is proved.

It follows from the claim that ed(X) ≥ cdim(P) + s. □

3. Canonical dimension of a subgroup of Br(F)

Let p be a prime integer and D a finite subgroup of Br_{p} F of dimension r over ℤ/pℤ. In this section we determine the canonical dimension cdim D of the class of common splitting fields of all elements of D. We say that a basis {a_{1}, a_{2}, …, a_{r}} of D is minimal if for any i = 1, …, r and any element d ∈ D outside of the subgroup generated by a_{1}, …, a_{i−1}, we have ind d ≥ ind a_{i}.

One can construct a minimal basis of D by induction as follows. Let a_{1} be a nonzero element of D of minimal index. If the elements a_{1}, …, a_{i−1} are already chosen for some i ≤ r, then we take for a_{i} an element of D of the minimal index among the elements outside of the subgroup generated by a_{1}, …, a_{i−1}.

In this section we prove the following

Theorem 3.1. Let p be a prime integer, D ⊂ Br_{p} F a subgroup of dimension r and {a_{1}, a_{2}, …, a_{r}} a minimal basis of D. Then

cdim(D) = (∑_{i=1}^{r} ind a_{i}) − r .

We prove Theorem 3.1 in several steps.

Let {a_{1}, a_{2}, …, a_{r}} be a minimal basis of D. For every i = 1, 2, …, r, let P_{i} be the Severi-Brauer variety of a central division F -algebra A_{i} representing the element a_{i} ∈ Br_{p} F. We write P for the product P_{1} × P_{2} × ⋯ × P_{r}. We have

dim P = ∑_{i=1}^{r} dim P_{i} = (∑_{i=1}^{r} ind a_{i}) − r.

Moreover, the classes of splitting fields of P and D coincide, hence cdim(D) = cdim(P). Thus, the statement of Theorem 3.1 is equivalent to the equality cdim(P) = dim(P).

Let 0 ≤ n_{1} ≤ n_{2} ≤ ⋯ ≤ n_{r} be integers and K = K(n_{1}, …, n_{r}) the subgroup of the polynomial ring ℤ[x] in r variables x = (x_{1}, x_{2}, …, x_{r}) generated by the monomials p^{e(j_{1},...,j_{r})}x_{1}^{j_{1}} … x_{k}^{j_{r}} for all j_{1}, …, j_{r} ≥ 0, where the exponent e(j_{1}, …, j_{r}) is 0 if all the j_{1}, …, j_{r} are divisible by p, otherwise e(j_{1}, …, j_{r}) = n_{k} with the maximum k such that j_{k} is not divisible by p. In fact, K is a subring of ℤ[x].

Remark 3.2. Let A_{1}, …, A_{r} be central division algebras over some field such that for any non-negative integers j_{1}, …, j_{r}, the index of the tensor product A_{1}^{⊗j_{1}} ⊗ ⋯ ⊗ A_{r}^{⊗j_{r}} is equal to p^{e(j_{1},...,j_{r})}. The group K can be interpreted as the colimit of the Grothendieck groups of the product over i = 1, …, r of the Severi-Brauer varieties of the matrix algebras M_{l_{i}}(A_{i}) over all positive integers l_{1}, …, l_{r}.

We set h = (h_{1}, …, h_{r}) with h_{i} = 1 − x_{i} ∈ ℤ[x].

Proposition 3.3. Let bh_{1}^{i_{1}} … h_{r}^{i_{r}} be a monomial of the lowest total degree of a polynomial f in the variables h lying in K. Assume that the integer b is not divisible by p. Then p^{n_{1}}|i_{1}, …, p^{n_{r}}|i_{r}.

Proof. We recast the proof for r = 1 given in [8, Lemma 2.1.2] to the case of arbitrary r.

We proceed by induction on m = r + n_{1} + ⋯ + n_{r}. The case m = 1 is trivial. If m > 1 and n_{1} = 0, then K = K(n_{2}, …, n_{r})[x_{1}] and we are done by induction applied to K(n_{2}, …, n_{r}). In what follows we assume that n_{1} ≥ 1.

Since K(n_{1}, n_{2}, …, n_{r}) ⊂ K(n_{1} − 1, n_{2}, …, n_{r}), by the induction hypothesis p^{n_{1}−1}|i_{1}, p^{n_{2}}|i_{2}, …, p^{n_{r}}|i_{r}. It remains to show that i_{1} is divisible by p^{n_{1}}.

Consider the additive operation φ: ℤ[x] → ℚ[x] which takes a polynomial g ∈ ℤ[x] to the polynomial p^{−1}x_{1} · g′, where g′ is the partial derivative of g with respect to x_{1}. We have

φ(K) ⊂ K(n_{1} − 1, n_{2} − 1, …, n_{r} − 1) ⊂ K(n_{1} − 1)[x_{2}, …, x_{r}]

and

φ(h_{1}^{j_{1}}h_{2}^{j_{2}} ⋯ h_{r}^{j_{r}}) = −p^{−1}j_{1}h_{1}^{j_{1}−1}h_{2}^{j_{2}} ⋯ h_{r}^{j_{r}} + p^{−1}j_{1}h_{1}^{j_{1}}h_{2}^{j_{2}} ⋯ h_{r}^{j_{r}}.

Since bh^{i_{1}} ⋯ h^{i_{r}} is a monomial of the lowest total degree of the polynomial f, it follows that −bp^{−1}i_{1}h_{1}^{i_{1}−1}h_{2}^{i_{2}} ⋯ h_{r}^{i_{r}} is a monomial of φ(f) considered as a polynomial in h. As

φ(f) ∈ K(n_{1} − 1)[x_{2}, …, x_{r}] ,

we see that −bp^{−1}i_{1}h_{1}^{i_{1}−1} is a monomial of a polynomial from K(n_{1} − 1). It follows that p^{−1}i_{1} is an integer and by Lemma 3.4 below, this integer is divisible by p^{n_{1}−1}. Therefore p^{n_{1}}|i_{1}. □

Lemma 3.4. Let g be a polynomial in h_{1} lying in K(m) for some m ≥ 0. Let bh_{1}^{i−1} be a monomial of g such that i is divisible by p^{m}. Then b is divisible by p^{m}.

Proof. We write h for h_{1} and x for x_{1}. Note that h^{i} ∈ K(m) since i is divisible by p^{m}. Moreover, the quotient ring K(m)/(h^{i}) is additively generated by p^{e(j)}x^{j} with j < i. Indeed, the polynomial x^{i} − (−h)^{i} = x^{i} − (x − 1)^{i} is a linear combination with integer coefficients of p^{e(j)}x^{j} with j < i. Consequently, for any k ≥ 0, multiplying by p^{e(k)}x^{k}, we see that the polynomial p^{e(i+k)}x^{i+k} = p^{e(k)}x^{i+k} modulo the ideal (h^{i}) is a linear combination with integer coefficients of the p^{e(j)}x^{j} with j < i + k.

Thus, K(m)/(h^{i}) is additively generated by p^{e(j)}(1 − h)^{j} with j < i. The only one generator p^{e(i−1)}(1 − h)^{i−1} = p^{m}(1 − h)^{i−1} has nonzero h^{i−1}-coefficient and that coefficient is divisible by p^{m}. □

Let Y be a scheme over the field F. We write CH(Y) for the Chow group of Y and set Ch(Y) = CH(Y)/p CH(Y). We define Ch(\overline{Y}) as the colimit of Ch(Y_{L}) where L runs over all field extensions of F. Thus for any field extension L/F, we have a canonical homomorphism Ch(Y_{L}) → Ch(\overline{Y}). This homomorphism is an isomorphism if Y = P, the variety defined above, and L is a splitting field of P.

We define \overline{Ch}(Y) as the image of the homomorphism Ch(Y) → Ch(\overline{Y}).

Proposition 3.5. We have \overline{Ch}^{j}(P) = 0 for any j > 0.

Proof. Let K_{0}(P) be the Grothendieck group of P. We write K_{0}(\overline{P}) for the colimit of K_{0}(P_{L}) taken over all field extensions L/F. The group K(\overline{P}) is canonically isomorphic to K_{0}(P_{E}) for any splitting field E of P. Each of the groups K_{0}(P) and K_{0}(\overline{P}) is endowed with the topological filtration. The subsequent factor groups G^{j}K_{0}(P) and G^{j}K_{0}(\overline{P}) of these filtrations fit into the commutative square

CH^{j}(\overline{P}) ⟶ G^{j}K_{0}(\overline{P})
↑  ↑
CH^{j}(P) ⟶ G^{j}K_{0}(P)

where the top map is an isomorphism. Therefore it suffices to show that the image of the homomorphism G^{j}K_{0}(P) → G^{j}K_{0}(\overline{P}) is divisible by p for any j > 0.

The ring K_{0}(\overline{P}) is identified with the quotient of the polynomial ring ℤ[h] by the ideal generated by h_{1}^{ind a_{1}}, …, h_{r}^{ind a_{r}}. Under this identification, the element h_{i} is the pull-back to P of the class of a hyperplane in P_{i} over a splitting field and the j-th term K_{0}(\overline{P})^{(j)} of the filtration is generated by the classes of monomials of degree at least j. The group G^{j}K_{0}(\overline{P}) is identified with the group of all homogeneous polynomials of degree j.

The group K_{0}(P) is isomorphic to the direct sum of K_{0}(B), where B = A_{1}^{⊗j_{1}} ⊗ ⋯ ⊗ A_{r}^{⊗j_{r}}, over all j_{i} with 0 ≤ j_{i} < ind a_{i} (cf. [15]). The image of the natural map K_{0}(B) → K_{0}(B_{L}) = ℤ, where L is a splitting field of B, is equal to ind(a_{1}^{j_{1}} ⋯ a_{r}^{j_{r}})ℤ. The image of the homomorphism K_{0}(P) → K_{0}(\overline{P}) (which is in fact an injection) is generated by

ind(a_{1}^{j_{1}} ⋯ a_{r}^{j_{r}})(1 − h_{1})^{j_{1}} ⋯ (1 − h_{r})^{j_{r}}

over all j_{1}, …, j_{r} ≥ 0.

We embed K_{0}(\overline{P}) into the polynomial ring ℤ[x] = ℤ[x_{1}, …, x_{r}] as a subgroup by identifying a monomial h_{1}^{j_{1}} ⋯ h_{r}^{j_{r}} where 0 ≤ j_{i} < ind a_{i} with the polynomial (1 − x_{1})^{j_{1}} ⋯ (1 − x_{r})^{j_{r}}. As the elements a_{1}, …, a_{r} form a minimal basis of D, the index ind(a_{1}^{j_{1}} ⋯ a_{r}^{j_{r}}) is a power of p with the exponent at least e(ind a_{1}, …, ind a_{r}). Therefore,

K_{0}(P) ⊂ K(ind a_{1}, …, ind a_{r}) ⊂ ℤ[x].

An element of K_{0}(P)^{(j)} with j > 0 is a polynomial f in h of degree at least j. The image of f in G^{j}K_{0}(\overline{P}) is the j-th homogeneous part f_{j} of f. As the degree of f with respect to h_{i} is less than ind a_{i}, it follows from Proposition 3.3 that all the coefficients of f_{j} are divisible by p. □

Let d = dim P and α ∈ CH^{d}(P × P). The first multiplicity mult_{1}(α) of α is the image of α under the push-forward map CH^{d}(P × P) → CH^{0}(P) = ℤ given by the first projection P × P → P (cf. [10]). Similarly, we define the second multiplicity mult_{2}(α).

Corollary 3.6. For any element α ∈ CH^{d}(P × P), we have

mult_{1}(α) ≡ mult_{2}(α) modulo p.

Proof. We follow the proof of [9, Th. 2.1]. The homomorphism

f: CH^{d}(P × P) → (ℤ/pℤ)^{2},

taking an α ∈ CH^{d}(P × P) to (mult_{1}(α), mult_{2}(α)) modulo p, factors through the group \overline{Ch}^{d}(P × P). Since for any i, any projection P_{i} × P_{i} → P_{i} is a projective bundle, the Chow group \overline{Ch}^{d}(P × P) is a direct some of several copies of \overline{Ch}^{i}(P) for some i’s and the value i = 0 appears once. By Proposition 3.5, the dimension over ℤ/pℤ of the vector space \overline{Ch}^{d}(P × P) is equal to 1 and consequently the dimension of the image of f is at most 1. Since the image of the diagonal class under f is (1, 1), the image of f is generated by (1, 1). □

Corollary 3.7. Any rational map P −−→ P is dominant.

Proof. Let α ∈ CH^{d}(P × P) be the class of the closure of the graph of a rational map P −−→ P. We have mult_{1}(α) = 1. Therefore, by Corollary 3.6, mult_{2}(α) ≠ 0, and it follows that the rational map is dominant. □

Example 1.4.2 then yields

Corollary 3.8. cdim P = dim P. □

The corollary completes the proof of Theorem 3.1.

Remark 3.9. Theorem 3.1 can be generalized to the case of any finite subgroup D ⊂ Br(F) consisting of elements of p-primary orders. Let {a_{1}, a_{2}, …, a_{r}} be elements of D such that their images {a′_{1}, a′_{2}, …, a′_{r}} in D/D^{p} form a minimal basis, i.e., for any i = 1, … r and any element d ∈ D with the class in D/D^{p} outside of the subgroup generated by a′_{1}, …, a′_{i−1}, the inequality ind d ≥ ind a_{i} holds. Then, as in Theorem 3.1, we have

cdim(D) = (∑_{i=1}^{r} ind a_{i}) − r .

Indeed, the quotient D/D^{p} has the same splitting fields as the group D itself. Therefore, cdim(D) = cdim(D/D^{p}) = cdim(P) for P = P_{1} ×⋯×P_{r}, where P_{i} for every i = 1, …, r is the Severi-Brauer variety of a central division algebra representing the element a_{i}. Corollaries 3.6, 3.7 and 3.8 hold for P since K_{0}(P) ⊂ K(ind a_{1}, …, ind a_{r}).

4. Main theorem

The main result of the paper is the following

Theorem 4.1. Let G be a p-group and F a field of characteristic different from p containing a primitive p-th root of unity. Then ed_{F} (G) coincides with the least dimension of a faithful representation of G over F.

The rest of the paper is devoted to the proof of the theorem. As was mentioned in the introduction, we have ed_{F} (G) ≤ dim(V) for any faithful representation of G over F. We shall construct a faithful representation of G over F with ed_{F} (G) ≥ dim(V).

Denote by C the subgroup of all central elements of G of exponent p and set H = G/C, so we have an exact sequence

(2)  1 → C → G → H → 1.

Let E → Spec F be an H-torsor and Spec F → BH be the corresponding morphism. Set X^{E} := BG ×_{BH} Spec F. Then X^{E} is a gerbe over F banded by C and its class in H^{2}(F, C) coincides with the image of the class of E under the connecting map H^{1}(F, H) → H^{2}(F, C) (cf. [14, Ch. 4, §2]). An object of X^{E} over a field extension L/F is a pair (E′, α), where E′ is a G-torsor over L and α : E′/C →^{∼} E_{L} is an isomorphism of H-torsors over L.

Alternatively. X^{E} = [E/G] (cf. [17]) with objects (over L) G-equivariant morphisms E′ → E_{L}, where E′ is a G-torsor over L.

The following lower bound for ed(G) was established in [4, Prop. 2.20]. We give a short proof here for completeness.

Theorem 4.2. For any H-torsor E over F, we have ed_{F} (G) ≥ ed(X^{E}).

Proof. Let L/F be a field extension and x = (E′, α) an object of X^{E}(L). Choose a subfield L_{0} ⊂ L over F such that tr. deg(L_{0}) = ed(E′) and there is a G-torsor E′_{0} over L_{0} with (E′_{0})_{L} ≃ E′.

Let Z be the (zero-dimensional) scheme of isomorphisms Iso_{L_{0}}(E′_{0}/C, E_{L_{0}}) of H-torsors over L_{0}. The image of the morphism Spec L → Z over L_{0} representing the isomorphism α is a one point set {z} of Z. The field extension L_{0}(z)/L_{0} is algebraic since dim Z = 0.

The isomorphism α descents to an isomorphism of the H-torsors E′/C and E over L_{0}(z). Hence the isomorphism class of x belongs to the image of the map X̂^{E}(L_{0}(z)) → X̂^{E}(L). Therefore,

ed_{F} (G) ≥ ed(E′) = tr. deg(L_{0}) = tr. deg(L_{0}(z)) ≥ ed(x).

It follows that ed_{F} (G) ≥ ed(X^{E}). □

Let C^{∗} := Hom(C, G_{m}) denote the character group of C. An H-torsor E → Spec F yields a homomorphism

β^{E} : C^{∗} → Br(F)

taking a character χ : C → G_{m} to the image of the class of E under the composition

H^{1}(F, H) ⟶^{∂} H^{2}(F, C) ⟶^{χ^{∗}} H^{2}(F, G_{m}) = Br(F),

where ∂ is the connecting map for the exact sequence (2). Note that as μ_{p} ⊂ F^{×}, the intersection of Ker(χ^{∗}) over all characters χ ∈ C^{∗} is trivial. It follows that the classes of splitting fields of the gerbe X^{E} and the subgroup Im(β^{E}) coincide. It follows that

(3)  cdim(X^{E}) = cdim(Im(β^{E})).

Let χ_{1}, χ_{2}, …, χ_{s} be a basis of C^{∗} over ℤ/pℤ such that {β^{E}(χ_{1}), …, β^{E}(χ_{r})} is a minimal basis of Im(β^{E}) for some r and β^{E}(χ_{i}) = 1 for i > r. By Theorem 3.1, we have

(4)  cdim(Im(β^{E})) = (∑_{i=1}^{r} ind β^{E}(χ_{i})) − r = (∑_{i=1}^{s} ind β^{E}(χ_{i})) − s.

In view of Theorem 4.2 and (3), we shall find an H-torsor E (over a field extension of F) so that the integer in (4) is as large as possible. We embed H into X := GL_{n} as a subgroup for some n and set Y := X/H. Let E be the generic fiber of the H-torsor π : X → Y. It is a “generic” H-torsor over the function field L := F (Y).

Let χ : C → G_{m} be a character. Let Rep^{(χ)}(G) be the category of all finite dimensional representations ρ of G such that ρ(c) is multiplication by χ(c) for any c ∈ C. Choose a representations ρ : G → GL(V) in Rep^{(χ)}(G). The conjugation action of G on B := End(V) factors through an H-action. By descent (cf. [14, Ch. 1, §2]), there is (a unique up to canonical isomorphism) Azumaya algebra A over Y and an H-equivariant algebra isomorphism π^{∗}(A) ≃ B_{X}. Let A be the generic fiber of A; it is a central simple algebra over L = F (Y).

Consider the homomorphism β^{E} : C^{∗} → Br(L).

Lemma 4.3. The class of A in Br(L) coincides with β^{E}(χ).

Proof. Consider the commutative diagram

1 ⟶ C ⟶ G ⟶ H  ⟶ 1
χ↓  ^{ρ}↓  ^{α}↓
1 ⟶ G_{m} ⟶ GL(V) ⟶ PGL(V) ⟶ 1

The image of the H-torsor π : X → Y under α is the PGL(V)-torsor

E′ := PGL(V)_{X}/H → Y

where PGL(V)_{X} := PGL(V) × X and H acts on PGL(V)_{X} by h(a, x) = (ah^{−1}, hx). The conjugation action of PGL(V) on B gives rise to an isomorphism between PGL(V)_{X} and the H-torsor Iso_{X}(B_{X}, End(V)_{X}) of isomorphisms between the (split) Azumaya O_{X}-algebras B_{X} and End(V)_{X}. Note that this isomorphism is H-equivariant if H acts by conjugation on B_{X} and trivially on End(V)_{X}. By descent,

E′ ≃ Iso_{Y} (A, End(V)_{Y} ).

Therefore, the image of the class of the torsor E′ → Y under the connecting map for the bottom row of the diagram coincides with the class of the Azumaya algebra A. Restricting to the generic fiber yields [A] = β^{E}(χ). □

Theorem 4.4. For any character χ ∈ C^{∗}, we have ind β^{E}(χ) = min dim(V) over all representations V in Rep^{(χ)}(G).

Proof. We follow the approach given in [13]. Let G act on a scheme Z over F. Denote by M(G, Z) the (abelian) category of left G-modules on Z that are coherent O_{Z}-modules (cf. [16, §1.2]). In particular, M(G, Spec F) = Rep(G), the category off all finite dimensional representations of G.

For a character χ : C → G_{m}, let M^{(χ)}(G, Z) be the full subcategory of M(G, Z) consisting of G-modules on which C acts via χ. For example, M^{(χ)}(G, Spec F) = Rep^{(χ)}(G).

We write K_{0}(G, Z) and K_{0}^{(χ)}(G, Z) for the Grothendieck groups of M(G, Z) and M^{(χ)}(G, Z) respectively.

By representation theory of diagonalizable group schemes, every M in M(G, Z) is a direct sum of unique submodules M^{(χ)} of M in M^{(χ)}(G, Z) over all characters χ of C. It follows that

K_{0}(G, Z) = ∐ K_{0}^{(χ)}(G, Z).

By Lemma 4.3, it suffices to show that ind(A) = gcd dim(V) over all representations V in Rep^{(χ)}(G).

The image of the map dim : K_{0}(A) → ℤ given by the dimension over L is equal to ind(A) · dim(V) · ℤ. To finish the proof of the theorem is suffices to construct a surjective homomorphism

(5)  K_{0}(Rep^{(χ)}(G)) → K_{0}(A)

so that the composition K_{0}(Rep^{(χ)}(G)) → K_{0}(A) ⟶^{dim} ℤ is given by the dimension times dim(V).

First of all we have

(6)  K_{0}(Rep^{(χ)}(G)) ≃ K_{0}^{(χ)}(G, Spec F).

Recall that H is embedded into X = GL_{n}. Next, we embed X into 𝔸^{m}, where m = n^{2}, as an open set in the standard way. The G-action on X extends to a linear G-action on 𝔸^{m}. By homotopy invariance in the equivariant Ktheory [16, Cor. 4.2],

K_{0}(G, Spec F) ≃ K_{0}(G, 𝔸^{m}).

It follows that

(7)  K_{0}^{(χ)}(G, Spec F) ≃ K_{0}^{(χ)}(G, 𝔸^{m}).

By localization [16, Th. 2.7], the restriction homomorphism

(8)  K_{0}^{(χ)}(G, 𝔸^{m}) → K_{0}^{(χ)}(G, X).

is surjective.

Denote by M^{(1)}(G, X, B_{X}) the category of left G-modules M on X that are coherent O_{X}-modules and right B_{X}-modules such that C acts trivially on M and the G-action on M and the conjugation G-action on B_{X} agree. The corresponding Grothendieck group is denoted by K_{0}^{(1)}(G, X, B_{X}). For any object L in M^{(χ)}(G, X), the group C acts trivially on L ⊗_{F} V^{∗} and B acts on the right on L ⊗_{F} V^{∗}. We have Morita equivalence

M^{(χ)}(G, X) →^{∼} M^{(1)}(G, X, B_{X})

given by L ↦ L ⊗_{F} V^{∗} (with the inverse functor M ↦ M ⊗_{B} V). Hence

(9)  K_{0}^{(χ)}(G, X) ≃ K_{0}^{(1)}(G, X, B_{X}).

Now, as C acts trivially on X and B_{X}, the category M^{(1)}(G, X, B_{X}) is equivalent to the category M(H, X, B_{X}). Hence

(10)  K_{0}^{(1)}(G, X, B_{X}) ≃ K_{0}(H, X, B_{X}).

Recall that Y = X/H. By descent, the category M(H, X, B_{X}) is equivalent to the category M(Y, A) of coherent O_{Y} -modules that are right A-modules. Hence

(11)  K_{0}(H, X, B_{X}) ≃ K_{0}(Y, A).

The restriction to the generic point of Y gives a surjective homomorphism

(12)  K_{0}(Y, A) → K_{0}(A).

Finally, the homomorphism (5) is the composition of (6), (7), (8), (9), (10),

(11) and (12). □

We can now complete the proof of Theorem 4.1. By Theorem 4.4, there are representations V_{i} in Rep^{(χ_{i})}(G) such that ind β^{E}(χ_{i}) = dim V_{i}, i = 1, …, s. Let V be the direct sum of all the V_{i}. By Theorem 4.2 (applied to the group G over L and the generic torsor E), Theorem 2.1, (3) and (4), we have

ed_{F} (G) ≥ ed_{L}(G) ≥ ed(X^{E}) = cdim(X^{E}) + s = cdim(Im(β^{E})) + s

= ∑_{i=1}^{s} ind β^{E}(χ_{i}) = ∑_{i=1}^{s} dim(V_{i}) = dim(V).

Since χ_{1}, χ_{2}, …, χ_{s} generate C^{∗}, the restriction of V on C is faithful. As every nontrivial normal subgroup of G intersects C nontrivially, the G-representation V is faithful. We have constructed a faithful representation V of G over F with ed_{F} (G) ≥ dim(V). The theorem is proved.

Remark 4.5. The proof of Theorem 4.1 shows how to compute the essential dimension of G over F. For every character χ ∈ C^{∗} choose a representation V_{χ} ∈ Rep^{(χ)}(G) of the smallest dimension. It appears as an irreducible component of the smallest dimension of the induced representation Ind_{C}^{G}(χ). We construct a basis χ_{1}, …, χ_{s} of C^{∗} by induction as follows. Let χ_{1} be a nonzero character with the smallest dim(V_{χ_{1}}). If the characters χ_{1}, …, χ_{i−1} are already constructed for some i ≤ s, then we take for χ_{i} a character with minimal dim(V_{χ_{i}}) among the characters outside of the subgroup generated by χ_{1}, …, χ_{i−1}. Then ed_{F} (G) = ∑_{i=1}^{s} dim(V_{χ_{i}}).

5. An application

Theorem 5.1. Let G_{1} and G_{2} be two p-groups and F a field of characteristic different from p containing a primitive p-th root of unity. Then

ed_{F} (G_{1} × G_{2}) = ed_{F} (G_{1}) + ed_{F} (G_{2}).

Proof. The index j in the proof takes two values 1 and 2. If V_{j} is a faithful representation of G_{j} then V_{1} ⊕V_{2} is a faithful representation of G_{1} ×G_{2}. Hence ed_{F} (G_{1} × G_{2}) ≤ ed_{F} (G_{1}) + ed_{F} (G_{2}) (cf. [5, Lemma 4.1(b)]).

Denote by C_{j} the subgroup of all central elements of G_{j} of exponent p. Set C = C_{1} × C_{2}. We identify C^{∗} with C_{1}^{∗} ⊕ C_{2}^{∗}.

For every character χ ∈ C^{∗} choose a representation ρ_{χ} : G_{1} × G_{2} → GL(V_{χ}) in Rep^{(χ)}(G_{1} × G_{2}) of the smallest dimension. We construct a basis {χ_{1}, χ_{2}, …, χ_{s}} of C^{∗} following Remark 4.5. We claim that all the χ_{i} can be chosen in one of the C_{j}^{∗}. Indeed, suppose the characters χ_{1}, …, χ_{i−1} are already constructed, and let χ_{i} be a character with minimal dim(V_{χ_{i}}) among the characters outside of the subgroup generated by χ_{1}, …, χ_{i−1}. Let χ_{i} = χ_{i}^{(1)} + χ_{i}^{(2)} with χ_{i}^{(j)} ∈ C_{j}^{∗}. Denote by ε_{1} and ε_{2} the endomorphisms of G_{1} × G_{2} taking (g_{1}, g_{2}) to (g_{1}, 1) and (1, g_{2}) respectively. The restriction of the representation ρ_{χ_{i}} ∘ ε_{j} on C is given by the character χ_{i}^{(j)}. We replace χ_{i} by χ_{i}^{(j)} with j such that χ_{i}^{(j)} does not belong to the subgroup generated by χ_{1}, …, χ_{i−1}. The claim is proved.

Let W_{j} be the direct sum of all the V_{χ_{i}} with χ_{i} ∈ C_{j}^{∗}. Then the restriction of W_{j} on C_{j} is faithful, hence so is the restriction of W_{j} on G_{j}. It follows that ed_{F} (G_{j}) ≤ dim(W_{j}). As W_{1} ⊕ W_{2} = V, we have

ed_{F} (G_{1}) + ed_{F} (G_{2}) ≤ dim(W_{1}) + dim(W_{2}) = dim(V) = ed_{F} (G_{1} × G_{2}). □

Denote by ξ_{p^{m}} a primitive p^{m}-th root of unity (in a separable closure of F).

Corollary 5.2. Let F be a field as in Theorem 5.1. Then

ed_{F} (ℤ/p^{n_{1}}ℤ × ℤ/p^{n_{2}}ℤ × ⋯ × ℤ/p^{n_{s}}ℤ) = ∑_{i=1}^{s} [F (ξ_{p^{n_{i}}}) : F ].

Proof. By Theorem 5.1, it suffices to consider the case s = 1. This case has been done in [6]. It is also covered by Theorem 4.1 as the natural representation of the group ℤ/p^{n}ℤ in the F -space F (ξ_{p^{n}}) is faithful irreducible of the smallest dimension. □

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Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie - Paris 6, 4 place Jussieu, F-75252 Paris CEDEX 05, FRANCE

E-mail address: karpenko@math.jussieu.fr

Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

E-mail address: merkurev@math.ucla.edu