ESSENTIAL DIMENSION OF SIMPLE ALGEBRAS WITH INVOLUTIONS

SANGHOON BAEK

Abstract. Let 1 ≤ m ≤ n be integers with m|n and Alg_{n,m} the class of central simple algebras of degree n and exponent dividing m. In this paper, we find upper bounds for the essential (2)-dimension of Alg_{n,2}. Moreover, we find a stronger upper bound for the essential 2-dimension of Alg_{n,2} over a field F of char(F) ≠ 2. As a result, we show that ed_{2}(Alg_{16,2}) = 24 over a field F of char(F) ≠ 2.

1. Introduction

Let T : Fields/F → Sets be a functor from the category Fields/F of field extensions over F to the category Sets of sets. For fields E, E′ ∈ Fields/F, a field homomorphism f : E → E′ over F and α ∈ T (E), we write α_{E′} for the image of α under the morphism T (f) : T (E) → T (E′).

Let E ∈ Fields/F and K ⊂ E a subfield over F. An element α ∈ T (E) is said to be defined over K and K is called a field of definition of α if there exists an element β ∈ T (K) such that β_{E} = α. The essential dimension of α is ed(α) = min{tr. deg_{F} (K)} over all fields of definition K of α. The essential dimension of the functor T is ed(T ) = sup{ed(α)}, where the supremum is taken over all fields E ∈ Fields/F and all α ∈ T (E). Hence, the essential dimension of an algebraic structure T measures the complexity of the structure in terms of the smallest number of parameters required to define the structure over a field extension of F.

Let p be a prime integer. The essential p-dimension of α is ed_{p}(α) = min{ed(α_{L})}, where L ranges over all field extensions of E of degree prime to p. In other words, ed_{p}(α) = min{tr. deg_{F} (K)}, where the minimum is taken over all field extensions L/E of prime to p and all subextensions K/F of L which are field of definition of α_{L}. Hence, ed(α) ≥ ed_{p}(α) for all p. The essential p-dimension of F is ed_{p}(T ) = sup{ed_{p}(α)}, where the supremum ranges over all fields E ∈ Fields/F and all α ∈ T (E).

Let G be an algebraic group over F. The essential dimension ed(G) (respectively, essential p-dimension ed_{p}(G)) of G is defined to be ed(H^{1}(−, G)) (respectively, ed_{p}(H^{1}(−, G))), where H^{1}(E, G) is the Galois cohomology set (equivalently, the set of isomorphism classes of G-torsors) over a field extension E of F.

For every integer n ≥ 1, a divisor m of n and any field extension E/F, let Alg_{n,m}(E) denote the set of isomorphism classes of central simple E-algebras of degree n and exponent dividing m. Then, for any field extension E/F, there is a natural bijection between H^{1}(E, GL_{n} /μ_{m}) and Alg_{n,m}(E) (see [1, Example 1.1]), thus ed(Alg_{n,m}) = ed(GL_{n} /μ_{m}) and ed_{p}(Alg_{n,m}) = ed_{p}(GL_{n} /μ_{m}).

In this paper, we compute upper bounds for the essential dimension of Alg_{n,2}. By a theorem of Albert, a central simple algebra has exponent dividing 2 if and only if it admits an involution of the first kind (see [3, Theorem 3.1]). Thus, any algebra A in Alg_{n,2}(K) for any field extension K/F has involutions of the first kind. Moreover, such A has involutions of both symplectic and orthogonal types (see [3, Corollary 2.8(2)]). By the primary decomposition theorem and [2, Section 6], we have ed(Alg_{n,2}) = ed(Alg_{2^{r},2}) and ed_{2}(Alg_{n,2}) = ed_{2}(Alg_{2^{r},2}), where 2^{r} is the largest power of 2 dividing n. Hence, we may assume that n is a power of 2.

By [2, Remark 8.2 and Corollary 8.3], ed_{2}(Alg_{4,2}) = ed(Alg_{4,2}) = 4 and ed_{2}(Alg_{8,2}) = ed(Alg_{8,2}) = 8 over a field F of char(F) ≠ 2. In general, by [2, Theorem], the following bounds were established over a field F of char(F) ≠ 2:

2^{r−1}(r − 1) ≤ ed_{2}(Alg_{2^{r},2}) ≤ 2^{r−1}(2^{r−1} + 1) for all r ≥ 2.

In the present paper, we find an upper bound n(n − 1)/2 for the essential dimension of Alg_{n,2} in Corollary 2.2. Moreover, we find an upper bound 2^{2r−2} for the essential 2-dimension of Alg_{2^{r},2} in Corollary 2.4. Both upper bounds are valid over an arbitrary field F. In particular, the bound 2^{2r−2} improves the bound 2^{2r−2} + 2^{r−1} as above.

Using involutions of the first kind, we further improve the upper bound 2^{2r−2} as follows: Theorem. Let F be a field of characteristic different from 2. Then, for any integers r ≥ 3,

ed_{2}(Alg_{2^{r},2}) ≤ 2^{r−1}(2^{r−3} + 1).

As a result, we find the essential 2-dimension of Alg_{16,2}: Corollary. Let F be a field of characteristic different from 2. Then

ed_{2}(Alg_{16,2}) = 24.

Remark 1.1. Recently, V. Chernousov and A. Merkurjev proved that

ed_{p}(SL_{p^{r}} /μ_{p^{s}}) = ed_{p}(Alg_{p^{r},p^{s}}) + 1

for any 0 ≠ s < r over a field of char(F) ≠ p (this result is communicated to the author by A. Merkurjev). Therefore, the computation of essential pdimension of split simple group of type A_{p^{r}−1} is reduced to the computation of ed_{p}(Alg_{p^{r},p^{s}}). In particular, we have ed_{2}(SL_{16} /μ_{2}) = 25 over a field of char(F) ≠ 2.

Acknowledgements: I am grateful to my advisor A. Merkurjev for many useful discussions and support and to Z. Reichstein for helpful comments. Section 2 of this paper is based on the author’s doctoral thesis at the University of California at Los Angeles.

2. Upper bounds for the essential dimension of Alg_{n,2}

Let G be a reductive algebraic group over F, let T be a maximal torus of G and let N be the normalizer of T in G. Then the canonical map

H^{1}(K, N) → H^{1}(K, G)

is surjective for any field extension K/F by Springer’s Lemma ([7, III.4 Lemma 6]). Therefore, we have

(1)  ed(G) ≤ ed(N)

by [4, Proposition 1.3].

For any integer n ≥ 2, consider a reductive group GL_{n} /μ_{2} and the maximal torus T_{n,2} := 𝔾_{m}^{n} /μ_{2} in the group.

Lemma 2.1. Let F be an arbitrary base field and S_{n} be the symmetric group on n elements. Then for any n ≥ 3, we have

ed(T_{n,2} ⋊ S_{n}) ≤ (n^{2} − n)/2.

Proof. Note that the character group (T_{n,2})^{∗} is isomorphic to

{(t_{1}, ⋯, t_{n}) ∈ ℤ^{n}| t_{1} + ⋯ + t_{n} = 0 in ℤ/2ℤ}.

Let e_{i,j} = (0, ⋯, 1, ⋯, −1, 0) be an element of (T_{n,2})^{∗}, where 1 and −1 are placed in the ith and jth positions respectively for 1 ≤ i ≠ j ≤ n and 0’s are placed in other positions. Similarly, let f_{i,j} = (0, ⋯, 1, ⋯, 1, 0), where 1’s are placed in the ith and jth positions for 1 ≤ i ≠ j ≤ n and 0’s are placed in other positions and let g_{k} = (0, ⋯, −2, ⋯, 0) as an element of (T_{n,2})^{∗}, where −2 is placed in the kth position for 1 ≤ k ≤ n and 0’s are placed in other positions.

Let X be a set consisting of f_{i,j} and g_{k} for all 1 ≤ i ≠ j ≤ n and all 1 ≤ k ≤ n. Then X is a S_{n}-invariant subset of (T_{n,2})^{∗} and |X| = |f_{i,j}| + |g_{k}| = (n^{2} − n)/2 + n.

It is clear that e_{i,j} and f_{i,j} generate (T_{n,2})^{∗} as an abelian group, as the indices i and j run over 1 to n. Since f_{i,j} + g_{j} = e_{i,j}, X generates (T_{n,2})^{∗} as an abelian group and hence we have a surjective S_{n}-equivariant homomorphism ν : ℤ[X] → (T_{n,2})^{∗} taking f_{i,j} and g_{k} to themselves.

We show that S_{n} acts faithfully on Ker(ν). Let σ be a nontrivial element of S_{n}. Then there exists 1 ≤ i_{0} ≤ n such that σ(i_{0}) ≠ i_{0}. Choose a 1 ≤ j_{0} ≤ n which is different from σ(i_{0}) and i_{0}. Then σ does not fix 2f_{i_{0},j_{0}} + g_{i_{0}} + g_{j_{0}} ∈ Ker(ν). By [5, Lemma 3.3], ed(T_{n,2} ⋊ S_{n}) ≤ (n^{2} − n)/2 + n − rank((T_{n,2})^{∗}) = (n^{2} − n)/2.

□

By (1), we have an upper bound for ed(Alg_{n,2}) as follows:

Corollary 2.2. Let F be an arbitrary base field. Then for any n ≥ 3,

ed(Alg_{n,2}) ≤ (n^{2} − n)/2.

Let P_{n} be a Sylow 2-subgroup of the symmetric group S_{n} on n elements. In the following Lemma, we compute an upper bound for the essential dimension of T_{2^{r},2} ⋊ P_{2^{r}}.

Lemma 2.3. Let F be an arbitrary base field. Then for any r ≥ 2, we have

ed(T_{2^{r},2} ⋊ P_{2^{r}}) ≤ 2^{2r−2},

where P_{2^{r}} is a Sylow 2-subgroup of S_{2^{r}}.

Proof. Note that a Sylow 2-subgroup P_{2^{r}} of S_{2^{r}} is isomorphic to (P_{2^{r−1}})^{2} ⋊ ℤ/2ℤ.

Consider the e_{i,j}, f_{i,j} and g_{k} as in the proof of Lemma 2.1. We divide the set of integers {1, 2, ⋯, 2^{r}} into two subsets Λ_{1} := {1, 2, ⋯, 2^{r−1}} and Λ_{2} := {2^{r−1} + 1, 2^{r−1} + 2, ⋯, 2^{r}}. Let X be a set consisting of f_{i,j} and g_{k} for all 1 ≤ i ≠ j ≤ 2^{r} such that i and j are placed in different Λ_{l}’s and all 1 ≤ k ≤ 2^{r}, where l is either 0 or 1. Then X is a P_{2^{r}}-invariant subset of (T_{2^{r},2})^{∗} and |X| = 2^{2r−2} + 2^{r}.

It is clear that e_{i,j} and f_{i,j} generate (T_{2^{r},2})^{∗} as an abelian group, as the indices i and j run over 1 to 2^{r}. Note that f_{i,j} = f_{i,k} + f_{j,k} + g_{k} for all i and j which are in the same Λ_{l}’s, where l is either 1 or 2. As

e_{i,j} = \begin{cases}f_{i,j} + g_{j} & if i and j are in different Λ_{l}’s, \\ f_{i,k} + f_{j,k} + g_{j} + g_{k} & otherwise, \end{cases}

X generates (T_{2^{r},2})^{∗} as an abelian group and hence we have a surjective P_{2^{r}}- equivariant homomorphism ν : ℤ[X] → (T_{2^{r},2})^{∗} taking f_{i,j} and g_{k} to themselves.

We show that P_{2^{r}} acts faithfully on Ker(ν). Note that the center of P_{2^{r}}, which is generated by σ := (1, 2)(3, 4) ⋯ (2^{r} − 1, 2^{r}) and it is enough to show that σ acts faithfully on Ker(ν). In fact, σ does not fix the non-zero element 2f_{1,2^{r−1}+1} + g_{1} + g_{2^{r−1}+1} ∈ ℤ[X]. By [5, Lemma 3.3], we have

ed(T_{2^{r},2} ⋊ P_{2^{r}}) ≤ 2^{2r−2} + 2^{r} − rank((T_{2^{r},2})^{∗}) = 2^{2r−2}.
□

As (2, [T_{2^{r},2}⋊S_{2^{r}} : T_{2^{r},2}⋊P_{2^{r}}]) = 1, we have ed_{2}(T_{2^{r},2}⋊S_{2^{r}}) = ed_{2}(T_{2^{r},2}⋊P_{2^{r}}) by [5, Lemma 4.1]. Therefore, by Lemma 2.3, we have the following Corollary:

Corollary 2.4. Let F be an arbitrary base field. Then for any r ≥ 2,

ed_{2}(Alg_{2^{r},2}) ≤ 2^{2r−2}.

3. Algebras with involutions

Let A be a central simple algebra over F. For any a ∈ A^{×}, we denote the inner automorphism of A by Int(a): Int(a)(x) = axa^{−1} for all x ∈ A. For any subalgebra B of A, we write C_{A}(B) for the centralizer of B in A. The following Lemma characterizes all involutions of the first kind on A.

Lemma 3.1. [3, Proposition 2.7] Let F be a field of char(F) ≠ 2, A be a central simple algebra over F and σ be an involution of the first kind on A. Then every involution σ′ of the first kind on A is of the form Int(a) ∘ σ for some a ∈ A^{×} uniquely determined up to a factor in F^{×}, such that σ(a) = ±a. Moreover, σ and σ′ are of the same type if and only if σ(a) = a.

We use the following Lemma for extension of involutions:

Lemma 3.2. [3, Theorem 4.14] Let F be a field of char(F) ≠ 2, A be a central simple algebra over F with an involution σ of the first kind, and B be a simple subalgebra of A with an involution τ such that τ|_{F} = σ|_{F}. Then A has involutions of both types whose restriction to B is τ, unless τ is of the first kind and deg(C_{A}(B)) is odd.

From now we assume that the base field F is 2-closed (i.e., every finite extension of F is separable of degree a power of 2) and is of characteristic different from 2.

Proposition 3.3. Let r ≥ 3 be an integer, F a 2-closed field such that char(F) ≠ 2 and D a division F -algebra of degree 2^{r} and exponent 2. Then for any biquadratic field extension K_{1}K_{2}/F in D with quadratic field extensions K_{1}/F and K_{2}/F there exists a quadratic extension K_{3}/F in D such that K_{1}K_{2}K_{3}/F is a triquadratic extension in D.

Proof. By [3, Theorem 3.1(1)], D has an involution of the first kind σ. Let τ_{1} and τ_{2} be two distinct nontrivial automorphisms of the field K_{1}K_{2}. As σ|_{F} = τ_{i}|_{F} for any i = 1, 2, there are two distinct involutions σ_{1} and σ_{2} of the same type on A such that σ_{i}|_{K_{1}K_{2}} = τ_{i} by Lemma 3.2.

By Lemma 3.1, there exists d ∈ D^{×} such that σ_{1} = Int(d) ∘ σ_{2} and σ_{i}(d) = d for all i = 1, 2. In particular, d^{2} commutes with K_{1} and K_{2} and F (d^{2})∩K_{1}K_{2} = F. If F (d^{2}) ≠ F, then F (d^{2}) contains a quadratic extension K_{3} over F by [6, Proposition 1.1]. Hence we a triquadratic extension K_{1}K_{2}K_{3} in D.

Suppose that d^{2} ∈ F. Then there exist quaternion subalgebras Q_{1} := (K_{1}, d^{2}) and Q_{2} := (K_{2}, d^{2}) of D. As ind(C_{D}(Q_{1} ⊗ Q_{2})) ≥ 2, C_{D}(Q_{1} ⊗ Q_{2}) contains a quadratic extension K_{3}/F by [6, Proposition 1.1]. Therefore, we have K_{1}K_{2}K_{3} = K_{1} ⊗ K_{2} ⊗ K_{3} ⊂ Q_{1} ⊗ Q_{2} ⊗ C_{D}(Q_{1} ⊗ Q_{2}) = D. □

Corollary 3.4. Let r ≥ 3 be an integer and F be a 2-closed field such that char(F) ≠ 2. Then for any division F -algebra D of degree 2^{r} and exponent 2 and an étale subalgebra K_{1}K_{2} := K_{1} ⊗ K_{2} of D such that dim_{F} (K_{i}) = 2 for i = 1, 2, there exists a maximal étale subalgebra K_{1}K_{2}K := K_{1} ⊗ K_{2} ⊗ K of D with dim_{F} (K) = 2^{r−2}.

Proof. By Proposition 3.3, there exists a triquadratic field extension K_{1}K_{2}K_{3} over F. Induction on r. If r = 3, then K = K_{3} satisfies the conclusion of Corollary. For r ≥ 3, the centralizer C_{D}(K_{3}) is a division K_{3}-algebra of degree 2^{r−1}. By the induction hypothesis with K_{1}K_{3}/K_{3} and K_{2}K_{3}/K_{3}, C_{D}(K_{3}) contains a subfield K/F with [K : K_{3}] = 2^{r−3}. Hence D contains a field extension K_{1}K_{2}K over F such that dim_{F} (K) = 2^{r−3} · 2. □

4. Essential 2-dimension of Alg_{n,2}

Let n ≥ 2 be an integer, G be a subgroup of S_{n} and X be a G-set of n elements (G acts on X by permutation). For any divisor m of n, we consider the surjective G-modules homomorphism ε̄: ℤ[X] → ℤ/mℤ, defined by ε̄(x) = ε(x) + mℤ, where ε : ℤ[X] → ℤ is the augmentation homomorphism given by ε(x) = 1 for all x ∈ X. Set J = Ker(ε̄). Then we have an exact sequence

(2)  0 → J → ℤ[X] ⟶^{ε̄} ℤ/mℤ → 0.

We shall need the following lemma (see also the proof of [2, Theorem 8.1]):

Lemma 4.1. Let F be a field of char(F) |̸ n and T = Spec F [J] be the split torus with the character group J. Then

H^{1}(F, T ⋊ G) = ∐_{Gal(E/F)=G} Br_{m}(E/F),

where the disjoint union is taken over all isomorphism classes of Galois Galgebras E/F.

Proof. Let T_{γ} (respectively, G_{γ}) be the twist of T (respectively, G) by the 1-cocycle γ ∈ Z^{1}(F, G). Then by [3, Proposition 28.11], there is a natural bijection between the fiber of H^{1}(F, T ⋊ G) → H^{1}(F, G) over [γ] and the orbit set of the group G_{γ}(F) in H^{1}(F, T_{γ}), i.e.,

(3)  H^{1}(F, T ⋊ G) ≃ ∐ H^{1}(F, T_{γ})/G_{γ}(F),

where the coproduct is taken over all [γ] ∈ H^{1}(F, G).

Let E be the corresponding Galois G-algebra over F to γ. From (2), we have the corresponding exact sequence of algebraic groups

1 → μ_{m} → 𝔾_{m}^{n} → T → 1

and then the exact sequence

(4)  1 → μ_{m} → R_{E/F} (𝔾_{m,E}) → T_{γ} → 1,

each term of which is twisted by γ. The exact sequence (4) induces an exact sequence of Galois cohomology

(5) 1 → H^{1}(F, T_{γ}) → H^{2}(F, μ_{m}) = Br_{m}(F) → H^{2}(E, 𝔾_{m,E}) = Br(E)

by Eckmann-Faddeev-Shapiro’s Lemma and Hilbert’s 90. The G-action on R_{E/F} (𝔾_{m,E}) restricts to the trivial action on the subgroup μ_{m}. Let σ ∈ G_{γ}(F) acts on T_{γ} = R_{E/F} (𝔾_{m,E})/μ_{m}. The action of σ and (5) induce the following diagram

[figure: H^{1}(F, T_{γ}); σ^{∗}; H^{2}(F, μ_{m}); H^{1}(F, T_{γ}); H^{2}(F, μ_{m}).]

Therefore, G_{γ}(F) acts trivially on H^{1}(F, T_{γ}), hence the result follows by (3). □

Let r ≥ 3 be an integer. Let G_{r} = S_{2} × S_{2} × S_{2^{r−2}} be a subgroup of the symmetric group S_{2^{r}} on 2^{r} elements and let H_{r} = S_{2} × S_{2} × P_{2^{r−2}} be a Sylow 2-subgroup of G_{r}, where P_{2^{r−2}} is a Sylow 2-subgroup of S_{2^{r−2}}. Let X_{r} be a G_{r}-set of 2^{r} elements (G_{r} acts on X_{r} by permutations). The action of H_{r} may be described as follows: we subdivide the integers 1, 2, ⋯, 2^{r} into four blocks B_{1}, B_{2}, B_{3}, B_{4} such that each block consists of 2^{r−2} consecutive integers. The P_{2^{r−2}} permutes the elements of B_{i} for all 1 ≤ i ≤ 4, S_{2} interchanges B_{2i−1} and B_{2i} for all i = 1, 2, and another S_{2} interchanges B_{1} ∪ B_{2} and B_{3} ∪ B_{4}.

We set J_{r} = Ker(ℤ[X_{r}] ⟶^{ε̄} ℤ/2ℤ), where ε̄ is the map with m = 2 as in (2). Applying Lemma 4.1 with n = 2^{r}, m = 2, G = G_{r}, X = X_{r}, J = J_{r}, and T = T_{r} := Spec(F [J_{r}]), we have a morphism

θ : H^{1}(−, T_{r} ⋊ G_{r}) → Alg_{2^{r},2}

defined by θ(N)([A]) = B for a field extension N over F, where [A] ∈ Br_{2}(L/N) for some field extension L/N with Gal(L/N) = G_{r} and B is the central simple N-algebra of degree 2^{r} such that [A] = [B] in Br_{2}(L/N).

We also have a morphism

(6)  Θ : H^{1}(−, T_{r} ⋊ G_{r}) ∐ ( ∐_{1≤i≤r−1} Alg_{2^{i},2}) → Alg_{2^{r},2}

defined by

[A] ↦ θ(N)([A]), A_{i} ↦ M_{2^{r−i}}(A_{i})

over a field extension N over F, where A_{i} ∈ Alg_{2^{i},2}(N) for 1 ≤ i ≤ r − 1.

Lemma 4.2. If the base field F is 2-closed and is of characteristic different 2, then Θ is surjective.

Proof. We show that Θ(N) is surjective for a field extension N/F. By the definition of Θ, we only need to check the surjectivity for a division N-algebra D of degree 2^{r} and exponent 2. By [6, Theorem 1.2], there exists an étale subalgebra K_{1}K_{2} in D such that dim_{N}(K_{i}) = 2 for i = 1, 2. By Corollary 3.4, there exists a maximal étale subalgebra K_{1}K_{2}K in D such that dim_{N}(K) = 2^{r−2}. Hence θ is surjective, so is Θ. □

Example 4.3. (see [2, Remark 3.10]) Let r = 3. Then G_{3} = H_{3} = S_{2} × S_{2} × S_{2} := ⟨τ_{1}⟩ × ⟨τ_{2}⟩ × ⟨τ_{3}⟩. As the action of H_{3} on X_{3} is simply transitive, X_{3} ≃ H_{3} as H_{3}-sets, hence J_{3} is generated by 2 and τ_{i} − 1 for i = 1, 2, 3. Set Λ_{3} := ℤ[H_{3}/⟨τ_{1}⟩] ⊕ ℤ[H_{3}/⟨τ_{2}⟩] ⊕ ℤ[H_{3}/⟨τ_{3}⟩] ⊕ ℤ[H_{3}/⟨τ_{1}τ_{2}⟩]. Define a map ρ : Λ_{3} → J_{3} by

ρ(\overline{x_{1}}, \overline{x_{2}}, \overline{x_{3}}, \overline{x_{4}}) = ∑_{i=1}^{3}(τ_{i} + 1)x_{i} + (τ_{1}τ_{2} + 1)x_{4}.

As 2 = (τ_{1}τ_{2} + 1) − τ_{1}(τ_{2} + 1) + (τ_{1} + 1), ρ is surjective. It is easy to check that H_{3} acts on Ker(ρ) faithfully. Therefore, by [5, Lemma 3.3] and [4, Corollary 4.2], ed_{2}(Alg_{8,2}) ≤ 4 + 4 + 4 + 4 − 2^{3} = 8.

For an x ∈ X_{r}, let H_{r,x} be the stabilizer of x in H_{r} = S_{2} × S_{2} × P_{2^{r−2}} := ⟨τ_{1}⟩ × ⟨τ_{2}⟩ × P_{2^{r−2}}. We set P_{2^{r−2}} = (P_{2^{r−3}})^{2} ⋊ ⟨τ_{r}⟩.

Lemma 4.4. For any r ≥ 3 and any x ∈ X_{r}, we have (1) H_{r,x} = H_{r−1,x} × P_{2^{r−3}}.

(2) H_{r} = ⟨τ_{1}, τ_{2}, τ_{r}, H_{r,x}⟩.

(3) J_{r} = ⟨2x, τ_{1}x − x, τ_{2}x − x, τ_{r}x − x⟩.

(4) τ_{r}H_{r,x}τ_{r} ∩ H_{r,x} = H_{r−1,x} × H_{r−1,x}.

Proof. (1) The stabilizer of x in H_{r} is the stabilizer of x under the action of P_{2^{r−2}} on the block B_{i} containing x for some i. As P_{2^{r−2}} = (P_{2^{r−3}})^{2} ⋊ ⟨τ_{r}⟩, the stabilizer of x in P_{2^{r−2}} is H_{r−1,x} × P_{2^{r−3}}.

(2) Induction on r. The case r = 3 comes from Example 4.3. By induction hypothesis we have P_{2^{r−3}} = ⟨τ_{r−1}, H_{r−1,x}⟩. As τ_{r−1} is generated by τ_{r} and P_{2^{r−3}}, the result follows immediately.

(3) As H_{r} acts on X_{r} transitively, the result follows from Lemma 4.4 (2) and the sequence (2).

(4) As τ_{r}H_{r,x}τ_{r} = H_{r,τ_{r}(x)}, the result follows from (1). □

Theorem 4.5. For any r ≥ 3, there exists a 2^{r−1}(2^{r−3}+3)-dimensional generically free representation for T_{r} ⋊ H_{r}. Hence ed_{2}(Alg_{2^{r},2}) ≤ 2^{r−1}(2^{r−3} + 1).

Proof. For r ≥ 3 and x ∈ X_{r}, we set

Λ_{r} := ℤ[H_{r}/⟨τ_{1}⟩ × H_{r,x}] ⊕ ℤ[H_{r}/⟨τ_{2}⟩ × H_{r,x}] ⊕ ℤ[H_{r}/⟨τ_{1}τ_{2}⟩ × H_{r,x}]
⊕ ℤ[H_{r}/(τ_{r}H_{r,x}τ_{r} ∩ H_{r,x}) ⋊ ⟨τ_{r}⟩].

Define a map ρ : Λ_{r} → J_{r} by taking a generator of the first component (respectively, the second component) of Λ_{r} to τ_{1}x + x (respectively, τ_{2}x + x), a generator of the third component of Λ_{r} to τ_{1}τ_{2}x + x, and a generator of the last component of Λ_{r} to τ_{r}x + x. By construction, this map is well defined. As 2x = (τ_{1}τ_{2}x + x) − τ_{1}(τ_{2}x + x) + (τ_{1}x + x), ρ is surjective by Lemma 4.4 (3).

As H_{r} acts faithfully on Ker(ρ) by Lemma 4.6, there exists a generically free representation for T_{r} ⋊ H_{r} by [5, Lemma 3.3]. Therefore, by [4, Corollary 4.2], we have

ed_{2}(T_{r} ⋊ H_{r}) ≤ rank(ℤ[H_{r}/⟨τ_{1}⟩ × H_{r,x}]) + rank(ℤ[H_{r}/⟨τ_{2}⟩ × H_{r,x}])
+ rank(ℤ[H_{r}/⟨τ_{1}τ_{2}⟩ × H_{r,x}]) + rank(ℤ[H_{r}/(τ_{r}H_{r,x}τ_{r} ∩ H_{r,x}) ⋊ ⟨τ_{r}⟩])
− rank(J_{r})
= 2^{r−1} + 2^{r−1} + 2^{r−1} + 2^{r+(r−1)−2−1} − 2^{r} (by Lemma 4.4(1),(4))
= 2^{r−1} + 2^{2r−4}.

By [5, Lemma 4.1], ed_{2}(T_{r} ⋊ G_{r}) = ed_{2}(T_{r} ⋊ H_{r}). As the morphism Θ in (6) is surjective by Lemma 4.2, we get

ed_{2}(Alg_{2^{r},2}) ≤ max{ed_{2}(T_{r} ⋊ G_{r}), ed_{2}(Alg_{2,2}), ⋯, ed_{2}(Alg_{2^{r−1},2})}.

By induction on r, we finally have ed_{2}(Alg_{2^{r},2}) ≤ ed_{2}(T_{r} ⋊ G_{r}) ≤ 2^{r−1} + 2^{2r−4}.

□

Lemma 4.6. Let ρ : Λ_{r} → J_{r} be the morphism in proof of Theorem 4.5. The action of H_{r} on Ker(ρ) is faithful.

Proof. Note that J_{r} ⊗ ℚ = ℚ[X_{r}] by the exact sequence (2). Hence, by the exact sequence

Ker(ρ) → Λ_{r} ⟶^{ρ} J_{r},

we have

ℚ[X_{r}] ⊕ (Ker(ρ))_{ℚ} = ℚ[H_{r}/⟨τ_{1}⟩ × H_{r,x}] ⊕ ℚ[H_{r}/⟨τ_{2}⟩ × H_{r,x}] ⊕ ℚ[H_{r}/⟨τ_{1}τ_{2}⟩ × H_{r,x}]

⊕ ℚ[H_{r}/(τ_{r}H_{r,x}τ_{r} ∩ H_{r,x}) ⋊ ⟨τ_{r}⟩]. (7)

By the actions of τ_{1} and τ_{2}, the natural map

i : ℤ[X_{r}] → ℤ[X_{r}/⟨τ_{1}⟩] ⊕ ℤ[X_{r}/⟨τ_{2}⟩] ⊕ ℤ[X_{r}/⟨τ_{1}τ_{2}⟩]

is injective, hence we get the exact sequence

0 → ℤ[X_{r}] ⟶^{i} ℤ[X_{r}/⟨τ_{1}⟩] ⊕ ℤ[X_{r}/⟨τ_{2}⟩] ⊕ ℤ[X_{r}/⟨τ_{1}τ_{2}⟩] → Coker(i) → 0

and

(8) ℚ[X_{r}] ⊕ (Coker(i))_{ℚ} = ℚ[X_{r}/⟨τ_{1}⟩] ⊕ ℚ[X_{r}/⟨τ_{2}⟩] ⊕ ℚ[X_{r}/⟨τ_{1}τ_{2}⟩].

By (7) and (8), we have

(Ker(ρ))_{ℚ} = (Coker(i))_{ℚ} ⊕ ℚ[H_{r}/(τ_{r}H_{r,x}τ_{r} ∩ H_{r,x}) ⋊ ⟨τ_{r}⟩],

thus it is enough to show that H_{r} = ⟨τ_{1}⟩×⟨τ_{2}⟩×[(P_{2^{r−3}})^{2} ⋊⟨τ_{r}⟩] acts faithfully on Y_{r} := ℚ[H_{r}/(τ_{r}H_{r,x}τ_{r} ∩ H_{r,x}) ⋊ ⟨τ_{r}⟩]:

Case 1: h = τ_{i}h′ or τ_{1}τ_{2}h′′ ∈ H_{r} for i = 1, 2 and h′, h′′ ∈ (P_{2^{r−3}})^{2} ⋊ ⟨τ_{r}⟩.

If h = τ_{i}h′ ∈ H_{r} for some h′ ∈ (P_{2^{r−3}})^{2} ⋊⟨τ_{r}⟩ and i = 1, 2, then h\overline{τ_{i}} = \overline{h′} ≠ \overline{τ_{i}} in Y_{r}. Similarly, if h = τ_{1}τ_{2}h′′ ∈ H_{r} for some h′′ ∈ (P_{2^{r−3}})^{2} ⋊ ⟨τ_{r}⟩, then h\overline{τ_{1}τ_{2}} = \overline{h′′} ≠ \overline{τ_{1}τ_{2}} in Y_{r}.

Case 2: h ∈ (P_{2^{r−3}})^{2} ⋊ ⟨τ_{r}⟩\(τ_{r}H_{r,x}τ_{r} ∩ H_{r,x}) ⋊ ⟨τ_{r}⟩.

If h ∈ (P_{2^{r−3}})^{2} ⋊ ⟨τ_{r}⟩\(τ_{r}H_{r,x}τ_{r} ∩ H_{r,x}) ⋊ ⟨τ_{r}⟩, then h\overline{1} = \overline{h} ≠ \overline{1} in Y_{r}.

Case 3: h ∈ (τ_{r}H_{r,x}τ_{r} ∩ H_{r,x}) ⋊ ⟨τ_{r}⟩\(τ_{r}H_{r,x}τ_{r} ∩ H_{r,x}).

Let h = h_{1}h_{2}τ_{r} ∈ (H_{r−1,x} × H_{r−1,x}) ⋊ ⟨τ_{r}⟩. We may assume that h_{1} ≠ 1. Choose a transposition δ ∈ P_{2^{r−3}} which does not fix x. Then h\overline{δ} = \overline{h_{1}h_{2}τ_{r}δ} = \overline{h_{1}h_{2}δ′τ_{r}} = \overline{δ′h_{1}h_{2}τ_{r}} = \overline{δ′} ≠ \overline{δ}, where δ′ is the transposition such that τ_{r}δ = δ′τ_{r}.

Case 4: h ∈ τ_{r}H_{r,x}τ_{r} ∩ H_{r,x}.

Let h = h_{1}h_{2} ∈ H_{r−1,x} × H_{r−1,x} = τ_{r}H_{r,x}τ_{r} ∩ H_{r,x}. We may assume that h_{1} ≠ 1. Let τ_{r−1} be the permutation which acts on the same block with h_{1}. Then h\overline{τ_{r−1}} = \overline{h_{1}τ_{r−1}} ≠ \overline{τ_{r−1}} in Y_{r}.

This completes the proof of faithfulness.

□

Corollary 4.7. Let F be a field of characteristic different 2. Then

ed_{2}(Alg_{16,2}) = 24.

Proof. The lower bound 24 ≤ ed_{2}(Alg_{16,2}) follows from [2, Theorem 6.1]. Therefore, the statement follows from Theorem 4.5. □

References

[1] S. Baek and A. Merkurjev, Invariants of simple algebras, Manuscripta Math, Vol. 129, No. 4 (2009), 409–421.

[2] S. Baek and A. Merkurjev, Essential dimension of central simple algebras, to appear in Acta Math.

[3] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, American Mathematical Society, Providence, RI, 1998, With a preface in French by J. Tits.

[4] A. S. Merkurjev, Essential dimension, Quadratic forms—algebra, arithmetic, and geometry, Contemp. Math., vol. 493, Amer. Math. Soc., Providence, RI, 2009, pp. 299–325.

[5] A. Meyer and Z. Reichstein, The essential dimension of the normalizer of a maximal torus in the projective linear group, Algebra and Number Theory 3 (2009), no. 4, 467– 487.

[6] L. H. Rowen and D. J. Saltman, Prime to p extensions of division algebra, Israel J. Math. 78 (1992), no. 2-3, 197–207.

[7] J.-P. Serre, Galois cohomology, Springer-Verlag, Berlin, 1997, Translated from the French by Patrick Ion and revised by the author.

Department of Mathematics and Statistics, University of Ottawa, Canada

E-mail address: sbaek@uottawa.ca