ESSENTIAL DIMENSION OF CENTRAL SIMPLE ALGEBRAS

SANGHOON BAEK AND ALEXANDER S. MERKURJEV

Abstract. Let p be a prime integer, 1 ≤ s ≤ r integers and F a field of characteristic different from p. We find upper and lower bounds for the essential p-dimension ed_{p}(Alg_{p^{r},p^{s}}) of the class Alg_{p^{r},p^{s}} of central simple algebras of degree p^{r} and exponent dividing p^{s}. In particular, we show that ed_{2}(Alg_{8,2}) = 8 and ed_{p}(Alg_{p^{2},p}) = p^{2} + p for p odd.

1. Introduction

Let F : Fields/F → Sets be a functor from the category Fields/F of field extensions over F to the category Sets of sets. Let E ∈ Fields/F and K ⊂ E a subfield over F. An element α ∈ F(E) is said to be defined over K (and K is called a field of definition of α) if there exists an element β ∈ F(K) such that α is the image of β under the map F(K) → F(E). The essential dimension of α, denoted ed^{F}(α), is the least transcendence degree tr. deg_{F} (K) over all fields of definition K of α. The essential dimension of the functor F is

ed(F) = sup{ed^{F}(α)},

where the supremum is taken over all fields E ∈ Fields/F and all α ∈ F(E) (see [3, Def. 1.2] or [8, Sec.1]). Informally, the essential dimension of F is the smallest number of algebraically independent parameters required to define F and may be thought of as a measure of complexity of F.

Let p be a prime integer. The essential p-dimension of α, denoted ed_{p}^{F}(α), is defined as the minimum of ed^{F}(α_{E′}), where E′ ranges over all field extensions of E of degree prime to p. The essential p-dimension of F is

ed_{p}(F) = sup{ed_{p}^{F}(α)},

where the supremum ranges over all fields E ∈ Fields/F and all α ∈ F(E). By definition, ed(F) ≥ ed_{p}(F) for all p.

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. Equivalently, Alg_{n,m}(E) is the subset of the m-torsion part Br_{m}(E) of the Brauer group of E consisting of all elements a such that ind(a) divides n. In particular, if n = m, then Alg_{n}(E) := Alg_{n,n}(E) is the set of isomorphism classes of central simple E-algebras of degree n. We view Alg_{n,m} and Alg_{n} as functors Fields/F → Sets.

Key words and phrases. Essential dimension, Brauer group, algebraic tori 2000 Mathematical Subject Classifications: primary 16K50, secondary 14L30, 20G15 .

The work of the first author has been supported by the Beckenbach Dissertation Fellowship at the University of California at Los Angeles.

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

In the present paper we give upper and lower bounds for ed_{p}(Alg_{n,m}) for a prime integer p. Let p^{r} (respectively, p^{s}) be the largest power of p dividing n (respectively, m). Then ed_{p}(Alg_{n,m}) = ed_{p}(Alg_{p^{r},p^{s}}) and ed_{p}(Alg_{n}) = ed_{p}(Alg_{p^{r}}) (see Section 6). Thus, we may assume that n and m are the ppowers p^{r} and p^{s} respectively.

Using structure theorems on central simple algebras, we can compute the essential (p)-dimension of Alg_{p^{r},p^{s}} for certain small values of r, s or p as follows. As every central simple algebra A of degree p is cyclic over a finite field extension of degree prime to p, A can be given by two parameters (see Section 2.1). In fact, ed_{p}(Alg_{p}) = 2 by [13, Lemma 8.5.7].

By Albert’s theorem, every algebra in Alg_{4,2} is biquaternion and hence can be given by 4 parameters. In fact, ed(Alg_{4,2}) = ed_{2}(Alg_{4,2}) = 4 (see Remark 8.2).

The upper and lower bounds for ed_{p}(Alg_{p^{r}}) can be found in [12] and [10] respectively. In this paper (see Sections 6 and 7), we establish the following upper and lower bounds for ed_{p}(Alg_{p^{r},p^{s}}): Theorem. Let F be a field and p a prime integer different from char(F). Then, for any integers r ≥ 2 and s with 1 ≤ s ≤ r,

2p^{2r−2} − p^{r} + p^{r−s} ≥ ed_{p}(Alg_{p^{r},p^{s}}) ≥ \begin{cases}(r − 1)2^{r−1} & if p = 2 and s = 1, \\ (r − 1)p^{r} + p^{r−s} & otherwise. \end{cases}

Corollary. (cf. [9]) Let p be a prime integer and F a field of characteristic different from p. Then

ed_{p}(Alg_{p^{2}}) = p^{2} + 1.

Corollary. Let p be an odd prime integer and F a field of characteristic different from p. Then

ed_{p}(Alg_{p^{2},p}) = p^{2} + p.

The corollary recovers a result in [21] that for p odd, there exists a central simple algebra of degree p^{2} and exponent p which is not decomposable as a tensor product of two algebras of degree p. Indeed, if every central simple algebra of degree p^{2} and exponent p is decomposable, then the essential pdimension of Alg_{p^{2},p} would be at most 4. Corollary. Let F be a field of characteristic different from 2. Then

ed_{2}(Alg_{8,2}) = ed(Alg_{8,2}) = 8.

The proof is given in Section 8. The corollary recovers a result in [1] that there is a central simple algebra of degree 8 and exponent 2 which is not decomposable as a tensor product of three quaternion algebras. Indeed, if every central simple algebra of degree 8 and exponent 2 is decomposable, then the essential 2-dimension of Alg_{8,2} would be at most 6.

2. Character, Brauer group and algebraic tori

2.1. Character and Brauer group. Let F be a field, F_{sep} a separable closure of F, Γ_{F} = Gal(F_{sep}/F). For a (discrete) Γ_{F} - module M, we write H^{n}(F, M) for the Galois cohomology group H^{n}(Γ_{F} , M).

If S is an algebraic group over F, we let H^{1}(F, S) denote the set H^{1}(Γ_{F} , S(F_{sep})) (see [18]).

The character group of F is defined by

Ch(F) := Hom_{cont}(Γ_{F} , ℚ/ℤ) = H^{1}(F, ℚ/ℤ) ≃ H^{2}(F, ℤ).

The n-torsion character group Ch_{n}(F) is identified with H^{1}(F, ℤ/nℤ). For a character χ ∈ Ch(F), set F (χ) = (F_{sep})^{Ker(χ)}. The field extension F (χ)/F is cyclic of degree ord(χ). If Ψ ⊂ Ch(F) is a finite subgroup, we set

F (Ψ) := (F_{sep})^{∩Ker(χ)},

where the intersection is taken over all χ ∈ Ψ. The Galois group G = Gal(F (Ψ)/F ) is abelian and Ψ is canonically isomorphic to the character group Hom(G, ℚ/ℤ) of G. Note that a character η ∈ Ch(F) is trivial over F (Ψ) if and only if η ∈ Ψ.

We write Br(F) for the Brauer group H^{2}(F, F_{sep}^{×}) of F. If L/F is a field extension and α ∈ Br(F), we let α_{L} denote the image of α under the natural map Br(F) → Br(L). We say that L is a splitting field of α if α_{L} = 0. The index ind(α) of α is the smallest degree of a splitting field of α. The exponent exp(α) is the order of α in Br(F). The integer exp(α) divides ind(α).

Let A be a central simple F -algebra. The degree of A in the square root of dim(A). We write [A] for the class of A in Br(F). The index of [A] divides deg(A). If α ∈ Br(F) and n is a positive multiple of ind(α), then there is a central simple F -algebra A of degree n with [A] = α.

The cup-product

Ch(F) ⊗ F^{×} = H^{2}(F, ℤ) ⊗ H^{0}(F, F_{sep}^{×}) → H^{2}(F, F_{sep}^{×}) = Br(F)

takes χ⊗b to the class χ∪(b) in Br(F) that is split by F (χ). A class α ∈ Br(F) is called n-cyclic if α = χ ∪ (b) for a character χ with nχ = 0. Such classes belong to Br_{n}(F). If n is prime to char(F), then Br_{n}(F) ≃ H^{2}(F, μ_{n}), where μ_{n} is the Γ_{F} -module of all n-th roots of unity in F_{sep}.

Let n be prime to char(F) and suppose that F contains a primitive n-th root of unity ξ. For any a ∈ F^{×}, let χ_{a} ∈ Ch(F) be a unique character with values in \frac{1}{n}ℤ/ℤ ⊂ ℚ/ℤ such that

γ(a^{1/n}) = ξ^{(nχ_{a}(γ))}a^{1/n}

for all γ ∈ Gal(F_{sep}/F). We write (a, b)_{n} for χ_{a} ∪ (b). The symbol (a, b)_{n} satisfies the following properties (see [17, Chap. XIV, Prop.4]):

(a, b)_{n} + (a′, b)_{n} = (aa′, b)_{n},

(a, b)_{n} = −(b, a)_{n},

(a, −a)_{n} = 0.

For a finite subgroup Φ ⊂ Ch(F) write Br(F (Φ)/F )_{dec} for the subgroup of decomposable elements in Br(F (Φ)/F ) generated by the elements χ∪(a) for all χ ∈ Φ and a ∈ F^{×}. The indecomposable relative Brauer group Br(F (Φ)/F )_{ind} is the factor group Br(F (Φ)/F )/ Br(F (Φ)/F )_{dec}. Similarly, if Φ ⊂ Ch_{n}(F) for some n, then Br_{n}(F (Φ)/F )_{ind} is the indecomposable n-torsion relative Brauer group defined as the factor group Br_{n}(F (Φ)/F )/ Br(F (Φ)/F )_{dec}.

Let E be a complete field with respect to a discrete valuation v and K its residue field. Let p be a prime integer different from char(K). There is a natural injective homomorphism Ch(K){p} → Ch(E){p} of the p-primary components of the character groups that identifies Ch(K){p} with the character group of an unramified field extension of E. For a character χ ∈ Ch(K){p}, we write χ̂ for the corresponding character in Ch(E){p}.

By [4, §7.9], there is an exact sequence

0 → Br(K){p} ⟶^{i} Br(E){p} ⟶^{∂_{v}} Ch(K){p} → 0.

If α ∈ Br(K){p}, then we write α̂ for the element i(α) in Br(E){p}. For example, if α = χ ∪ (ū) for some χ ∈ Ch(K){p} and a unit u ∈ E, then α̂ = χ̂ ∪ (u). In the case F contains a primitive n-th root of unity, where n is a power of p, if α = (ā, b̄)_{n} with a and b units in E, then α̂ = (a, b)_{n}.

If β = α̂ + (χ̂ ∪ (x)) for an element α ∈ Br(K){p}, χ ∈ Ch(K){p} and x ∈ E^{×} such that v(x) is not divisible by p, we have (cf. [19, Prop. 2.4])

(1)  ind(β) = ind(α_{K(χ)}) · ord(χ).

2.2. Representations of algebraic tori. Let T be an algebraic torus over a field F, L/F a finite Galois splitting field for T with Galois group G. The group G is called the decomposition group of T. The character group T^{∗} := Hom_{L}(T_{L}, 𝔾_{m}_{,L}) has the structure of a G-module. The torus T can be reconstructed from T^{∗} by

T = Spec(L[T^{∗}]^{G}).

A torus P over F split by L is called quasi-split if P^{∗} is a permutation Gmodule, i.e., if there exists a G-invariant ℤ-basis X for P^{∗}. The torus P is canonically isomorphic to the group of invertible elements of the étale F - algebra A = Map_{G}(X, L). The torus P acts linearly by multiplication on the vector space A over F making A a faithful P -space (a linear representation of P) of dimension dim(P). It follows that a homomorphism of algebraic tori ν : T → P with P a quasi-split torus yields a linear representation of T of dimension dim(P) that is faithful if ν is injective.

Let P be a split torus over F, and P^{∗} its character group. As above, the choice of a ℤ-basis X for P^{∗} allows us to identify P with the group of invertible elements of a split étale F -algebra A and make A a faithful P -space over F. Let ν : T → P be a homomorphism of split tori over F. Suppose a finite group G acts on T and P by tori automorphisms so that ν is a G-equivariant homomorphism. Then the map ν^{∗} : P^{∗} → T^{∗} is a G-module homomorphism.

Suppose that there is a G-invariant ℤ-basis X for P^{∗}, i.e., P^{∗} is permutation. Then G acts on the algebra A by F -algebra automorphisms. The torus T acts linearly on A via ν. It follows that the semidirect product T ⋊ G acts linearly on A making A a T ⋊ G-space.

Let L be a Galois G-algebra over F (for example, L/F is a Galois field extension with Galois group G). Then γ : Spec L → Spec F is a G-torsor. Twisting the split torus T by the torsor γ, we get the torus

T_{γ} = (T × Spec L)/G = Spec(L[T^{∗}]^{G})

that is split by L and T_{γ}^{∗} is isomorphic to T^{∗} as G-modules.

By [5, Prop. 28.11], the fiber of H^{1}(F, T ⋊ G) → H^{1}(F, G) over the class of γ is naturally bijective to the orbit set of the group G_{γ}(F) in H^{1}(F, T_{γ}), i.e.,

(2)  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).

2.3. Generic torsors. Let T be an algebraic torus split by a finite Galois field extension L/F with G = Gal(L/F). Let P be a quasi-split torus split by L and containing T as a subgroup. Set S = P/T. Then the canonical homomorphism γ : P → S is a T -torsor.

Proposition 2.1. The T -torsor γ is generic, i.e., for every field extension K/F with K infinite, every T -torsor γ′ : E → Spec K and every nonempty open subset W ⊂ S, there is a morphism s : Spec K → S over F with Im(s) ⊂ W such that the T -torsors γ′ and s^{∗}(γ) = γ ×_{S} Spec K over K are isomorphic.

Proof. As P is quasi-split, the last term in the exact sequence

P (K) ⟶^{γ_{K}} S(K) ⟶^{δ} H^{1}(K, T) → H^{1}(K, P)

is trivial. Then there is s ∈ S(K) with δ(s) = [γ′]. As K is infinite, the K-points of P are dense in P and we can modify s by an element in the image of γ_{K} so that s ∈ W (K), i.e., Im(s) ⊂ W. Then the T -torsor γ′ over K with the class δ(s) satisfies the required property. □

2.4. The algebraic tori P^{Φ}, S^{Φ}, T^{Φ}, U^{Φ} and V^{Φ}. Let 1 ≤ s ≤ r be integers, p a prime integer, F a field with char(F) ≠ p, Φ a subgroup of Ch_{p}(F) of rank r and L = F (Φ). Let G = Gal(L/F). Choose a basis χ_{1}, χ_{2}, …, χ_{r} for Φ. Each χ_{i} can be viewed as a character of G, i.e., as a homomorphism χ_{i} : G → ℚ/ℤ. Let σ_{1}, σ_{2}, …, σ_{r} be the dual basis for G, i.e.,

χ_{i}(σ_{j}) = \{ \begin{matrix}(1/p) + ℤ, & if i = j; \\ 0, & otherwise. \end{matrix}

Let R be the group ring ℤ[G]. Consider the surjective G-modules homomorphism ε̄: R → ℤ/p^{s}ℤ, defined by ε̄(x) = ε(x) + p^{s}ℤ, where ε : R → ℤ is the augmentation homomorphism given by ε(ρ) = 1 for all ρ ∈ G. Set J := Ker(ε̄), thus, we have an exact sequence

0 → J → R ⟶^{ε̄} ℤ/p^{s}ℤ → 0.

Moreover, the G-module J is generated by I and p^{s}, where I := Ker(ε) is the augmentation ideal in R.

Consider the G-module homomorphism h : R^{r+1} → R taking the i-th canonical basis element e_{i} to σ_{i} − 1 for 1 ≤ i ≤ r and e_{r+1} to p^{s}. The image of h coincides with J.

Set N := Ker(h) and write w_{i} = 1 + σ_{i} + σ_{i}^{2} + ⋯ + σ_{i}^{p−1} ∈ R for 1 ≤ i ≤ r. Consider the following elements in N:

e_{ij} = (σ_{i} − 1)e_{j} − (σ_{j} − 1)e_{i}, f_{i} = w_{i}e_{i}, and g_{i} = −p^{s}e_{i} + (σ_{i} − 1)e_{r+1}

for all 1 ≤ i, j ≤ r.

Lemma 2.2. The G-module N is generated by e_{ij}, f_{i} and g_{i}.

Proof. Consider the surjective morphism k : R^{r} → I taking e_{i} to σ_{i} − 1 and set N′ := Ker(k). Then we have the following commutative diagram

\xymatrix{N′ \ar@{^{(}->}[d] \ar@{^{(}->}[r] & R^{r} \ar@{^{(}->}[d] \ar[r]^{k} & I \ar@{^{(}->}[d] \\ N \ar@{->>}[d] \ar@{^{(}->}[r] & R^{r+1} \ar@{->>}[d] \ar@{->>}[r]^{h} & J \ar@{->>}[d]^{ε′} \\ I \ar@{^{(}->}[r] & R \ar@{->>}[r]^{ε} & ℤ}

where R^{r+1} → R is the projection morphism to the last coordinate and ε′ : J → ℤ is given by ε′(j) = ε(j)/p^{s}.

By the exactness of the first column of the diagram, N is generated by N′ and the liftings g_{i} of σ_{i} − 1 in N. The module N′ is generated by e_{ij} and f_{i} by [10, Lemma 3.5]. This completes the proof. □

Let ε_{i} : R^{r+1} → ℤ be the i-th projection followed by the augmentation map ε. It follows from Lemma 2.2 that ε_{i}(N) = pℤ for every i = 1, …, r. Moreover, the G-homomorphism

q : N → ℤ^{r}, x ↦ (ε_{1}(x)/p, …, ε_{r}(x)/p)

is surjective. Set M := Ker(q) and Q := R^{r+1}/M.

Let P^{Φ}, S^{Φ}, T^{Φ}, U^{Φ} and V^{Φ} be the algebraic tori over F with the character G-modules R^{r+1}, Q, M, J and N, respectively. The diagram of homomorphisms of G-modules with the exact columns and rows

(3)  [figure: M; M; N; q; R^{r+1}; h; J; ℤ^{r}; Q; J]

yields the following diagram of homomorphisms of the tori

(4)  [figure: T^{Φ}; T^{Φ}; V^{Φ}; P^{Φ}; γ; U^{Φ}; _{𝔾{r \atop m}}; S^{Φ}; U^{Φ}]

Let K/F be a field extension and set KL := K ⊗_{F} L. The commutative diagram

0 ⟶ I ⟶ R ⟶ ℤ ⟶ 0
↓  ∥  ↓
0 ⟶ J ⟶ R ⟶ ℤ/p^{s}ℤ ⟶ 0

induces the commutative diagram of homomorphisms of algebraic groups

(5)  \begin{aligned}1 & ⟶ & μ_{p^{s}} & ⟶ & R_{L/F} (𝔾_{m,L}) & ⟶ & U^{Φ} & ⟶ & 1 \\ & & ↓ & & ∥ & & ↓ \\ 1 & ⟶ & 𝔾_{m} & ⟶ & R_{L/F} (𝔾_{m,L}) & ⟶ & U^{′Φ} & ⟶ & 1 \end{aligned}

and then the commutative diagram

(6)  \begin{aligned}0 & ⟶ & H^{1}(K, U^{Φ}) & ⟶ & H^{2}(K, μ_{p^{s}}) & ⟶ & H^{2}(KL, 𝔾_{m}) \\ & & ↓ & & ↓ & & ∥ \\ 0 & ⟶ & H^{1}(K, U^{′Φ}) & ⟶ & H^{2}(K, 𝔾_{m}) & ⟶ & H^{2}(KL, 𝔾_{m}). \end{aligned}

Hence

(7)  H^{1}(K, U^{Φ}) ≃ Br_{p^{s}}(KL/K) and H^{1}(K, U′^{Φ}) ≃ Br(KL/K).

Lemma 2.3. The map H^{1}(K, U^{Φ}) → H^{1}(K, S^{Φ}) induces an isomorphism H^{1}(K, S^{Φ}) ≃ Br_{p^{s}}(KL/K)_{ind}.

Proof. Consider the following commutative diagram

1 ⟶ U^{Φ} ⟶ S^{Φ} ⟶ 𝔾_{m}^{r} ⟶ 1
↓  ↓  ∥
1 ⟶ U^{′Φ} ⟶ S^{′Φ} ⟶ 𝔾_{m}^{r} ⟶ 1,

where the bottom row is induced by the bottom row of the diagram (4) in [10]. This yields a commutative diagram

(K^{×})^{r} ⟶ H^{1}(K, U^{Φ}) ⟶ H^{1}(K, S^{Φ}) ⟶ 0
∥  ↓  ↓
(K^{×})^{r} ⟶^{λ} H^{1}(K, U^{′Φ}) ⟶ H^{1}(K, S^{′Φ}) ⟶ 0

with the exact rows. The homomorphism λ takes (x_{1}, …, x_{r}) to ∑_{i=1}^{r}((χ_{i})_{K} ∪ (x_{i})) by [10, Lemma 3.6], whence the result. □

3. Essential dimension of algebraic tori

Let S be an algebraic group over F. The essential dimension ed(S) (respectively, essential p-dimension ed_{p}(S)) of S is defined to be the essential (p-)dimension of the functor taking a field extension K/F to the set of isomorphism classes S-torsors(K) of S-torsors over K. Note that the functor S-torsors is isomorphic to the functor taking K to the set H^{1}(K, S).

Let S be an algebraic torus over F split by L with G = Gal(L/F). We assume that G is a group of order p^{r}, where p is a prime integer and r ≥ 2. Let X be the G-module of characters of S. Define the group \overline{X} := X/(pX + IX), where I is the augmentation ideal in R = ℤ[G]. For any subgroup H ⊂ G, consider the composition X^{H} ↪ X → \overline{X}. For every k, let V_{k} denote the subgroup generated by images of the homomorphisms X^{H} → \overline{X} over all subgroups H with [G : H] ≤ p^{k}. We have the sequence of subgroups

0 = V_{−1} ⊂ V_{0} ⊂ ⋯ ⊂ V_{r} = \overline{X}.

A p-presentation of X is a G-homomorphism P → X with P a permutation G-module and finite cokernel of order prime to p. A p-presentation with the smallest rank(P) is called minimal. The essential p-dimension of algebraic tori was determined in [7, Th. 1.4] in terms of a minimal p-presentation P → X:

(8)  ed_{p}(S) = rank(P) − dim(S).

We have the following explicit formula for the essential (p-)dimension of S (cf. [10, Th. 4.3]):

Theorem 3.1. Let S be a torus over a field F and p a prime integer different from char(F). If the decomposition group G of S is a p-group, then

ed(S) = ed_{p}(S) = ∑_{k=0}^{r}(rank V_{k} − rank V_{k−1})p^{k} − dim(S).

Proof. The second equality was proven in [10, Th. 4.3]. Let ν : P → X be a minimal p-presentation. By definition, the index [X : Im(ν)] is prime to p. Let T and U be algebraic groups of multiplicative type split by L with the character G-modules Im(ν) and X/ Im(ν), respectively, hence we have an exact sequence

1 → U → S → T → 1.

Let K/F be a field extension. By assumption, the group U(KL) = Hom(X/ Im(ν), KL^{×}) has order prime to p. We have an exact sequence

H^{1}(G, U(KL)) → H^{1}(G, S(KL)) → H^{1}(G, T (KL)) → H^{2}(G, U(KL)).

As the order of U(KL) is prime to p and G is a p-group, the groups H^{i}(G, U(KL)) are trivial for i ≥ 1, hence the homomorphism S → T induces an isomorphism of functors S-torsors →^{∼} T -torsors. It follows that ed(S) = ed(T). The surjection P → Im(ν) yields a generically free representation of T by [11, Lemma 3.3]. Hence, by [3, Prop. 4.11] and (8), we have

ed_{p}(S) ≤ ed(S) = ed(T) ≤ rank(P) − dim(T) = rank(P) − dim(S) = ed_{p}(S),

therefore, ed(S) = ed_{p}(S). □

Let F be a field, Φ a subgroup of Ch_{p}(F) of rank r ≥ 2, L = F (Φ) and G = Gal(L/F). In this section we compute the essential (p-)dimension of the algebraic tori U^{Φ} and S^{Φ} defined by (4). For any subgroup H of G, we write n_{H} := ∑_{τ∈H} τ in R = ℤ[G]. An element x ∈ R is decomposable if x = yz with y, z ∈ R, and ε(y), ε(z) ∈ pℤ.

Lemma 3.2. Let H ⊂ G be a nontrivial subgroup and x ∈ R such that ε(n_{H}x) ∈ p^{2}ℤ. Then n_{H}x is decomposable.

Proof. If |H| = p, then ε(x) ∈ pℤ and hence n_{H}x is decomposable. Otherwise H = H′ × H′′ for nontrivial subgroups H′ and H′′. As n_{H} = n_{H′} · n_{H′′}, the element n_{H} and therefore, n_{H}x is decomposable. □

Lemma 3.3. If x ∈ R is decomposable, then x ≡ ε(x) modulo pI + I^{2}.

Proof. Let y = ε(y) + u and z = ε(z) + v for some u, v ∈ I. Then we have yz − ε(yz) = (ε(y)v + ε(z)u) + uv ∈ pI + I^{2}. □

Lemma 3.4. The group V_{k} is generated by (1) the elements \overline{n_{H}x} such that |H| ≥ p^{r−k} and ε(n_{H}x) ∈ p^{s}ℤ if r − k < s,

(2) the elements \overline{n}_{H} such that |H| ≥ p^{r−k} if r − k ≥ s.

Proof. The statement follows from the equality J^{H} = R^{H} ∩ J = n_{H}R ∩ J. □

Lemma 3.5. If k < r − s, then V_{k} = 0.

Proof. By Lemma 3.4(2), V_{k} is generated by \overline{n}_{H} with |H| ≥ p^{r−k}. As n_{H} is decomposable and |H| > p^{s}, in view of Lemma 3.3, we have \overline{n}_{H} = \overline{ε(n_{H})} = \overline{|H|} = 0 as |H| ∈ pJ. □

Lemma 3.6. If s ≥ 2 and r − s ≤ k ≤ r − 1, then dim(V_{k}) = 1.

Proof. By Lemma 3.4, V_{k} is generated by \overline{n_{H}x} with H nontrivial and ε(n_{H}x) ∈ p^{s}ℤ. As s ≥ 2, the element n_{H}x is decomposable by Lemma 3.2. In view of Lemma 3.3, \overline{n_{H}x} = \overline{ε(n_{H}x)}, hence V_{k} is generated by \overline{p^{s}}. □

Lemma 3.7. If s = 1 and p is odd, then dim(V_{r−1}) = 1.

Proof. We claim that V_{r−1} is generated by \overline{p}. By Lemma 3.4(2), V_{r−1} is generated by \overline{n}_{H} with |H| ≥ p. If |H| ≥ p^{2}, then by Lemma 3.2, \overline{n}_{H} is decomposable and in view of Lemma 3.3, \overline{n}_{H} = \overline{ε(n_{H})} = 0.

Suppose |H| = p and let σ ∈ H be a generator. We have n_{H} −p = (σ −1)m, where m = ∑_{i=0}^{p−2}(p − 1 − i)σ^{i}, so ε(m) = p(p − 1)/2. As p is odd, ε(m) ∈ pℤ. Hence, m ∈ pR + I, therefore, n_{H} − p ∈ pI + I^{2} and \overline{n}_{H} = \overline{p} in \overline{J}. □

Lemma 3.8. If s = 1 and p = 2, then V_{r−1} = \overline{J}.

Proof. By Lemma 3.4(2), V_{r−1} is generated by \overline{n}_{H} with |H| ≥ 2. Take nontrivial elements σ ≠ τ in G. Then \overline{2} = \overline{(1 + στ)} − σ\overline{(1 + τ)} + \overline{(1 + σ)} ∈ V_{r−1}. Also, for any σ ∈ G, \overline{σ − 1} = \overline{1 + σ} − \overline{2} ∈ V_{r−1}. The group \overline{J} is generated by \overline{2} and \overline{σ − 1} over all σ ∈ G. □

Proposition 3.9. We have

ed(U^{Φ}) = ed_{p}(U^{Φ}) = \begin{cases}(r − 1)2^{r−1} & if p = 2 and s = 1, \\ (r − 1)p^{r} + p^{r−s} & otherwise. \end{cases}

Proof. Note that V_{r} = \overline{J}, rank(\overline{J}) = rank(V_{r}) = r + 1 and dim(U^{Φ}) = p^{r}. Case 1: p is odd or p = 2 and s ≥ 2. By Lemmas 3.5, 3.6 and 3.7, we have

rank V_{k} = \begin{cases}r + 1 & if k = r, \\ 1 & if r − s ≤ k < r, \\ 0 & if 0 ≤ k < r − s. \end{cases}

Since the decomposition group G of U^{Φ} is a p-group, by Theorem 3.1,

ed(U^{Φ}) = ed_{p}(U^{Φ}) = rp^{r} +p^{r−s} −dim(U^{Φ}) = rp^{r} +p^{r−s} −p^{r} = (r −1)p^{r} +p^{r−s}.

Case 2: p = 2 and s = 1. By Lemmas 3.5 and 3.8, we have

rank V_{k} = \begin{cases}r + 1 & if k = r − 1 or k = r, \\ 0 & if 0 ≤ k ≤ r − 2. \end{cases}

Again by Theorem 3.1,

ed(U^{Φ}) = ed_{2}(U^{Φ}) = (r + 1)2^{r−1} − dim(U^{Φ}) = (r − 1)2^{r−1}.  □

Remark 3.10. One can construct a surjective minimal p-presentation ν : P′ → J as follows.

Case 1: p is odd or p = 2 and s ≥ 2. Let H be a subgroup of G of order p^{s} and P′ := R^{r} ⊕ ℤ[G/H]. We define ν by

ν(x_{1}, …, x_{n}, y) = ∑_{i=1}^{r}(σ_{i} − 1)x_{i} + n_{H}y.

The image of ν contains I and n_{H}. As n_{H} ≡ p^{s} modulo I, we have p^{s} ∈ Im(ν), hence ν is surjective. Note that e_{ij} = (σ_{i} − 1)e_{j} − (σ_{j} − 1)e_{i} ∈ Ker(ν). As σ_{i}e_{ij} ≠ e_{ij} for every j ≠ i, the group G acts faithfully on Ker(ν).

Case 2: p = 2 and s = 1. Let H_{i} be the subgroup of G generated by σ_{i} and H = ⟨σ_{1}σ_{2}⟩. Set P′ = ∐_{i=1}^{r} ℤ[G/H_{i}] ⊕ ℤ[G/H]. We define ν by

ν(x_{1}, …, x_{n}, y) = ∑_{i=1}^{r}(σ_{i} + 1)x_{i} + (σ_{1}σ_{2} + 1)y.

The image of ν contains σ_{i} +1 and 2 = (σ_{1}σ_{2} +1)−σ_{1}(σ_{2} +1)+(σ_{1} +1), hence ν is surjective. Note that h_{ij} := (σ_{i} +1)e_{j} −(σ_{j} +1)e_{i} ∈ Ker(ν). As σ_{k}h_{ij} ≠ h_{ij} for distinct i, j and k, the group G acts faithfully on Ker(ν) if r ≥ 3. In fact, G acts trivially on Ker(ν) if r = 2.

Corollary 3.11. We have

ed(S^{Φ}) = ed_{p}(S^{Φ}) = \begin{cases}(r − 1)2^{r−1} − r & if p = 2 and s = 1, \\ (r − 1)p^{r} + p^{r−s} − r & otherwise. \end{cases}

Proof. By (8) and Proposition 3.9, there is a minimal p-presentation ν : P → J such that

(9)  rank(P) = \begin{cases}(r + 1)2^{r−1} & if p = 2 and s = 1, \\ rp^{r} + p^{r−s} & otherwise. \end{cases}

The exact sequence

0 → ℤ^{r} → Q → J → 0

in the bottom row of (3) yields an exact sequence

Hom_{G}(P, Q) → Hom_{G}(P, J) → Ext_{G}^{1}(P, ℤ^{r}).

As P and ℤ^{r} are permutation G-modules, Ext_{G}^{1}(P, ℤ^{r}) = 0, hence the homomorphism ν factors through a morphism ν′ : P → Q.

Recall that we write \overline{X} = X/(pX + IX) for a G-module X. As \overline{ℤ^{r}} ≃ (ℤ/pℤ)^{r} → \overline{Q} is zero map, the natural homomorphism \overline{Q} → \overline{J} is an isomorphism, hence ν′ is a minimal p-presentation of Q. Note that G is the decomposition group of S^{Φ} and dim(S^{Φ}) = p^{r} + r. By Theorem 3.1, ed(S^{Φ}) = ed_{p}(S^{Φ}) = rank(P) − dim(S^{Φ}), hence the result follows by (9). □

4. Degeneration

In this section we relate the essential p-dimensions of Alg_{p^{r},p^{s}} and of the torus S^{Φ} by means of the iterated degeneration (Proposition 4.1). The latter is a method of comparison of the essential p-dimension of an object (a central simple algebra in our case) over a complete discrete valued field and of its specialization over the residue field.

4.1. A simple degeneration. Let F be a field, p a prime integer different from char(F) and Φ ⊂ Ch_{p}(F) a finite subgroup. For integers k ≥ 0, s ≥ 1 and a field extension K/F, let

(10)  B_{k,s}^{Φ}(K) = {α ∈ Br(K){p} | ind(α_{K(Φ)}) ≤ p^{k}, exp(α) ≤ p^{s}}.

We say that two elements α and α′ in B_{k,s}^{Φ}(K) are equivalent if α − α′ ∈ Br(K(Φ)/K)_{dec}. Write B̃_{k,s}^{Φ}(K) for the set of equivalence classes in B_{k,s}^{Φ}(K). To simplify notation, we shall write α for the equivalence class of an element α ∈ B_{k,s}^{Φ}(K) in B̃_{k,s}^{Φ}(K). We view B_{k,s}^{Φ} and B̃_{k,s}^{Φ} as functors from Fields/F to Sets.

In particular, if k = 0, then B_{0,s}^{Φ}(K) and B̃_{0,s}^{Φ}(K) are bijective to Br_{p^{s}}(K(Φ)/K) and Br_{p^{s}}(K(Φ)/K)_{ind}, respectively. Hence, by (7) and Lemma 2.3,

(11)  B_{0,s}^{Φ} ≃ U^{Φ}- torsors and B̃_{0,s}^{Φ} ≃ S^{Φ}- torsors .

Moreover, if Φ = 0, then

(12)  B_{k,s}^{Φ} = B̃_{k,s}^{Φ} ≃ Alg_{p^{k},p^{s}}.

Let Φ′ ⊂ Φ be a subgroup of index p and η ∈ Φ \ Φ′, hence Φ = ⟨Φ′, η⟩. Let E/F be a field extension such that η_{E} ∉ Φ′_{E} in Ch(E). Choose an element a ∈ B_{k,s}^{Φ}(E), i.e., α ∈ Br(E){p} such that ind(α_{E(Φ)}) ≤ p^{k} and exp(α) ≤ p^{s}.

Let E′ be a field extension of F that is complete with respect to a discrete valuation v′ over F with residue field E and set

(13)  α′ := α̂ + (η̂_{E} ∪ (x)) ∈ Br(E′),

for some x ∈ E′^{×} such that v′(x) is prime to p. As η_{E(Φ′)} ≠ 0, it follows from (1) that

ind(α′_{E′(Φ′)}) = p · ind(α_{E(Φ)}) ≤ p^{k+1} and exp(α′) = lcm(exp(α), p) ≤ p^{s},

hence α′ ∈ B_{k+1,s}^{Φ′}(E′).

In the case the condition exp(α) ≤ p^{s} in (10) is dropped, the following proposition was proved in [10, Prop. 5.2]:

Proposition 4.1. Suppose that for any finite field extension N/E of degree prime to p and any character ρ ∈ Ch(N) of order p^{2} such that pρ ∈ Φ_{N} \ Φ′_{N}, we have ind(α_{N(Φ′,ρ)}) ≥ p^{k}. Then

edB̃_{k+1,s}^{Φ′} ; p(α′) ≥ edB̃_{k,s}^{Φ} ; p(α) + 1.

Proof. The proof of [10, Prop. 5.2] still works with the following modification.

Let M/E′ be a finite field extension of degree prime to p, M_{0} ⊂ M a subfield over F and α′_{0} ∈ B_{k+1,s}^{Φ′}(M_{0}) such that (α′_{0})_{M} = α′_{M} in B̃_{k+1,s}^{Φ′} and

tr. deg_{F} (M_{0}) = edB̃_{k+1,s}^{Φ′} ; p(α′). We extend the discrete valuation v′ on E′ to a (unique) discrete valuation v on M and let N be its residue field. Let n_{0} be the residue field of the restriction of v on M_{0}. It was shown in the proof of [10, Prop. 5.2] that there exist α_{0} ∈ Br(N_{0}){p} with ind(α_{0})_{N_{0}(Φ)} ≤ p^{k}, a prime element π_{0} in M_{0}, and η_{0} ∈ Ch_{p}(N_{0}) such that

(14)  (α′_{0})M̂_{0} = α̂_{0} + (η̂_{0} ∪ (π_{0})) in Br(M̂_{0})

and

(15)  α_{N} − (α_{0})_{N} ∈ Br(N(Φ)/N)_{dec}.

By (14), we have

exp(α_{0}) = exp(α̂_{0}) ≤ lcm(exp(α′_{0})M̂_{0}, p) ≤ lcm(exp(α′_{0}), p) ≤ p^{s},

hence α_{0} ∈ B_{k,s}^{Φ}(N_{0}). Therefore, the class of α_{N} in B̃_{k,s}^{Φ}(N) is defined over N_{0} by (15). It follows that

edB̃_{k+1,s}^{Φ′} ; p(α′) = tr. deg_{F} (M_{0}) ≥ tr. deg_{F} (N_{0}) + 1 ≥ edB̃_{k,s}^{Φ} ; p(α) + 1.  □

4.2. A technical lemma. In this subsection we prove Lemma 4.2 that will allow us to apply Proposition 4.1.

Until the end of this subsection we assume that the base field F contains a primitive p^{2}-th root of unity.

Let χ_{1}, χ_{2}, …, χ_{r} with r ≥ 2 be linearly independent characters in Ch_{p}(F) and Φ = ⟨χ_{1}, χ_{2}, …, χ_{r}⟩. Let E/F be a field extension such that rank(Φ_{E}) = r and let α ∈ Br(E){p} be an element that is split by E(Φ) and exp(α) ≤ p^{s}.

Let E_{0} = E, E_{1}, …, E_{r} be field extensions of F such that for any k = 1, 2, …, r, the field E_{k} is complete with respect to a discrete valuation v_{k} over F and E_{k−1} is its residue field. For any k = 1, 2, …, r, choose elements x_{k} ∈ E_{k}^{×} such that v_{k}(x_{k}) is prime to p and define the elements α_{k} ∈ Br(E_{k}){p} inductively by α_{0} := α and

α_{k} := α̂_{k−1} + ( ˆ(χ_{k})_{E_{k−1}} ∪ (x_{k})).

Let Φ_{k} be the subgroup of Φ generated by χ_{k+1}, …, χ_{r}. Thus, Φ_{0} = Φ, Φ_{r} = 0 and rank(Φ_{k}) = r − k. Note that the character (χ_{k})_{E_{k−1}(Φ_{k})} is not trivial. It follows from (1) that

ind(α_{k})_{E_{k}(Φ_{k})} = p · ind(α_{k−1})_{E_{k−1}(Φ_{k−1})}

for any k = 1, …, r. As ind α_{E(Φ)} = 1, we have ind(α_{k})_{E_{k}(Φ_{k})} = p^{k} for all k = 0, 1, …, r. Moreover, as exp(α) ≤ p^{s}, we have exp(α_{k}) = lcm(exp(α_{k−1}), p) ≤ p^{s}. Therefore, α_{k} ∈ B_{k,s}^{Φ_{k}}(E_{k}).

The followings lemma assures that under a certain restriction on the element α, the conditions of Proposition 4.1 are satisfied for the fields E_{k}, the groups of characters Φ_{k} and the elements α_{k}. This lemma is similar to [10, Lemma 5.4].

Lemma 4.2. Suppose that for any subgroup Ψ ⊂ Φ with [Φ : Ψ] = p^{2} and any field extension L/E(Ψ) of degree prime to p, the element α_{L} is not p^{2}- cyclic . Then for every k = 0, 1, …, r − 1, and any finite field extension N/E_{k} of degree prime to p and any character ρ ∈ Ch(N) of order p^{2} such that pρ ∈ (Φ_{k})_{N} \ (Φ_{k+1})_{N}, we have

(16)  ind(α_{k})_{N(Φ_{k+1},ρ)} ≥ p^{k}.

Proof. Let k, N and ρ satisfy the conditions of the lemma. We construct a new sequence of fields Ẽ_{0}, Ẽ_{1}, …, Ẽ_{r} such that each Ẽ_{i} is a finite extension of E_{i} of degree prime to p as follows. We set Ẽ_{k} = N. The fields Ẽ_{j} with j < k are constructed by descending induction on j. If we have constructed Ẽ_{j} as a finite extension of E_{j} of degree prime to p, then we extend the valuation v_{j} to Ẽ_{j} and let Ẽ_{j−1} to be its residue field. The fields Ẽ_{m} with m > k are constructed by ascending induction on m. If we have constructed Ẽ_{m} as a finite extension of E_{m} of degree prime to p, then let Ẽ_{m+1} be an extension of E_{m+1} of degree [Ẽ_{m} : E_{m}] with residue field Ẽ_{m}. Replacing E_{i} by Ẽ_{i} and α_{i} by (α_{i})Ẽ_{i}, we may assume that N = E_{k}.

We proceed by induction on r. The case r = 1 is obvious.

(r − 1) ⇒ r: First suppose that k < r − 1. Consider the fields F′ = F (χ_{r}), E′ = E(χ_{r}), E′_{i} = E_{i}(χ_{r}), the sequence of characters χ′_{i} = (χ_{i})_{F′}, and the sequence of elements α′_{i} := (α_{i})_{E′_{i}} ∈ Br(E′_{i}) for i = 0, 1, …, r − 1. Let Φ′ = ⟨χ′_{1}, χ′_{2}, …, χ′_{r−1}⟩ ⊂ Ch(F′), let Φ′_{i} be the subgroup of Φ′ generated by χ′_{i+1}, …, χ′_{r−1} and ρ′ = ρ_{E′_{k}}.

We check the conditions of the lemma for the new datum. Let Ψ′ be a subgroup of Φ′ of index p^{2}. Then the pre-image Ψ of Ψ′ under the map Ch(F) → Ch(F′) is a subgroup of Φ of index p^{2} and E′(Ψ′) = E(Ψ). Let L′/E′(Ψ′) be a field extension of degree prime to p. By assumption, the element α′_{L′} = α_{L′} is not p^{2}-cyclic. We also have pρ′ = pρ_{E′_{k}} ∈ (Φ_{k})_{E′_{k}} = (Φ′_{k})_{E′_{k}}. Suppose that pρ′ ∈ (Φ′_{k+1})_{E′_{k}}, i.e., pρ_{E′_{k}} = pρ′ = η_{E′_{k}} for some η ∈ (Φ_{k+1})_{E_{k}}. It follows that pρ − η ∈ Ker(Ch(E_{k}) → Ch(E′_{k})) = ⟨(χ_{r})_{E_{k}}⟩ and therefore, pρ ∈ (Φ_{k+1})_{E_{k}}, a contradiction, hence pρ′ ∈ (Φ′_{k})_{E′_{k}} \ (Φ′_{k+1})_{E′_{k}}.

By the induction hypothesis, the inequality (16) holds for α′_{k}, i.e,

ind(α′_{k})_{E′_{k}(Φ′_{k+1},ρ′)} ≥ p^{k}.

As

(α′_{k})_{E′_{k}(Φ′_{k+1},ρ′)} = (α_{k})_{E_{k}(Φ_{k+1},ρ)},

the inequality (16) holds for α_{k}. Therefore, it remains to show the inequality (16) in the case k = r − 1. Note that is this case pρ is a nonzero multiple of (χ_{r})_{E_{r−1}} and Φ_{k+1} = Φ_{r} = 0.

Case 1: The character ρ is unramified with respect to v_{r−1}, i.e., ρ = μ̂ for a character μ ∈ Ch(E_{r−2}) of order p^{2}. Note that pμ is a nonzero multiple of (χ_{r})_{E_{r−2}}.

By (1),

(17)  ind(α_{r−2})_{E_{r−2}(χ_{r−1},μ)} = ind(α_{r−1})_{E_{r−1}(ρ)}/p.

Consider the fields F′ = F (χ_{r−1}), E′ = E(χ_{r−1}), E′_{i} = E_{i}(χ_{r−1}), the new sequence of characters χ′_{1} = (χ_{1})_{F′}, …, χ′_{r−2} = (χ_{r−2})_{F′}, χ′_{r−1} = (χ_{r})_{F′}, the group of characters Φ′ = ⟨χ′_{1}, χ′_{2}, …, χ′_{r−1}⟩ and the elements α′_{i} ∈ Br(E′_{i}) for i = 0, 1, …, r − 1 defined by α′_{i} = (α_{i})_{E′_{i}} for i ≤ r − 2 and α′_{r−1} = α̂_{r−2} + (χ̂_{r} ∪ (x_{r−1})) over E′_{r−1}, and the character μ. The new datum satisfy the conditions of the lemma. By the induction hypothesis, the inequality (16) holds for α′_{r−2}, i.e,

ind(α′_{r−2})_{E′_{r−2}(μ)} ≥ p^{r−2}.

As

(α′_{r−2})_{E′_{r−2}(μ)} = (α_{r−2})_{E_{r−2}(χ_{r−1},μ)},

the inequality (16) holds for α_{r−1} in view of the equality (17).

Case 2: The character ρ is ramified. Assume that inequality (16) does not hold for α_{r−1}, i.e., we have

ind(α_{r−1})_{E_{r−1}(ρ)} ≤ p^{r−2}.

By [10, Lemma 2.3(2)], there exists a unit u ∈ E_{r−1} such that E_{r−2}(χ_{r}) = E_{r−2}(ū^{1/p}) and

ind(α_{r−2} − (χ_{r−1} ∪ (ū^{1/p})))_{E_{r−2}(χ_{r})} = ind(α_{r−1})_{E_{r−1}(ρ)} ≤ p^{r−2}.

By descending induction on j = 0, 1, …, r − 2 we show that there exist an element u_{j} in E_{j}^{×} and a subgroup Ψ_{j} ⊂ Φ of rank r − j − 2 such that ⟨χ_{1}, …, χ_{j}, χ_{r−1}, χ_{r}⟩ ∩ Ψ_{j} = 0, E_{j}(χ_{r}) = E_{j}(u_{j}^{1/p}) and

(18)  ind(α_{j} − (χ_{r−1} ∪ (u_{j}^{1/p})))_{E_{j}(Θ_{j})} ≤ p^{j},

where Θ_{j} := ⟨Ψ_{j}, χ_{r}⟩. We set Ψ_{r−2} = 0 and u_{r−2} = ū.

j ⇒ (j − 1): The field E_{j}(u_{j}^{1/p}) = E_{j}(χ_{r}) is unramified over E_{j}, hence v_{j}(u_{j}) is divisible by p. Modifying u_{j} by a p^{2}-th power, we may assume that u_{j} = vx_{j}^{mp} for a unit v ∈ E_{j}, x_{j} ∈ E_{j}^{×} and an integer m. Then

(α_{j} − (χ_{r−1} ∪ (u_{j}^{1/p})))_{E_{j}(Θ_{j})} = β̂ + (η̂ ∪ (x_{j}))_{E_{j}(Θ_{j})},

where η = χ_{j} −mχ_{r−1} and β = (α_{j−1}−(χ_{r−1}∪(u_{j−1}^{1/p})))_{E_{j−1}(Θ_{j})}, where u_{j−1} = v̄. As η is not contained in Θ_{j}, the character η_{E_{j−1}(Θ_{j})} is not trivial. Set Ψ_{j−1} = ⟨Ψ_{j}, η⟩. It follows from (1) and the induction hypothesis that

ind(β_{E_{j−1}(Θ_{j−1})}) = ind(α_{j} − (χ_{r−1} ∪ (u_{j}^{1/p})))_{E_{j}(Θ_{j})}/p ≤ p^{j−1}.

Applying the inequality (18) in the case j = 0, we have

α_{E(Θ_{0})} = (χ_{r−1} ∪ (w^{1/p}))_{E(Θ_{0})}

for an element w ∈ E^{×} such that E(w^{1/p}) = E(χ_{r}). Hence

α_{E(Ψ_{0})(w^{1/p^{2}})} = (α_{E(Θ_{0})})_{E(Θ_{0})(w^{1/p^{2}})} = 0 in Br(E(Ψ_{0})(w^{1/p^{2}})).

Since α_{E(Ψ_{0})} is split by a cyclic extension E(Ψ_{0})(w^{1/p^{2}})/E(Ψ_{0}) of degree p^{2}, α_{E(Ψ_{0})} is p^{2}-cyclic. As [Φ : Ψ_{0}] = p^{2}, this contradicts the assumption. Hence, the inequality (16) holds for α_{r−1}. □

5. Non-cyclicity of the generic element

The aim of this section is the technical Lemma 5.4 that will allow us to apply later Lemma 4.2 and Proposition 4.1.

In this section we assume that the base field F contains a primitive p^{3}-th root of unity. The choice of a primitive p^{2}-th root of unity ξ allows us to define the symbol (a, b)_{p^{2}} as in Section 2.1. As −1 is a p^{2}-th power in F^{×}, we have (a, −1)_{p^{2}} = 0, hence (a, a)_{p^{2}} = 0 for all a ∈ F^{×}. We shall write (a, b)_{p} for p(a, b)_{p^{2}} = (a^{p}, b)_{p^{2}}.

Lemma 5.1. Let E be a field extension of F that is complete with respect to a discrete valuation v with residue field K and α ∈ Br(K). Set β = α̂ + (a, x)_{p} for a unit a ∈ E and x ∈ E^{×} such that ā ∉ K^{×p} and v(x) is prime to p. If β is p^{2}-cyclic, then α = (ā, z)_{p^{2}} in Br(K) for some z ∈ K^{×}.

Proof. Suppose that β = (uπ^{i}, wπ^{j})_{p^{2}} and write x = tπ^{k} for a prime element π, integers i, j, k = v(x) and units u, w, t in E. Then we have

α̂ + (a^{p}, wπ^{k})_{p^{2}} = β = (uπ^{i}, vπ^{j})_{p^{2}} = (u, v)_{p^{2}} + (u^{j}/v^{i}, π)_{p^{2}}.

Applying the residue map ∂_{v}, we get ā^{pk} = ū^{j}/v̄^{i} in K^{×}/K^{×p^{2}} and

α = (ū, v̄)_{p^{2}} − (ā, w̄^{p})_{p^{2}}.

Suppose that i/j is a p-integer (the other case is similar). As k is not divisible by p and ā is not a p-th power in K^{×}, j is not divisible by p^{2}. It follows that ū ∈ ⟨ā, v̄⟩ in K^{×}/K^{×p^{2}} and then ū ∈ ā^{r}v̄^{s}K^{×p^{2}} for some r and s. Hence α = (ā, v̄^{r}/w̄^{p})_{p^{2}}. □

Corollary 5.2. Let x,y be independent variables over F and a, b ∈ F^{×}. If (a, b)_{p} ≠ 0 in Br(F), then for any field extension M/F (x, y) of degree prime to p, the element (a, x)_{p} + (b, y)_{p} in Br(M) is not p^{2}-cyclic.

Proof. Let M/F (x, y) be a field extension of degree prime to p and β = (a, x)_{p}+ (b, y)_{p} over M. As the degree of M/F (x, y) is prime to p, by [8, Lemma 6.1], there exists a field extension E of the fields F ((y))((x)) and M over F such that the degree of E/F ((y))((x)) is finite and prime to p. The discrete valuation v_{x} on the complete field F ((y))((x)) extends uniquely to a discrete valuation v of E. The ramification index of E/F ((y))((x)) is prime to p, hence v(x) is prime to p. The residue field K of v is an extension of F ((y)) of degree prime to p.

Let v′ be the valuation on K extending the discrete valuation v_{y} on F ((y)). The ramification index e′ of K/F ((y)) is prime to p. The residue field N of v′ is a finite extension of F of degree prime to p.

Let α = (b, y)_{p} over K, so β_{E} = α̂ + (a, x)_{p}. Suppose that β is p^{2}-cyclic over M. Then β_{E} is also p^{2}-cyclic. By Lemma 5.1, applied to β_{E} over E, we have α = (a, z)_{p^{2}} for some z ∈ K^{×}, hence (b^{p}, y)_{p^{2}} = (a, z)_{p^{2}}. Taking the cup product with (a)_{p^{2}} ∈ K^{×}/K^{×p^{2}}, we get

(a)_{p^{2}} ∪ (b^{p}, y)_{p^{2}} = (a)_{p^{2}} ∪ (a, z)_{p^{2}} = (a, a)_{p^{2}} ∪ (z)_{p^{2}} = 0.

Applying the residue map ∂_{v′}, we find that e′(a, b)_{p} = e′(a, b^{p})_{p^{2}} = 0 over N, hence (a, b)_{p} = 0 in Br(N). Taking the corestriction map Br(N) → Br(F), we see that (a, b)_{p} = 0 in Br(F), a contradiction. □

Lemma 5.3. For any integer r ≥ 2, there exist a field extension F′/F and a subgroup Φ ⊂ Ch_{p}(F′) of rank r such that for any subgroup Ψ ⊂ Φ of index p^{2}, there is an element β ∈ Br_{p}(F′(Φ)/F′) with the property that any field extension M/F′(Ψ) of degree prime to p, the element β_{M} is not p^{2}-cyclic.

Proof. Let a_{1}, a_{2}, …, a_{r}, x, y be independent variables over F and set F′ := F (a_{1}, a_{2}, …, a_{r}, x, y). For every i = 1, …, r, let χ_{i} ∈ Ch_{p}(F′) be a character such that F′(χ_{i}) = F′(a_{i}^{1/p}) and set Φ := ⟨χ_{1}, χ_{2}, …, χ_{r}⟩. Let Ψ be a subgroup of Φ of index p^{2}. Choose a basis η_{1}, η_{2}, …, η_{r} for Φ such that Ψ = ⟨η_{1}, η_{2}, …, η_{r−2}⟩ and the elements b_{1}, b_{2}, …, b_{r} in F′ such that F (η_{i}) = F (b_{i}^{1/p}) for all i = 1, …, r and F (b_{1}, b_{2}, …, b_{r}) = F (a_{1}, a_{2}, …, a_{r}). Clearly, b_{1}, b_{2}, …, b_{r} are algebraically independent over F and F′(Ψ) = L(x, y), where L := F (b_{1}^{1/p}, …, b_{r−2}^{1/p}, b_{r−1}, b_{r}) with the generators algebraically independent over F.

Let β = (b_{r−1}, x)_{p} +(b_{r}, y)_{p} in Br_{p}(F′(Φ)/F′) and M/F′(Ψ) a field extension of degree prime to p. As ∂_{v}((b_{r−1}, b_{r})_{p}) = b̄_{r−1}, where v is the discrete valuation on L associated with b_{r}, is nontrivial, we have (b_{r−1}, b_{r})_{p} ≠ 0 in Br(L). The result follows from Corollary 5.2. □

Let F′/F be the field extension and Φ ⊂ Ch_{p}(F′) the subgroup of rank r as in Lemma 5.3. Consider the algebraic tori P^{Φ}, S^{Φ}, T^{Φ}, U^{Φ} and V^{Φ} over F′ defined in Section 2.4. The morphism γ : P^{Φ} → V^{Φ} in the diagram (4) is a U^{Φ}-torsor. Denote by δ the image of the class of γ under the composition

H_{ét}^{1}(V^{Φ}, U^{Φ}) → H_{ét}^{1}(V^{Φ}, U^{′Φ}) → H_{ét}^{2}(V^{Φ}, 𝔾_{m}),

induced by the diagram (5). We write δ_{gen} for the image of δ under the homomorphism

H_{ét}^{2}(V^{Φ}, 𝔾_{m}) → H^{2}(F (V^{Φ}), 𝔾_{m}) = Br(F′(V^{Φ}))

induced by the generic point morphism Spec(F′(V^{Φ})) → V^{Φ}. It follows from (6) that δ_{gen} ∈ Br_{p^{s}}(F′(V^{Φ})).

Lemma 5.4. Let K = F′(V^{Φ}) and Ψ ⊂ Φ a subgroup with [Φ : Ψ] = p^{2}. Then for any field extension M/K(Ψ) of degree prime to p, the element (δ_{gen})_{M} is not p^{2}-cyclic.

Proof. Suppose that there exist a subgroup Ψ ⊂ Φ with [Φ : Ψ] = p^{2} and a field M/K(Ψ) of degree prime to p such that (δ_{gen})_{M} = χ ∪ (a) for some χ ∈ H^{2}(M, ℤ) = Ch(M) with p^{2}χ = 0 and a ∈ H^{0}(M, 𝔾_{m}) = M^{×}. Choose an integral scheme X over F′ such that F′(X) = M together with a dominant F′-morphism

f : X → V^{Φ}(Ψ) := (V^{Φ})_{F′(Ψ)}

of degree prime to p that induces the embedding of the function field K(Ψ) into M. Let h : X → V^{Φ} be the composition of f with the natural morphism g : V^{Φ}(Ψ) → V^{Φ}. Replacing X by a nonempty open set, we may assume that h^{∗}(δ) = χ_{0} ∪(a_{0}) for some χ_{0} ∈ H_{ét}^{2}(X, ℤ) with p^{2}χ_{0} = 0 and a_{0} ∈ H_{ét}^{0}(X, 𝔾_{m}).

By [8, Lemma 6.2], there is a nonempty open set W′ ⊂ V^{Φ}(Ψ) such that for every x′ ∈ W′ there exists a point x ∈ X with f(x) = x′ and the degree [F′(x) : F′(x′)] prime to p. Let Z = V^{Φ}(Ψ) \ W′. As g is finite, g(Z) ≠ V^{Φ}, hence the open set W := V^{Φ} \ g(Z) is not empty. We have g^{−1}(W) ⊂ W′.

Consider the element β ∈ Br_{p}(F′(Φ)/F′) constructed in Lemma 5.3. Let γ′ ∈ H^{1}(F′, U^{Φ}) be the corresponding class of U^{Φ}-torsors over F′ under the isomorphism H^{1}(F′, U^{Φ}) ≃ Br_{p^{s}}(F′(Φ)/F′) by (7). As γ is a generic U^{Φ}- torsor, there exists an F′-morphism v : Spec F′ → V^{Φ} such that v^{∗}(γ) = γ′ and Im(v) ⊂ W (see Section 2.3). From the commutativity of the diagram

H_{ét}^{1}(V^{Φ}, U^{Φ}) ⟶^{v^{∗}} H^{1}(F′, U^{Φ})
↓  ↓
H_{ét}^{2}(V^{Φ}, 𝔾_{m}) ⟶^{v^{∗}} H^{2}(F′, 𝔾_{m})

we find that v^{∗}(δ) = β.

Let v′ : Spec F′(Ψ) → V^{Φ}(Ψ) be the morphism v_{F′(Ψ)}. Note that Im(v′) ⊂ g^{−1}(W) ⊂ W′.

By the definition of W′, there is a point x ∈ X such that the degree of the field extension F′(x) over the residue field of (the only) point in Im(v′) is prime to p. By [8, Lemma 6.1], there exist a field extension M/F′(Ψ) of degree prime to p and a morphism w : Spec(M) → X such that the diagram

Spec(M) ⟶ Spec(F′(Ψ)) ⟶ Spec(F′)
w↓  _{v′}↓  ^{v}↓
X  ⟶^{f} V^{Φ}(Ψ) ⟶^{g} V^{Φ}

is commutative. It follows that

β_{M} = v^{∗}(δ)_{M} = w^{∗}h^{∗}(δ) = w^{∗}(χ_{0} ∪ (a_{0})) = w^{∗}(χ_{0}) ∪ w^{∗}(a_{0}),

i.e., β_{M} is p^{2}-cyclic. This contradicts Lemma 5.3. □

6. A lower bound for ed_{p}(Alg_{p^{r},p^{s}})

Let n ≥ 1 be an integer, m a divisor of n and p a prime integer. Let p^{r} (respectively, p^{s}) be the largest power of p dividing n (respectively, m). If A ∈ Alg_{n,m}(K) for some field extension K/F, then there is a finite field extension E/K of degree prime to p such that ind(A_{E}) is a p-power. Hence ind(A_{E}) divides p^{r} and exp(A_{E}) divides p^{s} as it divides m and ind(A_{E}), i.e., A_{E} ∈ Alg_{p^{r},p^{s}}(E). It follows that the embedding functor Alg_{p^{r},p^{s}} → Alg_{n,m} is p-surjective and hence ed_{p}(Alg_{n,m}) ≤ ed_{p}(Alg_{p^{r},p^{s}}) by [8, Sec. 1.3]. Conversely, if A ∈ Alg_{n,m}(K), then the p-primary component A_{p} of A satisfies A_{p} ∈ Alg_{p^{r},p^{s}}(K), hence the morphism of functors Alg_{n,m} → Alg_{p^{r},p^{s}}, taking A to A_{p} is surjective and therefore, ed_{p}(Alg_{n,m}) ≥ ed_{p}(Alg_{p^{r},p^{s}}). We proved that

ed_{p}(Alg_{n,m}) = ed_{p}(Alg_{p^{r},p^{s}}).

Theorem 6.1. Let F be a field and p a prime integer different from char(F). Then, for any integers r and s with 1 ≤ s ≤ r,

ed_{p}(Alg_{p^{r},p^{s}}) ≥ \begin{cases}(r − 1)2^{r−1} & if p = 2 and s = 1, \\ (r − 1)p^{r} + p^{r−s} & otherwise. \end{cases}

Proof. By [8, Prop.1.5], we can replace the base field by any field extension. Hence we may assume that F contains a primitive p^{3}-th root of unity. Moreover, we can replace F by the field F′ in Lemma 5.3. Let V^{Φ} be the algebraic torus constructed in Section 2.4. Set E = F (V^{Φ}) and let α := δ_{gen} ∈ Br_{p^{s}}(E(Φ)/E) be the element defined in Section 5. Let E_{k} be the fields and α_{k} ∈ B_{k,s}^{Φ_{k}}(E_{k}) the elements constructed in Section 4.2, so that E_{0} = E and α_{0} = α. By Lemma 5.4, α_{M} is not p^{2}-cyclic for any subgroup Ψ ⊂ Φ with [Φ : Ψ] = p^{2} and any field extension M/E(Ψ) of degree prime to p, hence α satisfies the condition of Lemma 4.2. It follows that we can apply Proposition 4.1. By the iterated application of this proposition, we have

(19) ed_{p}^{Alg_{p^{r},p^{s}}} (α_{r}) = edB̃_{r,s}^{Φr} ; p(α_{r}) ≥ edB̃_{r−1,s}^{Φ_{r−1}} ; p(α_{r−1}) + 1 ≥ …

≥ edB̃_{1,s}^{Φ_{1}} ; p(α_{1}) + (r − 1) ≥ edB̃_{0,s}^{Φ_{0}} ; p(α_{0}) + r = edB̃_{0,s}^{Φ} ; p(α) + r.

Consider the commutative diagram with exact rows:

1 ⟶ U^{Φ} ⟶ P^{Φ}  ⟶^{γ} V^{Φ} ⟶ 1
↓  ↓  ∥
1 ⟶ S^{Φ} ⟶ P^{Φ} × 𝔾_{m}^{r} ⟶^{γ′} V^{Φ} ⟶ 1,

where P^{Φ} → P^{Φ} × 𝔾_{m}^{r} takes x to (x, 1) and S^{Φ} ↪ P^{Φ} × 𝔾_{m}^{r} is the product of S^{Φ} ↪ P^{Φ} and S^{Φ} ↠ 𝔾_{m}^{r}.

The element α considered in B_{0,s}^{Φ}(E) corresponds to the generic fiber of the U^{Φ}-torsor γ under the bijection B_{0,s}^{Φ}(E) ≃ U^{Φ}- torsors(E) in (11). Hence, by the diagram, the class of α in B̃_{0,s}^{Φ}(E) corresponds to the generic fiber γ′_{gen} of the S^{Φ}-torsor γ′ under the bijection B̃_{0,s}^{Φ}(E) ≃ S^{Φ}- torsors(E). As P^{Φ} × 𝔾_{m}^{r} is a quasi-split torus, γ′ is a generic S^{Φ}-torsor by Proposition 2.1, hence

(20)  edB̃_{0,s}^{Φ} ; p(α) = ed_{p}^{S^{Φ}- torsors}(γ′_{gen}) = ed_{p}(S^{Φ})

by [8, Th. 2.9]. The essential p-dimension of S^{Φ} was calculated in Corollary 3.11. From (19),(20) and this corollary, we have

ed_{p}(Alg_{p^{r},p^{s}}) ≥ ed_{p}^{Alg_{p^{r},p^{s}}} (α_{r}) ≥ ed_{p}(S^{Φ})+r = \begin{cases}(r − 1)2^{r−1} & if p = 2 and s = 1, \\ (r − 1)p^{r} + p^{r−s} & otherwise. \end{cases}

This concludes the proof. □

7. An upper bound for ed_{p}(Alg_{p^{r},p^{s}})

Lemma 7.1. Let F be a field and p a prime. Then, for any integers r and s with 1 ≤ s ≤ r,

ed_{p}(Alg_{p^{r},p^{s}}) ≤ ed_{p}(Alg_{p^{r}}) + p^{r−s} − 1.

Proof. Let A ∈ Alg_{p^{r},p^{s}}(K) ⊂ Alg_{p^{r}}(K) for a field extension K/F. There exist a field extension K′/K of degree prime to p, a subfield K_{0} ⊂ K′ over F and B ∈ Alg_{p^{r}}(K_{0}) such that tr. deg_{F} (K_{0}) ≤ ed_{p}(Alg_{p^{r}}) and A ⊗_{K} K′ ≃ B ⊗_{K_{0}} K′.

By [16, Lemma 5.6], ind(B^{⊗p^{s}}) divides p^{r−s}. Choose a central simple algebra C of degree p^{r−s} over K_{0} in the Brauer class of B^{⊗p^{s}} in Br(K_{0}) and consider the Severi-Brauer variety X := SB(C) of C. Since exp(A) divides p^{s}, the algebra C is split over K′, hence X(K′) ≠ ∅. This implies that there exists x ∈ X such that K_{0}(x) ⊂ K′ and X(K_{0}(x)) ≠ ∅. Therefore, C_{K_{0}(x)} is split, hence exp(B_{K_{0}(x)}) divides p^{s}, i.e., B_{K_{0}(x)} ∈ Alg_{p^{r},p^{s}}(K_{0}(x)). Since dim(X) = p^{r−s}−1, we have

ed_{p}^{Alg_{p^{r},p^{s}}}(A) ≤ tr. deg_{F} (K_{0}(x)) = tr. deg_{F} (K_{0}) + tr. deg_{K_{0}}(K_{0}(x)) ≤

ed_{p}(Alg_{p^{r}}) + dim(x) ≤ ed_{p}(Alg_{p^{r}}) + (p^{r−s} − 1). □

By [12, Th.1.1],

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

if r ≥ 2, therefore, by Lemma 7.1, we have the following upper bound for ed_{p}(Alg_{p^{r},p^{s}}):

Theorem 7.2. Let F be a field and p a prime integer. Then, for any integers r ≥ 2 and s with 1 ≤ s ≤ r,

ed_{p}(Alg_{p^{r},p^{s}}) ≤ 2p^{2r−2} − p^{r} + p^{r−s}.

8. Essential dimension of Alg_{L/F}, Alg_{G} and ALG_{G}.

Let G be an elementary abelian group of order p^{r} and K/F a field extension. Consider the subset Alg_{G}(K) of Alg_{p^{r},p^{s}}(K) consisting of all classes that have a splitting Galois K-algebra E with Gal(E/K) ≃ G.

Let L/F be a Galois field extension with Gal(L/F) ≃ G. Consider the subset Alg_{L/F}(K) of Alg_{G}(K) consisting of all classes split by the field extension KL/K. We have the subfunctors of Alg_{p^{r},p^{s}}:

Alg_{L/F} ⊂ Alg_{G} ⊂ Alg_{p^{r},p^{s}}.

We write ALG_{G}(K) for the set of pairs (A, E), where A ∈ Alg_{G}(K) and E is a Galois G-algebra splitting A. We have an obvious surjective morphism of functors ALG_{G} → Alg_{G}.

Theorem 8.1. Let F be a field, p a prime integer different from char(F), G an elementary abelian group of order p^{r} with r ≥ 2, and L/F a Galois field extension with Gal(L/F) ≃ G. Let an integer s satisfy 1 ≤ s ≤ r. Suppose that r ≥ 3 if p = 2 and s = 1. Let F be one of the three functors: Alg_{L/F}, Alg_{G} or ALG_{G}. Then

ed_{p}(F) = ed(F) = \begin{cases}(r − 1)2^{r−1} & if p = 2 and s = 1, \\ (r − 1)p^{r} + p^{r−s} & otherwise. \end{cases}

Proof. Let Φ be a subgroup of Ch_{p}(F) of rank r such that L = F (Φ). By (7), we have Alg_{L/F} ≃ U^{Φ}- torsors. It follows from Proposition 3.9 that

ed_{p}(Alg_{L/F}) = ed(Alg_{L/F}) = d_{p,r,s} := \begin{cases}(r − 1)2^{r−1} & if p = 2 and s = 1, \\ (r − 1)p^{r} + p^{r−s} & otherwise. \end{cases}

Let α_{r} ∈ Br(E_{r}) be as in the proof of Theorem 6.1. By construction, α_{r} is split by E_{r}(Φ), hence α_{r} ∈ Alg_{G}(E_{r}). Note that ed_{p}^{B}(β) ≤ ed_{p}^{H}(β) for any subfunctor H of a functor B and any β ∈ H(K). Hence, by the proof of Theorem 6.1, we have

ed_{p}(Alg_{G}) ≥ ed_{p}^{Alg_{G}} (α_{r}) ≥ ed_{p}^{Alg_{p^{r},p^{s}}} (α_{r}) ≥ d_{p,r,s}.

Let J be the G-module defined in the Section 2.4 and T := Spec F [J] the split torus with the character group J. Consider the minimal surjective ppresentation ν : P′ → J as in Remark 3.10. As explained in Section 2.2, a choice of a G-invariant basis of P yields a linear T ⋊G-space V with dim(V) = rank(P′). By Remark 3.10, G acts faithfully on Ker(ν). It follows from [11, Lemma 3.3] that the action of T ⋊G on V is generically free in this case, hence, by [3, Prop. 4.11],

ed(T ⋊ G) ≤ dim(V) − dim(T ⋊ G)

= rank(P′) − rank(J)

= rank(Ker(ν))

= d_{p,r,s}.

Let γ ∈ H^{1}(F, G) and let L be the corresponding Galois G-algebra over F. Since G is an abelian group, we have G = G_{γ}. The G-action on R_{L/F} (𝔾_{m,L}) restricts to the trivial action on the subgroup μ_{p^{s}}. As T_{γ} = R_{L/F} (𝔾_{m,L})/μ_{p^{s}}, the connecting map

H^{1}(F, T_{γ}) → H^{2}(F, μ_{p^{s}}) = Br_{p^{s}}(F)

is injective, hence the group G_{γ}(F) = G acts trivially on H^{1}(F, T_{γ}). By (2),

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

where the disjoint union is taken over all isomorphism classes of Galois Galgebras E/F. Hence we have a surjective morphism of functors T ⋊G- torsors → ALG_{G}. As ALG_{G} surjects on Alg_{G}, we have

ed_{p}(Alg_{G}) ≤ (ed_{p}(ALG_{G}) or ed(Alg_{G})) ≤ ed(ALG_{G}) ≤ ed(T ⋊ G) ≤ d_{p,r,s}. □

Remark 8.2. Suppose that p = r = 2 and s = 1 and F is a field of characteristic different from 2. By [15, Th.1] or [2, Sec.2.4], there exists a nontrivial cohomological invariant of degree 4 for Alg_{G} over F (i), where i is a primitive 4-th root of unity. Hence, ed_{2}(Alg_{G}) ≥ ed_{2}(Alg_{G})_{F (i)} ≥ 4 by [13, Lemma 6.9]. Moreover, by the structure theorem on central simple algebras split by a biquadratic field extension [20, Cor.2.8], every (A, E) ∈ ALG_{G}(K) is of the form E = K(a^{1/2}, b^{1/2}) and [A] = (a, x)_{2} + (b, y)_{2} for some a, b, x, y ∈ K^{×}. Hence ed(ALG_{G}) ≤ 4. As ALG_{G} surjects on Alg_{G}, we have

4 ≤ ed_{2}(Alg_{G}) ≤ (ed_{2}(ALG_{G}) or ed(Alg_{G})) ≤ ed(ALG_{G}) ≤ 4,

hence the essential (2)-dimension of Alg_{G} and ALG_{G} is equal to 4. Corollary 8.3. Let F be a field of characteristic ≠ 2. Then

ed_{2}(Alg_{8,2}) = ed(Alg_{8,2}) = 8.

Proof. As any central simple algebra of degree 8 and exponent 2 has a triquadratic splitting field by [14], we have Alg_{8,2} = Alg_{G} for the elementary abelian group G of order 8, hence the statement follows from Theorem 8.1. Note that the inequality ed_{2}(Alg_{8,2}) ≥ 8 is also proven in Theorem 6.1 and the opposite inequality ed(Alg_{8,2}) ≤ 8 was shown in [2, Th.2.12]. □

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

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

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

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