ESSENTIAL DIMENSION OF SIMPLE ALGEBRAS

ALEXANDER S. MERKURJEV

1. Introduction

The essential dimension of an “algebraic structure” is a numerical invariant that measures its complexity. Informally, the essential dimension of an algebraic structure over a field F is the smallest number of algebraically independent parameters required to define the structure over a field extension of F (see [1] or [10]).

Let F : Fields/F → Sets be a functor (an “algebraic structure”) from the category Fields/F of field extensions of F and field homomorphisms over F to the category of sets. Let K ∈ Fields/F, α ∈ F(K) and K_{0} a subfield of K over F. We say that α is defined over K_{0} (and K_{0} is called a field of definition of α) if there exists an element α_{0} ∈ F(K_{0}) such that the image (α_{0})_{K} of α_{0} under the map F(K_{0}) → F(K) coincides with α. The essential dimension of α, denoted ed^{F}(α), is the least transcendence degree tr. deg_{F} (K_{0}) over all fields of definition K_{0} of α. The essential dimension of the functor F is

ed(F) = sup{ed^{F}(α)},

where the supremum is taken over fields K ∈ Fields/F and all α ∈ F(K).

Let p be a prime integer and α ∈ F(K). The essential p-dimension ed_{p}^{F}(α) of α is the minimum of ed^{F}(α_{K′}) over all finite field extensions K′/K of degree prime to p. The essential p-dimension ed_{p}(F) of F is the supremum of ed_{p}^{F}(α) over all fields K ∈ Fields/F and all α ∈ F(K) (see [14, §6]). Clearly, ed(F) ≥ ed_{p}(F) for all p.

Let G be an algebraic group over F. The essential dimension ed(G) (resp. essential p-dimension ed_{p}(G)) of G is the essential dimension (resp. essential pdimension) of the functor G- torsors taking a field K to the set of isomorphism classes of all G-torsors (principal homogeneous G-spaces) over K.

If G = PGL_{n} over F, the functor G- torsors is isomorphic to the functor Alg_{F} (n) taking a field K to the set of isomorphism classes of central simple K-algebras of degree n. Let p be a prime integer and let p^{r} be the highest power of p dividing n. Then ed_{p}(Alg_{F} (n)) = ed_{p}(Alg_{F} (p^{r})) [14, Lemma 8.5.5]. Every central simple E-algebra of degree p is cyclic over a finite field extension of degree prime to p, hence ed_{p}(Alg_{F} (p)) = 2 [14, Lemma 8.5.7]. It was proven in [11] that ed_{p}(Alg_{F} (p^{2})) = p^{2} + 1 and in general, ed_{p}(Alg_{F} (p^{r})) ≥ 2r for all r in [14, Th. 8.6].

We prove the following:

Theorem. Let F be a field and p an integer different from char(F). Then

ed_{p}(Alg_{F} (p^{r})) ≥ (r − 1)p^{r} + 1.

In other words, we have the following lower bound for the essential dimension of PGL_{F} (p^{r}):

ed(PGL_{F} (p^{r})) ≥ ed_{p}(PGL_{F} (p^{r})) ≥ (r − 1)p^{r} + 1.

2. Preliminaries

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

The character group Ch(F) of F is defined as

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

For a character χ ∈ Ch(F), set F (χ) = (F_{sep})^{Ker(χ)}. Then F (χ)/F is a cyclic field extension of degree ord(χ). If Φ ⊂ Ch(F) is a finite subgroup, we set

F (Φ) = (F_{sep})^{∩ Ker(χ)},

where the intersection is taken over all χ ∈ Φ. The Galois group G = Gal(F (Φ)/F ) is abelian and Φ is canonically isomorphic to the character group Ch(G) = Hom(G, ℚ/ℤ) of G.

If F′ ⊂ F is a subfield and χ ∈ Ch(F′), we write χ_{F} for the image of χ under the natural map Ch(F′) → Ch(F) and F (χ) for F (χ_{F}). If Φ ⊂ Ch(F) is a finite subgroup, then the character χ_{F (Φ)} is trivial if and only if χ ∈ Φ.

Lemma 2.1. Let Φ, Φ′ ⊂ Ch(F) be two finite subgroups. Suppose that for a field extension K/F, we have Φ_{K} = Φ′_{K} in Ch(K). Then there is a finite subextension K′/F in K/F such that Φ_{K′} = Φ′_{K′} in Ch(K′).

Proof. Choose a set of characters {χ_{1}, …, χ_{m}} generating Φ and a set of characters {χ′_{1}, …, χ′_{m}} generating Φ′ such that (χ_{i})_{K} = (χ′_{i})_{K} for all i. Let η_{i} = χ_{i} − χ′_{i}. As all η_{i} vanish over K, the finite field extension K′ := F (η_{1}, …, η_{m}) of F can be viewed as a subextension in K/F. As (χ_{i})_{K′} = (χ′_{i})_{K′}, we have Φ_{K′} = Φ′_{K′}. □

2.2. Brauer group. We write Br(F) for the Brauer group H^{2}(F, F_{sep}^{×}) of a field F. If a ∈ Br(F) and K/F is a field extension, then we write a_{K} for the image of a under the natural homomorphism Br(F) → Br(K). We write Br(K/F) for the relative Brauer group Ker(Br(F) → Br(K)). We say that K is a splitting field of a if a_{K} = 0, i.e., a ∈ Br(K/F). The index ind(a) of a is the smallest degree of a splitting field of a.

The cup-product

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

takes χ ⊗ a to the class χ ∪ (a) in Br(F) that is split by F (χ).

For a finite subgroup Φ ⊂ Ch(F) write Br_{dec}(F (Φ)/F ) 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_{ind}(F (Φ)/F ) is the factor group Br(F (Φ)/F )/ Br_{dec}(F (Φ)/F ).

2.3. Complete fields. 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

(1)  0 → Br(K){p} ⟶^{i} Br(E){p} ⟶^{∂_{v}} Ch(K){p} → 0.

If a ∈ Br(K){p}, then we write â for the element i(a) in Br(E){p}. For example, if a = χ ∪ (ū) for some χ ∈ Ch(K){p} and a unit u ∈ E, then â = χ̂ ∪ (u).

The following proposition was proved in [6, Th. 5.15(a)], [16, Prop. 2.4]) and [4, Prop. 8.2].

Proposition 2.2. Let E be a complete field with respect to a discrete valuation v and K its residue field of characteristic different from p. Then

(1) ind(â) = ind(a) for any a ∈ Br(K){p}.

(2) Let b = â + (χ ∪ (x)) for an element a ∈ Br(K){p}, χ ∈ Ch(K){p} and x ∈ E^{×} such that v(x) is not divisible by p. Then

ind(b) = ind(a_{K(χ)}) · ord(χ).

(3) Let E′/E be a finite field extension and v′ the discrete valuation on E′ extending v with residue field K′. Then for any b ∈ Br(E){p}, one has

∂_{v′}(b_{E′}) = e · ∂_{v}(b)_{K′},

where e is the ramification index of E′/E.

The choice of a prime element π in E provides with a splitting of the sequence (1) by sending a character χ to the class χ̂ ∪ (π) in Br(E){p}. Thus, any b ∈ Br(E){p} we can written in the form:

(2)  b = â + (χ̂ ∪ (π))

for χ = ∂_{v}(b) and a unique a ∈ Br(K){p}.

The homomorphism

s_{π} : Br(E){p} → Br(K){p},

defined by s_{π}(b) = a, where a is given by (2), is called a specialization map. For example, s_{π}(â) = a for any a ∈ Br(K){p} and s_{π}(χ̂∪(x)) = χ∪(ū), where χ ∈ Ch(K){p}, x ∈ E^{×} and u is the unit in E such that x = uπ^{v(x)}.

Moreover, if v is trivial on a subfield F ⊂ E and Φ ⊂ Ch(F){p} a finite subgroup, then

(3)  s_{π}(Br_{dec}(E(Φ)/E)) ⊂ Br_{dec}(K(Φ)/K).

We shall need the following technical Lemma. For an abelian group A we write _{p}A for the subgroup of all elements in A of exponent p.

Lemma 2.3. Let (E, v) be a complete discrete valued field with the residue field K of characteristic different from p containing a primitive p^{2}-th root of unity. Let η ∈ Ch(E) be a character of order p^{2} such that p · η is unramified, i.e., p · η = ν̂ for some ν ∈ Ch(K) of order p. Let χ ∈_{p} Ch(K) be a character linearly independent from ν. Let a ∈ Br(K) and set b = â+(χ̂∪(x)) ∈ Br(E), where x ∈ E^{×} is an element such that v(x) is not divisible by p. Then:

(1) If η is unramified, , i.e., η = μ̂ for some μ ∈ Ch(K) of order p^{2}, then ind(b_{E(η)}) = p · ind(a_{K(μ,χ)}).

(2) If η is ramified, then there exists a unit u ∈ E^{×} such that K(ν) = K(ū^{1/p}) and ind(b_{E(η)}) = ind(a − (χ ∪ (ū^{1/p})))_{K(ν)}.

Proof. (1) If η = μ̂ for some μ ∈ Ch(K), then K(μ) is the residue field of E(η) and we have

b_{E(η)} = â_{K(μ)} + (χ̂_{K(μ)} ∪ (x)).

As χ and ν are linearly independent, the character χ_{K(μ)} is nontrivial. The first statement follows from Proposition 2.2(2).

(2) Since p · η is unramified, the ramification index of E(η)/E is equal to p, hence E(η) = E((ux^{p})^{1/p^{2}}) for some unit u ∈ E. Note that K(ν) = K(ū^{1/p}) is the residue field of E(η). As u^{1/p}x is a p-th power in E(η), the class

b_{E(η)} = â_{K(ν)} − (χ̂_{K(ν)} ∪ (u^{1/p})) = â_{K(ν)} − ( ; χ_{K(ν)} ∪ (ū^{1/p}))

is unramified. It follows from Proposition 2.2(1) that the elements b_{E(η)} in Br(E(η)) and a_{K(ν)} − (χ_{K(ν)} ∪ (ū^{1/p})) in Br(K(ν)) have the same indices. □

3. Brauer group and algebraic tori

3.1. Torsors. Let G be an algebraic groups over F and let K/F be a field extension. The set of isomorphism classes of G-torsors (principal homogeneous spaces) over K is bijective to H^{1}(K, G) (see [15]).

Example 3.1. Let A be a central simple F -algebra of degree n and G = Aut(A). Then H^{1}(K, G) is the set of isomorphism classes of central simple K-algebras of degree n, or equivalently, the set of elements in Br(K) of index dividing n. If A = M_{n}(F) is the split algebra, then G = PGL_{n,F}.

Example 3.2. Let L be an étale F -algebra of dimension n. Consider the algebraic torus U = R_{L/F} (𝔾_{m,L})/ 𝔾_{m} over F. The exact sequence

1 → 𝔾_{m} → R_{L/F} (𝔾_{m,L}) → U → 1

and Hilbert Theorem 90 yield an isomorphism θ : H^{1}(F, U) →^{∼} Br(L/F). Note that if L is a subalgebra of a central simple F -algebra A of degree n, then U is a maximal torus in the group Aut(A).

Let α : G → GL(W) be a finite dimensional representation over F. Suppose that α is generically free, i.e., there is a non-empty open subset W′ ⊂ W and a G-torsor β : W′ → X for a variety X over F. The torsor β is versal, i.e., every G-torsor over a field extension K/F is the pull-back of β with respect to a K-point of X. The generic fiber of β is called a generic G-torsor. It is a torsor over the function field F (X) (see [4] and [13]).

Example 3.3. Let S be an algebraic torus over F. We embed S into the quasi-trivial torus P = R_{L/F} (𝔾_{m,L}), where L is an étale F -algebra (see [3]). Then S acts on the vector space L by multiplication, so that the action on the open subset P is regular. If T is the factor torus P/S, then the S-torsor P → T is versal.

3.2. The tori P^{Φ}, S^{Φ}, T^{Φ}, U^{Φ} and V^{Φ}. Let F be a field, Φ a subgroup of _{p} Ch(F) of rank r and L = F (Φ). Let G = Gal(L/F). Choose a basis χ_{1}, χ_{2}, …, χ_{r} for Φ. We can view each χ_{i} 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 homomorphism of G-modules k : R^{r} → R taking the i-th basis element e_{i} of R^{r} to σ_{i} − 1. The image of k is the augmentation ideal I = Ker(ε) in R, where ε : R → ℤ is defined by ε(ρ) = 1 for all ρ ∈ G.

Write N_{i} = 1 + σ_{i} + σ_{i}^{2} + ⋯ + σ_{i}^{p−1} ∈ R.

Set N := Ker(k). Consider the following elements in N:

e_{ij} := (σ_{i} − 1)e_{j} − (σ_{j} − 1)e_{i} and f_{i} = N_{i}e_{i}, i, j = 1, … r.

Lemma 3.4. The G-module N is generated by e_{ij} and f_{i}.

Proof. Let \overline{R} = ℤ[t_{1}, …, t_{r}] be the polynomial ring. Acyclicity of the Koszul complex for the homomorphism k̄ : (\overline{R})^{r} → \overline{R}, taking the i-th basis element ē_{i} to t_{i} − 1 (see [9, Th. 43]) implies that Ker(k̄) is generated by ē_{ij} := (t_{i} − 1)ē_{j} − (t_{j} − 1)ē_{i}.

The kernel J of the surjective homomorphism \overline{R} → R, taking t_{i} to σ_{i}, is generated by t_{i}^{p} − 1.

Let x := ∑ x_{i}e_{i} ∈ Ker(k). Lift every x_{i} to a polynomial x̄_{i} ∈ \overline{R} and consider x̄ := ∑ x̄_{i}ē_{i} ∈ (\overline{R})^{r}. We have k̄(x̄) ∈ J, hence

k̄(x̄) = ∑(t_{i} − 1)x̄_{i} = ∑(t_{i}^{p} − 1)h_{i} = ∑(t_{i} − 1)\overline{N}_{i}h_{i}

for some polynomials h_{i} ∈ \overline{R}, where \overline{N}_{i} = 1 + t_{i} + t_{i}^{2} + ⋯ + t_{i}^{p−1} ∈ R. Hence the element ∑(x̄_{i} − h_{i}\overline{N}_{i})ē_{i} belongs to the kernel of k̄ and therefore is a linear combination of ē_{ij}. It follows that x̄ is a linear combination of ē_{ij} and \overline{N}_{i}ē_{i}, hence x is a linear combination of e_{ij} and f_{i}. □

Let ε_{i} : R^{r} → ℤ be the i-th projection followed by the augmentation map ε. It follows from Lemma 3.4 that ε_{i}(N) = pℤ for every i. Moreover, the G-homomorphism

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

is surjective. Set M = Ker(l) and Q = R^{r}/M.

Lemma 3.5. The G-module M is generated by e_{ij}.

Proof. Let M′ be the submodule of N generated by e_{ij}. Clearly, M′ ⊂ M. Note also that (σ_{j} − 1)f_{i} = N_{i}e_{ij} ∈ M′, hence If_{i} ⊂ M′.

Suppose that m ∈ M. By Lemma 3.4, modifying m by an element in M′ we can assume that m = ∑_{i=1}^{r} x_{i}f_{i} for some x_{i} ∈ R. As l(m) = 0, we have ε(x_{i}) = 0, i.e., x_{i} ∈ I for all i, hence m ∈ ∑ If_{i} ⊂ 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}, Q, M, I and N, respectively. The diagram of homomorphisms of G-modules with exact columns and rows

(4)  [figure: M; M; N; l; R^{r}; k; I; ℤ^{r}; Q; I]

yields the following diagram of homomorphisms of the tori

(5)  [figure: U^{Φ}; S^{Φ}; 𝔾_{m}^{r}; U^{Φ}; P^{Φ}; V^{Φ}; T^{Φ}; T^{Φ}]

Let K/F be a field extension. Set KL := K ⊗_{F} L. The exact sequence of G-modules

(6)  0 → I → R → ℤ → 0

gives an exact sequence of the tori

1 → 𝔾_{m} → R_{L/F} (𝔾_{m,L}) → U → 1

and then an exact sequence

0 → H^{1}(K, U^{Φ}) → H^{2}(K, 𝔾_{m}) → H^{2}(KL, 𝔾_{m}).

Hence

(7)  H^{1}(K, U^{Φ}) ≃ Br(KL/K).

Lemma 3.6. The homomorphism (K^{×})^{r} → H^{1}(K, U^{Φ}) ≃ Br(KL/K) induced by the first row of the diagram (5) takes (x_{1}, …, x_{r}) to ∑_{i=1}^{r}((χ_{i})_{K} ∪ (x_{i})).

Proof. Consider the composition

(8) h : Hom_{G}(ℤ^{r}, ℤ) → Ext_{G}^{1}(I, ℤ) → Ext_{G}^{2}(ℤ, ℤ) = H^{2}(G, ℤ) = Ch(G),

where the first homomorphism is induced by the bottom row of the diagram (4) and the second one - by the exact sequence (6).

We claim that for any k, the image of the k-th projection p_{k} : ℤ^{r} → ℤ under the composition (8) coincides with χ_{k}. Consider the G-homomorphism R^{r} → ℚ, taking e_{k} to 1/p and e_{i} to 0 for all i ≠ k. By Lemma 3.5, this homomorphism vanishes on M and hence it factors through a map Q → ℚ. Thus, we have a commutative diagram

(9)  \begin{aligned}0 & ⟶ & ℤ^{r} & ⟶ & Q & ⟶ & I & ⟶ & 0 \\ & ^{p_{k}}↓ & & ↓ & & ^{f_{k}}↓ \\ 0 & ⟶ & ℤ & ⟶ & ℚ & ⟶ & ℚ/ℤ & ⟶ & 0 \end{aligned}

for the map f_{k} defined by f_{k}(σ_{k} −1) = 1/p+ℤ and f_{k}(σ_{i} −1) = 0 for all i ≠ k.

Let α be the image of the class of the top row of (9) under the map p_{k}^{∗} : Ext_{G}^{1}(I, ℤ^{r}) → Ext_{G}^{1}(I, ℤ). Then h(p_{k}) is the image of α under the second map in the composition (8). Hence h(p_{k}) is also the image of the class β of the sequence (6) under the connecting map H^{1}(G, I) = Ext_{G}^{1}(ℤ, I) → Ext_{G}^{2}(ℤ, ℤ) = H^{2}(G, ℤ) induced by the exact sequence representing the class α.

The diagram (9) yields a commutative diagram

H^{1}(G, I) ⟶^{∂} H^{2}(G, ℤ^{r})
f_{k}^{∗}↓  ^{p_{k}^{∗}}↓
H^{1}(G, ℚ/ℤ)  H^{2}(G, ℤ)

As we have shown, p_{k}^{∗}(∂(β)) = h(p_{k}). Therefore, it suffices to prove that f_{k}^{∗}(β) = χ_{k}. The cocycle β satisfies β(σ_{i}) = σ_{i} − 1. It follows that f_{k}^{∗}(β)(σ_{k}) = f_{k}(σ_{k} − 1) = 1/p + ℤ and f_{k}^{∗}(β)(σ_{i}) = 0 for all i ≠ k. This proves the claim.

Consider the commutative diagram

[figure: (K^{×})^{r} = Hom_{G}(ℤ^{r}, ℤ) ⊗ K^{×}; Ext_{G}^{1}(I, ℤ) ⊗ K^{×}; Ext_{G}^{2}(ℤ, ℤ) ⊗ K^{×}; (K^{×})^{r} = Hom_{G}(ℤ^{r}, KL^{×}); Ext_{G}^{1}(I, KL^{×}); Ext_{G}^{2}(ℤ, KL^{×}),]

where the vertical homomorphisms are given by the cup-products. By the claim, the image of the tuple (x_{1}, …, x_{r}) under the diagonal composition is equal to ∑_{i=1}^{r}((χ_{i})_{K} ∪ (x_{i})). On the other hand, the bottom composition coincides with (K^{×})^{r} → H^{1}(K, U^{Φ}) ≃ Br(KL/K). □

Corollary 3.7. The map H^{1}(K, U^{Φ}) → H^{1}(K, S^{Φ}) induces an isomorphism H^{1}(K, S^{Φ}) ≃ Br_{ind}(KL/K).

It follows from Corollary 3.7 the triviality of the group H^{1}(K, P^{Φ}) that we have a commutative diagram

(10)

V (K) ⟶ H^{1}(K, U^{Φ})  Br(KL/K)
↓  ↓  ↓
T (K) ⟶ H^{1}(K, S^{Φ})  Br_{ind}(KL/K)

with surjective homomorphisms.

3.3. The element a. Let a′ be the image of the generic point of V over K = F (V) in Br(L(V)/F (V)) in the diagram (10). Choose also an element a ∈ Br(L(T)/F (T)) corresponding to the generic point of T over F (T). The field F (T) is a subfield of F (V) and the classes a_{F (V)} and a′ are equal in Br_{ind}(L(V)/F (V)). It follows that pa_{F (V)} = pa′ in Br F (V).

The exact sequence of G-modules

0 → L^{×} ⊕ N → L(V)^{×} → Div(V_{L}) → 0

induces an exact sequence

H^{1}(G, Div(V_{L})) → H^{2}(G, L^{×}) ⊕ H^{2}(G, N) → H^{2}(G, L(V)^{×}).

As Div(V_{L}) is a permutation G-module, the first term in the sequence is trivial. Therefore, we get an injective homomorphism

φ : H^{2}(G, N) → Br F (V)/ Br(F).

Then (4) and (6) yield

H^{2}(G, N) ≃ H^{1}(G, I) ≃ Ĥ^{0}(G, ℤ) = ℤ/p^{r}ℤ,

thus, H^{2}(G, N) has a canonical generator ξ of order p^{r}.

Lemma 3.8. (cf., [11, Lemma 2.4]) We have φ(ξ) = −a′ + Br(F).

Proof. Consider the following diagram

\xymatrix{ &  & Hom_{G}(ℤ, ℤ) \ar[d] \\  & Hom_{G}(I, I) \ar[d] \ar[r] & Ext_{G}^{1}(ℤ, I) \ar[d] \\ Hom_{G}(N, N) \ar[d] \ar[r] & Ext_{G}^{1}(I, N) \ar[d]_{l} \ar[r] & Ext_{G}^{2}(ℤ, N) \ar[d] \\ Hom_{G}(N, L(V)^{×}) \ar[r] & Ext_{G}^{1}(I, L(V)^{×}) \ar[r] & Ext_{G}^{2}(ℤ, L(V)^{×})}

By [2, Ch. XIV], the images of 1_{ℤ} and −1_{I} agree in Ext_{G}^{1}(ℤ, I) and the images of 1_{N} and −1_{I} agree in Ext_{G}^{1}(I, N). It follows from [2, Ch. V, Prop. 4.1] that the upper square is anticommutative. The image of 1_{ℤ} is equal to φ(ξ) and the image of 1_{N} is equal to a′ + Br(F) in the right bottom corner. □

Corollary 3.9. If r ≥ 2, then the class p^{r−1}a in Br F (T) does not belong to the image of Br(F) → Br F (T).

Proof. The image of p^{r−1}a in Br F (V) coincides with p^{r−1}a′. Modulo the image of the map Br(F) → Br F (V), the class p^{r−1}a′ is equal to −φ(p^{r−1}ξ) and therefore, is nonzero as φ is injective. □

4. Essential dimension of algebraic tori

Let S be an algebraic torus over F with the splitting group G. We assume that G is a p-group of order p^{r}. Let X be the G-module of characters of S. A p-presentation of X is a G-homomorphism f : 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.

Essential p-dimension of algebraic tori was determined in [8, Th. 1.4]:

Theorem 4.1. Let S be an algebraic torus over F with the splitting p-group G, X the G-module of characters of S and f : P → X a minimal p-presentation of X. Then ed_{p}(S) = rank(Ker(f)).

Corollary 4.2. Suppose that X admits a surjective minimal p-presentation f : P → X. Then ed(S) = ed_{p}(S) = rank(Ker(f)).

Proof. As explained in Example 3.3, a surjective G-homomorphism f yields a generically free representation of S of dimension rank(P). By [13, §3],

ed_{p}(S) ≤ ed(S) ≤ rank(P) − dim(S) = rank(Ker(f)).  □

In this section we derive from 4.1 an explicit formula for the essential pdimension of algebraic tori.

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 image of the homomorphism

∐_{H⊂G} X^{H} → \overline{X},

where the coproduct is taken over all subgroups H with [G : H] ≤ p^{k}. We have the sequence of subgroups

(11)  0 = V_{−1} ⊂ V_{0} ⊂ ⋯ ⊂ V_{r} = \overline{X}.

Theorem 4.3. We have the following explicit formula for the essential pdimension of S:

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

Proof. Setb_{k} = rank(V_{k}). By Theorem 4.1, it suffices to prove that the smallest rank of the G-module P is a p-presentation of X is equal to ∑_{k=0}^{r}(b_{k} −b_{k−1})p^{k}.

Let f : P → X be a p-presentation of X and A a G-invariant basis of P. The set A is the disjoint union of the G-orbits A_{j}, so that P is the direct sum of the permutation G-modules ℤ[A_{j}].

The composition f̄: P → X → \overline{X} is surjective. As G acts trivially on \overline{X}, the rank of the group f̄(ℤ[A_{j}]) is at most 1 for all j and f̄(ℤ[A_{j}]) ⊂ V_{k} if |A_{j}| ≤ p^{k}. It follows that the group \overline{X}/V_{k} is generated by the images under

the composition P ⟶^{f̄} \overline{X} → \overline{X}/V_{k} of all ℤ[A_{j}] with |A_{j}| > p^{k}. Denote by c_{k} the number of such orbits A_{j}, so we have

c_{k} ≥ rank(\overline{X}/V_{k}) = b_{r} − b_{k}.

Set c′_{k} = b_{r} − c_{k}, so that b_{k} ≥ c′_{k} for all k and b_{r} = c′_{r}.

Since the number of orbits A_{j} with |A_{j}| = p^{k} is equal to c_{k−1} − c_{k}, we have

rank(P) = ∑_{k=0}^{r}(c_{k−1} − c_{k})p^{k} = ∑_{k=0}^{r}(c′_{k} − c′_{k−1})p^{k} =

c′_{r}p^{r} + ∑_{k=0}^{r−1} c′_{k}(p^{k} − p^{k+1}) ≥ b_{r}p^{r} + ∑_{k=0}^{r−1} b_{k}(p^{k} − p^{k+1}) = ∑_{k=0}^{r}(b_{k} − b_{k−1})p^{k}.

It remains to construct a p-presentation with P of rank ∑_{k=0}^{r}(b_{k} − b_{k−1})p^{k}. For every k ≥ 0 choose a subset X_{k} in X of the pre-image of V_{k} under the canonical map X → \overline{X} with the property that for any x ∈ X_{k} there is a subgroup H_{x} ⊂ G with x ∈ X^{H_{x}} and [G : H_{x}] = p^{k} such that the composition

X_{k} → V_{k} → V_{k}/V_{k−1}

yields a bijection between X_{k} and a basis of V_{k}/V_{k−1}. In particular, |X_{k}| = b_{k} − b_{k−1}. Consider the G-homomorphism

f : P := ∐_{k=0}^{r} ∐_{x∈X_{k}} ℤ[G/H_{x}] → X,

taking 1 in ℤ[G/H_{x}] to x in X.

By construction, the composition of f with the canonical map X → \overline{X} is surjective. As G is a p-group, the ideal pR_{(p)} +I of R_{(p)} is the Jacobson radical of the ring R_{(p)} := R ⊗ ℤ_{(p)}. By Nakayama Lemma, f_{(p)} is surjective. Hence the cokernel of f is finite of order prime to p. The rank of the permutation G-module P is equal to

∑_{k=0}^{r} ∑_{b∈B_{k}} p^{k} = ∑_{k=0}^{r} |B_{k}|p^{k} = ∑_{k=0}^{r}(b_{k} − b_{k−1})p^{k}.  □

4.1. Examples. Let F be a field, Φ a subgroup of_{p} Ch(F) of rank r, L = F (Φ) and G = Gal(L/F). Consider the torus U^{Φ} with the character group the augmentation ideal I defined in 3.2.

The middle row of (4) yields an exact sequence

\overline{N} → (\overline{R})^{r} → \overline{I} → 0.

It follows from Lemma 3.4 that N ⊂ pR^{r} + I^{r}, hence the first homomorphism in the sequence is trivial. The middle group is isomorphic to (ℤ/pℤ)^{r}, hence rank(\overline{I}) = r.

For any subgroup H ⊂ G, the Tate cohomology group Ĥ^{0}(H, I) ≃ Ĥ^{−1}(H, ℤ) is trivial. It follows that the group I^{H} is generated by N_{H}x for all x ∈ I, where N_{H} = ∑_{h∈H} h ∈ R. Since \overline{I} is of period p with the trivial G-action, the classes of the elements N_{H}x in \overline{I} are trivial if H is a nontrivial subgroup of G. It follows that the maps I^{H} → \overline{I} are trivial for all H ≠ 1. In the notation of (11), V_{0} = ⋯ = V_{r−1} = 0 and V_{r} = \overline{I}. By Theorem 4.3,

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

and the rank of the permutation module in a minimal p-presentation of I is equal to rp^{r}. Therefore, k : R^{r} → I is a minimal p-presentation of I that appears to be surjective. Therefore, by Corollary 4.2,

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

Let S^{Φ} be the torus with the character group Q defined in 3.2. As in (4), the homomorphism k factors through a surjective map R^{r} → Q that is then necessarily a minimal p-presentation of Q. According to Theorem 4.3 and Corollary 4.2,

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

5. Degeneration

In this section we study the behavior of the essential p-dimension under degeneration, i.e. we compare the essential p-dimension of an object over a complete discrete valued field and its specialization over the residue field (Proposition 5.2). The iterated degeneration (Corollary 5.4) connects a class in the Brauer group degree p^{r} over some (large) field and the elements of the indecomposable relative Brauer group that are torsors for a certain torus.

5.1. A simple degeneration. Let F be a field, p a prime integer different from char(F) and Φ ⊂ _{p}Ch(F) a finite subgroup. For an integer k ≥ 0 and a field extension K/F, let

B_{k}^{Φ}(K) = {a ∈ Br(K){p} such that ind a_{K(Φ)} ≤ p^{k}}.

Two elements a and a′ in B_{k}^{Φ}(K) are equivalent if a − a′ ∈ Br_{dec}(K(Φ)/K). Write F_{k}^{Φ}(K) for the set of equivalence classes in B_{k}^{Φ}(K). Abusing notation we shall write a for the equivalence class of an element a ∈ B_{k}^{Φ}(K) in F_{k}^{Φ}(K).

We view B_{k}^{Φ} and F_{k}^{Φ} as functors from Fields/F to Sets.

Example 5.1. (1) If Φ is the zero subgroup, then F_{k}^{Φ} = B_{k}^{Φ} ≃ Alg(p^{r}) ≃ PGL(p^{r})- torsors.

(2) The set B_{0}^{Φ}(K) is naturally bijective to Br(K(Φ)/K) and F_{0}^{Φ}(K) ≃ Br_{ind}(K(Φ)/K). By Corollary 3.7, the latter group is naturally isomorphic to H^{1}(K, S^{Φ}), where S^{Φ} is the torus defined in 3.2, thus, F_{0}^{Φ} ≃ S^{Φ}- torsors.

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}^{Φ}(E), i.e., a ∈ Br(E){p} and ind(a_{E(Φ)}) ≤ p^{k}.

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

(14)  a′ = â + (η̂_{E} ∪ (x)) ∈ Br(E′),

for some x ∈ E′^{×} such that v′(x) is not divisible by p. By Proposition 2.2(2), ind(a_{E′(Φ′)}) = p · ind(a_{E(Φ)}) ≤ p^{k+1}, hence a′ ∈ B_{k+1}^{Φ′}(E′).

Proposition 5.2. 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 p · ρ ∈ Φ_{N} \ Φ′_{N}, we have ind a_{N(Φ′,ρ)} > p^{k−1}. Then

ed_{p}^{F_{k+1}^{Φ′}}(a′) ≥ ed_{p}^{F_{k}^{Φ}}(a) + 1.

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

tr. deg_{F} (M_{0}) = ed_{p}^{F_{k+1}^{Φ′}}(a′). We have

(15)  a′_{M} − (a′_{0})_{M} ∈ Br_{dec}(M(Φ′)/M).

It follows from (14) that

(16)  a′_{M} = â_{N} + (η̂_{N} ∪ (x))

and ∂_{v′}(a′) = q · η_{E}, where q = v′(x) is relatively prime to p. We extend the discrete valuation v′ on E′ to a (unique) discrete valuation v on M. The ramification index e′ and inertia degree are both prime to p. Thus, the residue field N of v is a finite extension of E of degree prime to p. By Proposition 2.2(3),

(17)  ∂_{v}(a′_{M}) = e′ · ∂_{v′}(a′) = e′q · η_{E}.

Let v_{0} be the restriction of v to M_{0} and N_{0} its residue field. It follows from (15) that

(18)  ∂_{v}(a′_{M}) − ∂_{v}((a′_{0})_{M}) ∈ Φ′_{N}.

Recall that η_{E} ∉ Φ′_{E}. As [N : E] is not divisible by p, it follows that

(19)  η_{N} ∉ Φ′_{N}.

By (17), (18) and (19), ∂_{v}((a′_{0})_{M}) ≠ 0, i.e., (a′_{0})_{M} is ramified and therefore v_{0} is nontrivial, i.e., v_{0} is a discrete valuation on M_{0}.

Let η_{0} := ∂_{v_{0}}(a′_{0}) ∈ Ch(N_{0}){p}. By Proposition 2.2(3),

(20)  ∂_{v}((a′_{0})_{M}) = e · (η_{0})_{N},

where e is the ramification index of M/M_{0}, hence (η_{0})_{N} ≠ 0. It follows from (17),(18) and (20) that

(21)  e′q · η_{N} − e · (η_{0})_{N} ∈ Φ′_{N}.

As e′q is relatively prime to p,

(22)  η_{N} ∈ ⟨Φ′_{N}, (η_{0})_{N}⟩ in Ch(N).

Let p^{t} (t ≥ 1) be the order of (η_{0})_{N}. It follows from (19) and (21) that v_{p}(e) = t − 1 and

(23)  p^{t−1} · (η_{0})_{N} ∈ Φ_{N} \ Φ′_{N}.

Choose a prime element π_{0} in M_{0} and write

(24)  (a′_{0})M̂_{0} = â_{0} + (η̂_{0} ∪ (π_{0}))

in Br(M̂_{0}), where a_{0} ∈ Br(N_{0}){p}.

Applying the specialization homomorphism s_{π} : Br(M){p} → Br(N){p} (for a prime element π in M) to (15), (16) and (24), using (3) and (22), we get

(25)  a_{N} − (a_{0})_{N} ∈ Br_{dec}(N(Φ′, η_{0})/N).

It follows from (25) that

(26)  a_{N(Φ′,η_{0})} = (a_{0})_{N(Φ′,η_{0})}

in Br(N(Φ′, η_{0})).

By (24),

(a′_{0})_{M̂_{0}(Φ′)} = ˆ(a_{0})_{N_{0}(Φ′)} + (ˆ(η_{0})_{N_{0}(Φ′)} ∪ (π_{0})).

As no nontrivial multiple of (η_{0})_{N} belongs to Φ′_{N} by (23), the order of the character (η_{0})_{N_{0}(Φ′)} is at least p^{t}. It follow from Proposition 2.2(2) that

(27)  ind(a_{0})_{N_{0}(Φ′,η_{0})} = ind(a′_{0})_{M̂_{0}(Φ′)}/ ord(η_{0})_{N_{0}(Φ′)} ≤ p^{k+1}/p^{t} = p^{k−t+1}.

By (26) and (27),

(28)  ind(a_{N(Φ′,η_{0})}) ≤ p^{k−t+1}.

Suppose that t ≥ 2 and consider the character ρ = p^{t−2} · (η_{0})_{N} of order p^{2} in Ch(N). We have p · ρ = p^{t−1}(η_{0})_{N} ∈ Φ_{N} \ Φ′_{N} by (23). Moreover, the degree of the field extension N(Φ′, η_{0})/N(Φ′, ρ) is equal to p^{t−2}. Hence by (28),

ind(a_{N(Φ′,ρ)}) ≤ ind(a_{N(Φ′,η_{0})}) · p^{t−2} ≤ p^{k−t+1} · p^{t−2} = p^{k−1}.

This contradicts the assumption. Therefore, t = 1, i.e., ord(η_{0})_{N} = p. Then (e, p) = 1 and it follows from (21) that (η_{0})_{N} ∈ ⟨Φ′_{N}, η_{N}⟩. Moreover,

(29)  ⟨Φ′, η_{0}⟩_{N} = ⟨Φ′, η⟩_{N} = Φ_{N}.

By Lemma 2.1, there is a finite subextension N_{1}/N_{0} of N/N_{0} such that ⟨Φ′, η_{0}⟩_{N_{1}} = Φ_{N_{1}}. Replacing N_{0} by N_{1} and a_{0} by (a_{0})_{N_{1}}, we may assume that ⟨Φ′, η_{0}⟩_{N_{0}} = Φ_{N_{0}}. In particular, η_{0} is of order p in Ch(N_{0}).

Since by (27),

ind(a_{0})_{N_{0}(Φ)} = ind(a_{0})_{N_{0}(Φ′,η_{0})} ≤ p^{k},

we have a_{0} ∈ B_{k}^{Φ}(N_{0}).

It follows from (25) that

a_{N} − (a_{0})_{N} ∈ Br_{dec}(N(Φ)/N).

Hence the classes of a_{N} and (a_{0})_{N} are equal in F_{k}^{Φ}(N). The class of a_{N} in F_{k}^{Φ}(N) is then defined over N_{0}, therefore,

ed_{p}^{F_{k+1}^{Φ′}}(a′) = tr. deg_{F} (M_{0}) ≥ tr. deg_{F} (N_{0}) + 1 ≥ ed_{p}^{F_{k}^{Φ}}(a) + 1.  □

5.2. Multiple degeneration. In this subsection we assume that the base field F contains a primitive p^{2}-th root of unity.

Let Φ be a subgroup in _{p} Ch(F) of rank r. Choose a basis χ_{1}, χ_{2}, …, χ_{r} of Φ. Let E/F be a field extension such that rank(Φ_{E}) = r and let a ∈ Br(E){p} be an element that is split by E(Φ).

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 not divisible by p and define the elements a_{k} ∈ Br(E_{k}){p} inductively by a_{0} = a and a_{k} = â_{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 Proposition 2.2(2) that

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

for any k = 1, …, r. As ind a_{E(Φ)} = 1, we have ind(a_{k})_{E_{k}(Φ_{k})} = p^{k} for all k = 0, 1, …, r. In particular, a_{k} ∈ B_{k}^{Φ_{k}}(E_{k}).

The followings lemma assures that under a certain restriction on the element a, the conditions of Proposition 5.2 are satisfied for the fields E_{k}, the groups of characters Φ_{k} and the elements a_{k}.

Lemma 5.3. Suppose that p^{r−1}a ∉ Im(Br(F) → Br(E)). 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 p · ρ ∈ (Φ_{k})_{N} \ (Φ_{k+1})_{N}, we have

(30)  ind(a_{k})_{N(Φ_{k+1},ρ)} > p^{k−1}.

Proof. Induction on r. The case r = 1 is obvious. Suppose that the inequality (30) does not hold for some k = 1, …, r − 1, a finite field extension N/E_{k} and a character ρ ∈ Ch(N). Suppose first that k < r − 1. Consider the fields F′ = F (Φ_{k+1}), E′ = E(Φ_{k+1}), E′_{i} = E_{i}(Φ_{k+1}), N′ = N(Φ_{k+1}), the sequence of characters (χ_{i})_{F′} and the sequence of elements a′_{i} := (a_{i})_{E′_{i}} ∈ Br(E_{i}) for i = 0, 1, …, k + 1. As (a′_{k})_{N′(ρ)} = (a_{k})_{N(Φ_{k+1},ρ)}, the inequality (30) does not hold for the term a′_{k} of the new sequence, the field extension N′/E′_{k} and the character ρ_{N′}.

Note that p^{k}a_{E′} ∉ Im(Br(F′) → Br(E′)), because otherwise, taking the norm map for the extension F′/F of degree p^{r−k−1}, we would get p^{r−1}a ∈ Im(Br(F) → Br(E)). By induction, the inequality (30) holds for all the terms of the new sequence, in particular for a′_{k}, a contradiction.

Thus we can assume that k = r − 1. 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_{r−1} = N and let Ẽ_{r} be an unramified extension of E_{r} with the residue field Ẽ_{r−1}. The fields Ẽ_{j} with j < r − 1 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. Replacing E_{i} by Ẽ_{i} and a_{i} by (a_{i})Ẽ_{i}, we may assume that N = E_{r−1}.

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}. By Lemma 2.3(1), (31)

ind(a_{r−2})_{E_{r−2}(χ_{r−1},μ)} = ind(a_{r−1})_{E_{r−1}(ρ)}/p = ind(a_{r−1})_{E_{r−1}(Φ_{r},ρ)}/p ≤ p^{r−3}.

Consider the fields F′ = F (χ_{r−1}), E′ = E(χ_{r−1}), E′_{i} = E_{i}(χ_{r−1}), N′ = N(χ_{r−1}), the sequence of characters χ_{1}, …, χ_{r−2}, χ_{r} and the elements a′_{i} ∈ Br(E′_{i}) for i = 0, 1, …, r − 1 defined by a′_{i} = (a_{i})_{E′_{i}} for i ≤ r − 2 and a′_{r−1} = â_{r−2} + (χ̂_{r} ∪ (x_{r−1})) over E′_{r−1}. As (a′_{r−2})_{N′(μ)} = (a_{r−2})_{N(χ_{r−1},ρ)}, the inequality (31) shows that (30) does not hold for the term a′_{r−2} of the new sequence, the field extension N′/E′_{r−2} and the character μ_{N′}.

Note that p^{r−2}a_{E′} ∉ Im(Br(F′) → Br(E′)), as otherwise, taking the norm map for the extension F′/F of degree p, we get p^{r−1}a ∈ Im(Br(F) → Br(E)).

By induction, the inequality (30) holds for all the terms of the new sequence, in particular for a′_{r−2}, a contradiction.

Case 2: The character ρ is ramified. Note that p · ρ is a nonzero multiple of (χ_{r})_{E_{r−1}}. As the inequality (30) fails for a_{r−1}, we have

ind(a_{r−1})_{E_{r−1}(ρ)} ≤ p^{r−2}.

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

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

(32)  ind(a_{j} − (χ_{r−1} ∪ (ū_{j}^{1/p})))_{E_{j}(Θ_{j})} ≤ p^{j}.

If j = r − 2, we set u_{j} = u and Θ_{j} = {χ_{r}}.

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

(a_{j} − (χ_{r−1} ∪ (ū_{j}^{1/p})))_{E_{j}(Θ_{j})} = b̂ + (η̂ ∪ (x_{j}))_{E_{j}(Θ_{j})},

where η = χ_{j} − mχ_{r−1} and b = (a_{j−1} − (χ_{r−1} ∪ (ū_{j−1}^{1/p})))_{E_{j−1}(Θ_{j})}. As η is not contained in Θ_{j}, the character η_{E_{j−1}(Θ_{j})} is not trivial. Set Θ_{j−1} = ⟨Θ_{j}, η⟩. It follows from Proposition 2.2(2) that

ind(b_{E_{j−1}(Θ_{j−1})}) = ind(a_{j} − (χ_{r−1} ∪ (ū_{j}^{1/p})))_{E_{j}(Θ_{j})}/p ≤ p^{j−1}.

Applying the inequality (32) in the case j = 0, we get

a_{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}). The degree of the extension E(Θ_{0})/E is equal to p^{r−1} and E(w^{1/p}) ⊂ E(Θ_{0}). Taking the norm for the extension E(Θ_{0})/E, we get that p^{r−1}a is a multiple of χ_{r−1} ∪ (w). As the character χ_{r} is defined over F, we may assume that w ∈ F^{×}, hence p^{r−1}a ∈ Im(Br(F) → Br(E)), a contradiction. Thus, we have shown that the inequality (30) holds. □

By Example 5.1(2), we can view a as an S^{Φ}-torsor over E.

Corollary 5.4. Suppose that p^{r−1}a ∉ Im(Br(F) → Br(E)). Then

ed_{p}^{Alg(p^{r})}(a_{r}) ≥ ed_{p}^{S^{Φ}- torsors}(a) + r.

Proof. By iterated application of Proposition 5.2 and Example 5.1,

ed_{p}^{Alg(p^{r})}(a_{r}) = ed_{p}^{F_{r}^{Φr}} (a_{r}) ≥ ed_{p}^{F_{r−1}^{Φ_{r−1}}}(a_{r−1}) + 1 ≥ …

≥ ed_{p}^{F_{1}^{Φ_{1}}}(a_{1}) + (r − 1) ≥ ed_{p}^{F_{0}^{Φ_{0}}}(a_{0}) + r = ed_{p}^{S^{Φ}- torsors}(a) + r.

□

6. Proof of the main theorem

Theorem 6.1. Let F be a field and p an integer different from char(F). Then

ed_{p}(Alg_{F} (p^{r})) ≥ (r − 1)p^{r} + 1.

Proof. As ed_{p}(Alg_{F} (p^{r})) ≥ ed_{p}(Alg_{F′}(p^{r})) for any field extension F′/F by [10, Prop. 1.5], we can replace F by any field extension. In particular, we may assume that F contains a primitive p^{2}-th root of unity and there is a subgroup Φ of _{p} Ch(F) of rank r. Let T^{Φ} be the algebraic torus constructed in Section 3 for the field extension L = F (Φ) of F. Set E = F (T^{Φ}) and let a ∈ Br(EL/E) be the element defined in 3.3. Let a_{r} ∈ Br(E_{r}) be the element of index p^{r} constructed in 5.2. By Corollary 3.9, the class p^{r−1}a in Br(E) does not belong to the image of Br(F) → Br(E). It follows from Corollary 5.4 that

(33)  ed_{p}^{Alg(p^{r})}(a_{r}) ≥ ed_{p}^{S^{Φ}- torsors}(a) + r.

The S^{Φ}-torsor a is the generic fiber of the versal S^{Φ}-torsor P^{Φ} → S^{Φ} (see Example 3.3), hence a is a generic torsor. By [14, §6] or [10, Th. 2.9]

(34)  ed_{p}^{S^{Φ}- torsors}(a) = ed_{p}(S^{Φ}).

The essential p-dimension of S^{Φ} was calculated in (13):

(35)  ed_{p}(S^{Φ}) = (r − 1)p^{r} − r + 1.

Finally, it follows from (33), (34) and (35) that

ed_{p}(Alg_{F} (p^{r})) ≥ ed_{p}^{Alg(p^{r})}(a_{r}) ≥ ed_{p}^{S^{Φ}- torsors}(a) + r = (r − 1)p^{r} + 1. □

7. Remarks

Let K/F be a field extension and G an elementary abelian group of order p^{r}. Consider the subset Alg_{K}(G) of Alg_{K}(p^{r}) consisting of all classes admitting a splitting Galois K-algebra with the Galois group G. Equivalently, Alg_{K}(G) consists of all classes represented by crossed product algebras with the group G (see [5, §4.4]).

Write Pair_{K}(G) for the set of pairs (a, E), where a ∈ Alg_{K}(G) and E is a Galois G-algebra splitting a.

Finally, fix a Galois field extension L/F with Gal(L/F) ≃ G and consider the subset Alg_{K}(L/F) of Alg_{K}(G) consisting of all classes split by KL. Thus, Alg(L/F) is a subfunctor of Alg(G) and there is the obvious surjective morphism of functors Pair_{K}(G) → Alg_{K}(G).

Theorem 7.1. Let F be a field, p an integer different from char(F), G an elementary abelian group of order p^{r}, r ≥ 2, and L/F a Galois field extension with Gal(L/F) ≃ G. Let F be one of the three functors Alg(L/F), Alg(G) and Pair_{K}(G). Then

ed(F) = ed_{p}(F) = (r − 1)p^{r} + 1.

Proof. The functor Alg(L/F) is isomorphic to U^{Φ}- torsors by (7). It follows from (12) that

ed(Alg(L/F)) = ed_{p}(Alg(L/F)) = (r − 1)p^{r} + 1.

Let a_{r} be the element in Br(E_{r}) in the proof of Theorem 6.1. It satisfies ed_{p}^{Alg(p^{r})}(a_{r}) ≥ (r − 1)p^{r} + 1. By construction, a_{r} ∈ Alg_{E_{r}}(G). As Alg(G) is a subfunctor of Alg(p^{r}), we have

ed_{p}(Alg(G)) ≥ ed_{p}^{Alg(G)}(a_{r}) ≥ ed_{p}^{Alg(p^{r})}(a_{r}) ≥ (r − 1)p^{r} + 1.

The upper bound ed(Alg(G)) ≤ (r − 1)p^{r} + 1 was proven in [7, Cor. 3 10].

The split étale F -algebra E := Map(G, F) has the natural structure of a Galois G-algebra over F. The group G acts on the split torus U := R_{E/F} (𝔾_{m,E})/ 𝔾_{m}. Let A be the split F -algebra End_{F} (E). The semidirect product H := U ⋊ G acts naturally on A by F -algebra automorphisms. Moreover, by the Skolem-Noether Theorem, H is precisely the automorphism group of the pair (A, E). It follows that the functor Pair_{K}(G) is isomorphic to H- torsors.

The character group of U is G-isomorphic to the ideal I in R = ℤ[G]. By [12, §3], the G-homomorphism k : R^{r} → I constructed in 3.2 yields a representation W of the group H of dimension rp^{r}. As r ≥ 2, by Lemma 3.4, G acts faithfully on the kernel N of k. By [12, Lemma 3.3], the action of H on W is generically free, hence

ed(Pair(G)) = ed(H) ≤ dim(W) − dim(H) = (r − 1)p^{r} + 1.

Since Pair(G) surjects onto Alg(G), we have

ed(Pair(G)) ≥ ed_{p}(Pair_{K}(G)) ≥ ed_{p}(Alg(G)) = (r − 1)p^{r} + 1.  □

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

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