ESSENTIAL p-DIMENSION OF SPLIT SIMPLE GROUPS OF TYPE A_{n}

VLADIMIR CHERNOUSOV AND ALEXANDER MERKURJEV

Abstract. We compute the essential p-dimension of split simple groups of type A_{n−1} in terms of the functor Alg(n, m) of central simple algebras of degree n and exponent dividing m.

1. Introduction

Let F be a field and 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. We say that that K is a field of definition of α ∈ F(E) if α belongs to the image of 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 [5, 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 finite 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_{E}(n, m) denote the set of isomorphism classes of central simple E-algebras of degree n and exponent dividing m. We can identify Alg_{E}(n, m) with the subset of the m-torsion part Br_{m}(E) of the Brauer group of E consisting of all elements a such that the index ind(a) of a divides n. We view Alg(n, m) as a functor Fields/F → Sets. Upper and lower bounds for the essential pdimension ed_{p}(Alg(n, m)) for a prime integer p different from char(F) can be found in [2].

2000 Mathematics Subject Classification. 11E72, 16K50, 20G15, 14L30.

Key words and phrases. Linear algebraic group, torsor, central simple algebra, essential dimension, Brauer group.

The work of the first author has been supported in part by Canada Research Chairs Program and NSERC research grant.

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

Let G be an algebraic group scheme over F. Write F_{G} for the functor taking a field extension E/F to the set H^{1}(E, G) of isomorphism classes of principal homogeneous G-spaces (G-torsors) over E. The essential (p-)dimension of F_{G} is called the essential (p-)dimension of G and is denoted by ed(G) and ed_{p}(G).

A split simple algebraic group G of type A_{n−1} is isomorphic to SL_{n} /μ_{m} for a divisor m of n. In the present paper we compute the essential p-dimension of G in terms of the integer ed_{p}(Alg(n, m)).

Theorem 1.1. Let n be a natural number, m a divisor of n and p a prime integer. Let p^{r} and p^{s} be the largest powers of p dividing n and m respectively and let G = SL_{n} /μ_{m} be the algebraic group defined over a field F of the characteristic not p. Then

ed_{p}(G) = \{ \begin{matrix}0, & if s = 0; \\ ed_{p}(Alg(p^{r}, p^{r})), & if s = r; \\ ed_{p}(Alg(p^{r}, p^{s})) + 1, & if 0 < s < r. \end{matrix}

Using lower bounds for ed_{p}(Alg(p^{r}, p^{s})) obtained in [1] and [2], we get:

Corollary 1.2. If p is a prime integer then

(1) p^{2r−2} + p^{r−s} + 1 ≥ ed_{p}(SL_{p^{r}} /μ_{p^{s}}) ≥ (r − 1)p^{r} + p^{r−s} + 1 if 0 < s < r and p is odd in the case s = 1,

(2) 2^{2r−4} + 2^{r−1} + 1 ≥ ed_{2}(SL_{2^{r}} /μ_{2}) ≥ (r − 1)2^{r−1} + 1 if r ≥ 3.

(3) ed_{p}(SL_{p^{2}} /μ_{p}) = p^{2} + p + 1 if p is odd,

(4) ed_{2}(SL_{4} /μ_{2}) = 5,

(5) ed_{2}(SL_{8} /μ_{2}) = 9,

(6) ed_{2}(SL_{8} /μ_{4}) = 19,

(7) ed_{2}(SL_{16} /μ_{2}) = 25.

2. Unramified torsors

Let R be a commutative ring and let G be a group scheme defined over R. There is a bijection between the set of isomorphism classes of G-torsors over R and the pointed set H^{1}(R, G) of the first cohomology of G for the flat topology (see [4, Exp. XXIV]). If G is smooth, one can use the étale topology instead of flat topology.

Let K be a discrete valued field with valuation ring R ⊂ K and residue field \overline{K}. We write H^{1}(K, G)_{nr} for the image of the map

H^{1}(R, G) → H^{1}(K, G).

Let α ∈ H^{1}(K, G). If α ∈ H^{1}(K, G)_{nr} we say that α is unramified. Otherwise α is ramified.

If K is complete and G is smooth, the canonical map H^{1}(R, G) → H^{1}(\overline{K}, G) is a bijection [4, Exp. XXIV, Prop. 8.1], hence H^{1}(K, G)_{nr} ≃ H^{1}(\overline{K}, G).

If G is commutative, then H^{1}(K, G)_{nr} is a subgroup of H^{1}(K, G). We write H^{1}(K, G)_{ram} for the factor group.

Example 2.1. We have H^{1}(K, μ_{k}) = K^{×}/K^{×k}, H^{1}(K, μ_{k})_{nr} = R^{×}/R^{×k} and H^{1}(K, μ_{k})_{ram} = ℤ/kℤ.

Suppose that K is complete. Let T′ be a torus over R. We write T for T′⊗_{R}K and \overline{T} for T′⊗_{R}\overline{K}. Clearly, T and \overline{T} are tori over K and \overline{K} respectively. The character group \overline{T}^{∗} of \overline{T} is a module over the absolute Galois group Γ\overline{K}. The character group T^{∗} coincides with \overline{T}^{∗}, and Γ_{K} acts on T^{∗} via the canonical surjective homomorphism Γ_{K} → Γ\overline{K}.

We have the split exact sequence of Galois Γ\overline{K}-modules

1 → R_{nr}^{×} → K_{nr}^{×} ⟶^{v} ℤ → 0,

where R_{nr} and K_{nr} are maximal unramified extensions of R and K respectively. Tensoring this sequence with the Γ\overline{K}-module of co-characters \overline{T}_{∗}, the dual of \overline{T}^{∗}, and taking cohomology groups yields exact sequences

1 → T (R) → T (K) → \overline{T}_{∗}(\overline{K}) → 0,

0 → H_{et}^{1}(R, T) → H^{1}(K, T) ⟶^{v_{∗}} H^{1}(\overline{K}, \overline{T}_{∗}) → 0.

In particular, the group H^{1}(K, T)_{ram} is canonically isomorphic to H^{1}(\overline{K}, \overline{T}_{∗}).

3. Azumaya algebras and torsors

Let n = km and G = SL_{n} /μ_{m} over a field F, so we have an exact sequence

(1)  1 → μ_{k} → G → PGL_{n} → 1.

Let R be a commutative local F -algebra. The exact sequence (1) yields an exact sequence of pointed sets

H^{1}(R, G) ⟶^{α} H^{1}(R, PGL_{n}) ⟶^{∂} H^{2}(R, μ_{k}).

Moreover, the group H^{1}(R, μ_{k}) = R^{×}/R^{×k} acts on the set H^{1}(R, G) transitively in the fibers of the map α. For an element r ∈ R^{×} and ξ ∈ H^{1}(R, G) we write rξ for the result of the action of rR^{×k} on ξ.

Recall that there is a canonical bijection between H^{1}(R, PGL_{n}) and the set of isomorphism classes Alg_{R}(n) of Azumaya R-algebras of degree n, so we have the map H^{1}(R, G) → Alg_{R}(n) [6, Ch. IV].

The group H^{2}(R, μ_{k}) is identified with the subgroup Br_{k}(R) of the Brauer group Br(R) = H^{2}(R, G_{m}) of R and the map ∂ takes an algebra A to the class of A^{⊗m} in Br(R). Therefore, the image of an element ξ ∈ H^{1}(R, G) in H^{1}(R, PGL_{n}) yields a class A_{ξ} in Alg_{R}(n, m) ⊂ Alg_{R}(n) of algebras of exponent dividing m. Moreover, every class A ∈ Alg_{R}(n, m) is of the form A = A_{ξ} for some ξ ∈ H^{1}(R, G).

Twisting (1) by the class of an algebra A ∈ Alg_{R}(n) yields an exact sequence

1 → μ_{k} → G′ → PGL_{1}(A) → 1.

The connecting homomorphism

A^{×}/R^{×} = PGL_{1}(A)(R) → H^{1}(R, μ_{k}) = R^{×}/R^{×k}

takes the class aR^{×} to Nrd(a)R^{×k}, where Nrd : A^{×} → R^{×} is the reduced norm homomorphism. This yields:

Lemma 3.1. Let R be a commutative local F -algebra and A ∈ Alg_{R}(n, m). Then the factor group R^{×}/(R^{×k} · Nrd(A)) acts simply transitively on the fiber of the surjective map H^{1}(R, G) → Alg_{R}(n, m) over A.

Let K/F be a field extension with a discrete valuation v over F and a prime element π.

Lemma 3.2. Let ξ ∈ H^{1}(K, G) be an unramified element such that A_{ξ} ≠ 0 in Br(K). If πξ is unramified then k and ind(A_{ξ}) are relatively prime.

Proof. Let R ⊂ K be the valuation ring. By assumption, there are ζ, ζ′ ∈ H^{1}(R, G) such that ξ = ζ_{K} and πξ = ζ′_{K}. We have (A_{ζ})_{K} = A_{ξ} = A_{πξ} = (A_{ζ′})_{K}. As the map Br(R) → Br(K) is injective by [6, Ch. IV, Cor. 2.6], we have A_{ζ} = A_{ζ′}. It follows from Lemma 3.1 that ζ′ = λζ for some λ ∈ R^{×}. Then πξ = ζ′_{K} = λζ_{K} = λξ, therefore by Lemma 3.1 again, π ∈ λ(K^{×k} · Nrd(A_{ξ})). Therefore, 1 = v(π) ∈ kℤ + ind(A_{ξ})ℤ as v(Nrd(A_{ξ})) ⊂ ind(A_{ξ})ℤ by [7, Ch. XII, §2]. □

4. Tori

Let L/F be a separable field extension of degree n = p^{r}, where p is a prime integer and m = p^{s} a divisor of n. Consider the torus of norm one elements R_{L/F}^{(1)} (G_{m}_{,L}) for the extension L/F, the factor torus T = R_{L/F}^{(1)} (G_{m}_{,L})/μ_{m} and S = R_{L/F} (G_{m}_{,L})/ G_{m}. Then T and S can be viewed as maximal tori of G and PGL_{n} respectively and we have an exact sequence

(2)  1 → μ_{k} → T → S → 1.

Let R be a commutative local F -algebra. The group H^{1}(R, S) is identified with the relative Brauer group Br(LR/R) := Ker(Br(R) → Br(LR)), where we write LR for L ⊗_{F} R. The composition H^{1}(R, S) → H^{1}(R, PGL_{n}) ↪ Br(R) is identified with the inclusion of Br(LR/R) into Br(R). Comparing the exact sequences (1) and (2) we have:

Lemma 4.1. The image of H^{1}(R, T) → H^{1}(R, G) coincides with the set of all ξ such that A_{ξ} ∈ Br_{m}(LR/R).

Let Γ be the Galois group of a normal closure L′/F of L/F, so Γ is the decomposition group of the tori T and S. Let X be the Γ-set of all F -homomorphisms L → L′. We have |X| = n and R_{L/K}(G_{m}_{,L})^{∗} = ℤ[X].

Choose a point x_{0} ∈ X and let Γ_{0} be the stabilizer of x_{0} in Γ. As Γ acts transitively on X, we have, X ≃ Γ/Γ_{0} and [Γ : Γ_{0}] = n.

Let I be the augmentation ideal in ℤ[Γ]. Write I_{X} for the kernel of the augmentation map ε : ℤ[X] → ℤ. We have I_{X} = I · ℤ[X].

Write N_{X} = ∑_{x∈X} x ∈ ℤ[X], so ε(N_{X}) = n.

Let V = R_{L/K}(G_{m}_{,L})/μ_{m}. The character group J_{X} of V is identified with the subgroup of elements w ∈ ℤ[X] with ε(w) ∈ mℤ. Note that I_{X} ⊂ J_{X}.

Lemma 4.2. Suppose that r > s. Then N_{X} ∈ pJ_{X} + I · J_{X}.

Proof. The map

Γ → I/I^{2}, γ ↦ (γ − 1) + I^{2}

is a group homomorphism. It follows that if γ belongs to the commutator subgroup [Γ, Γ] of Γ, then

(3)  γ − 1 ∈ I^{2}.

Set Δ := [Γ, Γ]Γ_{0}. Suppose first that Δ contains Γ_{0} properly. Consider the sum u in ℤ[Γ] of all representatives of the set of left cosets Δ/Γ_{0} chosen in [Γ, Γ]. It follows from (3) that u is congruent to [Δ : Γ_{0}] modulo I^{2}.

The element N_{X} is divisible by u, i.e., there is M ∈ ℤ[X] such that N_{X} = uM. It follows that N_{X} is congruent to [Δ : Γ_{0}]M modulo I ·I_{X}. As [Δ : Γ_{0}] is divisible by p, we have [Δ : Γ_{0}]M = pR for some R ∈ ℤ[X] with ε(R) = n/p. Since r > s, n/p is divisible by m, hence we have R ∈ J_{X}. Overall N_{X} ∈ pJ_{X} + I · I_{X} ⊂ pJ_{X} + I · J_{X}.

Now suppose that Δ = Γ_{0}, i.e., Γ_{0} is normal in Γ. It follows that Γ_{0} = 1 and Γ is an abelian p-group of order n. Let Γ′ be a subgroup of Γ of order p and v = ∑_{γ∈Γ′} γ in ℤ[Γ]. Then N_{X} is divisible by v, i.e., there is M′ ∈ ℤ[X] such that N_{X} = vM′. Since ε(M′) = n/p, we have M′ ∈ J_{X}. As v is congruent to p modulo I, N_{X} is congruent to pM′ modulo I ·J_{X}, hence N_{X} ∈ pJ_{X} +I ·J_{X}. □

The exact sequence of tori

1 → T → V → G_{m} → 1

yields an exact sequence of Γ-modules of co-characters

(4)  0 → T_{∗} → V_{∗} → ℤ → 0.

Write θ_{T} for the image of 1 under the connecting homomorphism ℤ → H^{1}(Γ, T_{∗}) = H^{1}(F, T_{∗}).

Proposition 4.3. Suppose that r > s. Then θ_{T} is not divisible by p in H^{1}(F, T_{∗}).

Proof. Consider the exact sequence of Γ-modules

(5)  0 → ℤ ⟶^{f} J_{X} → T^{∗} → 0

dual to (4). The image ν of θ_{T} under the canonical isomorphisms

H^{1}(F, T_{∗}) ≃ Ext_{Γ}^{1}(ℤ, T_{∗}) ≃ Ext_{Γ}^{1}(T^{∗}, ℤ)

is the class of the sequence (5).

Suppose that ν is divisible by p. Then the image of ν under the map Ext_{Γ}^{1}(T^{∗}, ℤ) → Ext_{Γ}^{1}(T^{∗}, ℤ/pℤ) is trivial, i.e, the canonical homomorphism ℤ → ℤ/pℤ factors as ℤ ⟶^{f} J_{X} ⟶^{h} ℤ/pℤ for a Γ-homomorphism h. Note that f(1) = N_{X}, hence h(N_{X}) = 1 + pℤ.

The map h vanishes on pJ_{X} + I · J_{X}, hence by Lemma 4.2, h(N_{X}) = 0, a contradiction. □

5. The key proposition

Let K/F be a complete field with discrete valuation v over F and residue field \overline{K}. Let ξ ∈ H^{1}(K, G) be an element and L/K an unramified (separable) field extension of degree n splitting A_{ξ}. Let T = R_{L/K}^{(1)}(G_{m}_{,L})/μ_{m} be the torus as defined in Section 4. Note that T is actually defined over the valuation ring, so the residue torus \overline{T} is defined over \overline{K}. As A_{ξ} ∈ Br_{m}(L/K), the element ξ has a lifting to H^{1}(K, T) by Lemma 4.1.

Lemma 5.1. The image of the class xK^{×k} under the composition

K^{×}/K^{×k} = H^{1}(K, μ_{k}) → H^{1}(K, T) ⟶^{v_{∗}} H^{1}(\overline{K}, \overline{T}_{∗})

is equal to v(x)θ\overline{T}.

Proof. The commutativity of the diagram

1 ⟶ μ_{k} ⟶ G_{m} ⟶^{k} G_{m} ⟶ 1
↓  ↓  ∥
1 ⟶ T ⟶ V ⟶ G_{m} ⟶ 1

shows that the image of xK^{×k} in H^{1}(K, T) coincides with the image of xK^{×k} under the connecting homomorphism induced by the bottom sequence in the diagram.

The result follows from the commutativity of the diagram

K^{×} ⟶ H^{1}(K, T)
v↓  ↓^{v_{∗}}
ℤ ⟶ H^{1}(\overline{K}, \overline{T}_{∗}),

where the bottom map in the connecting homomorphism for the exact sequence (4). □

Lemma 5.2. Suppose that ξ is unramified and ind(A_{ξ}) ∈ kℤ. Then every ρ ∈ H^{1}(K, T) over ξ is unramified.

Proof. By assumption, the class A_{ξ} ∈ Alg_{K}(n, m) is unramified. By Lemma 4.1, there is an unramified element η ∈ H^{1}(K, T) over ξ. In view of Lemma 3.1, ρ = xη for some x ∈ K^{×}. It follows that ξ = xξ.

By Lemma 3.1, x ∈ Nrd(A_{ξ})K^{×k}. As v(Nrd(A_{ξ})) ⊂ ind(A)ℤ, by assumption, v(x) ∈ kℤ. Multiplying x by a kth power in K^{×} we may assume that x is a unit. Therefore, ρ is unramified. □

Let M be a field extension of K and let w be an extension on M of the discrete valuation v. We assume that M is complete. Write e for the ramification index of M/K and \overline{M} for the residue field of M.

Lemma 5.3. Suppose that ind((A_{ξ})_{M}) ∈ kℤ and ξ_{M} = xξ′, where x ∈ M^{×} and ξ′ is an unramified element in H^{1}(M, G). Then the element w(x)θ\overline{T} in H^{1}(\overline{M}, \overline{T}_{∗}) is divisible by e.

Proof. Choose an element ρ ∈ H^{1}(K, T) over ξ. The image of ρ′ := x^{−1}ρ_{M} in H^{1}(M, G) is equal to ξ′ and hence is unramified. By Lemma 5.2, applied to the field M, ρ′ is unramified.

Consider the following commutative diagram:

H^{1}(K, T) ⟶^{v_{∗}} H^{1}(\overline{K}, \overline{T}_{∗})
↓  ^{e}↓
H^{1}(M, T) ⟶^{w_{∗}} H^{1}(\overline{M}, \overline{T}_{∗})

where the right vertical map is e times the canonical map. Hence the image of ρ in H^{1}(\overline{M}, \overline{T}_{∗}) is divisible by e. On the other hand, ρ_{M} = xρ′ and ρ′ is unramified. Hence by Lemma 5.1, the image of ρ in H^{1}(\overline{M}, \overline{T}_{∗}) coincides with w(x)θ\overline{T}. □

Proposition 5.4. Let M/K be an extension of complete fields with discrete valuations, let ξ ∈ H^{1}(K, G) be such that ind((A_{ξ})_{M}) = n = p^{r}. Suppose that ξ_{M} = πξ′ for a prime element π ∈ M and an unramified element ξ′ ∈ H^{1}(M, G). If s < r then the ramification index of the extension M/K is not divisible by p.

Proof. Let L/K be an unramified spitting field for A_{ξ} of degree n and let T be the torus as above. By Lemma 5.3, θ\overline{T} in H^{1}(\overline{M}, \overline{T}_{∗}) is divisible by the ramification index e and by Proposition 4.3 applied to the torus T_{M} over M, θ\overline{T} is not divisible by p in H^{1}(\overline{M}, \overline{T}_{∗}). Hence, p does not divide e. □

6. Proof of the theorem

We prove Theorem 1.1. Write n = p^{r}n′, m = p^{s}m′ and k′ = n′/m′. Consider the groups H = SL_{p^{r}} /μ_{p^{s}} and G′ = SL_{n′} /μ_{m′}. We have a natural group homomorphism H × G′ → G. For a field extension E/F take algebras B ∈ Alg_{E}(p^{r}, p^{s}), A′ ∈ Alg_{E}(n′, m′) and A := B ⊗A′ ∈ Alg_{E}(n, m). By Lemma 3.1, the fiber of the natural surjection H^{1}(F, H) → Alg_{R}(p^{r}, p^{s}) over B is a principal homogeneous space under C := E^{×}/(E^{×p^{r−s}} · Nrd(B)). Similarly, the fibers of the natural surjections H^{1}(E, G′) → Alg_{E}(n′, m′) and H^{1}(E, G) → Alg_{E}(n, m) over A′ and A are principal homogeneous spaces under D′ := E^{×}/(E^{×k′} · Nrd(A′)) and D := E^{×}/(E^{×k} · Nrd(A)) respectively.

The tensor product yields a bijection

Alg_{R}(p^{r}, p^{s}) × Alg_{R}(n′, m′) → Alg_{R}(n, m).

There is a natural isomorphism C × D′ → D. It follows that the natural map

H^{1}(E, H) × H^{1}(E, G′) → H^{1}(E, G)

is a bijection.

This bijection yields a surjection F_{G} → F_{H} and a p-surjective map F_{H} → F_{G}. By [5, Sec. 1.3], ed_{p}(G) = ed_{p}(H).

Replacing G by H we may assume that n = p^{r} and m = p^{s}. If s = 0 then G = SL_{n} and ed_{p}(G) = 0 as G is special, i.e., all G-torsors over fields are trivial. If s = r, G = PGL_{n} and F_{G} = Alg(p^{r}, p^{r}), hence ed_{p}(G) = ed_{p}(Alg(p^{r}, p^{r})).

We may assume that 0 < s < r. By Lemma 3.1, for any field E, the natural map H^{1}(E, G) → Alg(p^{r}, p^{s})(E) is surjective and the fibers are homogeneous sets under E^{×}. It follows that

ed_{p}(G) ≤ ed_{p}(Alg(p^{r}, p^{s})) + 1.

To prove the opposite inequality choose a field E/F and a (generic) algebra A in Alg(p^{r}, p^{s})(E) such that

(6)  ed_{p}(Alg(p^{r}, p^{s})) = ed_{p}(A).

Note that as s > 0, the index of A is equal to p^{r}. Choose an element η ∈ H^{1}(E, G) with A_{η} = A.

Consider the field of formal Laurent series E((t)) and set ξ′ := η_{E((t))} ∈ H^{1}(E((t)), G). We have A_{ξ′} = A_{E((t))}. Choose a finite field extension M/E((t)) of degree prime to p and a subfield K ⊂ M over F such that tr. deg_{F} (K) = ed_{p}(tξ′) and there is an element ξ ∈ H^{1}(K, G) with ξ_{M} = tξ′_{M}.

Let w be the extension of the discrete valuation of E((t)) on M. The ramification index of M/E((t)) is not divisible by p. The degree of the residue field \overline{M} over E is also not divisible by p.

Note that the element ξ_{M} is ramified by Lemma 3.2, hence the restriction on K of the discrete valuation of M is nontrivial. We have tr. deg_{F} (K) ≥ tr. deg_{F} (\overline{K}) + 1, therefore,

(7)  ed_{p}(tξ′) ≥ tr. deg_{F} (\overline{K}) + 1.

Write K̂ for the completion of K. As M is complete we may assume that K̂ is a subfield of M. Since 0 < s < r, by Proposition 5.4, the ramification index e of M/K̂ is not divisible by p.

As char(F) is not equal to p, there is the residue homomorphism [7, Ch. XII]

∂ : Br_{n}(K̂) → H^{1}(\overline{K}, ℤ/nℤ).

Let χ̄ = ∂(A_{ξ}) ∈ H^{1}(\overline{K}, ℤ/nℤ). As (A_{ξ})_{M} = A_{tξ′} = A_{ξ′} = A_{M} is unramified, we have e · χ̄\overline{M} = 0 and hence χ̄\overline{M} = 0 as e is not divisible by p. Hence we can view the cyclic extension \overline{K}(χ̄) of \overline{K} given by χ̄ as a subfield of \overline{M}.

Let χ ∈ H^{1}(K̂, ℤ/nℤ) be the lift of χ̄. The field K̂(χ) is a subfield of M. Therefore, the algebra B := (A_{ξ})_{K̂(χ)} is unramified and its residue \overline{B} satisfies \overline{B} ∈ Alg(p^{r}, p^{s})(\overline{K}(χ̄)) and (\overline{B})\overline{M} = A\overline{M}.

Thus, the algebra A\overline{M} is defined over \overline{K}(χ̄), hence

(8)  tr. deg_{F} (\overline{K}) = tr. deg_{F} (\overline{K}(χ̄)) ≥ ed_{p}(A).

We have by (6), (7) and (8):

ed_{p}(G) ≥ ed_{p}(tξ′) ≥ tr. deg_{F} (\overline{K}) + 1 ≥ ed_{p}(A) + 1 = ed_{p}(Alg(p^{r}, p^{s})) + 1.

References

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Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

  E-mail address: chernous@math.ualberta.ca

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

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