Power-Central Elements in Tensor Products of Symbol Algebras

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Power-Central Elements in Tensor Products of Symbol Algebras

Demba Barry

To cite this version:

Demba Barry. Power-Central Elements in Tensor Products of Symbol Algebras. 2013. �hal-00875041�

SYMBOL ALGEBRAS

DEMBA BARRY

Abstract. Let A be a central simple algebra over a field F. Let k1, . . . , kr

be cyclic extensions of F such thatk1⊗_F · · · ⊗_F kr is a field. We investigate conditions under whichAis a tensor product of symbol algebras where eachkiis in a symbolF-algebra factor of the same degree aski. As an application, we give an example of an indecomposable algebra of degree 8 and exponent 2 over a field of 2-cohomological dimension 4.

Keywords Central simple algebra. Symbol algebra, Armature, Valuation, Co- homological dimension

Mathematics Subject Classification (2010) 16K20, 16W60, 12E15, 12G10, 16K50

1. Introduction

LetF be a field and letkbe a Galois extension ofF of degreenwith cyclic group generated by σ. For a∈F^×, we let (k, σ, a) denote the cyclic F-algebra generated overkby a single elementywith defining relationycy⁻¹ =σ(c) forc∈kandyⁿ=a.

If F contains a primitive n-th root of unityζ, it follows from Kummer theory that one may write k in the form k = F(√ⁿ

b) for some b ∈ F^×. The algebra (k, σ, a) is then isomorphic to the symbol algebra (a, b)_n over F, that is, a central simple F-algebra generated by two elementsi andj satisfying iⁿ=a,jⁿ=band ij =ζji (see for instance [P, §15. 4]). In the case n = 2, ζ = −1, one gets a quaternion algebra overF that will be denoted (a, b).

A central division algebra decomposes into a tensor product of symbol algebras (of degree 2 in [ART] and of degree an arbitrary prime p in [Ro]) if and only if it contains a set whose elements satisfy some commuting properties: q-generating set in [ART], p-central set in [Ro], and a set of representatives of an armature in [T₁].

Our approach is based on these notions.

The main goal of this paper is to further investigate the decomposability of cen- tral simple algebras; the study of power central-elements is a constant tool. Let A be a central simple algebra over F and let k₁, . . . , k_r be cyclic extensions of F, of respective degree n₁, . . . , n_r, contained in A such thatk₁⊗F · · · ⊗F k_r is a field. It is natural to ask: when does there exist a decomposition of A into a tensor prod- uct of symbol algebras in which each k_i is in a symbol F-subalgebra factor of A of degree n_i? Starting from A, we construct a division algebra E whose center is

1

the iterated Laurent series with r indeterminates over F and show that A admits such a decomposition if and only if E is a tensor product of symbol algebras (see Theorem 3.1 and Corollary 3.2 for details). Note that Corollary 3.2 is very close to a result of Tignol [T₂, Prop. 2.10]. In contrast with Tignol’s result (which is stated in terms of Brauer equivalence and only for prime exponent), Corollary 3.2 is stated in terms of isomorphism classes and is valid for any exponent. Moreover, our ap- proach is completely different. We will give an example, pointed out by Merkurjev, of a division algebra which is a tensor product of three quaternion algebras, and containing a quadratic field extension which is in no quaternion subalgebra (Corol- lary 4.6). Using valuation theory, we give a general method for constructing tensor products of quaternion algebras containing a quadratic field extension which is in no quaternion subalgebra. As an application, let A be a central simple algebra of degree 8 and exponent 2 over F, and containing a quadratic field extension which is in no quaternion subalgebra. We use Corollary 3.2 to associate with A an example of an indecomposable algebra of degree 8 and exponent 2 over a field of rational functions in one variable over a field of 2-cohomological dimension 3 (see Theorem 4.8). This latter field is obtained by an inductive process pioneered by Merkurjev [M].

We next recall some results related to our main question: let A be a 2-power dimensional central simple algebra overF and letF(√

d₁,√

d₂)⊂Abe a biquadratic field extension of F. If A is a biquaternion algebra, it follows from a result of Albert that A ≃(d₁, d^′₁)⊗F (d₂, d^′₂) whered^′₁, d^′₂ ∈ F^× (see for instance [Ra]). As observed above, this is not true anymore in higher degree. More generally, if A is of degree 8 and exponent 2 and F(√

d) ⊂ A is a quadratic field extension, there exists a cohomological criterion associated with the centralizer ofF(√

d) inAwhich determines whether F(√

d) lies in a quaternion F-subalgebra of A (see [Ba, Prop.

4.4]). In the particular case where the 2-cohomological dimension of F is 2 and A is a division algebra of exponent 2 over F, the situation is more favorable: it is shown in [Ba, Thm. 3.3] that there exits a decomposition ofAinto a tensor product of quaternion F-algebras in which each F(√

d_i) (for i = 1,2) is in a quaternion F-subalgebra.

An outline of this article is the following: in Section 2 we collect from [T₁] and [TW] some results on armatures of algebras that will be used in the proofs of the main results. Section 3 is devoted to the statements and the proofs of the main results. The particular case of exponent 2 is analyzed (in more details) in Section 4.

All algebras considered in this paper are associative and finite-dimensional over their center. A central simple algebra A over a field F is decomposable if A ≃ A₁⊗F A₂ for two central simpleF-algebras A₁ and A₂ both non isomorphic to F; otherwise A is called indecomposable.

Throughout this article, we shall use freely the standard terminology and nota- tion from the theory of finite-dimensional algebras and the theory of valuations on division algebras. For these, as well as background information, we refer the reader to Pierce’s book [P].

2. Armatures of algebras

Armatures in central simple algebras are a major tool for the next section. The goal of this section is to recall the notion of an armature and gather some preliminary results that will be used in the sequel.

We write |H|for the cardinality of a setH. LetAbe a central simpleF-algebra.

For a ∈ A^×/F^×, we fix an element x_a of A whose image in A^×/F^× is a, that is, a=x_aF^×. For a finite subgroup Aof A^×/F^×,

F[A] =n X

a∈A

c_ax_a|c_a∈Fo

denotes the F-subspace of A generated by {x_a|a∈ A}. Note that this subspace is independent of the choice of representatives x_a fora∈ A. SinceA is a group,F[A] is the subalgebra of A generated by {x_a|a∈ A}. As was observed in [T₁], if A is a finite abelian subgroup of A^×/F^× there is an associated pairing h,i on A × A defined by

ha, bi =xax_bx⁻_a¹x⁻_b¹.

This definition is independent of the choice of representativesx_a, x_bfora, bandha, bi belongs to F^× asA is abelian. Hence, ha, bi is central inA, and it follows that the pairingh,i is bimultiplicative. It is also alternating, obviously. Thus, asAis finite, the image of h,i is a finite subset ofµ(F) (whereµ(F) denotes the group of roots of unity of F). For any subgroupH of Alet

H^⊥={a∈ A | ha, hi= 1 for allh∈ H},

a subgroup of A. The subgroup H^⊥ is called the orthogonal of H with respect to h,i. The radical of A, rad(A), is defined to be A^⊥. The pairing h,i is called nondegenerate on Aif rad(A) ={1_A}.

For g ∈ A, we denote by (g) the cyclic subgroup of A generated byg . The set {g₁, . . . , g_r} is called a base ofA if A is the internal direct product

A= (g₁)× · · · ×(g_r).

Ifh,iis nondegenerate thenAhas asymplectic base with respect to h,i, i.e, a base {g1, h1, . . . , gn, hn} such that for alli, j

hg_i, h_ii=c_i, where ord(g_i) = ord(h_i) = ord(c_i) hg_i, g_ji=hh_i, h_ji= 1 and, ifi6=j, hg_i, h_ji= 1 (see [T1, 1.8]).

Definition 2.1. For any finite-dimensional F-algebra A, a subgroup A of A^×/F^× is anarmature of Aif A is abelian, |A|= dim_FA, and F[A] =A.

If A={a₁, . . . , a_n} is an armature of A, the above definition shows that the set {xa1, . . . , xan} is an F-base of A. The notion of an armature was introduced by Tignol in [T₁] for division algebras. The definition given here, slightly different from that given in [T₁], comes from [TW]. This definition allows armatures in algebras

other than division algebras. The following examples will be used repeatedly in the next section.

Examples 2.2. (a) ([T1]) LetA=A1⊗^F · · · ⊗^F Ar be a tensor product of symbol F-algebras where A_k is a symbol subalgebra of degree n_k. Suppose F contains a primitive n_k-th root ζ_k of unity for k= 1, . . . , r. So,A_k is isomorphic to a symbol algebra (a_k, b_k)_ζk of degree n_k for some a_k, b_k ∈ F^×. For each k, let i_k, j_k be a symbol generator of (a_k, b_k)_ζk. The imageA inA^×/F^× of the set

{i^α₁¹j₁^β1. . . i^α_r^rj_r^βr|0≤α_k, β_k≤n_k−1}

is an armature of A isomorphic to (Z/n₁Z)² × · · · ×(Z/n_rZ)². We observe that for all 1 6= a ∈ A there exists b ∈ A such that ha, bi 6= 1; that is the pairing h,i is nondegenerate onA. Furthermore, {i₁F^×, j₁F^×, . . . , i_rF^×, j_rF^×}is a symplectic base ofA.

(b) Let M be a finite abelian extension of a fieldF and letGbe the Galois group of M over F. Let ℓ be the exponent of G. If F contains an ℓ-th primitive root of unity, the extension M/F is called aKummer extension. Let

S ={x∈M^× |x^ℓ ∈F^×} and Kum(M/F) =S/F^×.

It follows from Kummer theory (see for instance [J₁, p.119-123]) that Kum(M/F) is a subgroup of M^×/F^× and is dual to G by the nondegenerate Kummer pairing G×Kum(M/F)→µ(F) given by (σ, b) =σ(x_b)x⁻_b¹, forσ ∈Gand b∈Kum(M/F).

Whence,Kum(M/F) is isomorphic (not canonically in general) toG. As observed in [TW, Ex. 2.4], the subgroupKum(M/F) is the only armature of M with exponent dividingℓ.

LetA be an armature of a central simpleF-algebra A and let{a₁, b₁, . . . , a_n, b_n} be a symplectic base ofAwith respect toh,i. We shall denote byF[(a_k)×(b_k)] the subalgebra ofAgenerated by the representativesx_ak andx_bk ofa_kandb_k. It is clear that F[(a_k)×(b_k)] is a symbol subalgebra of A of degree n_k = ord(a_k) = ord(b_k) and generated by xak and x_bk. It is shown in [TW, Lemma 2.5] thatA ≃F[(a1)× (b₁)]⊗F · · · ⊗F F[(a_n)×(b_n)].

Actually, the notion of an armature is a generalization and refinement of the notion of a quaternion generating set (q-generating set) introduced in [ART]. Indeed, a central simple algebraA over F has an armature if and only ifA is isomorphic to a tensor product of symbol algebras over F (see [TW, Prop. 2.7]).

Note that if the exponent ofAis a prime p, we may considerAas a vector space over the field with p elements F_p. Identifying the group of p-th roots of unity with F_p by a choice of a primitive p-th root of unity, we may suppose the pairing has values inF_p. So, two elements a, b∈ A are orthogonal if and only if ha, bi = 0. We need the following proposition:

Proposition 2.3. Let V be a vector space over F_p of dimension 2n and let h,i be a nondegenerate alternating pairing on V. Let {e₁, . . . , e_r} be a base of a totally

isotropic subspace ofV with respect toh,i. There aref₁, . . . , f_r, e_r+1, f_r+1, . . . , e_n, f_n in V such that {e1, f1, . . . , en, fn} is a symplectic base of V.

Proof. We argue by induction on the dimension of the totally isotropic subspace spanned by e1, . . . , er. Ifr = 1, since the pairing is nondegenerate, there is f1 ∈V such thathe₁, f₁i 6= 0 . Denote byU = span(e₁, f₁) the subspace spanned by e₁, f₁. We have V =U ⊥U^⊥ since the restriction ofh,i to U is nondegenerate. We take for{e2, f2, . . . , en, fn}a symplectic base of U^⊥.

Assume the statement for a totally isotropic subspace of dimension r−1. Let W = span(e₂, . . . , e_r); we have W ⊂ W^⊥. First, we find f₁ ∈ W^⊥ such that he₁, f₁i 6= 0. For this, consider the induced pairing, also denoted by h,i, onW^⊥/W defined by hx+W, y +Wi = hx, yi for x, y ∈ W^⊥. It is well-defined, and non- degenerate since (W^⊥)^⊥ = W. The element e₁ +W being non-zero in W^⊥/W, there is f₁ +W ∈ W^⊥/W such that 0 6= he₁, f₁i = he₁ +W, f₁ +Wi. Letting U = span(e1, f1), we have V = U ⊥ U^⊥ and e2, . . . , er ∈ U^⊥. Induction yields f₂, . . . , f_r, e_r+1, f_r+1, . . . , e_n, f_n ∈ U^⊥ such that {e₂, f₂, . . . , e_n, f_n} is a symplectic base ofU^⊥. Then{e₁, f₁, . . . , e_n, f_n}is a symplectic base of V.

3. Decomposability

Let A be a central simple algebra overF and let t₁, . . . , t_r be independent inde- terminates over F. For i = 1, . . . , r, let k_i be a cyclic extension of F of degree n_i contained inA. We assume thatM =k₁⊗F· · · ⊗Fk_ris a field and denote byGthe Galois group ofM over F. So [M :F] =n₁. . . n_r and G=hσ₁i × · · · × hσ_ri, where hσ_ii is the Galois group ofk_i over F, and the order of Gis n₁. . . n_r. Every element σ ∈ G can be expressed as σ = σ^m₁ ¹. . . σ^m_r^r (0 ≤ m_i < n_i). We shall denote by C =CAM the centralizer of M in A. Lett1, . . . , tr be independent indeterminates over F. Consider the fields

L^′ =F(t₁, . . . , t_r) and L=F((t₁)). . .((t_r)) and the following central simple algebras over L^′ and Lrespectively

N^′= (k1⊗^F L^′, σ1⊗id, t1)⊗L^′· · · ⊗L^′ (kr⊗^F L^′, σr⊗id, tr) and

N = (k1⊗^F L, σ1⊗id, t1)⊗^L· · · ⊗^L(kr⊗^F L, σr⊗id, tr).

We let

R^′ =A⊗F N^′ and R=A⊗F N.

In this section our goal is to prove the following results:

Theorem 3.1. Let M be a finite group with a nondegenerate alternating pairing M×M →µ(F). SupposeCis a division subalgebra ofA, andF contains a primitive exp(M)-th root of unity. Then, the following are equivalent:

(i) The division algebra Brauer equivalent toR^′ (respectivelyR) has an armature isomorphic toM.

(ii) The algebraAhas an armature isomorphic toMand containingKum(M/F) as a totally isotropic subgroup.

In the particular case whereAis of degreepⁿand exponentp(for a prime number p) and each k_i is a cyclic extension ofF of degree p, we have:

Corollary 3.2. Assume that A is of degree pⁿ and exponent p. Suppose C is a division subalgebra of A, F contains a primitive p-th root of unity, and n_i =p for i= 1, . . . , r. Then, the following are equivalent:

(i) The division algebra Brauer equivalent toR^′ (respectivelyR) is decomposable into a tensor product of symbol algebras of degree p.

(ii) The algebra A decomposes as

A≃(k₁, σ₁, δ₁)⊗F · · · ⊗F (k_r, σ_r, δ_r)⊗F A_r+1⊗F · · · ⊗F A_n

for some δ₁, . . . , δ_r ∈ F^× and some symbol algebras A_r+1, . . . , A_n of degree p.

Moreover, if these conditions are satisfied then the division algebras Brauer equiva- lent to R^′ andR decompose respectively as

(k₁⊗F L^′, σ₁⊗id, δ₁t₁)⊗L^′· · · ⊗L^′(k_r⊗F L^′, σ_r⊗id, δ_rt_r)⊗F A_r+1⊗F · · · ⊗F A_n and

(k₁⊗F L, σ₁⊗id, δ₁t₁)⊗L· · · ⊗L(k_r⊗F L, σ_r⊗id, δ_rt_r)⊗F A_r+1⊗F · · · ⊗F A_n. As opposed to part (i), it is not enough in part (ii) of Theorem 3.1 to assume simply that A has an armature to get an armature in the division algebra Brauer equivalent toR^′orR. The following example shows that the existence of an armature and the existence of an armature containing Kum(M/F) are different.

Example 3.3. Denote by Q₂ the field of 2-adic numbers, and let A be a division algebra of degree 4 and exponent 4 over F = Q₂(√

−1). Such an algebra is a symbol (see [P, Th., p. 338]). Note that M = F(√

2,√

5) is a field since the set {−1,2,5} forms a Z/2Z-basis of Q^×₂/Q^×₂² (see for instance [L, Lemma 2.24, p.

163]). It also follows by [P, Prop., p. 339] thatA⊗M is split. Hence, since [M :F] divides deg(A), we deduce that M ⊂ A. Moreover, it is clear that the centralizer of M in A is M. The algebra A being a symbol, it has an armature A isomorphic to (Z/4Z)² (see Example 2.2). Assume that A contains Kum(M/F) ≃ (Z/2Z)². Since A² = {a²|a ∈ A} = Kum(M/F), the algebra A is a symbol of the form A= (c, d)₄ withc.F^×2,d.F^×2 ∈ {2.F^×2,5.F^×2,2.5.F^×2}. It follows that A⊗A is either Brauer equivalent to the quaternion algebra (2,5) or (2,2.5) or (5,2.5). Since (2,5) ≃ (−1,−1) over Q₂, the algebra (2,5) is split over F; that is A⊗A is split.

Therefore the exponent of A must be 2; impossible. Therefore A has no armature containing Kum(M/F).

Now, let E be the division algebra Brauer equivalent toA⊗(2, t₁)⊗(5, t₂). As we will see soon (Lemma 3.4 and Lemma 3.5), deg(E) = exp(E) = 4. But E has

no armature, that is, E is not a symbol. Indeed, suppose E has an armature B. If B ≃ (Z/2Z)⁴ then E is a biquaternion algebra, so exp(E) = 2; contradiction.

Therefore B is isomorphic to (Z/4Z)². It follows then by Theorem 3.1 that A has an armature containing the amature Kum(M/F) of M. This is impossible as we showed above; thereforeE has no armature.

3.1. Brauer classes of R^′ and R. For the proof of the results above, we need an explicit description of the division algebras Brauer equivalent toR^′ andR. First, we fix some notation: recall that we denoted byG=hσ₁i × · · · × hσ_ri the Galois group ofM overF. By the Skolem-Noether Theorem, for eachσ_ithere existsz_i ∈A^×such thatσ_i(b) =z_ibz_i⁻¹for allb∈M. Notice thatz_iⁿⁱ =:c_i ∈C andz_iz_jz_i⁻¹z⁻_j¹ =:u_ij ∈ C for all i, j. For all σ =σ^m₁¹. . . σ_r^mr ∈G, we set z_σ =z₁^m1. . . z^m_r^r (0 ≤ m_i < n_i).

Settingc(σ, τ) =z_σz_τ(z_στ)⁻¹, a simple observation shows for allσ, τ (3.1) c(σ, τ)∈C and z_σb=σ(b)z_σ for allb∈M.

In fact,c(σ, τ) can be calculated from the elementsu_ij and c_i.

Let y_i be a generator of (k_i ⊗L^′, σ_i ⊗1, t_i), that is an element satisfying the relations y_i(d⊗l)y⁻_i¹ = σ_i(d) ⊗l for all d ∈ k_i and l ∈ L^′ and yⁿ_iⁱ = t_i. For σ =σ₁^m1. . . σ_r^mr ∈ G with 0≤ m_i < n_i, we set y_σ = y^m₁ ¹. . . y_r^mr. The algebras N^′ and N being crossed products, we may write

N^′=M

σ∈G

(M⊗L^′)y_σ and N =M

σ∈G

(M⊗L)y_σ and they_σ satisfy

(3.2) y_σy_τ(y_στ)⁻¹∈M⊗L and y_σ(b⊗l) = (σ(b)⊗l)y_σ

for allb∈M andl∈L. Set f(σ, τ) =y_σy_τ(y_στ)⁻¹. The elementsf(σ, τ) are in fact inL^′. Indeed, let y_σ =y^α₁¹. . . y_r^αr and y_τ =y₁^β1. . . y_r^βr with 0≤α_i, β_i< n_i . We get (3.3) f(σ, τ) =y_σy_τ(y_στ)⁻¹ =t^ε₁¹. . . t^ε_r^r

where εi = 0 if αi+βi <ord(σi) = ni and εi = 1 if αi+βi ≥ord(σi). Note that f(σ, τ) =f(τ, σ) for all σ, τ ∈G.

Now, let e be the separability idempotent of M, that is, the idempotent e ∈ M ⊗F M determined uniquely by the conditions that

(3.4) e·(x⊗1) =e·(1⊗x) for allx∈M

and the multiplication map M⊗F M → M carries eto 1 (see for instance [P, §14.

3]). Forσ ∈G, let

e_σ = (id⊗σ)(e)∈M ⊗F M.

The elements (eσ)σ∈Gform a family of orthogonal primitive idempotents ofM⊗^FM (see [P, §14. 3] ) and it follows, by applying id⊗σ to each side of (3.4), that (3.5) e_σ·(x⊗1) =e_σ·(1⊗σ(x)) forx∈M.

We need the following lemma:

Lemma 3.4. Notations are as above.

(1) The elementsz_σ⊗y_σ are subject to the following rules (i) For all σ, τ ∈G, there existsu∈CL such that

(z_σ⊗y_σ)(z_τ ⊗y_τ) = (u⊗1)(z_στ ⊗y_στ).

(ii) (z_σ ⊗y_σ)(c⊗1) = ((z_σcz⁻_σ¹)⊗1)(z_σ ⊗y_σ) for all c ∈ C. Moreover, z_σcz_σ⁻¹∈C for allσ ∈G.

(2) The sums

E^′ =X

σ∈G

C_L′z_σ⊗y_σ and E= X

σ∈G

C_Lz_σ⊗y_σ

are direct and are central simple subalgebras ofR^′ andR respectively. More- over, the algebras R^′ and R are Brauer equivalent to E^′ and E respectively, anddegE^′ = degE = degA.

Proof. (1) The statement of (i) follows from relations (3.1), (3.2) and the fact that f(σ, τ) ∈ L^′. Since the elements 1⊗yσ centralize CL⊗1, the statement of (ii) is clear.

(2) Suppose P

σ∈Gc_σz_σ⊗y_σ = 0 with c_σ ∈ C_L′ (or C_L). Pick such a sum with a minimal number of non-zero terms. There are at least two non-zero elements, say c_ρz_ρ⊗y_ρ,c_τz_τ⊗y_τ, in the sum. Letb∈M be such thatρ(b)6=bandτ(b) =b. One has

(b⊗1) X

σ∈G

c_σz_σ⊗y_σ

(b⊗1)⁻¹−X

σ∈G

c_σz_σ⊗y_σ = 0

and the number of non-zero terms is nontrivial and strictly smaller; contradiction.

Whence we have the direct sumsE^′ =L

σ∈GC_L′z_σ⊗y_σ andE =L

σ∈GC_Lz_σ⊗y_σ. It is clear thatE^′ ⊂R^′andE ⊂R. On the other hand, the computation rules of the part (1) show thatE^′ and E are generalized crossed products (see [A, Th. 11.11] or [J₂,§1.4]). Hence, the same arguments as for the usual crossed products show that E^′ and E are central simple algebras over L^′ and L respectively. Moreover, since dimC = _n1...n¹

r dimA, we have degE^′ = degE= degA.

Now, it remains to show that R^′ and R are respectively Brauer equivalent to E^′ and E. For this we work over L; the same arguments apply over L^′. Forzσ⊗yσ as above, consider the inner automorphism

Int(zσ⊗yσ) : R −→ R.

Notice that Int(z_σ⊗y_σ)(e_τ) is in M⊗M and is a primitive idempotent for allτ ∈G (since e_τ is primitive). Moreover, for x∈M,

Int(zσ⊗yσ)(eτ)·(x⊗1) = (zσ⊗yσ)eτ(z_σ⁻¹⊗y_σ⁻¹)·(x⊗1)

= (z_σ⊗y_σ)e_τ ·(σ⁻¹(x)⊗1)(z⁻_σ¹⊗y⁻_σ¹)

= (z_σ⊗y_σ)e_τ ·(1⊗τ σ⁻¹(x))(z⁻_σ¹⊗y⁻_σ¹)

= (1⊗τ(x))·Int(zσ ⊗yσ)(eτ).

Therefore Int(z_σ⊗y_σ)(e_τ) =e_τ by comparing with the definition ofeand the relation (3.5). Hence, each e_τ centralizes E in R. On the other hand, since the degree of

the centralizer C_RE of E in R isn₁. . . n_r and (e_τ)_τ∈G⊂C_RE, the algebra C_RE is

split. So R is Brauer equivalent toE.

Lemma 3.5. Notations are as in Lemma 3.4. If C is a division algebra then there exists a unique valuation on E which extends the (t₁, . . . , t_r)-adic valuation on L.

Consequently E^′ and E are division algebras.

Proof. The (t₁, . . . , t_r)-adic valuation on L being Henselian, it extends to a unique valuation to each division algebra overL(see for instance [W1]). It follows that there is a valuation on C_L extending the (t₁, . . . , t_r)-adic valuation on L. More precisely this valuation is constructed as follows: writing C_L as

C_L= (

X

i1∈Z

· · ·X

ir∈Z

c_i1...irtⁱ₁¹. . . tⁱ_r^r

c_i1...ir ∈C and

{(i₁, . . . , i_r)|c_i1...ir 6= 0 is well-ordered for the right-to-left lexicographic ordering}

) ,

computations show that the map v:C_L^×→Z^r defined by v X

i¹∈Z

· · ·X

ir∈Z

c_i1...irtⁱ₁¹. . . tⁱ_r^r

= min{(i₁, . . . , i_r)|c_i1...ir 6= 0}

is a valuation. Clearlyv extends the (t₁, . . . , t_r)-adic valuation onL. Recall thatN is a division algebra overL(see e.g. [W₂, Ex. 3.6]). We also denote by vthe unique extension of the (t1, . . . , tr)-valuation toN. Sinceyⁿ_iⁱ =ti, we have

v(y_i) = (0, . . . ,0, 1

n_i,0, . . . ,0)∈ 1

n₁Z×. . .× 1 n_rZ.

Hence, for y_σ = y₁^mi. . . y^m_r ^r where σ = σ^m₁¹. . . σ_r^mr (with 1 ≤ m_i < n_i), we have v(yσ) = (^m_n1¹, . . . ,^m_n^r

r). It then follows that

(3.6) v(y_σ)6≡v(y_τ) modZ^r if σ6=τ.

Now, define a mapw : E^× → _n¹1Z×. . .×_n¹_rZas follows: for anyσ ∈Gand any c∈C_L^×, set

w(cz_σ ⊗y_σ) =v(c) +v(y_σ).

For anys∈E^×,shas a unique representations=P

σ∈Gc_σz_σ⊗y_σ with thec_σ ∈C_L and some c_σ 6= 0. Define

w(s) = min

σ∈G{w(c_σz_σ⊗y_σ)|c_σ 6= 0}.

It follows by (3.6) that w(cσzσ ⊗yσ) 6= w(cτzτ ⊗yτ) for σ 6= τ. Thus, there is a unique summand c_ιz_ι⊗y_ι of ssuch thatw(s) =w(c_ιz_ι⊗y_ι); this c_ιz_ι⊗y_ι is called theleading term of s.

We are going to show that wis a valuation onE. Lets^′ =P

σ∈Gd_σz_σ⊗y_σ ∈E^× with d_σ ∈C_L and s+s^′ 6= 0. Let (c_ρ+d_ρ)z_ρ⊗y_ρ be the leading term of s+s^′. If

c_ρ6= 0 and d_ρ6= 0, we have

w((c_ρ+d_ρ)z_ρ⊗y_ρ) = v(c_ρ+d_ρ) +v(y_ρ)

≥ min(v(cρ) +v(yρ), v(dρ) +v(yρ))

= min(w(c_ρz_ρ⊗y_ρ), w(d_ρz_ρ⊗y_ρ))

≥ min(w(s), w(s^′)).

Thus,w(s+s^′) =w((cρ+dρ)zρ⊗yρ))≥min(w(s), w(s^′)). This inequality still holds if c_ρ= 0 or d_ρ= 0.

By the usual argument, we also check that

(3.7) if w(s)6=w(s^′) thenw(s+s^′) = min(w(s), w(s^′)).

It remains to show that w(ss^′) =w(s) +w(s^′). For σ, τ ∈G, recall that (z_σ⊗y_σ)(z_τ ⊗y_τ) =c(σ, τ)f(σ, τ)(z_στ ⊗y_στ)

for somec(σ, τ)∈C^× and some f(σ, τ)∈L^×. It follows that, forc_σ, d_τ ∈C_L^×, w((c_σz_σ ⊗y_σ)(d_τz_τ⊗y_τ)) = w(c_σ(z_σd_τz_σ⁻¹)(z_σ⊗y_σ)(z_τ ⊗y_τ)) (Lemma 3.4)

= w(c_σ(z_σd_τz_σ⁻¹)c(σ, τ)f(σ, τ)(z_στ ⊗y_στ))

= v(c_σ) +v(d_τ) +v(f(σ, τ)y_στ)

= v(cσ) +v(yσ) +v(dτ) +v(yτ)

= v(c_σz_σ⊗y_σ) +v(d_τz_τ⊗y_τ).

(3.8)

Hence, we have

w(ss^′) = w X

σ,τ

(c_σz_σ⊗y_σ)(d_τz_τ⊗y_τ)

≥ min

σ,τ {w((cσzσ⊗yσ)(dτzτ ⊗yτ))|cσ, dτ 6= 0}

= min

σ,τ {w(c_σz_σ⊗y_σ) +w(d_τz_τ ⊗y_τ)|c_σ, d_τ 6= 0}

≥ w(s) +w(s^′).

(3.9)

Let cρzρ⊗yρ and dιzι⊗yι be the leading terms of s and s^′ respectively. Sets1 = s−c_ρz_ρ⊗y_ρ and s^′₁ =s^′−d_ιz_ι⊗y_ι. So,

w(s) =w(c_ρz_ρ⊗y_ρ)< w(s₁) and w(s^′) =w(d_ιz_ι⊗y_ι)< w(s^′₁).

Writing

ss^′ = (c_ρz_ρ⊗y_ρ)(d_ιz_ι⊗y_ι) +s₁(d_ιz_ι⊗y_ι) + (c_ρz_ρ⊗y_ρ)s^′₁+s₁s^′₁,

it follows by (3.8) and (3.9) that the first summand in the right side of the above equality has valuation strictly smaller than the other three. Hence, by (3.7) and (3.8), one has ss^′ 6= 0 and

w(ss^′) =w((c_ρz_ρ⊗y_ρ)(d_ιz_ι⊗y_ι)) =w(s) +w(s^′).

Therefore w is a valuation onE. Since E =E^′⊗L^′ L, the restriction of w to E^′ is also a valuation. The uniqueness of w follows from its existence by [W₁].

Let D be a division algebra with a valuation. Theresidue division algebra of D is denoted byD.

We keep the notations above. Now, suppose C is a division subalgebra of A and denote by Γ_E and Γ_L the corresponding value groups of E and Lrespectively.

Furthermore, assume that E has an armature A. The diagram 1 ^//L^{× //}

v

E^{× //}

w

E^×/L^{× //}

1

0 ^//Γ_L ^//Γ_E ^//Γ_E/Γ_L ^//0 induces a homomorphism

w^′ : A ⊂E^×/L^× −→ Γ_E/Γ_L.

Put A0 = kerw^′ and let a ∈ A0. The above diagram shows that there exists a representative xa ofasuch that w(xa) = 0. Define

¯ : A⁰ −→ E^×/L^×=C^×/F^× by

a=x_aL^× 7−→ a¯= ¯x_aF^×

where x_a is such that w(x_a) = 0 and ¯x_a is the residue of x_a. If y_a is another representative of a such that w(y_a) = 0, there is h ∈L^× with w(h) = 0 such that ya=xah. Hence ¯ya= ¯xa¯h, that is ¯ya= ¯xaF^×, so ¯ is well defined. We have:

Proposition 3.6. Assume thatC is a division algebra and A is an armature of E as above. Then

(1) The map¯ : A0 −→ C^×/F^× is an injective homomorphism.

(2) The image A0 of A0 is an armature of C over F.

Proof. (1) Let a=x_aL^× and b=x_bL^× be such that w(x_a) =w(x_b) = 0. Choosing x_ab = x_ax_b, we have w(x_ab) = 0. Therefore ¯x_ab = ¯x_ax¯_b; this shows that ¯ is a homomorphism.

Let c ∈ ker¯ with c 6= 1 and let x_c ∈ E^× be a representative of c such that w(x_c) = 0. Let ¯x_c = α ∈ F^×; then x_c =α+x^′_c for some x^′_c ∈ E with w(x^′_c) > 0.

The pairing h,i being nondegenerate on A, there exists d∈ A such that hd, ci=ζ for some 16=ζ ∈µ(F). Thus,

x_dx_cx⁻_d¹ =ζx¯_c =ζα.

On the other hand,

x_dx_cx⁻_d¹ =x_dαx⁻_d¹+x_dx^′_cx⁻_d¹ =α+x_dx^′_cx⁻_d¹. Hence, we get

x_dx_cx⁻_d¹ =α since w(x_dx^′_cx⁻_d¹)>0; contradiction.

Therefore the map ¯ is injective. Consequently, we have |A0|=|A0|.

(2) We first show that |A0| ≤ dim_F C: since C = E, it suffices to prove that (¯xa)a∈A⁰ are linearly independent over F. LetP

a∈A⁰λax¯a= 0, with λa ∈F, be a zero linear combination such that the set

S={a∈ A⁰ |λa6= 0}

is not empty and of least cardinality. For s ∈S, let x_s be a representative of s in E^× such that w(x_s) = 0. We have

¯ x_s X

a∈A⁰

λ_ax¯_a

¯

x⁻_s¹ = X

a∈A⁰

ha, siλ_ax¯_a= 0 = X

a∈A⁰

λ_ax¯_a.

Then the linear combination P

a∈A⁰(1− ha, si)λax¯a is zero and the number of non- zero coefficients is less than the cardinality of S because hs, si = 1. Therefore, ha, si = 1 for all a, s ∈ S; this implies that ¯x_s and ¯x_s′ commute for all s, s^′ ∈ S. It follows from [T₁, Lemma 1.5] that the elements ¯x_s, for s ∈ S, are linearly independent; contradicting the fact that S is not empty. Combining with the part (1), we get

|A0|=|A0| ≤dimC = 1

n1. . . nr|A|. On the other hand, since |w^′(A)|= ^|A|

|A⁰|, we have |w^′(A)| ≥ n₁. . . n_r. We already know that |w^′(A)| ≤n₁. . . n_r because w^′(A) ⊂Γ_E/Γ_L and |Γ_E/Γ_L|=n₁. . . n_r. It follows that|w^′(A)|=n₁. . . n_r and |A0|= dim_FC. Since we showed that (¯x_a)_a∈A0

are linearly independent, the subgroup A⁰ is an armature of C over F. 3.2. Proof of the main result. Recall that the division algebras Brauer equivalent to R^′ and R are respectively E^′ and E by Lemma 3.4.

Proof of Theorem 3.1. (i) ⇒ (ii): we give the proof for E; the proof for E^′ follows because if E^′ has an armature then E = E^′ ⊗L^′ L has an isomorphic armature.

Assume thatE has an armatureA. Letc∈ Aand letc_ρz_ρ⊗y_ρbe the leading term of a representativex_cofcinE. That is,x_c =c_ρz_ρ⊗y_ρ+x^′_cwithw(x_c) =w(c_ρz_ρ⊗y_ρ) and w(x^′_c)> w(x_c). Define the map

ν : A −→ A^×/F^× by

c=xc.L^× 7−→ cρzρ.F^×.

If y_c is another representative of c, we have y_c = ℓx_c for some ℓ ∈ L^×. Note that the leading term of y_c is the leading term of ℓ multiplied by c_ρz_ρ ⊗y_ρ (see the proof of Lemma 3.5); moreover, the leading term of ℓlies in F^×. One deduces that ν(x_c.L^×) =ν(y_c.L^×). Soν is well-defined.

We show that ν(A) is an armature of A: first, we claim that ν is an injective homomorphism. Indeed, leta, b∈ Awith respective representatives x_a and x_b. Let c_σz_σ⊗y_σ and d_τz_τ⊗y_τ be the leading terms ofx_a and x_b respectively. As showed

in the proof of Lemma 3.5, the leading term of x_ax_b is (c_σz_σ ⊗y_σ)(d_τz_τ ⊗y_τ). On the other hand, it follows by (3.1), (3.2), (3.3) and Lemma 3.4 that

(c_σz_σ⊗y_σ)(d_τz_τ⊗y_τ) =c_σ(z_σd_τz_σ⁻¹)f(σ, τ)c(σ, τ)z_στ ⊗y_στ for somef(σ, τ)∈L^× and some c(σ, τ)∈C^×, andz_σd_τz_σ⁻¹ ∈C.

Sincex_ab=x_ax_bmodL^×(becausea, b∈ A), we may takex_ax_bas a representative of ab, so

x_ab =x_ax_b =c_σ(z_σd_τz⁻_σ¹)c(σ, τ)z_στ ⊗y_στ +x^′_ab with w(x^′_ab)> w(x_ab).

Hence, it follows by the definition of ν that

ν(ab) = c_σ(z_σd_τz⁻_σ¹)c(σ, τ)z_στ modF^×

= (c_σz_σ)(d_τz_τ) modF^×

= ν(a)ν(b).

Therefore, ν is a homomorphism.

For the injectivity, we start out by proving that the pairing h,i is an isometry forν: by definition

ha, bi=x_ax_bx⁻_a¹x⁻_b¹=ζ for some ζ ∈µ(F).

Hence

xax_b = (cσzσ⊗yσ)(dτzτ⊗yτ)+ (xax_b)^′ =ζx_bxa=ζ(dτzτ⊗yτ)(cσzσ⊗yσ)+ζ(x_bxa)^′, withw((x_ax_b)^′) =w((x_bx_a)^′)> w(x_ax_b). Sincey_σ and y_τ commute, it follows that

(c_σz_σ)(d_τz_τ) =ζ(d_τz_τ)(c_σz_σ).

Therefore

hν(a), ν(b)i= (c_σz_σ)(d_τz_τ)(c_σz_σ)⁻¹(d_τz_τ)⁻¹=ζ.

The pairing h,i is then an isometry for ν. Now, to see that ν is an injection, let a∈ A be such that ν(a) = 1. Sinceh,i is an isometry forν, for all b∈ A, one has

1 =hν(a), ν(b)i=ha, bi.

We infer that a = 1 because the pairing is nondegenerate on A. It follows that ν(A) is an abelian subgroup ofA^×/F^× isomorphic to A. Consequently, ν(A) is an armature of A^×/F^× isomorphic to M(≃ Aby hypothesis) since degE= degA by Lemma 3.4.

Now, we prove that Kum(M/F) ⊂ ν(A). Since A0 is an armature of C by Proposition 3.6, it follows by [TW, Lemma 2.5] that rad(A0) is an armature of the center ofC which isM. The extensionM/F being a Kummer extension, Examples 2.2 (b) indicates that rad(A⁰) = Kum(M/F). Therefore, we have Kum(M/F) ⊂ ν(A) sinceν is the identity on A0.

(ii)⇒(i): letBbe an armature ofAisomorphic toMand containingKum(M/F) as a totally isotropic subgroup. We construct an isomorphic armature in E^′. Note that if E^′ has an armature, then E also has an armature. For each σ∈G, set

Bσ ={a∈ B | ha, bi=σ(x_b)x⁻_b¹ for allb∈Kum(M/F)}.