A REMARK ON TRANSFERS AND

HYPERBOLICITY

M.G. Mahmoudi

Laboratoire de Mathématiques, Université de Franche-Comté, 25030 Besançon CEDEX, FRANCE

                    E-mail: mahmoudi@math.univ-fcomte.fr.

Abstract In this note, we prove that the intersection of the kernels of two transfers maps, involved in an exact sequence of Parimala, Sridharan and Suresh consists of the elements with order at most two. As an application, a new proof of a result of Scharlau stating that the torsion part of the Witt group of a central simple algebra with an involution of any kind is 2-primary is given.

1 Introduction

Let K/k be a quadratic extension of fields of characteristic other than two. Let D be a central K-division algebra with an involution τ, with the assumption that there are λ, μ ∈ D such that τ(λ) = −λ, τ(μ) = −μ, λμ = −μλ and K(λ)/K is a quadratic extension. Criteria for assuring the “existence” of the above mentioned λ and μ by tensoring in a suitable odd degree extension of k, under some assumptions, are presented in [2] (see section 2 below).

One can consider D̃, the centralizer of L = K(λ) in D. On D̃, we have two natural involutions. First τ_{1} = τ|D̃ and second τ_{2} = Int(μ^{−1})|D̃ ∘ τ_{1}, where Int(μ^{−1}) is the inner automorphism of D induced by μ^{−1}. Each element of D can be uniquely written as x + μy where x, y ∈ D̃. We have two projections:

\begin{matrix}π_{1} : D ⟶ D̃, & π_{2} : D ⟶ D̃, \\ x + μy ↦^{π_{1}} x & x + μy ↦^{π_{2}} y \end{matrix}

12000 Mathematics Subject Classification. 16K20; 11E39.

2Key words and phrases. Witt group, hermitian form, central simple algebra with involution.

These projections induce the homomorphisms of Witt groups:

\begin{matrix}π_{1}^{ε} : W^{ε}(D, τ) ⟶ W^{ε}(D̃, τ_{1}), \\ π_{2}^{ε} : W^{ε}(D, τ) ⟶ W^{−ε}(D̃, τ_{2}). \end{matrix}

These maps are involved in an exact sequence of Witt groups by Parimala-Sridharan-Suresh (cf. [2, Appendix 2]).

One may ask what are the elements of the group N = ker π_{1}^{ε} ∩ ker π_{2}^{ε}? As an example let us consider the Hamiltonian quaternions and its canonical involution. Here, we may have λ = i, μ = j, where i and j are generators of this algebra with i^{2} = j^{2} = −1 and ij = −ji = k. The one dimensional skew hermitian form < j > belongs to both ker π_{1}^{−1} and ker π_{2}^{−1}. So in general N ≠ 0. We will show that N consists of the elements with order at most two in W^{ε}(D, τ) (cf. Theorem 3.2).

Using this result we will be able to give a new proof (see 4.1) of a theorem of Scharlau ([6]) which states that the torsion of Witt group of a central simple algebra with an involution of any kind is two-primary. This result is an hermitian analogue for a classical result of Pfister in quadratic forms which states that the torsion of Witt group of a field is twoprimary. Scharlau’s proof uses some techniques from representation theory of finite groups and algebraic K-theory. In [4], Lewis and Unger complement Scharlau’s result by showing that an element of Witt group of a central simple algebra with involution is torsion if and only if its signature is zero for all orderings of the ground field.

We use the following notation and terminology: The characteristic of all fields is supposed to be different from 2. We denote hyperbolic forms in Witt groups by 0 and isometry of forms will be denoted by ≃. For a form h and an integer n, the form n × h is the orthogonal sum h⊕⋯⊕h. For a ring A, the invertible elements of A are denoted by A^{∗}. For a K-central simple algebra A, a K/k-involution σ on A is an involution where k is the fixed field of the restriction of this involution in K. For ε = ±1, we denote the Witt group of ε-hermitian forms over (A, σ) by W^{ε}(A, σ) and the semigroup of isometry classes of ε-hermitian forms by S^{ε}(A, σ). See the book of Scharlau [7] for these basic notions.

2 Extensions of odd degree

Lemma 2.1. ([2], lemma 3.1.1) Let A be a central simple algebra with a K/k-involution σ. Suppose that there exists a subfield L ⊂ A which is a quadratic extension of K stable by σ. Suppose that the restriction of σ to L is identity if σ is of first kind. Then there exists μ ∈ A^{∗} such that σ(μ) = −μ and Int(μ) restricts to the non-trivial automorphism σ_{0} of L/K.

Lemma 2.2. ([2], lemma 3.3.1) Let D be a division algebra with a K/k-involution σ of second kind. Then there exists an odd extension N of k such that D_{N} = D ⊗_{k} N has exponent a power of 2.

Lemma 2.3. ([2], lemma 3.3.2) Let E/k be an extension of 2−power degree then there exists an extension M/k of odd degree such that EM/M can be filtered by degree two extensions.

Lemma 2.4. ([2], lemma 3.3.3) Assume that D is a division algebra with a K/k-involution σ of second kind and assume that the degree of D is a power of 2 and D ≠ K. Then there exists an odd extension M/k such that D_{M} has a subfield L with [L : KM] = 2 and σ(L) = L.

Proposition 2.5. Let D be a noncommutative K-division algebra and σ a K/k-involution on D. Suppose that the degree of D is a 2 − power. Then there exists an extension M/k of odd degree such that D_{M} = D ⊗_{k} M contains the elements λ, μ such that τ(λ) = −λ and τ(μ) = −μ and λμ = −μλ and [F (λ) : F] = 2 where F = KM = K ⊗_{k} M and τ is the involution σ ⊗ id when σ is of second kind or σ is symplectic and D is a quaternion algebra otherwise τ = Int(μ) ∘ (σ ⊗ id).

Proof. Suppose that σ is of the first kind, i.e., K = k. If D is a quaternion algebra and σ symplectic, D already contains the elements μ and λ with the required properties.

Suppose that D is a K-quaternion algebra with an involution σ of orthogonal type or a division algebra with dimension greater than 4 over K. In this case there exists x ∈ D^{∗} such that σ(x) = x and x ∉ K^{∗}. Now consider the extension K(x)/K. This is an extension which its degree is a power of two so by 2.3, there exists an extension M/K of odd degree such that K(x) ⊗_{K} M/M can be filtered by quadratic extensions. Let L be a quadratic extension of M in M(x) = K(x) ⊗_{K} M. We can find λ ∈ L such that L = M(λ) and λ^{2} ∈ M. Since τ_{0} = σ ⊗ id |_{M(x)} is identity thus the restriction τ_{0}|_{L} is also identity. By 2.1 there exists μ ∈ D_{M}^{∗} such that τ_{0}(μ) = −μ and Int(μ) restricts a non-trivial automorphism σ_{0} of L/M. Put τ = Int(μ) ∘ τ_{0}, we have τ(λ) = −λ, τ(μ) = −μ and λμ = −μλ. These are the elements we were looking for.

Now suppose that σ is of second kind. Put τ = σ ⊗ id. Lemma 2.4 gives an extension M/k of odd degree such that D_{M} contains a commutative subfield L such that [L : KM] = 2 and τ(L) = L. By 2.2 we can find μ ∈ D_{M}^{∗} such that τ(μ) = −μ and Int(μ) restricts to the nontrivial automorphism σ_{0} of L/KM. It is easy to find λ ∈ L with L = KM(λ) and λ^{2} ∈ KM such that τ(λ) = −λ. Now λ and μ are the required.

3 Hyperbolicity under quadratic descent

In this section D is a division algebra with a K/k-involution τ. Assume that there exist λ, μ ∈ D^{∗} such that τ(λ) = −λ, τ(μ) = −μ, λμ = −μλ. Let L = K(λ) and D̃ be the centeralizer of L in D. We have μD̃ = D̃μ, μ^{2} ∈ D̃, τ(D̃) = D̃ and D = D̃ ⊕ μD̃. Let τ_{1} = τ|D̃ and τ_{2} = Int(μ^{−1}) ∘ τ_{1}. We have two projections:

\begin{matrix}π_{1} : D ⟶ D̃, & π_{2} : D ⟶ D̃, \\ d_{1} + μd_{2} ↦^{π_{1}} d_{1} & d_{1} + μd_{2} ↦^{π_{2}} d_{2} \end{matrix}

The maps π_{1} and π_{2} induce homomorphisms between the isometry classes of hermitian forms as well as homomorphisms between Witt groups:

\begin{matrix}π_{1}^{ε} : S^{ε}(D, τ) ⟶ S^{ε}(D̃, τ_{1}) & π_{1}^{ε} : W^{ε}(D, τ) ⟶ W^{ε}(D̃, τ_{1}) \\ π_{2}^{ε} : S^{ε}(D, τ) ⟶ S^{−ε}(D̃, τ_{2}) & π_{2}^{ε} : W^{ε}(D, τ) ⟶ W^{−ε}(D̃, τ_{2}). \end{matrix}

Proposition 3.1. With the previous notations,

(i) Assume that h ∈ S^{ε}(D, τ) is anisotropic. If π_{1}^{ε}h is hyperbolic then h ≃< μc_{1}, ⋯, μc_{n} > for suitable choices of c_{1}, ⋯, c_{n} ∈ D̃. (ii) The image of the one dimensional ε-hermitian form h =< d >∈ S^{ε}(D, τ) under π_{2}^{ε} is isometric to

( \begin{matrix}d_{2} & μ^{−1}d_{1}μ \\ −d_{1} & −μd_{2}μ \end{matrix} )

where d = d_{1} + μd_{2}, d_{1}, d_{2} ∈ D̃. (iii) The image of the one dimensional ε-hermitian form h =< d >∈ S^{ε}(D, τ) under π_{1}^{ε} is isometric to

( \begin{matrix}d_{1} & μd_{2}μ \\ −μ^{2}d_{2} & −μd_{1}μ \end{matrix} )

where d = d_{1} + μd_{2} d_{1}, d_{2} ∈ D̃.

Proof. (i) Take an isotropic vector x ≠ 0 for π_{1}^{ε}h, so π_{1}^{ε}h(x, x) = 0, therefore h(x, x) ∈ μD̃. So h(x, x) = μc for some c ∈ D̃ − {0}. This means that h contains a sub-form isometric to < μc >, by repeating this process and with an induction on dimension and using Witt decomposition, we conclude that h ≃< μc_{1}, ⋯, μc_{n} > where c_{i} ∈ D̃. For (ii) and (iii) take the D̃-basis {1, μ} of D. We have

\begin{matrix}h(1, 1) = d_{1} + μd_{2}, & h(1, μ) = d_{1}μ + μd_{2}μ, \\ h(μ, 1) = −μd_{1} − μ^{2}d_{2}, & h(μ, μ) = −μd_{1}μ − μ^{2}d_{2}μ. \end{matrix}

These relations imply that the representing matrices of π_{1}^{ε}h and π_{2}^{ε}h in this basis are those mentioned in the statement of the proposition.

Theorem 3.2. If h is an ε-hermitian form over (D, τ) such that π_{1}^{ε}h and π_{2}^{ε}h are hyperbolic, then 2 × h = 0 in W^{ε}(D, τ).

Proof. We may suppose that h is anisotropic. Since π_{1}^{ε}h = 0 by 3.1 (i) we have h ≃< μc_{1}, ⋯, μc_{n} > for some c_{i} ∈ D̃. By 3.1(ii) we obtain

\begin{aligned}π_{2}^{ε}h & ≃< c_{1}, −μc_{1}μ, ⋯, c_{n}, −μc_{n}μ > \\ & ≃< c_{1}, ⋯, c_{n} > ⊕ < −μc_{1}μ, ⋯, −μc_{n}μ > . \end{aligned}

Since π_{2}^{ε}h = 0, we conclude that < c_{1}, ⋯, c_{n} >≃< μc_{1}μ, ⋯, μc_{n}μ > over (D̃, τ_{2}). Put

B = ( \begin{matrix}μc_{1}μ & & & & 0 \\ & . & . & . \\ 0 & & & & μc_{n}μ \end{matrix} ) , C = ( \begin{matrix}c_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & c_{n} \end{matrix} ) .

There exists an invertible matrix A over D̃ such that τ_{2}(A)^{t}BA = C. On other hand we have

h ≃< μc_{1}, ⋯, μc_{n} >≃< −μ^{2}c_{1}μ, ⋯, −μ^{2}c_{n}μ >

so −μB is also a representing matrix for h over (D, τ). But we have

τ(A)^{t}(−μB)A = τ_{1}(A)^{t}(−μB)A = −μ(μ^{−1}τ_{1}(A)^{t}μBA)
= −μ(τ_{2}(A)^{t}BA)
= −μC

Thus h ≃< −μc_{1}, ⋯, −μc_{n} >. Therefore h ≃ −h, in other words 2 × h = 0.

Remark 3.3. The proof of the previous result, in a way, uses and generalizes the idea behind the proof of proposition 1.11 in Lewis’s paper [3].

Corollary 3.4. In the situation of 3.2, h is weakly hyperbolic if and only if π_{1}^{ε}h and π_{2}^{ε}h are so.

4 Torsion of Witt group

Theorem 4.1. (Scharlau) Let A be a central simple algebra with a K/k-involution τ. Then for ε = ±1 the torsion part of W^{ε}(A, τ) is 2-primary.

Proof. By Morita equivalence we may suppose that D := A is a division algebra. We prove this statement by induction on n = dim_{K} D.

If n = 1, D is commutative field. We have two cases, either τ = id or τ =¯ is the nontrivial automorphism of K/k. In the first case for ε = −1 , W^{ε}(D, τ) = W^{−1}(K) = 0 and there is nothing to prove. If ε = 1 we have W^{ε}(D, τ) = W (K) and by a classical theorem of Pfister the torsion part of W (K) is 2-primary. (See for instance [5], Theorem 3.4, Ch.2).

If τ =¯ then we have W (K,¯ ) ≃ W^{−1}(K,¯ ) so it is sufficient to prove the statement for ε = 1. Since we have an injection W (K,¯ ) ↪ W (k) so again by Pfister’s result the torsion part of W (K,¯ ) is 2-primary.

Now we may suppose that n > 1. Suppose that τ is of second kind. By 2.2, there exists an extension N/k of odd degree such that D ⊗_{k} N has exponent a power of 2. Let D_{1} be a division algebra which is Brauer equivalent to D ⊗_{k} N. We know that D_{1} admits an involution σ of the same kind, (i.e. the second kind), as τ ⊗ id on D ⊗_{k} N (cf. [7], Ch.8, Corollary 8.3). By Morita equivalence we have W^{ε}(D_{1}, σ) ≃ W^{ε}(D ⊗_{k} N, τ ⊗id). Since N/k is an odd degree extension, by a result of Bayer-Fluckiger–Lenstra (cf. [1]; Tignol’s paper [8] contains also another proof of this result) we have an injection W^{ε}(D, τ) ↪ W^{ε}(D_{1}, σ). Thus it is enough to prove the statement for D_{1}.

So we may suppose that in both cases of involution (first or second kind) the degree of D is a 2−power. (In the case of involution of first kind the degree of D is a priori a 2−power). Now by 2.5 there exists an odd degree extension M/k such that D_{M} = D ⊗_{k} M contains the elements λ and μ such that τ′(λ) = −λ, τ′(μ) = −μ, λμ = −μλ and F (λ)/F is a quadratic extension where F = K ⊗_{k} M and τ′ = τ ⊗ id when D is a quaternion algebra with a symplectic involution or when τ is of second kind otherwise τ′ = Int(μ) ∘ (τ ⊗ id). Since there exists an injection W^{ε}(D, τ) ↪ W^{ε}(D_{M}, τ ⊗ id) we obtain an injection W^{ε}(D, τ) ↪ W^{±ε}(D_{M}, τ′). Here +ε occurs for the case that D is a quaternion algebras with a symplectic involution or when τ is of second kind and −ε occurs otherwise. So in order to prove the theorem we may replace D with D_{M}.

Therefore we may suppose that there exist two elements μ and λ in D, having the properties discussed in §3. Now we can consider the projections π_{1}^{ε} and π_{2}^{ε} on W^{ε}(D, τ). Suppose that there exists an hermitian form h′ with order l = 2^{r}k where k is odd and k > 1. Then h = 2^{r} × h′ has odd order k. We may suppose that h is anisotropic. The order of π_{1}^{ε}h and π_{2}^{ε}h divide k. On the other hand by induction the orders of π_{1}^{ε}h and π_{2}^{ε}h in W^{ε}(D̃, τ_{1}) and W^{−ε}(D̃, τ_{2}) (resp.) are 2-powers, therefore π_{1}^{ε}h = 0 and π_{2}^{ε}h = 0. By 3.2 we conclude that 2 × h is hyperbolic but (k, 2) = 1 so h is hyperbolic, this is a contradiction because h is anisotropic.

ACKNOWLEDGMENTS

The author gratefully acknowledges support from the RT Network “K-theory, Linear Algebraic Groups and Related Structures” (contract HPRN-CT-2002-00287).

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