Exact Sequences of Witt Groups

N. Grenier-Boley M.G. Mahmoudi

Abstract

An exact octagon of Witt groups of central simple algebras with involution is constructed, extending an exact sequence of Parimala, Sridharan and Suresh and motivated by exact sequences obtained by Lewis. From this, we derive relations between the cardinality of certain Witt groups. An exact octagon of equivariant Witt groups is also obtained, thus generalizing a similar octagon constructed by Lewis for quaternion algebras.

1 Introduction

Base change is an important tool in the algebraic theory of quadratic forms and of hermitian forms over division algebras. For a field extension L/K (of characteristic different from 2), we can consider base change from K to L; if moreover the extension has finite degree, then we also have the Scharlau transfer. The situation is especially well understood when L/K is of odd degree or a quadratic extension: see the book of Scharlau [13] for these basic notions and results. Another classical result, due to Kneser and Springer, concerns hermitian forms over quaternion algebras (see the appendix written by Springer in [6] ).

The Witt group (and Witt ring for quadratic forms) gives a very useful way to study quadratic and hermitian forms. The above results can be expressed very efficiently in this framework. One of the basic result in the theory of quadratic forms is a theorem of Pfister which determines the kernel of the restriction map r_{L/K}^{∗} : W (K) → W (L) for a quadratic extension L/K. More precisely, this kernel is the ideal generated by the form < 1, −δ > where L = K(\sqrt{δ}). One can express this result by the exactness of the sequence:

W (K) ⟶^{t} W (K) ⟶^{r_{L/K}^{∗}} W (L)  (1)

12000 Mathematics Subject Classification. Primary 11E39; Secondary 11E81.

2Key words and phrases. Witt group, hermitian form, central simple algebra with involution, exact sequence of Witt groups, isotropy, equivariant Witt group, Morita equivalence.

3The authors gratefully acknowledge support from the RT Network ”K-theory, Linear Algebraic Groups and Related Structures” (contract HPRN-CT-2002-00287)

where t is multiplication by the 2-dimensional form < 1, −δ >. By a result of Elman-Lam, the Scharlau transfer map s_{∗} : W (L) ⟶ W (K), can be used to embed (1) in the following exact triangle (cf. [13, Ch. 2, 5.10]):

\xymatrix{W (K) \ar[rr]^{r^{∗}} &  & W (L) \ar[dl]^{s_{∗}} \\  & W (K) \ar[ul]^{t}}  (2)

For a quadratic extension L/K (resp. a quaternion division algebra (a, b)_{K} = D) with nontrivial automorphism − (resp. with canonical involution −), one can consider the trace map W (L, −) → W (K) (resp. W (D, −) → W (K)). By a result of Jacobson, these maps are injective (cf. [13, Ch. 10, 1.1, 1.2, 1.7] and [5]).

In [12, Appendix 2], Milnor and Husemoller construct the following exact sequence of W (K)- modules:

0 ⟶ W (L, −) ⟶ W (K) ⟶ W (L)  (3)

where − is the nontrivial automorphism of the quadratic extension L/K. The results concerning hermitian forms over quaternion division algebras are given in the papers of Lewis [8], [9]. He found the following exact sequence:

0 ⟶ W (D, −) ⟶ W (L, −) ⟶ W^{−1}(D, −) ⟶ W (L)  (4)

where L = K(\sqrt{a}) ⊂ D is stable by −. In fact, in this sequence, D can be also split (cf. [13, Ch. 10, 3.2]).

Remark 1.1. Note that Lewis uses W (D, ∧) instead of W^{−1}(D, −), where ∧ is the orthogonal involution of D defined by î = −i and ĵ = j where {1, i, j, ij} are usual generators of the quaternion algebra D. In fact W (D, ∧) ≃ W^{−1}(D, −): this isomorphism is induced by the map which associates to a hermitian form h over (D, ∧), the skew-hermitian form ih over (D, −).

In [2, Appendix 2], Parimala, Sridharan and Suresh obtain this crucial exact sequence of Witt groups:

W^{ε}(A, σ) ⟶^{π_{1}^{ε}} W^{ε}(Ã, σ_{1}) ⟶^{ρ_{1}^{ε}} W^{−ε}(A, σ) ⟶^{π_{2}^{−ε}} W^{ε}(Ã, σ_{2})  (5)

where A is a central simple algebra with an involution σ of any kind and ε = ±1 (for the notations, see the beginning of section 2). The exact sequence (4) is a particular case of (5). In general the map π_{1}^{ε} is not injective: in fact, 5.3 characterizes the cases where π_{1}^{ε} is injective. In [2], (5) is used to prove Serre’s conjecture II for classical groups. In [9], Lewis has found a longer exact sequence:

\begin{matrix}0 → W (D, −) → W (L, −) → W^{−1}(D, −) → W (L) → W^{−1}(D, −) → \\ → W (L, −) → W (D, −) → 0 \end{matrix}  (6)

This sequence was one of our main motivations in order to embed the exact sequence (5) in an even longer one. In fact we obtain an exact octagon:

Theorem 1.2. There is an exact sequence of Witt groups (in fact of W (K, σ|_{K})-modules):

[figure: W^{ε}(A, σ); π_{1}^{ε}; W^{ε}(Ã, σ_{1}); ρ_{1}^{ε}; W^{−ε}(Ã, σ_{2}); ρ_{2}^{−ε}; W^{−ε}(A, σ); π_{2}^{−ε}; W^{ε}(A, σ); π_{2}^{ε}; W^{ε}(Ã, σ_{2}); ρ_{2}^{ε}; W^{ε}(Ã, σ_{1}); ρ_{3}^{ε}; W^{−ε}(A, σ); π_{3}^{−ε}]

(for the definition of the maps, see section 2).

Remark 1.3. Note that the exact sequence (6) is a particular case of 1.2.

Remark 1.4. One can find a similar exact octagon of Witt groups of Clifford algebras of quadratic forms with their canonical involution in [11].

The proof of 1.2 can be found in section 5. Because of the nature of Witt groups, the above octagon does not give, a priori, much information about the behavior of these maps with respect to isotropy in the semigroups of isometry classes of nondegenerate hermitian forms: in section 3, theorem 3.4 describes this behavior in the case of division algebras. We also deduce an alternative proof of 1.2 in the case of division algebras.

Section 4 contains the relation between the notion of isotropy over a central simple algebra A and isotropy over the division algebra D Brauer equivalent to A via Morita equivalence as well as the Witt decomposition. In fact we point out that there exists a Witt decomposition over a central simple algebra with involution. In particular, this implies the existence of an anisotropic form in every class of the Witt group.

In [10], Lewis constructs an exact octagon of Witt groups of forms invariant under the action of a finite group G (Witt groups of equivariant forms) for quaternion division algebras. in section 6, we show likewise that the octagon of 1.2 is exact if we replace Witt groups by equivariant Witt groups. More precisely:

Theorem 1.5. We suppose that A satisfies the same hypotheses as at the beginning of section 2. If we replace W^{±ε}(A, σ) by W^{±ε}(G, A, σ) and W^{±ε}(Ã, σ_{i}) by W^{±ε}(G, Ã, σ_{i}) (for i = 1, 2) in theorem 1.2, we obtain an exact octagon of W (K, σ|_{K})-modules.

In section 7, by using the exact octagon, we obtain the following result:

Corollary 1.6. Let A be a K-central simple algebra with an involution σ of the first kind. Then we have |W^{ε}(A, σ)||W^{−ε}(A, σ)| = |W (K)|. In particular W (K) is finite if and only if W^{ε}(A, σ) and W^{−ε}(A, σ) are finite.

This result was well known for quaternion algebras: see [9].

We thank Eva Bayer-Fluckiger, Detlev Hoffmann and Emmanuel Lequeu for useful conversations about this work and for their comments on earlier versions of this paper.

2 Notation and Definition of the Maps

Let K be a field of characteristic different from 2. All the modules in this paper are supposed to be right modules which are finitely generated and all the ε-hermitian forms are supposed to be nondegenerate. Let A be a central simple algebra over K with an involution σ (of any kind). For ε = ±1, let S^{ε}(A, σ) denote the semigroup of isometry classes of ε-hermitian forms over (A, σ) and let W^{ε}(A, σ) be the Witt group of (A, σ) (i.e., the quotient of the Grothendieck group corresponding to S^{ε}(A, σ) by the subgroup generated by metabolic forms, an ε-hermitian form (V, h) being metabolic if there exists an A-submodule W of V such that W = W^{⊥} for h).

Remark 2.1. As A is simple, there is of course no difference between the notions of metabolic and hyperbolic hermitian forms. We use any of these two notions subsequently (except in section 6).

First, we define the different maps involved in 1.2.As in [2], we suppose that there exist λ, μ ∈ A^{∗} such that σ(λ) = −λ, σ(μ) = −μ, μλ = −λμ and such that L = K(λ) is a quadratic extension of K. We write Ã for the commutant of L in A: this is a central simple algebra over L . One can easily verify that μÃ = Ãμ, μ^{2} ∈ Ã, σ(Ã) = Ã and A = Ã ⊕ μÃ. We define two involutions on Ã in the following way: let σ_{1} = σ|Ã and let σ_{2} = Int(μ^{−1}) ∘ σ_{1} (where Int(μ^{−1})(x) = μ^{−1}xμ).

⋄ \underline{Definition of π_{1}^{ε} and π_{2}^{ε}} We have two L-linear projections: π_{1} : A → Ã: a_{1} + μa_{2} ↦ a_{1} and π_{2} : A → Ã: a_{1} + μa_{2} ↦ a_{2}. If h : V ×V → A is an ε-hermitian space over (A, σ), we define (for i = 1, 2) π_{i}^{ε}(h) : V ×V → Ã by π_{i}^{ε}(h)(x, y) = π_{i}(h(x, y)). One readily verifies that π_{1}^{ε}(h) is an ε-hermitian space over (Ã, σ_{1}) and that π_{2}^{ε}(h) is a −ε-hermitian space over (Ã, σ_{2}). In order to see that π_{1}^{ε} and π_{2}^{ε} induce homomorphisms of Witt groups we have to prove that these maps respect regularity, isometry classes, orthogonality and hyperbolicity. All of these properties come from the fact that:

h(x, y) = 0 ∀y ⟺ π_{1}^{ε}(h)(x, y) = 0 ∀y ⟺ π_{2}^{ε}(h)(x, y) = 0 ∀y

Hence, π_{1}^{ε} and π_{2}^{ε} induce homomorphisms of semigroups of isometry classes of nondegenerate hermitian forms and homomorphisms of Witt groups (again denoted by π_{1}^{ε} and π_{2}^{ε})

\begin{matrix}π_{1}^{ε} : S^{ε}(A, σ) → S^{ε}(Ã, σ_{1}); & π_{1}^{ε} : W^{ε}(A, σ) → W^{ε}(Ã, σ_{1}) \\ π_{2}^{ε} : S^{ε}(A, σ) → S^{−ε}(Ã, σ_{2}); & π_{2}^{ε} : W^{ε}(A, σ) → W^{−ε}(Ã, σ_{2}) \end{matrix}

⋄ \underline{Definition of ρ_{1}^{ε}} Let (V, f) be an ε-hermitian space over (Ã, σ_{1}). We associate to it (V ⊗Ã A, ρ_{1}^{ε}(f)) where ρ_{1}^{ε}(f)(x ⊗ α, y ⊗ β) = σ(α)λf(x, y)β for x, y ∈ V and α, β ∈ A. We can easily verify that ρ_{1}^{ε} is well defined and that (V ⊗Ã A, ρ_{1}^{ε}(f)) is a −ε-hermitian space over (A, σ). Moreover ρ_{1}^{ε} induces homomorphisms:

ρ_{1}^{ε} : S^{ε}(Ã, σ_{1}) → S^{−ε}(A, σ); ρ_{1}^{ε} : W^{ε}(Ã, σ_{1}) → W^{−ε}(A, σ)

⋄ \underline{Definition of ρ_{2}^{ε}} Let (V, f) be an ε-hermitian space over (Ã, σ_{2}). We associate to it (V ⊗Ã A, ρ_{2}^{ε}(f)) where ρ_{2}^{ε}(f)(x ⊗ α, y ⊗ β) = σ(α)λμf(x, y)β for x, y ∈ V and α, β ∈ A. One can verify that ρ_{2}^{ε} is well defined, that (V ⊗Ã A, ρ_{2}^{ε}(f)) is a −ε-hermitian space over (A, σ) and that ρ_{2}^{ε} induces homomorphism:

ρ_{2}^{ε} : S^{ε}(Ã, σ_{2}) → S^{−ε}(A, σ); ρ_{2}^{ε} : W^{ε}(Ã, σ_{2}) → W^{−ε}(A, σ)

⋄ \underline{Definition of π_{3}^{ε}} We define π_{3}^{ε} to be λπ_{1}^{ε}, so we obtain homomorphisms:

π_{3}^{ε} : S^{ε}(A, σ) → S^{−ε}(Ã, σ_{1}); π_{3}^{ε} : W^{ε}(A, σ) → W^{−ε}(Ã, σ_{1})

⋄ \underline{Definition of ρ_{3}^{ε}} We define: ρ_{3}^{ε}(f) = ρ_{1}^{−ε}(λ^{−1}f), i.e., ρ_{3}^{ε}(f)(x⊗α, y⊗β) = σ(α)f(x, y)β for x, y ∈ V and α, β ∈ A. By a straightforward verification we obtain homomorphisms:

ρ_{3}^{ε} : S^{ε}(Ã, σ_{1}) → S^{ε}(A, σ); ρ_{3}^{ε} : W^{ε}(Ã, σ_{1}) → W^{ε}(A, σ)

Remark 2.2. Note that, in these definitions, ε is arbitrary so for example π_{1}^{−ε} will be a homomorphism of Witt groups from W^{−ε}(A, σ) to W^{−ε}(Ã, σ_{1}) and so on for the other maps.

A summary of these definitions can be found in table 8.1.

3 The Behavior of the Maps for Division Algebras

In this section (D, τ) is a division algebra with an involution τ of any kind. We assume that (D, D̃, τ, τ_{1}, τ_{2}) satisfies the same hypotheses as the ones mentioned in section 2 for (A, Ã, σ, σ_{1}, σ_{2}).

Proposition 3.1. (i) If h =< δ > is a one dimensional ε-hermitian form over (D, τ) (with δ = d_{1} + μd_{2}, d_{1}, d_{2} ∈ D̃) then the matrix of π_{1}^{ε}(h) over (D̃, τ_{1}) with respect to the basis {1, μ} is

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

(ii) If P is the matrix of an ε-hermitian form f over (D̃, τ_{1}) with respect to a basis B, then λP is the matrix of ρ_{1}^{ε}(f) over (D, τ) with respect to the basis B ⊗ 1. (iii) If h =< δ > is a one dimensional −ε-hermitian form over (D, τ) (with δ = d_{1} + μd_{2}, d_{1}, d_{2} ∈ D̃) then the matrix of π_{2}^{−ε}(h) over (D̃, τ_{2}) with respect to the basis {1, μ} is

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

(iv) If P is the matrix of an ε-hermitian form f over (D̃, τ_{2}) with respect to a basis B, then λμP is the matrix of ρ_{2}^{ε}(f) over (D, τ) with respect to the basis B ⊗ 1.

(v) For a −ε-hermitian form h over (D, τ), the matrix of π_{3}^{−ε}(h) with respect to a basis B is λ times the matrix of π_{1}^{ε}(h) with respect to the basis B ∪ μB. (vi) If P is the matrix of an ε-hermitian form f over (D̃, τ_{1}) with respect to a basis B, then P is the matrix of ρ_{3}^{ε}(f) over (D, τ) with respect to the basis B ⊗ 1.

Proof. (i) We have : 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} and h(μ, μ) = −μd_{1}μ − μ^{2}d_{2}μ. As d_{1}, d_{2} ∈ D̃, we can find the given matrix by applying π_{1}^{ε} to these relations. (ii) If (V, f) is an ε-hermitian space over (D̃, τ_{1}) and if {e_{1}, …, e_{n}} is a basis of V over D̃ then {e_{1} ⊗ 1, …, e_{n} ⊗ 1} is a basis of V ⊗D̃ D over D. With the definition of ρ_{1}^{ε}, we have: ρ_{1}^{ε}(f)(e_{i} ⊗ 1, e_{j} ⊗ 1) = λf(e_{i}, e_{j}) ∀ i, j = 1, …, n. (iii) By applying π_{2}^{−ε} to the four relations found in (i), we deduce the given matrix for π_{2}^{−ε}(h). (iv) If (V, f) is an ε-hermitian space over (D̃, τ_{2}) and if {e_{1}, …, e_{n}} is a basis of V over D̃ then {e_{1} ⊗ 1, …, e_{n} ⊗ 1} is a basis of V ⊗D̃ D over D. With the definition of ρ_{2}^{ε}, we have: ρ_{2}^{ε}(f)(e_{i} ⊗ 1, e_{j} ⊗ 1) = λμf(e_{i}, e_{j}) ∀ i, j = 1, …, n.

(v) This is obvious by the definition of π_{3}^{−ε}. (vi) If (V, f) is an ε-hermitian space over (D̃, τ_{1}) and if {e_{1}, …, e_{n}} is a basis of V over D̃ then {e_{1} ⊗ 1, …, e_{n} ⊗ 1} is a basis of V ⊗D̃ D over D. By the definition of ρ_{3}^{ε}, we have: ρ_{3}^{ε}(f)(e_{i} ⊗ 1, e_{j} ⊗ 1) = f(e_{i}, e_{j}) ∀ i, j = 1, …, n. □

The summary of the previous proposition can be found in table 8.2.

Lemma 3.2. If D̃ is commutative then D is a quaternion division algebra, say D = (a, b)_{K}, D̃ = L = K(λ) with λ^{2} = a ∈ K and μ^{2} = b ∈ K.

Proof. D̃ is the commutant of L in D, so it is a central simple algebra of center L. As D̃ is commutative, we have D̃ = L and the lemma readily follows. □

Lemma 3.3. If E is a noncommutative central simple algebra and ⋆ an involution over E, then for ε = ±1, the set of nonzero ε-hermitian elements with respect to ⋆ is nonempty.

Proof. This is a consequence of [13, Ch. 8, 7.5]. □

The following theorem determines completely the behavior of the maps π_{1}, ρ_{1}, π_{2}, ρ_{2}, π_{3} and ρ_{3} with respect to isotropy.

Theorem 3.4. (i) Let h ∈ S^{ε}(D, τ). If D̃ is commutative and ε = 1, then π_{1}^{ε}(h) ∈ S^{ε}(D̃, τ_{1}) is isotropic if and only if h is isotropic. Otherwise, π_{1}^{ε}(h) is isotropic if and only if there exists c ∈ D̃^{∗} with τ_{2}(c) = −εc such that h contains a subform isometric to < μc >. (ii) Suppose that f ∈ S^{ε}(D̃, τ_{1}). If D̃ is commutative, ε = 1 and dim(f) = 2, then ρ_{1}^{ε}(f) ∈ S^{−ε}(D, τ) is isotropic if and only if f is isotropic or f is isometric to the two dimensional anisotropic form < c, −bc > where c ∈ K^{∗} and b = μ^{2} (see 3.2). Otherwise ρ_{1}^{ε}(f) is isotropic if and only if there exist d_{1}, d_{2} ∈ D̃ such that τ_{1}(d_{1}) = εd_{1}, τ_{2}(d_{2}) = −εd_{2} and f = f_{1}⊥f_{2} for some nondegenerate

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

(iii) For h ∈ S^{−ε}(D, τ), π_{2}^{−ε}(h) ∈ S^{ε}(D̃, τ_{2}) is isotropic if and only if there exists c ∈ D̃^{∗} with τ_{1}(c) = −εc such that h contains a subform isometric to < c >. (iv) For f ∈ S^{ε}(D̃, τ_{2}), ρ_{2}^{ε}(f) ∈ S^{−ε}(D, τ) is isotropic if and only if there exist d_{1}, d_{2} ∈ D̃ such that τ_{1}(d_{1}) = −εd_{1}, τ_{2}(d_{2}) = εd_{2} and f = f_{1}⊥f_{2} for some nondegenerate

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

(v) Let h ∈ S^{−ε}(D, τ). If D̃ is commutative and ε = −1, then π_{3}^{−ε}(h) ∈ S^{ε}(D̃, τ_{1}) is isotropic if and only if h is isotropic. Otherwise, π_{3}^{−ε}(h) is isotropic if and only if there exists c ∈ D̃^{∗} with τ_{2}(c) = εc such that h contains a subform isometric to < μc >. (vi) Suppose that f ∈ S^{ε}(D̃, τ_{1}). If D̃ is commutative, ε = −1 and dim(f) = 2, then ρ_{3}^{ε}(f) ∈ S^{ε}(D, τ) is isotropic if and only if f is isotropic or f is isometric to the two dimensional anisotropic form < λc, −λbc > where c ∈ K^{∗} and b = μ^{2} (see 3.2). Otherwise ρ_{3}^{ε}(f) is isotropic if and only if there exist d_{1}, d_{2} ∈ D̃ such that τ_{1}(d_{1}) = −εd_{1}, τ_{2}(d_{2}) = εd_{2} and f = f_{1}⊥f_{2} for some nondegenerate

f_{1} ≃ ( \begin{matrix}λd_{1} & λμd_{2}μ \\ −λμ^{2}d_{2} & −λμd_{1}μ \end{matrix} ) .

Proof. Let IH_{ε} denote an ε-hyperbolic plane.

(i) First, suppose that D̃ is commutative and ε = 1. By 3.2, we know that D is a quaternion division algebra. The equivalence comes from the fact that, in this case, the trace form of h is isotropic if and only if h is isotropic.

Next, suppose that the previous case is excluded. If h is anisotropic and π_{1}^{ε}(h) is isotropic then we can find x ∈ V − {0} such that π_{1}^{ε}(h(x, x)) = 0, i.e., h(x, x) = μc with c ∈ D̃^{∗}. As τ(h(x, x)) = εh(x, x), we conclude that τ_{2}(c) = −εc. It readily follows that h contains < μc >. Now consider the case where h is isotropic. We have IH_{ε} ≃< μc, −μc > for all c ∈ D̃^{∗} with τ_{2}(c) = −εc, provided such a c exists. If ε = −1 then we can take c = 1. If ε = 1 then we only have to show that there exists c ∈ D̃ such that τ_{2}(c) = −c. By assumption D̃ is noncommutative, and one can apply 3.3 to conclude. Conversely, suppose that h contains a subform isometric to < μc > as in the assertion. By applying 3.1 (i) to the form < μc >, we easily deduce that π_{1}^{ε}(h) is isotropic. (ii) First suppose that D̃ is commutative, ε = 1 and dim(f) = 2. We are in the situation of 3.2. If f is isotropic or f ≃< c, −bc >, it is obvious that ρ_{1}^{ε}(f) is isotropic. Conversely suppose that for a two dimensional form f =< c, d >, ρ_{1}^{ε}(f) =< λc, λd > is isotropic. So there exists q ∈ D^{∗} such that

τ(q)λcq + λd = 0.  (7)

Write q = z_{1} + μz_{2} with z_{1}, z_{2} ∈ D̃. By replacing q with z_{1} + μz_{2} in (7), by using the fact that {1, μ} is an L-basis of D and by remarking that τ_{2} = id_{L}, we obtain the following system:

\{ \begin{matrix}τ_{1}(z_{1})cz_{1} + τ_{1}(z_{2})bcz_{2} + d = 0 \\ z_{1}z_{2} = 0 \end{matrix}

If z_{2} = 0, then τ_{1}(z_{1})cz_{1} + d = 0; this means that f is isotropic. If z_{1} = 0 then d = −τ_{1}(z_{2})bcz_{2}, so f ≃< c, −τ_{1}(z_{2})bcz_{2} >≃< c, −bc >.

Next, suppose that the previous case is excluded. If f is anisotropic and ρ_{1}^{ε}(f) is isotropic, let z = x_{1} ⊗1+y_{1} ⊗μ be a nonzero isotropic vector for ρ_{1}^{ε}(f). By the definition of ρ_{1}^{ε}, we obtain (λf(x_{1}, x_{1}) − μλf(y_{1}, y_{1})μ) + (λf(x_{1}, y_{1})μ − μλf(y_{1}, x_{1})) = 0 and so we obtain the following system:

\{ \begin{matrix}f(x_{1}, x_{1}) + μf(y_{1}, y_{1})μ = 0 \\ μf(y_{1}, x_{1}) + f(x_{1}, y_{1})μ = 0 \end{matrix}  (8)

As f is anisotropic, thanks to this system, we can suppose that both x_{1} and y_{1} are nonzero. Moreover, x_{1} and y_{1} are linearly independent over D̃. In fact, if x_{1} and y_{1} are linearly dependent then x_{1} = y_{1}d with d ∈ D̃^{∗} and by replacing x_{1} with y_{1}d in (8), we obtain the following system:

\{ \begin{matrix}τ(d)f(y_{1}, y_{1})d + μf(y_{1}, y_{1})μ = 0 \\ μf(y_{1}, y_{1})d + τ(d)f(y_{1}, y_{1})μ = 0 \end{matrix}  (9)

From the second equation of (9), we obtain that τ(d)f(y_{1}, y_{1}) = −μf(y_{1}, y_{1})dμ^{−1}. By replacing τ(d)f(y_{1}, y_{1}) by −μf(y_{1}, y_{1})dμ^{−1} in the first equation of (9), we obtain μf(y_{1}, y_{1})(−dμ^{−1}d+μ) = 0. As the second factor is non zero for all d ∈ D̃, f(y_{1}, y_{1}) = 0 which is a contradiction with the anisotropy of f. Now, y_{1} and x_{1} span a two dimensional subspace W over D̃ and if we denote d_{1} = f(y_{1}, y_{1}), d_{2} = μ^{−1}f(y_{1}, x_{1})μ^{−1}, the matrix M of f|_{W} with respect to the basis {y_{1}, x_{1}} is exactly the one given in the proposition. As f_{1} = f|_{W} is nondegenerate (since f is anisotropic), we can write f = f_{1}⊥f_{2} so f contains the given form. Now consider the case where f is isotropic. If D̃ is noncommutative, we take d_{1} = 0 and we can find d_{2} ∈ D̃ such that τ_{2}(d_{2}) = −εd_{2} and it is obvious that

IH_{ε} ≃ ( \begin{matrix}0 & μd_{2}μ \\ −μ^{2}d_{2} & 0 \end{matrix} ) ,

so f contains the given form. If D̃ is commutative and ε = −1, we take d_{1} = 0 and d_{2} = 1 and IH_{−1} is isometric to the matrix given in the proposition. If D̃ is commutative, ε = 1 and dim(f) ≥ 3 then f ≃ IH_{1}⊥f_{1} with dim(f_{1}) ≥ 1. If f_{1} ≃< a, ... > then f ≃< μaμ, −μaμ, a, ... > so, for d_{1} = a, d_{2} = 0, f contains the given form. Conversely, with the same notations as (ii), 3.1 (ii) and a straightforward calculation show that (μ, 1) is an isotropic vector for ρ_{1}^{ε}(f_{1}). (iii) If h is anisotropic and π_{2}^{−ε}(h) is isotropic then we can find x ∈ V − {0} such that π_{2}^{−ε}(h(x, x)) = 0, that is h(x, x) = c ∈ D̃^{∗}. We conclude as in (i). If h is isotropic and ε = 1 then h ≃< λ, −λ > ⊥h_{1} and we can take c = λ. If h is isotropic and ε = −1, h ≃ IH_{1}⊥h_{1} and all we have to do is to find c ∈ D̃^{∗} such that τ_{1}(c) = c; we can take c = 1.

The converse is an easy consequence of 3.1 (iii). (iv) If f is anisotropic and ρ_{2}^{ε}(f) is isotropic, let z = x_{1} ⊗ 1 + y_{1} ⊗ μ be an isotropic vector for ρ_{2}^{ε}(f). With a straightforward computation we find the same system as in the proof of (ii). Proceeding in the same way, we can suppose that x_{1} and y_{1} are nonzero and span a two dimensional subspace W over D̃. If d_{1} = −f(x_{1}, y_{1}) and d_{2} = f(y_{1}, y_{1}) then the matrix of f|_{W} with respect to the basis {y_{1}, x_{1}} is exactly the one given in the proposition. Now consider the case where f is isotropic. If ε = 1, we take d_{2} = 0 and d_{1} = λ. If ε = −1, we take d_{2} = 0 and d_{1} = 1.

Conversely, (μ, 1) is an isotropic vector for ρ_{2}^{ε}(f_{1}).

(v) If D̃ is commutative and ε = −1 then the equivalence between π_{3}^{−ε}(h) being isotropic and h being isotropic readily comes from (i).

Next, we suppose that the previous case is excluded. If h is anisotropic and π_{3}^{−ε}(h) is isotropic we can conclude as in (i) and (iii). If h is isotropic then h ≃< μc, −μc > ⊥h_{1} for all c ∈ D̃^{∗} such that τ_{2}(c) = εc and we only have to find such a c. If D̃ is noncommutative this is clear. If D̃ is commutative then τ_{2} = id_{L}. As ε = 1 we can take c = 1. Conversely, we only have to apply 3.1 (v). (vi) First, suppose that D̃ is commutative, ε = −1 and dim(f) = 2 then we are in the situation of 3.2. The proof goes as in (ii). We leave it to the reader.

Next suppose that the previous case is excluded. If f is anisotropic and ρ_{3}^{ε}(f) is isotropic, let z = x_{1} ⊗ 1 + y_{1} ⊗ μ be an isotropic vector for ρ_{3}^{ε}(f). We have the following system:

\{ \begin{matrix}f(x_{1}, x_{1}) − μf(y_{1}, y_{1})μ = 0 \\ f(x_{1}, y_{1})μ − μf(y_{1}, x_{1}) = 0 \end{matrix}

Now, let d_{1} = λ^{−1}f(y_{1}, y_{1}) and d_{2} = μ^{−1}λ^{−1}f(y_{1}, x_{1})μ^{−1}. Let W be the two dimensional D̃- subspace generated by x_{1} and y_{1} (the proof of the fact that x_{1} and y_{1} are linearly independent over D̃ is similar to (ii)). The matrix of f|_{W} with respect to the basis {y_{1}, x_{1}} is exactly the one given in the proposition (the form f|_{W} is nondegenerate because f is anisotropic). Now consider the case where h is isotropic. If D̃ is noncommutative, we take d_{1} = 0 and d_{2} ∈ D̃ such that τ_{2}(d_{2}) = εd_{2}. If D̃ is commutative and ε = 1, f ≃ IH_{1}⊥f_{1} and we take d_{2} = 1 and d_{1} = 0. If D̃ is commutative, ε = −1 and dim(f) ≥ 3, we conclude as in (ii). Conversely, (μ, 1) is an isotropic vector for ρ_{3}^{ε}(f_{1}). □

In particular, if f and h are anisotropic, we obtain table 8.3. From 3.1 and 3.4, we obtain the following result:

Corollary 3.5. The octagon of 1.2 is exact when A is a division algebra.

4 Isotropy of Hermitian Forms over Central Simple Algebras

The goal of this section is to prove that there exists a Witt decomposition for nondegenerate ε-hermitian forms over a central simple algebra with involution (A, σ), i.e., for any form h, h ≃ h_{1}⊥h_{2} with h_{1} hyperbolic and h_{2} anisotropic (a notion that we have to define) unique up to isometry.

The notion of isotropy that we are going to use is the following:

Definition 4.1. A nondegenerate ε-hermitian space (V, h) over (A, σ) is said to be isotropic if there exists x ∈ V − {0} such that h(x, x) = 0.

Remark 4.2. In the case where A is a division algebra, this notion is exactly the one usually given in the literature.

Now, we recall a crucial notion: Morita-equivalence. The following definition is based upon [4].

Definition 4.3. Let (A, σ) and (B, τ) be two central simple algebras over K such that σ and τ are two K/k-involutions. Let δ = 1 if σ and τ are of the second kind or of the first kind and of the same type and δ = −1 if σ and τ are of the first kind and of different type. A δ-Morita equivalence ((A, σ), (B, τ), M, N, f, g, ν) between the algebra with involutions (A, σ) and (B, τ) is a tuple consisting of :

• an (A, B)-bimodule M (i.e., a left A-module and a right B-module with compatible structures);

• a (B, A)-bimodule N;

• two nonzero bimodule homomorphisms f : M ⊗_{B} N → A and g : N ⊗_{A} M → B which are associative, i.e., f(m ⊗ n).m′ = m.g(n ⊗ m′) et g(n ⊗ m).n′ = n.f(m ⊗ n′) for all m, m′ ∈ M, n, n′ ∈ N;

• a linear bijective map ν : M → N which verifies ν(amb) = τ(b)ν(m)σ(a) for all a ∈ A, m ∈ M, b ∈ B.

Remark 4.4. Note that we do not suppose that σ and τ are of the same type as in [4]:that is why we call this notion δ-Morita equivalence.

Remark 4.5. In fact one can prove that f (resp. g) is a bimodule isomorphism between M ⊗_{B} N and A (resp. N ⊗_{A} M and B), see [4, §1.1].

Now, we suppose that B = D denotes the division algebra Brauer equivalent to A. By Albert’s theorem, we know that there exists an involution τ over D such that τ is of the same kind as σ. By [4, §1.4], one can find M, N, f, g and ν as in 4.3 and δ ∈ {±1} such that ((A, σ), (D, τ), M, N, f, g, ν) is a δ-Morita equivalence. We can define semigroup homomorphisms:

F : S^{ε}(A, σ) → S^{δε}(D, τ); (V, h) ↦ (V ⊗_{A} M, b_{0}h)

G : S^{δε}(D, τ) → S^{ε}(A, σ); (W, ϕ) ↦ (W ⊗_{D} N, b0^{′}ϕ)

where:

\begin{aligned}(b_{0}h)(v ⊗ m, v′ ⊗ m′) & = & g(ν(m) ⊗ h(v, v′)m′) & ∀v, v′ ∈ V, m, m′ ∈ M \\ (b′_{0}ϕ)(w ⊗ n, w′ ⊗ n′) & = & f(ν^{−1}(n) ⊗ ϕ(w, w′)n′) & ∀w, w′ ∈ W, n, n′ ∈ N. \end{aligned}

In fact, we can prove that F is a semigroup isomorphism and G is its inverse and that they induce isomorphisms of Witt groups. The details of proofs can be found in [7, I.9, 3.5].

Remark 4.6. From now on, when we will use an argument involving Morita theory, we will implicitly refer to [7, I.9].

We have:

Lemma 4.7. (i) M is a simple left A-module and N is a simple right A-module. (ii) The maps F and G respect the rank of hermitian spaces (recall that the rank of a hermitian space (V, h) over (A, σ), where V is a right (resp. left) A-module, is defined to be the positive integer n such that V ≃ T^{n} where T is a simple right (resp. left) A-module).

Proof. (i) We prove it for N, the proof for M being similar. As A is a simple algebra, N is a semisimple right A-module and we can write N ≃ T^{n} where n ∈ IN − {0} and T is a simple right A-module. But, we know from 4.5 that N ⊗_{A} M ≃ D as (D, D)-bimodules so:

D ≃^{g^{−1}} N ⊗_{A} M ≃ (T ⊗_{A} M)^{n}

as right D-vector spaces. A dimension argument shows that n = 1 and N is simple as a right A-module. (ii) We prove it for F, the proof for G is similar. If (V, h) is a right A-module of rank n then we have V ≃ T^{n} where T is a simple right A-module; by (i), we can take T = N. As D is a division algebra, D is a simple right D-module and we have:

V ⊗_{A} M ≃ (N ⊗_{A} M)^{n} ≃^{g} D^{n}.

So we deduce that the rank of V ⊗_{A} M is n. □

Now, we can prove (for fixed A, σ and D as before):

Proposition 4.8. (V, h) is isotropic over (A, σ) if and only if for every δ-Morita equivalence ((A, σ), (D, τ), M, N, f, g, ν), F (V, h) is isotropic over (D, τ).

Proof. Let x ∈ V − {0} be such that h(x, x) = 0 and ((A, σ), (D, τ), M, N, f, g, ν) be a δ-Morita equivalence. We can easily see that there exists m ∈ M such that x ⊗ m ≠ 0; in fact if x ⊗ m = 0 for all m ∈ M then we have V_{1} ⊗_{A} M = 0 where V_{1} = xA ≠ 0. Now N is a simple right A-module by 4.7, so we have V_{1} = ⊕_{i=1}^{d}N, d ⩾ 1. Therefore we conclude that 0 = V_{1} ⊗_{A} M ≃ ⊕_{i=1}^{d}(N ⊗_{A} M) ≃^{g} D^{d} which is a contradiction. Now x⊗m is clearly an isotropic vector for b_{0}h so F (V, h) is isotropic. If F (V, h) is isotropic and if ((A, σ), (D, τ), M, N, f, g, ν) is a δ-Morita equivalence, let y ≠ 0 be an isotropic vector for b_{0}h. By the same argument as before one can find n ∈ N such that y ⊗ n ≠ 0. Using the definition of F^{−1}(= G) we see that y ⊗ n is an isotropic vector for b′_{0}b_{0}h for all n ∈ N. But (V ⊗_{A} M ⊗_{D} N, b′_{0}b_{0}h) is isometric to (V, h) so we can conclude the existence of an x ∈ V − {0} such that h(x, x) = 0. □

Using this proposition and the fact that the Witt decomposition exists over (D, τ), we conclude the existence of a Witt decomposition over (A, σ). Namely, if (V, h) is an ε-hermitian space over (A, σ), then F (V, h) ≃ ϕ_{1}⊥ϕ_{2} where ϕ_{1} is hyperbolic and ϕ_{2} anisotropic over (D, τ). We have :

(V, h) ≃ (G ∘ F)(V, h) ≃ G(ϕ_{1})⊥G(ϕ_{2})

By the previous proposition, G(ϕ_{1}) is hyperbolic and we can show that G(ϕ_{2}) is anisotropic. By the same type of argument, we can show that this decomposition is unique up to isometry because it is the case over (D, τ). So:

Corollary 4.9. (i) There exists a Witt decomposition over (A, σ). (ii) For all [h] ∈ W^{ε}(A, σ), there exists an anisotropic form h_{0} over (A, σ) such that [h] = [h_{0}] in W^{ε}(A, σ).

Remark 4.10. This result should be well known, but we could not find an explicit statement of it in the literature.

5 Exactness of the Octagon for Central Simple Algebras

Let (A, σ) be a central simple algebra over K (char K ≠ 2) with involution. We make the same hypotheses as in 2: namely we suppose that there exist λ, μ ∈ A^{∗} such that σ(λ) = −λ, σ(μ) = −μ, μλ = −λμ and that L = K(λ) is a quadratic extension of K. We write Ã for the commutant of L in A. We have μÃ = Ãμ, μ^{2} ∈ Ã, σ(Ã) = Ã and A = Ã ⊕ μÃ. We define two involutions on Ã in this way: let σ_{1} = σ|Ã and let σ_{2} = Int(μ^{−1}) ∘ σ_{1}.

Proof of theorem 1.2: thanks to [2, Appendix 2], we already know that im π_{1}^{ε} = ker ρ_{1}^{ε} and im ρ_{1}^{ε} = ker π_{2}^{−ε}. First, we prove that this sequence is a complex from π_{2}^{−ε} up to ρ_{3}^{ε}.

ρ_{2}^{ε}π_{2}^{−ε} = 0 : let (V, h) be a (−ε)-hermitian space over (A, σ), so π_{2}^{−ε}(h) is an ε-hermitian form over (Ã, σ_{2}) and ρ_{2}^{ε}π_{2}^{−ε}(h) will be a (−ε)-hermitian form over (A, σ). It is enough to find a self orthogonal right A-submodule of V ⊗Ã A with respect to ρ_{2}^{ε}π_{2}^{−ε}(h). Let

W = {x.μ ⊗ 1 + x ⊗ μ| x ∈ V }.  (10)

Now, W is readily seen to be a right A-submodule of VÃ ⊗Ã A and an easy calculation shows that this space is a totally isotropic subspace of ρ_{2}^{ε}π_{2}^{−ε}(h). By dimension count over K, we have W = W^{⊥} (with respect to ρ_{2}^{ε}π_{2}^{−ε}(h)) and so ρ_{2}^{ε}π_{2}^{−ε}(h) is hyperbolic.

π_{3}^{−ε}ρ_{2}^{ε} = 0 : let (V, h) be an ε-hermitian space over (Ã, σ_{2}). Let

W′ = {x ⊗ 1| x ∈ V } ⊂ V ⊗Ã A.  (11)

Then W′ is a Ã-submodule of V_{A} ⊗Ã A and it is a totally isotropic subspace for π_{3}^{−ε}ρ_{2}^{ε}(h). By a dimension argument one has W′ = W′^{⊥} and so π_{3}^{−ε}ρ_{2}^{ε}(h) is hyperbolic.

ρ_{3}^{ε}π_{3}^{−ε} = 0 : it is obvious from the definition that ρ_{3}^{ε}π_{3}^{−ε} = 0. Next we prove that the sequence is exact from π_{2}^{−ε} up to π_{1}^{ε}.

ker(ρ_{2}^{ε}) ⊂ im(π_{2}^{−ε}) : let (W, f) be an ε-hermitian form over (Ã, σ_{2}) such that ρ_{2}^{ε}(f) is hyperbolic. We may assume that f is anisotropic thanks to 4.9. There exists an A-submodule W_{1} of W ⊗Ã A such that W_{1}^{⊥} = W_{1} (with respect to ρ_{2}^{ε}(f)). Let W ⊗ μ = {w ⊗ μ|w ∈ W }. Let w_{1} ∈ W_{1} ∩ (W ⊗ μ) (w_{1} = w ⊗ μ with w_{1} ∈ W_{1} and w ∈ W). As ρ_{2}^{ε}(f)(w_{1}, w_{1}) = 0, f(w, w) = 0 and so w_{1} = w⊗μ = 0 since f is anisotropic. Moreover dim_{K} W_{1} = \frac{1}{2} dim_{K} W ⊗ÃA = dim_{K} W ⊗ μ so W ⊗Ã A = W_{1} ⊕ (W ⊗ μ) as Ã-modules. This implies that for all w ∈ W, there exists w′ ∈ W such that w⊗1+w′⊗μ ∈ W_{1}. Since A is a free Ã-module, w′ is uniquely determined by w and we write J(w) := w′. By definition of J, we have J^{2}(w) = wμ^{−2} and J(wa) = J(w)μaμ^{−1} for all w ∈ W and a ∈ Ã. As W_{1} = W_{1}^{⊥}, ρ_{2}^{ε}(f)(x ⊗ 1 + J(x) ⊗ μ, y ⊗ 1 + J(y) ⊗ μ) = 0 for all x, y ∈ W and we obtain the following system:

\{ \begin{matrix}f(x, y) + μf(J(x), J(y))μ = 0 \\ f(x, J(y))μ + μf(J(x), y) = 0 \end{matrix}  (12)

By means of J, we define an A-module structure over W by

w.μ = J(w)μ^{2}

for all w ∈ W. We denote by W_{J} the A-module W equipped with this new action. Let h be the map defined by

h(x, y) = μf(x, J(y))μ + μf(x, y)  (13)

for all x, y ∈ W_{J}. By the definition of J and (12), we conclude that (W_{J}, h) is a (−ε)-hermitian space over (A, σ). Let us show that h is sesquilinear on the left with respect to σ: h is clearly biadditive, so it suffices to show this fact for μ and for elements of Ã. We have:

h(x.μ, y) = h(J(x)μ^{2}, y)
= μf(J(x)μ^{2}, J(y))μ + μf(J(x)μ^{2}, y)
= μ^{3}f(J(x), J(y))μ + μ^{3}f(J(x), y)
= −μ^{2}f(x, y) − μ^{2}f(x, J(y))μ (using (12))
= σ(μ)h(x, y)

Sesquilinearity on the left for elements of Ã and linearity on the right are done in the same way. Let us prove that h is −ε-hermitian with respect to σ:

h(y, x) = μf(y, J(x))μ + μf(y, x)
= εσ(f(J(x), y))μ^{2} + εσ(f(x, y))μ
= −εσ(μ^{−1}f(x, J(y))μ)μ^{2} − εσ(μf(x, y)) (using (12))
= −εσ(h(x, y))

If h is degenerate, by (13), there exists a non zero x ∈ W such that f(x, y) = 0 for all y ∈ W. In particular, this implies that f is isotropic which is a contradiction. Now, (W_{J}, h) is the antecedent of (W, f) by π_{2}^{−ε}, i.e. ((W_{J})Ã, π_{2}^{−ε}(h)) is isometric to (W, f) (the isometry is given by the identity map). We conclude that ker(ρ_{2}^{ε}) ⊂ im(π_{2}^{−ε}).

ker(π_{3}^{−ε}) ⊂ im(ρ_{2}^{ε}): let (V, h) be a (−ε)-hermitian space over (A, σ) such that π_{3}^{−ε}(h) is hyperbolic. We can assume that h is anisotropic thanks to 4.9. There exists an Ã-submodule W of VÃ such that W^{⊥} = W (with respect to π_{3}^{−ε}(h)). This implies that h(x, y) ∈ μÃ, for all x, y ∈ W. We define a map f : W × W → Ã where:

f(x, y) = μ^{−1}λ^{−1}h(x, y)  (14)

for x, y ∈ W. Since h is anisotropic, f is nondegenerate, we can easily see that (W, f) is an ε-hermitian form over (Ã, σ_{2}) and (W ⊗Ã A, ρ_{2}^{ε}(f)) is isometric to (V, h) via Φ(w ⊗ a) = wa for all w ∈ W, a ∈ A. We conclude that ker π_{3}^{−ε} ⊂ im ρ_{2}^{ε}. The exactness of

W^{−ε}(A, σ) →^{π_{3}^{−ε}} W^{ε}(Ã, σ_{1}) →^{ρ_{3}^{ε}} W^{ε}(A, σ)

follows from theexactness of W^{ε}(A, σ) →^{π_{1}^{ε}} W^{ε}(Ã, σ_{1}) →^{ρ_{1}^{ε}} W^{−ε}(A, σ); in fact we have:

ker(ρ_{3}^{ε}) = λ ker(ρ_{1}^{−ε}) = λ im(π_{1}^{−ε}) = im(π_{3}^{−ε}).

The exactness of

W^{ε}(Ã, σ_{1}) →^{ρ_{3}^{ε}} W^{ε}(A, σ) →^{π_{2}^{ε}} W^{−ε}(Ã, σ_{2})

follows from the exactness of

W^{ε}(Ã, σ_{1}) →^{ρ_{1}^{ε}} W^{−ε}(A, σ) →^{π_{2}^{−ε}} W^{ε}(Ã, σ_{2}).

The exactness at W^{−ε}(Ã, σ_{2}) follows from the exactness at W^{ε}(Ã, σ_{2}). Finally, we have ker π_{1}^{ε} = λ^{−1} ker π_{3}^{ε} = λ^{−1} im ρ_{2}^{−ε} = im ρ_{2}^{−ε} which shows the exactness of

W^{−ε}(Ã, σ_{2}) →^{ρ_{2}^{−ε}} W^{ε}(A, σ) →^{π_{1}^{ε}} W^{ε}(Ã, σ_{1})

thus completing the proof. □

Remark 5.1. The proof of 1.2 is based on the one given in [10] for quaternion algebras. Nevertheless, one can also give a proof of it by an induction argument as in [2, Appendix 2].

We need the following trivial but useful lemma:

Lemma 5.2. Suppose that B is a central simple algebra over K with an involution τ. Then (i) W (B, τ) = 0 if and only if B is split and τ is of the first kind and of symplectic type. (ii) W^{−1}(B, τ) = 0 if and only if B is split and τ is of the first kind and of orthogonal type.

Proof. For (i), if B is split and τ is of the first kind and of symplectic type it is clear that W (B, τ) = 0. Conversely let suppose that W (B, τ) = 0. First we show that B is split. In fact let B_{0} be the Brauer-equivalent division algebra of B. By Albert’s theorem there exists an involution τ_{0} on B_{0} of the same kind than τ. By Morita theory, W (B, τ) ≃ W^{ε}(B_{0}, τ_{0}), where ε = 1 if τ and τ_{0} are both of the second kind or of the first kind and of the same type, otherwise ε = −1. So we conclude that W^{ε}(B_{0}, τ_{0}) = 0. If B is not split then the same holds for B_{0}. But in this case, 3.3 implies that the set of the nondegenerate ε-hermitian forms of dimension 1 is nonempty, in particular W^{ε}(B_{0}, τ_{0}) ≠ 0 which is a contradiction. So B is split. Now we prove that τ is of the first kind. In fact if τ is of the second kind then by Morita theory we have W (B, τ) ≃ W (K, τ|_{K}). So we conclude that W (K, τ|_{K}) = 0 which is impossible. Thus τ is of the first kind. Finally τ is of symplectic type, otherwise by Morita theory we obtain 0 = W (B, τ) ≃ W (K) which is impossible. The proof of (ii) is similar. □

Corollary 5.3. The map π_{1}^{ε} is injective if and only if L is a splitting field of A, σ is of the first kind and either ε = 1 and σ is symplectic, or ε = −1 and σ is orthogonal.

Proof. First, one can easily see that σ and σ_{2} are of the same kind and of different type when they are of the first kind. If L is a splitting field of A, then by [13, Ch. 8, 5.4], Ã is split. If ε = 1 and σ is of the first kind and of symplectic type or ε = −1 and σ is of the first kind and of orthogonal type then, by Morita theory, we conclude that W^{−ε}(Ã, σ_{2}) = 0. By the exactness of the octagon at W^{ε}(A, σ), π_{1}^{ε} becomes injective.

Conversely if π_{1}^{ε} is injective then again by the exactness of the octagon at W^{ε}(A, σ), im ρ_{2}^{−ε} = 0. We show that W^{−ε}(Ã, σ_{2}) = 0. If W^{−ε}(Ã, σ_{2}) ≠ 0, and as δ-Morita equivalence preserves the rank by 4.7, we can find a nondegenerate −ε-hermitian form of rank one over (Ã, σ_{2}) by considering the division algebra Brauer-equivalent to Ã. As ρ_{2}^{−ε} preserves the rank we conclude that there exists a nondegenerate ε-hermitian form of rank one in the image of ρ_{2}^{−ε}, which is a contradiction, since the image of ρ_{2}^{−ε} in W^{ε}(A, σ) is zero. So W^{−ε}(Ã, σ_{2}) = 0. Now the result follows from 5.2. □

Remark 5.4. If A is a split quaternion algebra and σ is symplectic, then the exact octagon of 1.2 becomes:

0 → W (L, −) → W (K) → W (L) → W (K) → W (L, −) → 0

where L/K is a quadratic extension with a nontrivial automorphism − = σ|_{L}. If A is a quaternion division algebra and σ is symplectic then 1.2 becomes:

\begin{matrix}0 → W (A, σ) → W (L, σ|_{L}) → W^{−1}(A, σ) → W (L) → W^{−1}(A, σ) → \\ → W (L, σ|_{L}) → W (A, σ) → 0 \end{matrix}

where L is a maximal subfield of A which is stable by the involution. These sequences can be found in [9].

6 Exact octagon of Witt groups of equivariant forms

Let K be a field of characteristic different from 2 and G be a finite group. Let A be a central simple algebra over K with an involution σ. We denote by A[G] the group algebra of G over A. Let ε = ±1.

Definition 6.1. We say that an ε-hermitian space (M, h) over (A, σ) is a G-space if:

• G acts on M on the right and for all g ∈ G, the map M → M : m ↦ m.g is A-linear on the right,

• we have h(m.g, n.g) = h(m, n) for all m, n ∈ M and for all g ∈ G. In this case, we will say that h is a G-form. It is obvious that M is a right A[G]-module.

We say that two ε-hermitian G-spaces (M, h) and (M′, h′) are isometric if there exists an isomorphism Φ : M → M′ of right A[G]-modules such that

h′(Φ(m), Φ(n)) = h(m, n),

for all m, n ∈ M. If M is a right A[G]-module, M^{∗} = Hom_{A}(M, A) has a natural structure of right A[G]-module: (f.g)(m) = f(m.g^{−1}) if f ∈ M^{∗}, g ∈ G and m ∈ M. A G-space (M, h) over (A, σ) is said to be nondegenerate if M → M^{∗} : x ↦ h(x, −) is an isomorphism of A-modules. One can define the hyperbolic ε-hermitian G-space associated to such an M by (M ⊕ M^{∗}, Ih_{M}), where

Ih_{M}(m ⊕ f, m′ ⊕ f′) = f(m′) + εσ(f′(m)),

for all m, m′ ∈ M and f, f′ ∈ M^{∗}.

Remark 6.2. If char K ∤ |G| then, by Maschke’s theorem, the group algebra A[G] is semisimple. Thanks to that, we can show that an ε-hermitian G-space (M, h) is hyperbolic if and only if it is metabolic (i.e. if there exists a right A[G]-submodule N of M such that N = N^{⊥} for h): the proof can be adapted from [1, Corollary 1.4]. This fact will not be used here.

Now, one can construct a group (as for ε-hermitian forms) called the Witt group of ε-hermitian G-forms (also called the Witt group of equivariant forms) which will be denoted by W^{ε}(G, A, σ) (i.e., the quotient of the Grothendieck group corresponding to isometry classes of nondegenerate ε-hermitian G-forms by the subgroup generated by metabolic forms). An element of this group is denoted by [(M, h)] where (M, h) is an ε-hermitian G-space over (A, σ).

We can easily see that the maps involved in the octagon 1.2 induce group homomorphisms between the corresponding Witt groups of hermitian G-forms: if W is an Ã[G]-module then W ⊗Ã A is an A[G]-module, where G acts on W ⊗Ã A by (w ⊗ a).g = w.g ⊗ a for w ∈ W, a ∈ A, g ∈ G.

The notion of anisotropy for G-forms that we will use is the following (as in [3, p. 29]):

Definition 6.3. An ε-hermitian G-space (M, h) over (A, σ) is said to be anisotropic if for all A[G]-submodules N of M, we have N ∩ N^{⊥} = 0 (for h).

Remark 6.4. Note that this notion of anisotropy coincides with the usual notion of anisotropy (see section 4) when (M, h) is an ε-hermitian space over a simple algebra with involution. But in the case of ε-hermitian G-forms, this notion of anisotropy is weaker than the usual one. For example, let q : ℂ × ℂ → ℂ be the quadratic form defined by q(x, y) = x^{2} + y^{2} and G = {1, θ} where θ is the reflection in the hyperplane orthogonal to (1, 0). Then, q is a quadratic G-form which is isotropic as a quadratic form but anisotropic as a G-form in the sense of 6.3.

Now, one can prove a proposition analogous to [3, proposition 2]:

Proposition 6.5. If [(M, h)] ∈ W^{ε}(G, A, σ), [(M, h)] ≠ 0 then we can find an anisotropic ε-hermitian G-space (M′, h′) over (A, σ) such that [(M, h)] = [(M′, h′)].

Proof. The proof goes as in [3, proposition 2]. □

Now, we prove theorem 1.5: Theorem We suppose that A satisfies the same hypotheses as at the beginning of section 2. If we replace W^{±ε}(A, σ) by W^{±ε}(G, A, σ) and W^{±ε}(Ã, σ_{i}) by W^{±ε}(G, Ã, σ_{i}) (for i = 1, 2) in theorem 1.2, we obtain an exact octagon of W (K, σ|_{K})-modules.

Proof. We only have to adapt the proof of 1.2. Let us keep the same notations. The fact that the considered octagon is a complex readily follows from the proof of 1.2 as we can easily verify that there exist right G-modules structures over W and W′ (see (10) and (11) for the definition of these spaces).

To show that this sequence is exact, we use proposition 6.5 to exhibit an anisotropic representative (in the sense of definition 6.3) of each form in the considered kernel.

Let us show that ker(ρ_{2}^{ε}) ⊂ im(π_{2}^{−ε}). We only point at the changes to the original proof. So let (W, f) be an anisotropic ε-hermitian G-space (in the sense of 6.3) lying in the kernel of ρ_{2}^{ε}. Thanks to 6.2, we know that there exists an A[G]-submodule W_{1} of W ⊗ÃA such that W_{1} = W_{1}^{⊥}. To show that W_{1}∩(W ⊗μ) = 0, we consider the following subspace: V = {w ∈ W |w⊗μ ∈ W_{1}}. It is an Ã-submodule of W and if w ∈ W, g ∈ G we have: w.g ⊗ μ = (w ⊗ μ).g ∈ W_{1} as W_{1} is a right G-module. So, V is an Ã[G]-submodule of W. Now, if v, v′ ∈ V then ρ_{2}^{ε}(f)(v ⊗ μ, v′ ⊗ μ) = 0 and we have f(v, v′) = 0. So V ⊂ V^{⊥} and as (W, f) is anisotropic, we deduce that V = 0 and that W_{1} ∩(W ⊗μ) = 0. Now, W ⊗ÃA = W_{1} ⊕(W ⊗μ) as Ã[G]-modules and this implies that for all w ∈ W, there exists w′ ∈ W such that w ⊗ 1 + w′ ⊗ μ ∈ W_{1}. One can define the map J : W → W as in the proof of 1.2 by J(w) = w′(the map J is well-defined because f is anisotropic). Thanks to the previous uniqueness, we have J(w.g) = J(w).g for all w ∈ W, g ∈ G. Thanks to that, we easily show that W_{J} is an A[G]-module. We define h as in (13). If h is degenerate then there exists x ≠ 0 in W_{J} such that h(x, y) = 0 for all y ∈ W_{J}. We deduce that f(x, y) = 0 for all y ∈ W and this shows that x ∈ W ∩W^{⊥} which is a contradiction to the anisotropy of (W, f). We can conclude as in 1.2.

Now let us show that ker(π_{3}^{−ε}) ⊂ im(ρ_{2}^{ε}). Let (V, h) be an anisotropic ε-hermitian G-space lying in the kernel of π_{3}^{−ε}. Then there exists an Ã[G]-submodule W of VÃ such that W = W^{⊥}. As in the proof of 1.2, we define a map f as in (14) and all we have to do is to show that the εhermitian form f is nondegenerate. If f is degenerate, let U be the right A[G]-module generated by W. Then U is an A[G]-submodule of V. Now there exists x ∈ W such that h(x, y) = 0 for all y ∈ W and this implies that h(x, y) = 0 for all y ∈ U and we have x ∈ U ∩ U^{⊥} which is a contradiction to the anisotropy of (V, h) thus completing the proof. □

Remark 6.6. If the group G is trivial, then 1.2 is a special case of 1.5.

7 Order of Witt group

Let L/K be a finite extension. One can ask for the relation between the orders of W (K) and W (L). If L/K is an extension of odd order then by the weak version of Springer’s theorem there is a canonical injection W (K) ↪ W (L) so the finiteness of W (L) implies that of W (K). If L/K is an extension of even degree, then this property fails. However, for a quadratic extension L/K the finiteness of W (K) implies that of W (L); what is easy to see is that by the exact triangle of Elman-Lam (2) one has: |W (L)| ⩽ |W (K)|^{2}. In [9] the defect of this inequality is calculated; in fact, in [9] the following relation has been proved:

|W (L)||W (L, −)|^{2} = |W (K)|^{2}  (15)

where − is the nontrivial automorphism of L/K. We have the same situation for a quaternion algebra Q with its symplectic involution −. In this case the finiteness of W (K) implies that of W (Q, −) by the exact sequence of Jacobson. In fact by [9] we have

|W^{ε}(Q, −)||W^{−ε}(Q, −)| = |W (K)|  (16)

More generally, as stated in 1.6, we have: Corollary Let A be a K-central simple algebra with an involution σ of the first kind. Then we have |W^{ε}(A, σ)||W^{−ε}(A, σ)| = |W (K)|. In particular W (K) is finite if and only if W^{ε}(A, σ) and W^{−ε}(A, σ) are finite.

Proof. By Merkurjev’s theorem A is similar to a multi-quaternion algebra, say A ∼ Q_{1} ⊗ … ⊗ Q_{n}. By Morita theory we have W^{ε}(A, σ) ≃ W^{ε′}(Q_{1} ⊗ … ⊗ Q_{n}, σ_{1} ⊗ … ⊗ σ_{n}) where ε′ = ε or ε′ = −ε and σ_{i} is the canonical involution of Q_{i} for 1 ⩽ i ⩽ n. So in order to prove the statement, we can suppose that:

(A, σ) = (Q_{1} ⊗ … ⊗ Q_{n}, σ_{1} ⊗ … ⊗ σ_{n}).

We proceed by induction on n. If n = 0, i.e., A is split, the statement becomes |W^{ε}(K)||W^{−ε}(K)| = |W (K)| which is true because {W^{ε}(K), W^{−ε}(K)} = {W (K), 0}. If n = 1 the statement is a consequence of (16). Suppose that n > 1. Suppose that Q_{n} = (a, b)_{K} where (a, b)_{K} is the quaternion algebra generated by i and j with i^{2} = a, j^{2} = b and ij = −ji where a, b ∈ K. Take the split quaternion algebra Q′_{n} = (a, 1)_{K} generated by i′ and j′ with i′^{2} = a, j′^{2} = 1 and i′j′ = −j′i′. If we compare the exact octagon of 1.2 for (A, σ) = (Q_{1} ⊗ … ⊗ Q_{n−1} ⊗ Q_{n}, σ_{1} ⊗ … ⊗ σ_{n−1} ⊗ σ_{n}) with λ = 1 ⊗ … ⊗ 1 ⊗ i and μ = 1 ⊗ … ⊗ 1 ⊗ j and also for (A′, σ′) = (Q_{1} ⊗ … ⊗ Q_{n−1} ⊗ Q′_{n}, σ_{1} ⊗ … ⊗ σ_{n−1} ⊗ σ′_{n}) (σ′_{n} is the canonical involution of Q′_{n}) with λ′ = 1 ⊗ … ⊗ 1 ⊗ i′ and μ′ = 1 ⊗ … ⊗ 1 ⊗ j′ we deduce that |W^{ε}(A, σ)||W^{−ε}(A, σ)| = |W^{ε}(A′, σ′)||W^{−ε}(A′, σ′)|. By Morita theory we have W^{ε}(A′, σ′) ≃ W^{−ε}(A′′, σ′′) and W^{−ε}(A′, σ′) ≃ W^{ε}(A′′, σ′′) where

(A′′, σ′′) = (Q_{1} ⊗ … ⊗ Q_{n−1}, σ_{1} ⊗ … ⊗ σ_{n−1})

So by induction hypothesis we obtain |W^{ε}(A, σ)||W^{−ε}(A, σ)| = |W (K)|. □

Remark 7.1. If A is a quaternion algebra over K, and σ is the canonical involution of A, then for K = ℚ_{p}, both groups W^{±1}(A, σ) are finite. For K = ℝ, the group W^{−1}(A, σ) is finite and W^{1}(A, σ) is infinite.

Corollary 7.2. Let A be a quaternion algebra over K with an involution involution σ of the second kind. Then |W (A, σ)| = |W (K, σ|_{K})|.

Proof. By a theorem of Albert [13, Ch. 8, 11.2] (A, σ) = (A_{0} ⊗_{k} K, σ_{0} ⊗ σ|_{K}) where k is the fixed field of σ in K, A_{0} is quaternion algebra over k and σ_{0} is its canonical involution. Suppose that A_{0} = (a, b)_{k} where (a, b)_{k} is the quaternion algebra generated by i and j with i^{2} = a, j^{2} = b and ij = −ji where a, b ∈ k. Take the split quaternion algebra A′_{0} = (a, 1)_{k} generated by i′ and j′ with i′^{2} = a, j′^{2} = 1 and i′j′ = −j′i′. Let σ′_{0} be the canonical involution of A′_{0}. If we compare the exact octagon of 1.2 for (A_{0} ⊗_{k} K, σ_{0} ⊗ σ|_{K}) with λ = i ⊗ 1 and μ = j ⊗ 1 and for (A′, σ′) := (A′_{0} ⊗_{k} K, σ′_{0} ⊗ σ|_{K}) with λ′ = i′ ⊗ 1 and μ′ = j′ ⊗ 1, we deduce that |W^{ε}(A, σ)||W^{−ε}(A, σ)| = |W^{ε}(A′, σ′)||W^{−ε}(A′, σ′)|. As W^{ε}(A, σ) ≃ W^{−ε}(A, σ) ≃ W (A, σ) and W^{ε}(A′, σ′) ≃ W^{−ε}(A′, σ′) ≃ W (A′, σ′) we deduce that |W (A, σ)| = |W (A′, σ′)|. By Morita theory W (A′, σ′) ≃ W (K, σ|_{K}) because A′ is split. This implies the result. □

Corollary 7.3. Let A = Q_{1} ⊗_{K} … ⊗_{K} Q_{n} be a a multi-quaternion algebra over K with the involution σ = σ_{1} ⊗ … ⊗ σ_{n} where σ_{i} is an involution of Q_{i} of the second kind for i = 1, …, n. Then |W (A, σ)| = |W (K, σ|_{K})|.

Proof. The argument is similar to 1.6: we use an induction on n and the case n = 1 has already been proved in 7.2. □

The previous corollary gives the motivation to ask the following question for which we do not know the answer.

Question 7.4. Let A be a K-central simple algebra with an involution σ of the second kind. Is it true that |W (A, σ)| = |W (K, σ|_{K})|?

Remark 7.5. Using 1.5 and applying the same type of arguments, we can show that, if A is a central simple algebra with an involution σ of the first kind, then |W^{ε}(G, A, σ)||W^{−ε}(G, A, σ)| = |W (G, K)||W^{−1}(G, K)|. In 7.3 one can replace the Witt groups by equivariant Witt groups.

8 APPENDIX

Table 8.1.

Map  Definition
S^{ε}(A, σ) →^{π_{1}^{ε}} S^{ε}(Ã, σ_{1}) (V_{A}, h) ↦ (VÃ, π_{1}^{ε}(h))  π_{1}^{ε}(h)(x, y) = π_{1}(h(x, y))
S^{ε}(A, σ) →^{π_{2}^{ε}} S^{−ε}(Ã, σ_{2}) (V_{A}, h) ↦ (VÃ, π_{2}^{ε}(h))  π_{2}^{ε}(h)(x, y) = π_{2}(h(x, y))
S^{ε}(A, σ) →^{π_{3}^{ε}} S^{−ε}(Ã, σ_{1}) (V_{A}, h) ↦ (VÃ, π_{3}^{ε}(h))  π_{3}^{ε}(h)(x, y) = λπ_{1}^{ε}(h(x, y))
S^{ε}(Ã, σ_{1}) →^{ρ_{1}^{ε}} S^{−ε}(A, σ) (W, f) ↦ (W ⊗Ã A, ρ_{1}^{ε}(f)) ρ_{1}^{ε}(f)(x ⊗ α, y ⊗ β) = σ(α)λf(x, y)β
S^{ε}(Ã, σ_{2}) →^{ρ_{2}^{ε}} S^{−ε}(A, σ) (W, f) ↦ (W ⊗Ã A, ρ_{2}^{ε}(f)) ρ_{2}^{ε}(f)(x ⊗ α, y ⊗ β) = σ(α)λμf(x, y)β
S^{ε}(Ã, σ_{1}) →^{ρ_{3}^{ε}} S^{ε}(A, σ) (W, f) ↦ (W ⊗Ã A, ρ_{3}^{ε}(f)) ρ_{3}^{ε}(f)(x ⊗ α, y ⊗ β) = σ(α)f(x, y)β

Table 8.2.

Map  Form  Conditions  Image
S^{ε}(D, τ) →^{π_{1}^{ε}} S^{ε}(D̃, τ_{1})  \begin{matrix}< δ_{1}, …, δ_{n} >; δ_{i} = d_{2i−1} + μd_{2i}, \\ d_{2i−1}, d_{2i} ∈ D̃ \end{matrix}  \begin{matrix}τ_{1}(d_{2i−1}) = εd_{2i−1} \\ τ_{2}(d_{2i}) = −εd_{2i} \end{matrix}  ⨁_{i=1}^{n} ( \begin{matrix}d_{2i−1} & μd_{2i}μ \\ −μ^{2}d_{2i} & −μd_{2i−1}μ \end{matrix} )
S^{ε}(D̃, τ_{1}) →^{ρ_{1}^{ε}} S^{−ε}(D, τ)  \begin{matrix}< γ_{1}, …, γ_{n} >; \\ γ_{i} ∈ D̃ \end{matrix}  τ_{1}(γ_{i}) = εγ_{i}  < λγ_{1}, …, λγ_{n} >
S^{−ε}(D, τ) →^{π_{2}^{−ε}} S^{ε}(D̃, τ_{2})  \begin{matrix}< δ_{1}, …, δ_{n} >; δ_{i} = d_{2i−1} + μd_{2i}, \\ d_{2i−1}, d_{2i} ∈ D̃ \end{matrix}  \begin{matrix}τ_{1}(d_{2i−1}) = −εd_{2i−1} \\ τ_{2}(d_{2i}) = εd_{2i} \end{matrix}  ⨁_{i=1}^{n} ( \begin{matrix}d_{2i} & μ^{−1}d_{2i−1}μ \\ −d_{2i−1} & −μd_{2i}μ \end{matrix} )
S^{ε}(D̃, τ_{2}) →^{ρ_{2}^{ε}} S^{−ε}(D, τ)  \begin{matrix}< γ_{1}, …, γ_{n} >; \\ γ_{i} ∈ D̃ \end{matrix}  τ_{2}(γ_{i}) = εγ_{i}  < λμγ_{1}, …, λμγ_{n} >
S^{−ε}(D, τ) →^{π_{3}^{−ε}} S^{ε}(D̃, τ_{1})  \begin{matrix}< δ_{1}, …, δ_{n} >; δ_{i} = d_{2i−1} + μd_{2i}, \\ d_{2i−1}, d_{2i} ∈ D̃ \end{matrix}  \begin{matrix}τ_{1}(d_{2i−1}) = −εd_{2i−1} \\ τ_{2}(d_{2i}) = εd_{2i} \end{matrix}  ⨁_{i=1}^{n} ( \begin{matrix}λd_{2i−1} & λμd_{2i}μ \\ −λμ^{2}d_{2i} & −λμd_{2i−1}μ \end{matrix} )
S^{ε}(D̃, τ_{1}) →^{ρ_{3}^{ε}} S^{ε}(D, τ)  \begin{matrix}< γ_{1}, …, γ_{n} >; \\ γ_{i} ∈ D̃ \end{matrix}  τ_{1}(γ_{i}) = εγ_{i}  < γ_{1}, …, γ_{n} >
Table 8.3. ^{1}
π_{1}^{ε}(h) is hyperbolic ⇒ h ≃< μγ_{1}, …, μγ_{n} >  τ_{2}(γ_{i}) = −εγ_{i}  S^{ε}(D, τ) →^{π_{1}^{ε}} S^{ε}(D̃, τ_{1})
ρ_{1}^{ε}(f) is hyperbolic ⇒ f ≃ ⨁_{i=1}^{n} ( \begin{matrix}d_{2i−1} & μd_{2i}μ \\ −μ^{2}d_{2i} & −μd_{2i−1}μ \end{matrix} )  \begin{matrix}τ_{1}(d_{2i−1}) = εd_{2i−1} \\ τ_{2}(d_{2i}) = −εd_{2i} \end{matrix}  S^{ε}(D̃, τ_{1}) →^{ρ_{1}^{ε}} S^{−ε}(D, τ)
π_{2}^{−ε}(h) is hyperbolic ⇒ h ≃< γ_{1}, …, γ_{n} >  τ_{1}(γ_{i}) = −εγ_{i}  S^{−ε}(D, τ) →^{π_{2}^{−ε}} S^{ε}(D̃, τ_{2})
ρ_{2}^{ε}(f) is hyperbolic ⇒ f ≃ ⨁_{i=1}^{n} ( \begin{matrix}d_{2i} & μ^{−1}d_{2i−1}μ \\ −d_{2i−1} & −μd_{2i}μ \end{matrix} )  \begin{matrix}τ_{1}(d_{2i−1}) = −εd_{2i−1} \\ τ_{2}(d_{2i}) = εd_{2i} \end{matrix} S^{ε}(D̃, τ_{2}) →^{ρ_{2}^{ε}} S^{−ε}(D, τ)
π_{3}^{−ε}(h) is hyperbolic ⇒ h ≃< μγ_{1}, …, μγ_{n} >  τ_{2}(γ_{i}) = εγ_{i}  S^{−ε}(D, τ) →^{π_{3}^{−ε}} S^{ε}(D̃, τ_{1})
ρ_{3}^{ε}(f) is hyperbolic ⇒ f ≃ ⨁_{i=1}^{n} ( \begin{matrix}λd_{2i−1} & λμd_{2i}μ \\ −λμ^{2}d_{2i} & −λμd_{2i−1}μ \end{matrix} ) \begin{matrix}τ_{1}(d_{2i−1}) = −εd_{2i−1} \\ τ_{2}(d_{2i}) = εd_{2i} \end{matrix} S^{ε}(D̃, τ_{1}) →^{ρ_{3}^{ε}} S^{ε}(D, τ)

1provided that f and h are anisotropic.

References

[1] E. Bayer-Fluckiger, H. W. Lenstra Jr. Forms in odd degree extensions and self-dual normal bases. Amer. J. Math. 112 (1990), no. 3, 359–373.

[2] E. Bayer-Fluckiger, R. Parimala, Galois cohomology of the classical groups over fields of cohomological dimension ≦ 2. Invent. Math. 122 (1995), no. 2, 195–229.

[3] C. Cibils, Groupe de Witt d’une algbre avec involution. Enseign. Math. (2) 29 (1983), no. 1-2, 27–43.

[4] I. Dejaiffe, Somme orthogonale d’algèbres à involution et algèbre de Clifford. Comm. Algebra 26 (1998), no. 5, 1589–1612.

[5] N. Jacobson, A note on hermitian forms. Bull. Amer. Math. Soc. 46 (1940). 264–268.

[6] M. Kneser, Lectures on Galois cohomology of classical groups. Tata Lectures Notes, 47 (1969).

[7] M.-A. Knus, Quadratic and Hermitian forms over rings, Springer, Berlin, 1991.

[8] D. W. Lewis, A note on Hermitian and quadratic forms. Bull. London Math. Soc. 11 (1979), no. 3, 265–267.

[9] D. W. Lewis, New improved exact sequences of Witt Groups. J. Algebra 74 (1982), no. 1, 206–210.

[10] D. W. Lewis, Exact sequences of Witt groups of equivariant forms. Enseign. Math. (2) 29 (1983), no. 1-2, 45–51.

[11] D. W. Lewis, Periodicity of Clifford algebras and exact octagons of Witt groups. Math. Proc. Cambridge Philos. Soc. 98 (1985), no. 2, 263–269.

[12] J. Milnor, D. Husemoller, Symmetric bilinear forms. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. Springer-Verlag, New York-Heidelberg, 1973.

[13] W. Scharlau, Quadratic and hermitian forms. Grundlehren der Mathematischen Wissenschaften, 270. Springer-Verlag, Berlin, 1985.

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

grenier@math.univ-fcomte.fr
mahmoudi@math.univ-fcomte.fr