SERRE’S CONJECTURE II FOR CLASSICAL GROUPS OVER IMPERFECT FIELDS

GRÉGORY BERHUY, CHRISTOPH FRINGS, AND JEAN-PIERRE TIGNOL

Introduction

In his book “Cohomologie galoisienne,” Serre formulates the following conjecture:

Conjecture II: ([22, §3.1]) For every simply connected semisimple linear algebraic group G defined over a perfect field F of cohomological dimension at most 2, the Galois cohomology set H^{1}(F, G) is trivial.

Every simply connected semisimple group G over a field F is isomorphic to a product of Weil transfers

G = ∏_{i=1}^{n} R_{K_{i}/F} (G_{i})

where G_{i} is a simply connected absolutely simple group over a finite separable extension K_{i} of F, and Shapiro’s lemma yields

H^{1}(F, G) = ∏_{i=1}^{n} H^{1}(K_{i}, G_{i})

(see for instance [15, (26.8), (29.6)]). Since the cohomological dimension does not change under finite separable extensions, it suffices to consider Conjecture II for simply connected absolutely simple groups. This conjecture was proved for groups of type ^{1}A_{n} by Merkurjev–Suslin [25, Theorem 24.8] and for groups of type ^{2}A_{n}, B_{n}, C_{n}, D_{n} (with the exception of trialitarian D_{4}), F_{4} and G_{2} by Bayer-Fluckiger– Parimala [4].

In his Bourbaki talk [23], Serre proposed a stronger version of his Conjecture II, taking into account imperfect fields. To state this stronger version, define for any prime number p and any field F the p-separable dimension sd_{p} F as follows (see [12, §1.1], where sd_{p} F is denoted dim_{p}^{sep} F):

- if char F ≠ p, let sd_{p} F = cd_{p} F, the p-cohomological dimension of F;

- if char F = p, consider the p-cohomology groups H_{p}^{r} (see [14], [6]) and let sd_{p} F be the least integer r ≥ 0 such that H_{p}^{r+1}(F′) = 0 for every finite separable extension F′/F.

Date: September 15, 2006.

The authors acknowledge the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2002-00287, KTAGS. The third author was partially funded by IRCSET Basic Research Grant SC/02/265 and by the National Fund for Scientific Research (Belgium). He gratefully acknowledges the hospitality of UCD Dublin where parts of this paper were written and thanks David Lewis who helped to make possible a productive and enjoyable time in Dublin.

Conjecture II (Strong version): ([23, §5.5]) Let G be a simply connected absolutely simple group G defined over a field F. If sd_{p} F ≤ 2 for every torsion prime of the root system of G, then H^{1}(F, G) = 1.

For the reader’s convenience, we quote from [23, §2.2] the list of torsion primes:

Type  Torsion primes
A_{n}  2, prime divisors of n + 1
B_{n}, C_{n}, D_{n} (n ≠ 4), G_{2}  2
D_{4}, E_{6}, E_{7}, F_{4}  2, 3
E_{8}  2, 3, 5.

Note that the hypothesis on the separable dimension of F can be translated in more elementary terms by a theorem of Gille [12, Théorème 7]: sd_{p} F ≤ 2 if and only if for every finite separable extension E/F, the reduced norm map of every p-primary central simple E-algebra is surjective. It is mostly through this characterization that the hypothesis on separable dimensions is used in this work.

The strong version of Conjecture II was proved by Serre for groups of type G_{2} [23, Théorème 11], and by Gille for groups of type ^{1}A_{n} and F_{4} [12, Théorème 7, Théorème 9] and for quasi-split groups of any type except E_{8} [13, Théorème 4]. Our main result is the following:

Theorem. The strong version of Conjecture II holds for groups of type ^{2}A_{n}, B_{n}, C_{n}, and D_{n}, except perhaps for trialitarian groups D_{4}.

The proof is obtained by a case-by-case analysis in Corollaries 2.6, 3.12, 4.5, and 5.5. However, the pattern is the same in all cases: we consider an isogeny π: G → G^{ad} and derive from classification theorems for quadratic forms, involutions or quadratic pairs that the image of the induced map π^{1} : H^{1}(F, G) → H^{1}(F, G^{ad}) is trivial. On the other hand, a theorem of Gille [13, Théorème 6] readily shows that ker π^{1} = 1, hence H^{1}(F, G) = 1.

As this sketch of proof suggests, the essential part of our work in this paper goes to the proof of classification theorems, which roughly say that hermitian forms of various types, (generalized) quadratic forms, involutions and quadratic pairs are classified by their “classical” invariants if the separable dimension of the base field is at most 2. (The precise statements are given in Theorem 2.1 and Corollaries 2.3, 2.4, 3.9, 3.10, 4.3, 4.4, 5.2, and 5.3.) We thus follow the same approach as Bayer-Fluckiger and Parimala, whose arguments in [4] also involved classification theorems for hermitian forms. (The corresponding classification of involutions in characteristic different from 2 was derived in [16].) Our main contribution is the inclusion of the characteristic 2 case in these classification theorems. Actually, as compared to [4], we shift the emphasis from hermitian forms to involutions and quadratic pairs on central simple algebras, which allows us to give proofs valid in all characteristics. We thus recover a significant part of the results in [4], by a method which avoids Morita equivalence, and which therefore seems more transparent^{1} (at least to us).

In a first section, we give the main technical tools used in the classification theorems. These tools revolve around the notion of Witt equivalence for involutions and quadratic pairs. In the next sections, we successively tackle groups of type B_{n}, 2A_{n}, C_{n}, and D_{n}. For background information on involutions and quadratic pairs, we refer to [15], whose notation is used throughout the paper. In particular, if K/F is a separable quadratic extension of fields (of arbitrary characteristic), and if a ∈ F^{×}, we denote by (K, a)_{F} the quaternion algebra K ⊕Kz where conjugation by z restricts to the nontrivial automorphism of K/F and z^{2} = a. If σ is an involution on a central simple algebra A, we let

1For instance, we avoid the delicate justification that the form h_{0} can be chosen of the same rank as h in the proofs of Theorems 4.2.1, 4.3.1, and 4.4.1 of [4].

Sym(σ) = {a ∈ A | σ(a) = a},
Symd(σ) = {a + σ(a) | a ∈ A},
Skew(σ) = {a ∈ A | σ(a) = −a}.

Appendix A yields detailed proofs of some results on Witt kernels which do not appear in the literature in the required generality. In Appendix B, we recall the basic notions of flat cohomology for the reader’s convenience.

1. Witt equivalence of involutions and quadratic pairs

1.1. Orthogonal sums. Let A be a central simple algebra over an arbitrary field K and let σ be an involution (of any type) on A. Suppose A contains nonzero idempotents e_{1}, e_{2} such that e_{1} + e_{2} = 1 and σ(e_{1}) = e_{1}, σ(e_{2}) = e_{2}. The Kalgebras A_{1} = e_{1}Ae_{1} and A_{2} = e_{2}Ae_{2} are central simple and Brauer-equivalent to A, and σ restricts to involutions σ_{1}, σ_{2} on A_{1} and A_{2}. Note that A_{1}, A_{2} are not subalgebras of A since their unity elements e_{1}, e_{2} are not the unity 1 of A. If A = End_{D} V for some division K-algebra D and some D-vector space V, the subspaces V_{1} = e_{1}(V) and V_{2} = e_{2}(V) satisfy V = V_{1} ⊕ V_{2}, and there are canonical identifications

A_{1} = End_{D} V_{1}, A_{2} = End_{D} V_{2}.

By [15, (4.2)], there is an involution θ on D and a hermitian or skew-hermitian form h on V with respect to θ such that σ is the adjoint involution ad_{h} of h, in the sense that

h(x, a(y)) = h(σ(a)(x), y) for x, y ∈ V and a ∈ A.

Since σ(e_{1}) = e_{1} and σ(e_{2}) = e_{2}, it follows that V_{1} and V_{2} are orthogonal. Following Dejaiffe [7], we call (A, σ) an orthogonal sum of (A_{1}, σ_{1}) and (A_{2}, σ_{2}). More precisely, we set the following definition:

Definition 1.1. A central simple K-algebra with involution (A, σ) is an orthogonal sum of central simple K-algebras with involution (A_{1}, σ_{1}) and (A_{2}, σ_{2}) if there are orthogonal idempotents e_{1}, e_{2} ∈ A such that e_{1}+e_{2} = 1 and σ(e_{1}) = e_{1}, σ(e_{2}) = e_{2}, and K-algebra isomorphisms

φ_{1} : A_{1} →^{∼} e_{1}Ae_{1}, φ_{2} : A_{2} →^{∼} e_{2}Ae_{2}

such that φ_{i} ∘ σ_{i} = σ ∘ φ_{i} for i = 1, 2. Using φ_{1} and φ_{2} to identify A_{1} and A_{2} to subsets of A, we denote

(A, σ) = (A_{1}, σ_{1}) ⊞ (A_{2}, σ_{2}).

It is important to observe that an orthogonal sum is not uniquely determined up to isomorphism by its summands. Indeed, for λ_{1}, λ_{2} ∈ K^{×}, the involution σ′ on A defined by

σ′(a) = (λ_{1}e_{1} + λ_{2}e_{2})σ(a)(λ_{1}^{−1}e_{1} + λ_{2}^{−1}e_{2})

is different from σ if λ_{1} ≠ λ_{2}, but has the same restriction as σ to A_{1} and A_{2}.

Proposition 1.2. If a central simple K-algebra with involution (A, σ) is an orthogonal sum

(A, σ) = (A_{1}, σ_{1}) ⊞ (A_{2}, σ_{2}),

then A_{1}, A_{2} are Brauer-equivalent to A and σ, σ_{1}, σ_{2} have the same restriction to K,

σ|_{K} = σ_{1}|_{K} = σ_{2}|_{K}.

If σ|_{K} = Id, then σ, σ_{1}, σ_{2} have the same type (orthogonal or symplectic). Moreover,

deg A = deg A_{1} + deg A_{2}.

Proof. As observed before Definition 1.1, we may identify (A, σ) = (End_{D} V, ad_{h}) and (A_{1}, σ_{1}) = (End_{D} V_{1}, ad_{h_{1}}), (A_{2}, σ_{2}) = (End_{D} V_{2}, ad_{h_{2}}) for some D-vector space V = V_{1} ⊕ V_{2} and some hermitian or skew-hermitian form h = h_{1} ⊥ h_{2}. The proposition follows. □

Proposition 1.3. Let A_{1}, A_{2} be Brauer-equivalent central simple K-algebras and let σ_{1}, σ_{2} be involutions of the same type on A_{1}, A_{2} respectively. If σ_{1} and σ_{2} are unitary, assume moreover σ_{1}|_{K} = σ_{2}|_{K}. Then there exists a central simple K-algebra A with involution σ such that

(A, σ) = (A_{1}, σ_{1}) ⊞ (A_{2}, σ_{2}).

Proof. Let D be a central division K-algebra Brauer-equivalent to A_{1} and A_{2}. There are D-vector spaces V_{1}, V_{2} such that A_{1} = End_{D} V_{1} and A_{2} = End_{D} V_{2}. By [15, (4.2)], there is an involution θ on D and hermitian or skew-hermitian forms h_{1}, h_{2} on V_{1}, V_{2} respectively such that σ_{1} = ad_{h_{1}} and σ_{2} = ad_{h_{2}}. Since σ_{1} and σ_{2} have the same type, we may assume that h_{1} and h_{2} are both hermitian or both skew-hermitian. Then (End_{D}(V_{1} ⊕ V_{2}), ad_{h_{1}⊥h_{2}}) is an orthogonal sum of (A_{1}, σ_{1}) and (A_{2}, σ_{2}). □

To discuss orthogonal groups in characteristic 2, we need to use also orthogonal sums of quadratic pairs. Recall from [15, (5.4)] that a quadratic pair on a central simple K-algebra A of degree n (of arbitrary characteristic) is a pair (σ, f) consisting of an involution σ of the first kind such that dim Sym(σ) = \frac{1}{2}n(n + 1), and f: Sym(σ) → K is a linear map related to the reduced trace Trd_{A} by the following condition:

f(x + σ(x)) = Trd_{A}(x) for x ∈ A.

If char K ≠ 2, the conditions imply that σ is an orthogonal involution and

f(x) = \frac{1}{2} Trd_{A}(x) for x ∈ Sym(σ).

The map f is thus determined by σ; it does not carry any additional structure in this case.

If char K = 2, the involution σ is symplectic since the conditions imply that the trace of every symmetric element is zero.

Definition 1.4. A central simple K-algebra with quadratic pair (A, σ, f) is an orthogonal sum of central simple K-algebras with quadratic pair (A_{1}, σ_{1}, f_{1}) and (A_{2}, σ_{2}, f_{2}) if

(A, σ) = (A_{1}, σ_{1}) ⊞ (A_{2}, σ_{2})

and, identifying A_{1} and A_{2} with their images in A,

f|_{Sym(σ_{1})} = f_{1}, f|_{Sym(σ_{2})} = f_{2}.

We then denote

(A, σ, f) = (A_{1}, σ_{1}, f_{1}) ⊞ (A_{2}, σ_{2}, f_{2}).

(See [11, p. 379] or [10].)

As for involutions, the direct sum is not uniquely determined by its summands.

Proposition 1.5. Let A_{1}, A_{2} be Brauer-equivalent central simple K-algebras and let (σ_{1}, f_{1}), (σ_{2}, f_{2}) be quadratic pairs on A_{1} and A_{2} respectively. There is a central simple K-algebra A with quadratic pair (σ, f) such that

(A, σ, f) = (A_{1}, σ_{1}, f_{1}) ⊞ (A_{2}, σ_{2}, f_{2}).

Proof. We may mimic the arguments in the proof of Proposition 1.3, using generalized quadratic forms as in [11]. Alternatively, form an orthogonal sum

(A, σ) = (A_{1}, σ_{1}) ⊞ (A_{2}, σ_{2})

and, identifying A_{1} = e_{1}Ae_{1}, A_{2} = e_{2}Ae_{2} for some symmetric orthogonal idempotents e_{1}, e_{2} such that e_{1} + e_{2} = 1, define

f(x) = f_{1}(e_{1}xe_{1}) + f_{2}(e_{2}xe_{2}) for x ∈ Sym(σ).
□

Remark. In the notation of the proof, the map f is uniquely determined by the condition that f|_{Sym(σ_{1})} = f_{1} and f|_{Sym(σ_{2})} = f_{2}, since every x ∈ Sym(σ) decomposes as

x = e_{1}xe_{1} + e_{2}xe_{2} + e_{1}xe_{2} + e_{2}xe_{1},

and f(e_{1}xe_{2} + e_{2}xe_{1}) = Trd_{A}(e_{1}xe_{2}) = 0.

1.2. Discriminant of an orthogonal sum. We may compare the invariants of an orthogonal sum with the invariants of the summands; see [7] and [8] for results in this direction. For our purposes, we need to consider only the discriminant algebras of unitary involutions on central simple algebras of even degree. The following result is due to Tamagawa [26, Theorem 3]. We include a proof (different from Tamagawa’s) for completeness.

Proposition 1.6. Let σ be a unitary involution on a central simple K-algebra A and let F ⊂ K be the subfield of σ-invariant elements. Suppose (A, σ) is an orthogonal sum of central simple K-algebras with involution of even degree,

(A, σ) = (A_{1}, σ_{1}) ⊞ (A_{2}, σ_{2}).

Then the Brauer classes of the discriminant algebras are related by

[D(A, σ)] = [D(A_{1}, σ_{1})] + [D(A_{2}, σ_{2})]

in the Brauer group Br F.

Proof. Let F (X) be the function field of the Weil transfer of the Severi-Brauer variety of A. By [19], the scalar extension map Br F → Br F (X) is injective. Therefore, it suffices to prove the claim after scalar extension to F (X). We may thus assume that A (hence also A_{1} and A_{2}) is split, hence

(A, σ) = (End_{K} V, ad_{h}),
(A_{1}, σ_{1}) = (End_{K} V_{1}, ad_{h_{1}}), (A_{2}, σ_{2}) = (End_{K} V_{2}, ad_{h_{2}})

for some K-vector space V = V_{1} ⊕ V_{2} and some hermitian form h = h_{1} ⊥ h_{2} on V. The discriminant algebras of σ, σ_{1} and σ_{2} are Brauer-equivalent to quaternion F -algebras,

[D(A, σ)] = [(K, disc h)_{F}],
[D(A_{1}, σ_{1})] = [(K, disc h_{1})_{F}], [D(A_{2}, σ_{2})] = [(K, disc h_{2})_{F}].

The lemma follows by the additivity of the quaternion symbol, since

disc(h_{1} ⊥ h_{2}) = (disc h_{1})(disc h_{2}).
□

1.3. Centralizers of quadratic subfields. A major tool in our proof of the classification theorems is a reduction to centralizers of quadratic subfields. In this subsection, we consider this reduction from the viewpoint of orthogonal sums.

Let (A, σ) be a central simple K-algebra with involution (of any type) and let L/K be a separable quadratic extension with nontrivial automorphism ι. Suppose (A, σ) is an orthogonal sum

(A, σ) = (A_{1}, σ_{1}) ⊞ (A_{2}, σ_{2})

and there are embeddings

ε_{1} : (L, ι) ↪ (A_{1}, σ_{1}), ε_{2} : (L, ι) ↪ (A_{2}, σ_{2}),

i.e. K-algebra embeddings of L in A_{1}, A_{2} respectively such that

ε_{1} ∘ ι = σ_{1} ∘ ε_{1}, ε_{2} ∘ ι = σ_{2} ∘ ε_{2}.

The involutions σ_{1}, σ_{2} then restrict to unitary involutions σ̃_{1}, σ̃_{2} on the centralizers Ã_{1} = Cent_{A_{1}} L, Ã_{2} = Cent_{A_{2}} L of L in A_{1} and A_{2}.

Proposition 1.7. The embeddings ε_{1}, ε_{2} induce an embedding

ε_{1} ⊞ ε_{2} : (L, ι) ↪ (A, σ).

Moreover, letting σ̃ denote the restriction of σ to the centralizer Ã of L in A,

(Ã, σ̃) = (Ã_{1}, σ̃_{1}) ⊞ (Ã_{2}, σ̃_{2}).

Proof. Let e_{1}, e_{2} ∈ A be orthogonal σ-symmetric idempotents such that e_{1} +e_{2} = 1 and

φ_{1} : A_{1} →^{∼} e_{1}Ae_{1}, φ_{2} : A_{2} →^{∼} e_{2}Ae_{2}

be K-algebra isomorphisms under which σ_{1}, σ_{2} correspond to the restriction of σ. For x ∈ L, define

(ε_{1} ⊞ ε_{2})(x) = φ_{1} ∘ ε_{1}(x) + φ_{2} ∘ ε_{2}(x) ∈ A.

This map is an embedding (L, ι) ↪ (A, σ). The idempotents e_{1}, e_{2} centralize the image of L and φ_{1}, φ_{2} restrict to isomorphisms

φ̃_{1} : Ã_{1} →^{∼} e_{1}Ãe_{1},  φ̃_{2} : Ã_{2} →^{∼} e_{2}Ãe_{2},

so

(Ã, σ̃) = (Ã_{1}, σ̃_{1}) ⊞ (Ã_{2}, σ̃_{2}).
□

1.4. Witt decomposition and cancellation. Let A be a central simple algebra over an arbitrary field K, and let σ be an involution of symplectic or unitary type on A. Recall from [15, §6] that σ is called hyperbolic if A contains an idempotent e such that σ(e) = 1 − e; it is called isotropic if A contains a non-zero right ideal I such that σ(I)I = 0. To rephrase these conditions in terms of hermitian or skewhermitian forms, choose a representation of A by endomorphisms of a vector space V over a division algebra D. The involution σ is then adjoint to a hermitian or skew-hermitian form h on V,

(1)  (A, σ) = (End_{D} V, ad_{h}).

Then σ is hyperbolic (resp. isotropic) if and only if h is hyperbolic (resp. isotropic), see [15, (6.2), (6.7)]. In particular, dim_{D} V is even in this case, hence ind A (= deg D) divides \frac{1}{2} deg A (= \frac{1}{2} deg D dim_{D} V). Note that if σ is symplectic, we may choose a representation where h is alternating (i.e. even skew-hermitian), see [15, (4.2)]. If σ is unitary, h may be chosen hermitian with respect to a unitary involution on D. In both cases, a “Witt decomposition theorem” asserts that h decomposes into an orthogonal sum of an anisotropic and a hyperbolic form, see [21, Corollary 9.2, p. 268]. This observation yields the first part of the following proposition:

Proposition 1.8. (1) Every central simple algebra A with symplectic or unitary involution σ has an orthogonal sum decomposition

(A, σ) = (A_{0}, σ_{0}) ⊞ (A_{1}, σ_{1})

where σ_{0} is anisotropic and σ_{1} is hyperbolic.

(2) Suppose a central simple algebra with symplectic or unitary involution σ has an orthogonal sum decomposition

(A, σ) = (A_{0}, σ_{0}) ⊞ (A_{1}, σ_{1})

where σ_{1} is hyperbolic. If σ is hyperbolic, then σ_{0} is hyperbolic.

Proof. As observed above, the first part follows from the Witt decomposition theorem for hermitian or skew-hermitian forms. To prove the second part, choose a representation (1). The decomposition of (A, σ) yields an orthogonal decomposition of h,

(V, h) = (V_{0}, h_{0}) ⊕^{⊥} (V_{1}, h_{1})

where h_{1} is hyperbolic. If h is hyperbolic, then h_{0} is hyperbolic by Witt cancellation, see [21, Corollary 9.2, p. 268]. □

Remark. The second part of the proposition above may be regarded as a “Witt cancellation theorem” for involutions. Note that only hyperbolic involutions can be cancelled; Elomary has shown in [10] that general involutions do not admit cancellation.

Definition 1.9. Two central simple algebras with symplectic or unitary involutions (A, σ), (A′, σ′) are called Witt-equivalent if there are orthogonal sum decompositions

(A, σ) = (A_{0}, σ_{0}) ⊞ (A_{1}, σ_{1}), (A′, σ′) = (A′_{0}, σ′_{0}) ⊞ (A′_{1}, σ′_{1})

where σ_{1}, σ′_{1} are hyperbolic, and an isomorphism of algebras with involution

(A_{0}, σ_{0}) ≃ (A′_{0}, σ′_{0}).

From Proposition 1.8, it follows that σ is hyperbolic if (A, σ) is Witt-equivalent to a central simple algebra with hyperbolic involution.

Similar notions are defined for quadratic pairs. A quadratic pair (σ, f) on a central simple F -algebra A is called hyperbolic if A contains an idempotent e such that

(2)  f(x) = Trd_{A}(ex)  for x ∈ Sym(σ),

see [15, (6.14)]. It is called isotropic if A contains a nonzero right ideal I such that

σ(I)I = 0  and  f(I ∩ Sym(σ))= 0,

see [15, (6.5)].

Remark 1.10. Condition (2) implies σ(e) = 1 − e. More precisely, for any ℓ ∈ A the following conditions are equivalent:

(a) Trd_{A}(ℓx) = f(x) for all x ∈ Symd(σ),

(b) σ(ℓ) = 1 − ℓ.

Indeed, for y ∈ A we have f(y + σ(y)) = Trd_{A}(y), hence (a) is equivalent to

Trd_{A}(ℓy) + Trd_{A}(ℓσ(y)) = Trd_{A}(y) for all y ∈ A.

Since Trd_{A}(ℓσ(y)) = Trd_{A}(σ(ℓ)y), the last equation can be rewritten as

Trd_{A}((ℓ + σ(ℓ))y) = Trd_{A}(y) for all y ∈ A.

It is equivalent to (b) because the bilinear form Trd_{A}(xy) is regular.

The condition for isotropy or hyperbolicity of a quadratic pair can be translated in terms of generalized quadratic forms, see [11, Proposition 1.7 and Corollary 1.8]. Since hyperbolic quadratic pairs are adjoint to hyperbolic quadratic forms, it follows that ind A divides \frac{1}{2} deg A if A carries a hyperbolic quadratic pair. In view of the Witt decomposition and cancellation theorems for generalized quadratic forms (see [21, Corollary 9.2, p. 268]), the following result can be proved along the same lines as Proposition 1.8:

Proposition 1.11. (1) Every central simple algebra A with quadratic pair (σ, f) has an orthogonal sum decomposition

(A, σ, f) = (A_{0}, σ_{0}, f_{0}) ⊞ (A_{1}, σ_{1}, f_{1})

where (σ_{0}, f_{0}) is anisotropic and (σ_{1}, f_{1}) is hyperbolic.

(2) Suppose a central simple algebra with quadratic pair (σ, f) has an orthogonal sum decomposition

(A, σ, f) = (A_{0}, σ_{0}, f_{0}) ⊞ (A_{1}, σ_{1}, f_{1})

where (σ_{1}, f_{1}) is hyperbolic. If (σ, f) is hyperbolic, then (σ_{0}, f_{0}) is hyperbolic.

Definition 1.12. Two central simple F -algebras with quadratic pair (A, σ, f), (A′, σ′, f′) are called Witt-equivalent if there are orthogonal sum decompositions

(A, σ, f) = (A_{0}, σ_{0}, f_{0}) ⊞ (A_{1}, σ_{1}, f_{1}), (A′, σ′, f′) = (A′_{0}, σ′_{0}, f′_{0}) ⊞ (A′_{1}, σ′_{1}, f′_{1})

where (σ_{1}, f_{1}), (σ′_{1}, f′_{1}) are hyperbolic, and an isomorphism of algebras with quadratic pairs

(A_{0}, σ_{0}, f_{0}) ≃ (A′_{0}, σ′_{0}, f′_{0}).

1.5. The Witt kernel of odd-degree and quadratic extensions. A key tool to prove the classification theorems in Sections 3, 4, and 5 is the description of involutions and quadratic pairs which become hyperbolic under odd-degree or quadratic extensions. For odd-degree extensions, we have the following analogues of the weak version of Springer’s theorem:

Theorem 1.13. Let K be an arbitrary field. Let (A, σ) be a central simple Kalgebra with involution of symplectic or unitary type, and let F ⊂ K be the subfield of σ-invariant elements (so F = K if σ is symplectic). Let E/F be an odd-degree field extension and A_{E} = A ⊗_{F} E, σ_{E} = σ ⊗ Id_{E}. If σ_{E} is hyperbolic, then σ is hyperbolic.

Theorem 1.14. Let (A, σ, f) be a central simple algebra with quadratic pair over an arbitrary field F, and let E/F be an odd-degree field extension. Let A_{E} = A ⊗_{F} E, σ_{E} = σ ⊗ Id_{E}, f_{E} = f ⊗ Id_{E}. If the quadratic pair (σ_{E}, f_{E}) is hyperbolic, then (σ, f) is hyperbolic.

Theorem 1.13 is easily derived from the analogous statement for hermitian forms, due to Bayer-Fluckiger and Lenstra [3, Proposition 1.2]. The transfer argument used in [3] can be adapted to generalized quadratic forms to yield Theorem 1.14. For completeness, we spell out in Appendix A (see Section A.2) detailed proofs along the lines of [3, Proposition 1.2], focusing on involutions and quadratic pairs instead of hermitian forms and generalized quadratic forms.

For (separable) quadratic extensions, the key results are the following:

Theorem 1.15. Let K be an arbitrary field, let (A, σ) be a central simple K-algebra with involution of symplectic or unitary type, and let F ⊂ K be the subfield of σinvariant elements (so F = K if σ is symplectic). Let L/F be a separable quadratic extension with non-trivial automorphism ι and let A_{L} = A ⊗_{F} L, σ_{L} = σ ⊗ Id_{L}. The involution σ_{L} on A_{L} is hyperbolic if and only if there is an embedding (L, ι) ↪ (A, σ).

Theorem 1.16. Let (A, σ, f) be a central simple algebra with quadratic pair over an arbitrary field F, and let L/F be a separable quadratic extension. Let A_{L} = A⊗_{F} L, σ_{L} = σ ⊗ Id_{L} and f_{L} = f ⊗ Id_{L}. If there is an embedding ε: L ↪ A such that

(3)  Trd_{A}(ε(ℓ)x) = T_{L/F} (ℓ)f(x)  for all x ∈ Sym(σ) and ℓ ∈ L,

then (σ_{L}, f_{L}) is hyperbolic. The converse holds except if A is split and (σ, f) is adjoint to a quadratic form of odd Witt index.

Remark. Note that (3) implies ε ∘ ι = σ ∘ ε, since for x = y + σ(y) it yields

Trd_{A}(ε(ℓ)(y + σ(y))) = T_{L/F} (ℓ) Trd_{A}(y).

Expanding the left side as

Trd_{A}(ε(ℓ)y) + Trd_{A}(y · σε(ℓ)) = Trd_{A}(y(ε(ℓ) + σε(ℓ))),

we obtain

Trd_{A}(yT_{L/F} (ℓ)) = Trd_{A}(y(ε(ℓ) + σε(ℓ))) for y ∈ A and ℓ ∈ L.

Since the reduced trace bilinear form is nonsingular, it follows that ε(ℓ) + σε(ℓ) = T_{L/F} (ℓ), hence σε(ℓ) = ει(ℓ) for ℓ ∈ L.

If char F ≠ 2, (3) is in fact equivalent to ε ∘ ι = σ ∘ ε, as can be seen by reversing the steps in the above argument with y = \frac{1}{2}x.

Particular cases of Theorems 1.15 and 1.16 are known: if char F ≠ 2, Theorem 1.16 and the symplectic case of Theorem 1.15 are proved in [5, Theorem 3.3], and the unitary case of Theorem 1.15 in [17, Theorem 3.6]. If char F = 2, the symplectic case of Theorem 1.15 and Theorem 1.16 can be found in [11, Propositions 4.1 and 4.2], except that the converse implication is proved under the additional hypothesis that σ or (σ, f) is anisotropic. Since none of the quoted references establishes Theorems 1.15 and 1.16 in the generality we require, we give a complete proof, featuring a characteristic-free approach, in Appendix A (see Section A.3).

2. Quadratic forms

Let (V, q) be a quadratic space over an arbitrary field F and let b: V × V → F be the polar symmetric bilinear form, defined by

b(x, y) = q(x + y) − q(x) − q(y) for x, y ∈ V.

The radical of b is

rad b = {x ∈ V | b(x, y) = 0 for all y ∈ V }.

The quadratic form q is called regular if either rad b = {0} or char F = 2 and dim rad b = 1. The latter case occurs only if dim V is odd, since b is alternating when char F = 2. Henceforth, all the quadratic forms are assumed to be regular.

2.1. Classification. The classical invariants of quadratic forms—the discriminant and the Clifford invariant—are recalled next.

If char F ≠ 2, we denote by disc q ∈ F^{×}/F^{×2} the (signed) discriminant of q (see for instance [21, p. 36]). If char F = 2, the discriminant is defined as follows: if dim V is odd (hence dim rad b = 1), pick a non-zero vector x ∈ rad b and let

disc q = q(x)F^{×2} ∈ F^{×}/F^{×2};

if dim V is even, disc q is the Arf invariant of q (see [21, p. 340]),

disc q = Arf(q) ∈ F/℘(F),

where ℘(F) = {λ^{2} − λ | λ ∈ F }.

The Clifford invariant c(q) is defined independently of the characteristic as the Brauer class of the Clifford algebra or the even Clifford algebra,

c(q) = \begin{cases}[C(V, q)] ∈ Br(F) & if dim V is even, \\ [C_{0}(V, q)] ∈ Br(F) & if dim V is odd. \end{cases}

We can now state the classification theorem for quadratic forms:

Theorem 2.1. If the reduced norm map of every quaternion F -algebra is surjective, then quadratic forms over F are classified up to isometry by their dimension, discriminant and Clifford invariant.

Proof. The result was proved by Elman and Lam [9, Theorem 3.11] for fields of characteristic different from 2 and by Sah [20, Theorem 3] for quadratic forms of even dimension over fields of characteristic 2. For the rest of the proof,^{2} we assume char F = 2 and consider forms of odd dimension. We use the following notation for quadratic forms: for a, b ∈ F,

[a] = ax^{2}  and  [a, b] = ax^{2} + xy + by^{2}.

2We are indebted to Detlev Hoffmann for the following arguments.

Since

x^{2} + (y^{2} + yz + az^{2}) = (x + y)^{2} + (y + az)z,

the change of variables x′ = x + y, y′ = y + az, z′ = z yields

(4)  [1] ⊥ [1, a] ≃ [1] ⊥ [0, 0].

Let q, q′ be odd-dimensional quadratic forms over F with

dim q = dim q′,  disc q = disc q′,  c(q) = c(q′).

Let a ∈ F^{×} be a representative of disc q. The restriction q_{0} of q (resp. q′_{0} of q′) to an orthogonal complement of the radical is a regular even-dimensional quadratic form such that

q ≃ [a] ⊥ q_{0}, q′ ≃ [a] ⊥ q′_{0}.

Let d (resp. d′) be a representative of disc(a · q_{0}) (resp. disc(a · q′_{0})). It is easy to see that

c(q) = c(a · q_{0}) = c([1, d] ⊥ a · q_{0}), c(q′) = c(a · q′_{0}) = c([1, d′] ⊥ a · q′_{0}),

see for instance [18, Lemma 2, Proposition 5]. By the classification theorem for even-dimensional forms it follows that

[1, d] ⊥ a · q_{0} ≃ [1, d′] ⊥ a · q′_{0}.

Adding [1] to each side and using (4), we obtain

[1] ⊥ [0, 0] ⊥ a · q_{0} ≃ [1] ⊥ [0, 0] ⊥ a · q′_{0}.

Since even-dimensional regular forms can be cancelled by [1, Folgerung, p. 160], it follows that

[1] ⊥ a · q_{0} ≃ [1] ⊥ a · q′_{0}.

Multiplying each side by a, we obtain q ≃ q′. □

Remark 2.2. The reduced norm map of every quaternion F -algebra is surjective if sd_{2} F ≤ 2: this is a special case of a theorem of Merkurjev–Suslin if char F ≠ 2 and of Gille if char F = 2, see [12, Théorème 7]. Therefore, Theorem 2.1 applies in particular when sd_{2} F ≤ 2.

The classification of hermitian forms over separable quadratic extensions or quaternion algebras is easily derived from Theorem 2.1 by means of a transfer argument due to Jacobson (see [21, p. 348]), as we proceed to show.

Suppose first K/F is a separable quadratic extension. Let N(K/F) ⊂ F^{×} be the image of the norm map N_{K/F} : K^{×} → F^{×}. For every hermitian form h on a K-vector space V of dimension n, the discriminant disc h is defined by choosing a basis (e_{i})_{1≤i≤n} of V and letting

disc h = (−1)^{n(n−1)/2} det(h(e_{i}, e_{j}))_{1≤i,j≤n} · N(K/F) ∈ F^{×}/N(K/F),

see [15, p. 114].

Corollary 2.3. If the reduced norm map of every quaternion F -algebra is surjective (for instance if sd_{2} F ≤ 2), then hermitian forms over K are classified up to isometry by their dimension and discriminant.

Proof. For every hermitian form h on a K-vector space V, let q_{h}: V → F be the quadratic form defined by

q_{h}(x) = h(x, x) for x ∈ V .

If h and h′ are hermitian forms such that

dim h = dim h′, disc h = disc h′,

then the formulas^{3} in [21, p. 350] show that

dim q_{h} = dim q_{h′},  disc q_{h} = disc q_{h′},  c(q_{h}) = c(q_{h′}).

Therefore, Theorem 2.1 yields q_{h} ≃ q_{h′}, and it follows from [21, Theorem 10.1.1] that h ≃ h′. □

Now, let Q be a quaternion division F -algebra and let V be a finite-dimensional right Q-vector space. A hermitian form h on V (for the conjugation involution on Q) is even (or trace-valued) if h(x, x) ∈ F for x ∈ V. If char F ≠ 2, every hermitian form is even since only F is fixed under the conjugation involution. If char F = 2, even hermitian forms are called alternating or even skew-hermitian in [15, §4].

Corollary 2.4. If the reduced norm map of every quaternion F -algebra is surjective (for instance if sd_{2} F ≤ 2), then even hermitian forms over Q are classified up to isometry by their dimension.

Proof. For every even hermitian form h on a Q-vector space V, let q_{h}: V → F be the quadratic form defined by

q_{h}(x) = h(x, x) for x ∈ V.

Using an orthogonal basis of h, it is easily seen that

dim q_{h} = 4 dim h, disc q_{h} = \begin{cases}1 & if char F ≠ 2, \\ 0 & if char F = 2, \end{cases}

c(q_{h}) = \begin{cases}0 & if dim h ≡ 0 mod 2, \\ [Q] & if dim h ≡ 1 mod 2. \end{cases}

Therefore, if h and h′ are even hermitian forms with dim h = dim h′, then

dim q_{h} = dim q_{h′},  disc q_{h} = disc q_{h′},  c(q_{h}) = c(q_{h′}).

By Theorem 2.1, it follows that q_{h} ≃ q_{h′}. The same arguments as in [21, Theorem 10.1.1] then show that h ≃ h′. □

2.2. Conjecture II for groups of type B_{n}. The simply connected absolutely simple groups of type B_{n} are isomorphic to spin groups Spin(V, q), where (V, q) is a quadratic space of dimension 2n + 1. Let O^{+}(V, q) be the special orthogonal group of (V, q) and let χ: Spin(V, q) → O^{+}(V, q) be the vector representation. We have an exact sequence (see for instance [15, §23])

(5)  1 → μ_{2} → Spin(V, q) ⟶^{χ} O^{+}(V, q) → 1.

3The formula for c(q_{h}) in [21, p. 350] should be (δ, disc h) instead of (δ, det h).

It yields the following exact sequence in (flat) cohomology (see Appendix B):

\begin{aligned}O^{+}(V, q)(F) ⟶^{δ^{0}} H^{1}(F, μ_{2}) → & H^{1}(F, Spin(V, q)) ⟶^{χ^{1}} \\ & → H^{1}(F, O^{+}(V, q)) ⟶^{δ^{1}} H^{2}(F, μ_{2}). \end{aligned} ; (6)

Recall from [15, §29.E] that the elements in H^{1}(F, O^{+}(V, q)) can be identified with the isometry classes of quadratic forms q′ on V such that disc q′ = disc q. The following lemma dates back to Springer [24] for the case where char F ≠ 2. We include its short proof (in arbitrary characteristic) for the reader’s convenience.

Lemma 2.5. For q′ ∈ H^{1}(F, O^{+}(V, q)),

δ^{1}(q′) = c(q′) − c(q) ∈ _{2} Br(F) = H^{2}(F, μ_{2}).

Proof. Consider the even Clifford algebra C_{0}(V, q) and the associated group schemes GL_{1}(C_{0}(V, q)) of invertible elements and Aut_{F} (C_{0}(V, q)) of F -algebra automorphisms. By the Skolem-Noether theorem, there is an exact sequence

(7)  1 → G_{m} → GL_{1}(C_{0}(V, q)) ⟶^{Int} Aut_{F} (C_{0}(V, q)) → 1

where the homomorphism Int maps every invertible element to the corresponding inner automorphism.

By the functorial property of Clifford algebras, every isometry in O^{+}(V, q) induces an automorphism of the even Clifford algebra C_{0}(V, q), hence there is a commutative diagram relating (5) and (7), where the middle vertical map is the inclusion:

1 ⟶ μ_{2} ⟶ Spin(V, q) ⟶^{χ} O^{+}(V, q) ⟶ 1
↓  ↓  ↓^{ρ}
1 ⟶ G_{m} ⟶ GL_{1}(C_{0}(V, q)) ⟶^{Int} Aut(C_{0}(V, q)) ⟶ 1.

It yields the following commutative diagram, where the vertical map on the right is the inclusion _{2} Br(F) ↪ Br(F):

H^{1}(F, O^{+}(V, q)) ⟶^{δ^{1}} H^{2}(F, μ_{2})
ρ^{1}↓  ↓
H^{1}(F, Aut(C_{0}(V, q))) ⟶^{∂^{1}} H^{2}(F, G_{m}).

Elements in H^{1}(F, Aut_{F} (C_{0}(V, q))) can be identified with isomorphism classes of central simple F -algebras of the same degree as C_{0}(V, q), and for q′ ∈ H^{1}(F, O^{+}(V, q)) the image ρ^{1}(q′) is represented by C_{0}(V, q′). Therefore,

δ^{1}(q′) = ∂^{1}ρ^{1}(q′) = c(q′) − c(q).
□

Corollary 2.6. If sd_{2} F ≤ 2, then H^{1}(F, Spin(V, q)) = 1 for every quadratic space (V, q) of odd dimension.

Proof. Consider the exact sequence (6). If sd_{2} F ≤ 2, a theorem of Gille [13, Théorème 6] shows that δ^{0} is surjective, hence ker χ^{1} = 1. Therefore, it suffices to show that the image of χ^{1} is trivial or, equivalently, that ker δ^{1} = 1.

Lemma 2.5 shows that the elements in ker δ^{1} are the isometry classes of quadratic forms q′ on V such that disc q′ = disc q and c(q′) = c(q). By Theorem 2.1 and Remark 2.2 these forms are isometric to q if sd_{2} F ≤ 2, hence ker δ^{1} = 1. □

3. Unitary involutions

In this section, B is a central simple algebra over a field K of arbitrary characteristic, and τ is a unitary involution on B. We denote by F the subfield of τ-invariant elements in K. Our main goal is to show how the discriminant algebra D(B, τ) controls the hyperbolicity of τ when sd_{2} F ≤ 2. We shall then proceed to prove classification results for hermitian forms over division algebras with unitary involution.

Lemma 3.1. Suppose deg B is even.

(1) If τ is hyperbolic, then the discriminant algebra D(B, τ) is split.

(2) Let M ⊂ B be a subfield of symmetric elements which is a separable extension of F, let B̃ = Cent_{B} M be the centralizer of M and τ̃ = τ|B̃ be the restriction of τ to B̃. The algebra D(B, τ) is Brauer-equivalent to the norm (i.e. corestriction) of D(B̃, τ̃),

[D(B, τ)] = [N_{M/F} (D(B̃, τ̃))] in Br F.

(3) Let C be a central simple K-algebra and let θ be a unitary involution on C such that θ|_{K} = τ|_{K}. Then

[D(B ⊗_{K} C, τ ⊗ θ)] = \begin{cases}0 & if deg C is even, \\ [D(B, τ)] & if deg C is odd. \end{cases}

Proof. The first part is due to Tamagawa [26, Theorem 4]. We include a proof^{4} for the reader’s convenience.

As in the proof of Proposition 1.6, we may find a field F (X) in which F is algebraically closed, which splits B and such that the scalar extension map Br F → Br F (X) is injective. Therefore, it suffices to prove the lemma in the case where B is split. Choose a representation

(B, τ) = (End_{K} V, ad_{h})

where h is a hermitian form on a K-vector space V. Then

[D(B, τ)] = (K, disc h)_{F} ,

see [15, (10.35)]. If τ is hyperbolic, then disc h = 1 hence D(B, τ) is split, and (1) is proved. If B contains a field M consisting of symmetric elements, then V carries a KM-vector space structure. Let s: M → F be a non-zero F -linear map and let s_{K}: KM → K be its K-linear extension to KM. By [15, (4.10)], we may assume h is the transfer of a hermitian form h′ on the KM-vector space V with respect to the non-trivial automorphism of KM/M,

h = s_{K∗}(h′)  and  (B̃, τ̃) = (End_{KM} V, ad_{h′}).

Then [D(B̃, τ̃)] = (KM, disc h′)_{M}. Since disc h = N_{M/F} (disc h′) by [21, Theorem 5.12, p. 51], the projection formula

N_{M/F} (KM, disc h′)_{M} = (K, N_{M/F} (disc h′))_{F}

4Tamagawa’s proof is different.

completes the proof of (2).

To prove (3), we may use the same scalar extension argument to reduce to the case where C is split as well as B. Choosing a representation

(C, θ) = (End_{K} W, ad_{k})

for some hermitian form k on a K-vector space W, we have

(B ⊗_{K} C, τ ⊗ θ) = (End_{K}(V ⊗_{K} W), ad_{h⊗k}).

With respect to K-bases of V and W, the matrix of h ⊗ k is a Kronecker product of the matrices of h and k. Therefore,

disc(h ⊗ k) = \begin{cases}1 & if dim_{K} W is even, \\ disc h & if dim_{K} W is odd. \end{cases}

Part (3) follows, since [D(B ⊗_{K} C, τ ⊗ θ)] = (K, disc(h ⊗ k))_{F}. □

The main result of this section is the following:

Theorem 3.2. Let B be a central simple K-algebra of even degree with a unitary involution τ, and let F ⊂ K be the subfield fixed under τ. If τ is hyperbolic, then ind B divides \frac{1}{2} deg B and D(B, τ) is split. The converse holds if sd_{2} F ≤ 2.

Proof. If τ is hyperbolic, it was already observed in Section 1.4 that ind B divides \frac{1}{2} deg B. Moreover, D(B, τ) is split by Lemma 3.1, hence the first part is clear. In the next two subsections, we prove the second part by induction on ind B.

For the convenience of exposition, we define the following property of a field F:

U(d): A unitary involution τ on a central simple algebra B of index d and even degree over a separable quadratic extension of F is hyperbolic if d divides \frac{1}{2} deg B and D(B, τ) is split.

We thus have to show that fields with sd_{2} F ≤ 2 satisfy U(d) for all d ≥ 1.

3.1. Reduction to 2-power index. To achieve index reduction, we use the following result:

Lemma 3.3. Let (B, τ) be as in Theorem 3.2.

(1) If ind B is divisible by an odd prime, then there is an odd-degree separable field extension E/F such that ind(B ⊗_{F} E) < ind B.

(2) If ind B = 2^{k} with k ≥ 1, then there is an odd-degree separable field extension E/F and a separable quadratic extension L/E linearly disjoint from K such that ind(B ⊗_{F} L) = 2^{k−1}.

Proof. Let D be a central division K-algebra Brauer-equivalent to B, and let D_{1}, D_{2} ⊂ D be central division K-algebras such that D = D_{1} ⊗_{K} D_{2}, with deg D_{1} a power of 2 (possibly deg D_{1} = 1) and deg D_{2} odd. Since B carries a unitary involution, a theorem of Albert–Riehm–Scharlau [15, (3.1)] shows that

[N_{K/F} (D)] = 0 in Br(F),

hence N_{K/F} (D_{1}) ⊗_{F} N_{K/F} (D_{2}) is split. Therefore, N_{K/F} (D_{1}) and N_{K/F} (D_{2}) are both split, since their degrees are coprime,

deg N_{K/F} (D_{1}) = (deg D_{1})^{2}  and  deg N_{K/F} (D_{2}) = (deg D_{2})^{2}.

It follows by the Albert–Riehm–Scharlau theorem that D_{1} and D_{2} carry unitary involutions θ_{1}, θ_{2} such that

F = K ∩ Sym(θ_{1}) = K ∩ Sym(θ_{2}).

In case (1), we have D_{2} ≠ K. The set of elements in D_{2} which are separable of degree deg D_{2} over K is a Zariski-open subset U_{2} defined by the condition that the discriminant of the reduced characteristic polynomial does not vanish. Scalar extension to an algebraic closure of F shows that the open subset U_{2} ∩ Sym(θ_{2}) of Sym(θ_{2}) is not empty. Since D_{2} ≠ K, the field F is infinite, hence the rational points in Sym(θ_{2}) are dense. We may therefore find an element x ∈ Sym(θ_{2}) which is separable of degree deg D_{2} over K. Since θ_{2}(x) = x, the coefficients of the reduced characteristic polynomial of x are in F, hence F (x)/F is a separable field extension of degree deg D_{2}. Moreover, K(x) is a maximal subfield of D_{2}, hence scalar extension to F (x) splits D_{2} and reduces the index of D. Part (1) of the lemma is thus proved, with E = F (x).

In case (2), we have D_{2} = K and D = D_{1} ≠ K. Arguing as above, we may find x ∈ Sym(θ_{1}) such that F (x)/F is a separable field extension of degree deg D. Let R be a Galois closure of F (x)/F and let E ⊂ R be the subfield fixed under a 2-Sylow subgroup of Gal(R/F). Then E/F is a separable field extension of odd degree, and R/E is a Galois extension whose Galois group is a 2-group. It follows that E(x)/E is a separable 2-extension, hence we may find a subfield L ⊂ E(x) such that L/E is a separable quadratic extension. Since the dimension of K(x)/F is a power of 2, the extensions K(x)/F and E/F are linearly disjoint. Moreover,

K(x) ⊗_{F} E = K ⊗_{F} F (x) ⊗_{F} E = K ⊗_{F} E(x),

so K/F is linearly disjoint from E(x)/F hence also from L/F. As

L ⊂ F (x) ⊗_{F} E ⊂ D ⊗_{F} E,

we have ind(D ⊗_{F} L) = \frac{1}{2} ind(D ⊗_{F} E). The conditions are thus fulfilled, since ind(D ⊗_{F} E) = ind D = ind B as the degree of E/F is prime to the degree of D. □

Corollary 3.4. Let d ≥ 1 be an integer. Suppose every separable odd-degree extension of a field F satisfies U(k) for every k < d. If d has an odd prime factor, then U(d) holds for F.

Proof. Let τ be a unitary involution on a central simple K-algebra B of index d and even degree over a separable quadratic extension K of F, such that τ|_{F} = Id_{F}. Suppose d divides \frac{1}{2} deg B and D(B, τ) is split. By Lemma 3.3, there is a separable odd-degree extension E/F such that ind(B ⊗_{F} E) < d. Since U(ind(B ⊗_{F} E)) holds for E, it follows that τ ⊗ Id_{E} is a hyperbolic involution on B ⊗_{F} E. By Theorem 1.13, τ is hyperbolic. □

3.2. The 2-power index case. In this subsection, we complete the proof of Theorem 3.2. The key tool is the following:

Lemma 3.5. Let (B, τ) be as in Theorem 3.2. Suppose deg B ≡ 0 mod 4, ind B divides \frac{1}{2} deg B, and let deg B = 4n. Assume that for every quaternion F -algebra Q split by K there exists a central simple K-algebra B_{0} of even degree with unitary involution τ_{0} such that

[B_{0}] = [B] in Br K  and  [D(B_{0}, τ_{0})] = [Q] in Br F .

Assume moreover that there is a separable quadratic extension L/F, linearly disjoint from K, with non-trivial automorphism ι, such that

(L, ι) ↪ (B, τ)

and that the centralizer B̃ = Cent_{B} L satisfies n[B̃] = 0 in Br KL. If D(B, τ) is split, then there exists a central simple K-algebra with unitary involution (B′, τ′) Witt-equivalent to (B, τ), of degree divisible by 4, endowed with an embedding (L, ι) ↪ (B′, τ′), satisfying the following conditions:

(1) for the centralizer B̃′ = Cent_{B′} L and τ̃′ = τ′|_{B′}, the discriminant algebra D(B̃′, τ̃′) is split;

(2) ind B′ divides \frac{1}{2} deg B′.

Proof. The centralizer B̃ has center KL and degree \frac{1}{2} deg B = 2n. Let M ⊂ KL be the subfield fixed under τ̃. By [15, (10.30)] we have

[D(B̃, τ̃) ⊗_{M} KL] = n[B̃] in Br KL,

hence the hypotheses imply that D(B̃, τ̃) is split by KL. It is therefore Brauerequivalent to a quaternion M-algebra Q_{0} containing K. By Lemma 3.1(2), we have

[N_{M/F} (D(B̃, τ̃))] = [D(B, τ)] in Br F.

Since we assume that D(B, τ) is split, it follows by the Albert–Riehm–Scharlau theorem [15, (3.1)] that Q_{0} has unitary involutions fixing F. By [15, (4.14)], we may find such a unitary involution which restricts to the non-trivial automorphism of K/F. A theorem of Albert [15, (2.22)] then yields a quaternion F -algebra Q containing K such that

Q_{0} ≃ Q ⊗_{F} M.

By hypothesis, we may find a central simple K-algebra B_{0} of even degree with a unitary involution τ_{0} such that B_{0} is Brauer-equivalent to B and D(B_{0}, τ_{0}) to Q. Let J be the (unique) symplectic involution on End_{F} L and set

B_{1} = B_{0} ⊗_{F} End_{F} L, τ_{1} = τ_{0} ⊗ J.

Clearly, B_{1} is Brauer-equivalent to B, and τ_{1} is hyperbolic since J is hyperbolic. Through the regular representation L ↪ End_{F} L, we may embed

(L, ι) ↪ (End_{F} L, J) ↪ (B_{1}, τ_{1}).

Let B̃_{1} = Cent_{B_{1}} L and τ̃_{1} = τ_{1}|B̃_{1}. We have

B̃_{1} = B_{0} ⊗_{F} L = B_{0} ⊗_{F} M  and  τ̃_{1} = τ_{0} ⊗ ι = τ_{0} ⊗ Id_{M},

hence

(8)  [D(B̃_{1}, τ̃_{1})] = [D(B_{0}, τ_{0}) ⊗_{F} M] = [Q ⊗_{F} M] = [D(B̃, τ̃)].

Since B and B_{1} are Brauer-equivalent and τ|_{K} = τ_{1}|_{K}, we may consider an orthogonal sum (B′, τ′) of (B, τ) and (B_{1}, τ_{1}), see Proposition 1.3. It is a central simple K-algebra with involution which is Witt-equivalent to (B, τ), since τ_{1} is hyperbolic. The sum of the embeddings of (L, ι) in (B, τ) and (B_{1}, τ_{1}) is an embedding

(L, ι) ↪ (B′, τ′)

such that, using the˜ notation for the centralizer,

(B̃′, τ̃′) = (B̃, τ̃) ⊞ (B̃_{1}, τ̃_{1}),

by Proposition 1.7. It follows from Proposition 1.6 and (8) that

[D(B̃′, τ̃′)] = [D(B̃, τ̃)] + [D(B̃_{1}, τ̃_{1})] = 0.

To complete the proof, we compute the degree of B′,

deg B′ = deg B + deg B_{1} = 4n + 2 deg B_{0}.

Since ind B′ = ind B = ind B_{0} divides deg B_{0} and 2n, it also divides \frac{1}{2} deg B′. □

To enable us to use Lemma 3.5, we need examples where the hypothesis on the existence of B_{0} holds.

Lemma 3.6. Let (B, τ) be as in Theorem 3.2. Suppose ind B is a power of 2. If sd_{2} F ≤ 2, then for every quaternion F -algebra Q split by K there exists a central simple K-algebra B_{0} of even degree and a unitary involution τ_{0} on B_{0} such that τ_{0}|_{F} = Id_{F} and

[B_{0}] = [B] in Br K, [D(B_{0}, τ_{0})] = [Q] in Br F.

Proof. Let Q = (K, y)_{F} for some y ∈ F^{×}. By a theorem of Gille [12, Théorème 7], the condition sd_{2} F ≤ 2 implies that the reduced norm map Nrd_{B} is surjective. Let x ∈ B^{×} be such that Nrd_{B}(x) = y. A theorem of Yanchevskiĭ [28, Theorem 1] then shows that x lies in the subgroup of B^{×} generated by the invertible elements in Sym(τ). Therefore, we may find invertible elements x_{1}, …, x_{n} ∈ Sym(τ) such that

Nrd_{B}(x_{1} … x_{n}) = y.

Adjoining some x_{i} = 1 if necessary, we may assume n is even. Let B_{0} = M_{n}(B) = M_{n}(F) ⊗_{F} B and let θ = t ⊗ τ, where t is the transpose involution on M_{n}(F). Clearly, B_{0} is Brauer-equivalent to B. Consider the following diagonal matrices in B_{0}:

Δ_{0} = diag(x_{1}, …, x_{n}), Δ_{1} = diag(1, −1, …, 1, −1)

and the unitary involutions τ_{1}, τ_{0} defined by

τ_{1}(x) = Δ_{1}θ(x)Δ_{1}^{−1},  τ_{0}(x) = Δ_{0}τ_{1}(x)Δ_{0}^{−1}  for x ∈ B_{0}.

The involution τ_{1} is hyperbolic, hence D(B_{0}, τ_{1}) is split, by Lemma 3.1(1). On the other hand, [15, (10.36)] yields

[D(B_{0}, τ_{0})] = [D(B_{0}, τ_{1})] + (K, Nrd_{B_{0}}(Δ_{0}))_{F} ,

hence [D(B_{0}, τ_{0})] = [Q]. □

Corollary 3.7. Fields F with sd_{2} F ≤ 2 satisfy U(2^{k}) for all k ≥ 0.

Proof. We argue by induction on k. The case k = 0 follows from Corollary 2.3. Let B be a central simple algebra with ind B = 2^{k} ≥ 2 over a separable quadratic extension K of a field F with sd_{2} F ≤ 2, and let τ be a unitary involution on B such that τ|_{F} = Id_{F}. Assume moreover deg B ≡ 0 mod 2^{k+1} and D(B, τ) is split. We have to show τ is hyperbolic.

By Lemma 3.3(2), we may find a separable odd-degree extension E of F and a separable quadratic extension L of E such that

ind(B ⊗_{F} L) = 2^{k−1}.

By the induction hypothesis, τ ⊗ Id_{L} is hyperbolic. Let ι denote the non-trivial automorphism of L/E and let B_{E} = B ⊗_{F} E, τ_{E} = τ ⊗ Id_{E}. By Theorem 1.15, there is an embedding of E-algebras with involution

(L, ι) ↪ (B_{E}, τ_{E}).

The centralizer B̃_{E} = Cent_{B_{E}} L is Brauer-equivalent to

B_{E} ⊗_{E} L = B ⊗_{F} L.

Therefore, ind B̃_{E} = 2^{k−1} and

(\frac{1}{4} deg B)[B̃_{E}] = 0 in Br KE.

Since D(B_{E}, τ_{E}) = D(B, τ) ⊗_{F} E is split and sd_{2} E ≤ 2, we may apply Lemmas 3.5 and 3.6 to find a central simple KE-algebra with unitary involution (B′, τ′) Wittequivalent to (B_{E}, τ_{E}) and an embedding (L, ι) ↪ (B′, τ′) such that ind B′ divides \frac{1}{2} deg B′ and D(B̃′, τ̃′) is split. The algebra B̃′ is Brauer-equivalent to B′⊗_{E}L hence also to B ⊗_{F} L since B′ is Brauer-equivalent to B_{E}. Therefore, ind B̃′ = 2^{k−1}. Since sd_{2} L ≤ 2, the induction hypothesis shows that τ̃′ is hyperbolic. We may therefore find an idempotent e ∈ B̃′ such that τ̃′(e) = 1 − e. Since B̃′ ⊂ B′ and τ̃′ = τ′|B̃′, this idempotent lies in B′ and satisfies τ′(e) = 1 − e, hence τ′ is hyperbolic. By Witt cancellation (Proposition 1.8), it follows that τ_{E} is hyperbolic, since (B′, τ′) is Witt-equivalent to (B_{E}, τ_{E}). Now, E/F is an odd-degree extension, so Theorem 1.13 shows that τ is hyperbolic. □

Conclusion of the proof of Theorem 3.2. To establish that fields with sd_{2} F ≤ 2 satisfy U(d) for all d ≥ 1, it now suffices to use induction and Corollaries 3.4 and 3.7: if d is a power of 2 the result follows from Corollary 3.7; if d is not a power of 2 then it has an odd prime factor, hence induction and Corollary 3.4 prove that U(d) holds for F. □

3.3. Classification of hermitian forms. In this subsection, we use Theorem 3.2 to obtain a classification result for hermitian forms over a division algebra with unitary involution.

Let D be a central division algebra over an arbitrary field K. Suppose D carries a unitary involution θ, and let F ⊂ K be the subfield of θ-invariant elements in K. Let also ι = θ|_{K} be the non-trivial automorphism of K/F. The discriminant disc h of a hermitian form h on a D-vector space V with respect to θ is defined as in the case where D = K (see Section 2): considering a D-basis (e_{i})_{1≤i≤n} of V, set m = n deg D = deg End_{D} V and

disc h = (−1)^{m(m−1)/2} Nrd_{M_{n}(D)}(h(e_{i}, e_{j}))_{1≤i,j≤n} · N(K/F) ∈ F^{×}/N(K/F).

Even though this element lies in the factor group F^{×}/N(K/F), the quaternion F -algebra (K, disc h)_{F} is well-defined.

Lemma 3.8. If m is even,

[D(End_{D} V, ad_{h})] = \begin{cases}[(K, disc h)_{F}] & if dim_{D} V is even, \\ [(K, (−1)^{m/2} disc h)_{F}] + [D(D, θ)] & if dim_{D} V is odd. \end{cases}

Proof. Using the basis (e_{i})_{1≤i≤n} of V, we may identify

End_{D} V = M_{n}(D) = M_{n}(F) ⊗_{F} D.

Under this identification, the involution ad_{h} is given by

ad_{h}(x) = Δ^{−1} · (t ⊗ θ)(x) · Δ for x ∈ M_{n}(D),

where t is the transpose involution on M_{n}(F) and

Δ = (h(e_{i}, e_{j}))_{1≤i,j≤n} ∈ M_{n}(D).

By [15, (10.36)], it follows that

[D(End_{D} V, ad_{h})] = [(K, Nrd Δ)_{F}] + [D(M_{n}(D), t ⊗ θ)].

The last term on the right side is computed via Lemma 3.1(3):

[D(M_{n}(D), t ⊗ θ)] = \begin{cases}[D(M_{n}(K), t ⊗ ι)] & if deg D is odd and n is even, \\ 0 & if deg D and n are even, \\ [D(D, θ)] & if deg D is even and n is odd. \end{cases}

To complete the proof, observe that

(M_{n}(K), t ⊗ ι) ≃ (End_{K} K^{n}, ad_{⟨1,...,1⟩}),

hence if n is even

[D(M_{n}(K), t ⊗ ι)] = [(K, (−1)^{n/2})_{F}].
□

Corollary 3.9. Suppose sd_{2} F ≤ 2. Let h, h′ be hermitian forms on a D-vector space V with respect to θ. The forms h, h′ are isometric if and only if disc h = disc h′.

Proof. Since Witt cancellation holds for hermitian forms with respect to unitary involutions by [21, Corollary 7.9.2], it suffices to prove that h ⊥ −h′ is hyperbolic if and only if disc h = disc h′.

A computation yields

disc(h ⊥ −h′) = disc h disc h′,

hence Lemma 3.8 shows that disc h = disc h′ if and only if

[D(End_{D}(V ⊕ V), ad_{h⊥−h′})] = 0.

By Theorem 3.2, this equation holds if and only if ad_{h⊥−h′} is hyperbolic, i.e. h ⊥ −h′ is hyperbolic. □

3.4. Classification of unitary involutions.

Corollary 3.10. Let B be a central simple algebra over an arbitrary field K, and let τ, τ′ be unitary involutions on B such that τ|_{K} = τ′|_{K}. Let F ⊂ K be the fixed subfield of τ|_{K} and τ′|_{K}. Suppose sd_{2} F ≤ 2. If deg B is odd, then τ and τ′ are conjugate. If deg B is even, then τ and τ′ are conjugate if and only if D(B, τ) ≃ D(B, τ′).

Proof. Choose a representation B = End_{D} V for some vector space V over a central division K-algebra D, and a unitary involution θ on D fixing F. By [15, (4.2)], there are hermitian forms h, h′ on V with respect to θ such that

τ = ad_{h}, τ′ = ad_{h′} .

If deg B is even, Lemma 3.8 shows that D(B, τ) ≃ D(B, τ′) if and only if disc h = disc h′. This condition is equivalent to h ≃ h′ by Corollary 3.9. Therefore, it implies that τ and τ′ are conjugate.

If deg B is odd, let δ, δ′ ∈ F^{×} be such that

disc h = δN(K/F), disc h′ = δ′N(K/F).

Then disc(δh) = disc(δ′h′) = 1, hence δh ≃ δ′h′ by Corollary 3.9. On the other hand, ad_{δh} = ad_{h} = τ and ad_{δ′h′} = ad_{h′} = τ′, so τ and τ′ are conjugate. □

3.5. Conjecture II for groups of outer type A_{n}. Every simply connected absolutely simple group of type ^{2}A_{n} over a field F is isomorphic to SU(B, τ) for some central simple algebra B of degree n + 1 over a quadratic extension K of F and some unitary involution τ on B fixing F. Denote by μ the center of SU(B, τ). There is an exact sequence

1 → μ → SU(B, τ) ⟶^{π} PGU(B, τ) → 1

inducing the following exact sequence in (flat) cohomology:

(9) PGU(B, τ)(F) ⟶^{δ^{0}} H^{1}(F, μ) → H^{1}(F, SU(B, τ)) ⟶^{π^{1}} H^{1}(F, PGU(B, τ)).

Recall from [15, §29.D] that H^{1}(F, PGU(B, τ)) can be identified with the set of isomorphism classes of triples (B′, τ′, φ) consisting of a central simple algebra B′ of degree n + 1 over a quadratic extension K′ of F, a unitary involution τ′ on B′ fixing F and an F -algebra isomorphism φ: K′ → K.

Lemma 3.11. If n is even, the image of π^{1} is the set of isomorphism classes of triples (B, τ′, Id_{K}) where τ′ is a unitary involution on B fixing F. If n is odd, the image of π^{1} is the set of isomorphism classes of triples (B, τ′, Id_{K}) where τ′ is a unitary involution such that D(B, τ′) ≃ D(B, τ).

Proof. By [15, §29.D], the set H^{1}(F, SU(B, τ)) is in one-to-one correspondence with the set of equivalence classes

{(s, z) ∈ Sym(τ) × K^{×} | Nrd_{B}(s) = N_{K/F} (z)}/≈

where (s, z) ≈ (s′, z′) if and only if there exists b ∈ B^{×} such that

s′ = bsτ(b)  and  z′ = Nrd_{B}(b)z.

The image under π^{1} of the class of (s, z) is represented by the triple (B, τ′, Id_{K}) where τ′ = Int(s) ∘ τ. If n is odd, [15, (10.36)] yields

(10)  [D(B, τ′)] = [D(B, τ)] + [(K, Nrd_{B}(s))_{F}]  in Br(F).

Since Nrd_{B}(s) = N_{K/F} (z), the last term vanishes and therefore D(B, τ′) ≃ D(B, τ). The image of π^{1} is therefore contained in the set described in the statement of the lemma.

To prove the reverse inclusion, let τ′ be an arbitrary unitary involution on B fixing F. By [15, (2.18)], there exists a unit s ∈ Sym(τ) such that τ′ = Int(s) ∘ τ. If n is odd and D(B, τ′) ≃ D(B, τ), (10) shows that the quaternion algebra (K, Nrd_{B}(s))_{F} splits, hence there exists z ∈ K^{×} such that Nrd_{B}(s) = N_{K/F} (z). The isomorphism class of (B, τ′, Id_{K}) is the image under π^{1} of the class of (s, z).

If n is even, we have

Nrd_{B}(Nrd_{B}(s)s) = Nrd_{B}(s)^{n+2} = N_{K/F} (Nrd_{B}(s)^{(n/2)+1}).

The isomorphism class of (B, τ′, Id_{K}) is the image under π′ of the class of the pair (Nrd_{B}(s)s, Nrd_{B}(s)^{(n/2)+1}). □

Corollary 3.12. If sd_{p} F ≤ 2 for p = 2 and for every prime factor of n + 1, then H^{1}(F, SU(B, τ)) = 1.

Proof. A theorem of Gille [13, Théorème 6] shows that the map δ^{0} in (9) is surjective, hence ker π^{1} = 1. On the other hand, Corollary 3.10 and Lemma 3.11 show that im π^{1} = 1. □

4. Symplectic involutions

4.1. Classification. The main result of this section is the following:

Theorem 4.1. Let A be a central simple algebra of even degree over an arbitrary field F, and let σ be a symplectic involution on A. If σ is hyperbolic, then ind A divides \frac{1}{2} deg A. The converse holds if sd_{2} F ≤ 2.

The fact that ind A divides \frac{1}{2} deg A if σ is hyperbolic was already observed in Section 1.4. The proof of the converse follows the same general pattern as Theorem 3.2, which is the corresponding result for unitary involutions: we use induction on ind A to restrict to the centralizer of a quadratic extension, and thus reduce to the unitary case. The main technical tool is the following lemma:

Lemma 4.2. Let (A, σ) be a central simple F -algebra with symplectic involution. Assume deg A ≡ 0 mod 8 and A contains a separable quadratic extension L of F such that σ|_{L} is the non-trivial automorphism ι of L/F, i.e.

(L, ι) ↪ (A, σ).

If the reduced norm map Nrd_{A}: A^{×} → F^{×} is surjective, there exists a central simple F -algebra with symplectic involution (A′, σ′) and an embedding (L, ι) ↪ (A′, σ′) with the following properties:

(1) deg A′ ≡ 0 mod 4;

(2) (A′, σ′) is Witt-equivalent to (A, σ);

(3) for Ã′ = Cent_{A′} L and σ̃′ = σ′|Ã′, the discriminant algebra D(Ã′, σ̃′) is

split. Moreover, if ind A divides \frac{1}{2} deg A, then ind A′ divides \frac{1}{2} deg A′.

Proof. Let Ã = Cent_{A} L and σ̃ = σ|Ã. We have deg Ã = \frac{1}{2} deg A, so deg Ã is even. We may thus consider the discriminant algebra D(Ã, σ̃). By [15, (10.30)],

[D(Ã, σ̃) ⊗_{F} L] = (\frac{1}{2} deg Ã)[Ã],

hence the hypothesis that deg A ≡ 0 mod 8 implies that D(Ã, σ̃) is split by L. Therefore, we may find y ∈ F^{×} such that

(11)  [D(Ã, σ̃)] = (L, y)_{F} .

Let D be a central division F -algebra which is Brauer-equivalent to A and let ρ be an orthogonal involution on D. Since Nrd(A^{×}) = Nrd(D^{×}) = F^{×}, there exists x ∈ D^{×} such that Nrd_{D}(x) = y. The following dimension count argument due to Dieudonné [15, p. 266] shows that x is the product of two symmetric elements: since dim Sym(ρ) > \frac{1}{2} dim D,

Sym(ρ) ∩ (x Sym(ρ)) ≠ {0},

hence we may find non-zero elements s_{1}, s_{2} ∈ Sym(ρ) such that s_{1} = xs_{2}, i.e. x = s_{1}s_{2}^{−1}.

Let Δ = diag(s_{1}, −s_{2}^{−1}) ∈ M_{2}(D) = M_{2}(F) ⊗_{F} D. Define an orthogonal involution σ_{1} on M_{2}(D) by

σ_{1}(x) = Δ · (t ⊗ ρ)(x) · Δ^{−1} for x ∈ M_{2}(D),

where t is the transpose involution on M_{2}(F). By [15, (7.3)],

(12)  disc σ_{1} = Nrd_{M_{2}(D)}(Δ)F^{×2} = Nrd_{D}(s_{1}s_{2}^{−1})F^{×2} = yF^{×2}.

Let J be the (unique) symplectic involution on End_{F} L ≃ M_{2}(F). Let also A_{0} = M_{2}(D) ⊗_{F} End_{F} L and σ_{0} = σ_{1} ⊗ J, a symplectic involution on A_{0} which is hyperbolic since J is hyperbolic. The regular representation L → End_{F} L is an embedding

(L, ι) ↪ (End_{F} L, J) ↪ (A_{0}, σ_{0}).

Using the˜ notation for the centralizer, we have

Ã_{0} = M_{2}(D) ⊗_{F} L  and  σ̃_{0} = σ_{1} ⊗ ι.

Therefore, [15, (10.33)] yields

[D(Ã_{0}, σ̃_{0})] = (L, disc σ_{1})_{F} + (\frac{1}{2} deg Ã_{0})[D].

Since deg A_{0} = 4 deg D, the last term on the right side vanishes. By (11) and (12), it follows that

(13)  [D(Ã_{0}, σ̃_{0})] = [D(Ã, σ̃)].

Since A_{0} and A are Brauer-equivalent and σ_{0}, σ have the same type, we may consider an orthogonal sum

(A′, σ′) = (A, σ) ⊞ (A_{0}, σ_{0}),

see Proposition 1.3. This sum is Witt-equivalent to (A, σ) since σ_{0} is hyperbolic, and

(14)  deg A′ = deg A + deg A_{0} ≡ 0 mod 4.

The direct sum of the embeddings of L in A and A_{0} is an embedding

(L, ι) ↪ (A′, σ′)

such that, by Proposition 1.7,

(Ã′, σ̃′) = (Ã, σ̃) ⊞ (Ã_{0}, σ̃_{0}).

By Proposition 1.6 and (13),

[D(Ã′, σ̃′)] = [D(Ã, σ̃)] + [D(Ã_{0}, σ̃_{0})] = 0.

Since ind A = ind A′ and deg A_{0} = 4 deg D = 4 ind A, it follows from (14) that ind A′ divides \frac{1}{2} deg A′ if ind A divides \frac{1}{2} deg A. □

Proof of Theorem 4.1. We argue by induction on ind A, considering separately the cases where ind A = 1 or 2. If ind A = 1 the result is clear (without any hypothesis on sd_{2} F) since symplectic involutions on split algebras are hyperbolic. If ind A = 2, then A is Brauer-equivalent to a quaternion division F -algebra D, and we may choose a representation

(A, σ) = (End_{D} V, ad_{h})

where h is an even hermitian form on the even-dimensional D-vector space V with respect to the conjugation involution on D. By Corollary 2.4, h (hence also σ) is hyperbolic.

Now, suppose ind A = 2^{k} with k ≥ 2. Since ind A divides \frac{1}{2} deg A, it follows that deg A ≡ 0 mod 8. We show that σ is hyperbolic by induction on k.

The same arguments as in Lemma 3.3(2) (or [4, Lemma 3.3.3]) yield a separable odd-degree extension E/F and a separable quadratic extension L/E such that ind(A ⊗_{F} L) = 2^{k−1}. By the induction hypothesis, the involution σ_{L} = σ ⊗ Id_{L} on A_{L} = A ⊗_{F} L is hyperbolic. Therefore, Theorem 1.15 yields an embedding of E-algebras with involution

(L, ι) ↪ (A_{E}, σ_{E}).

Since sd_{2} F ≤ 2, a theorem of Gille [12, Théorème 7] shows that the reduced norm map on A_{E} is surjective. We may therefore apply Lemma 4.2 to find a central simple E-algebra with symplectic involution (A′, σ′) which is Witt-equivalent to (A_{E}, σ_{E}) and such that D(Ã′, σ̃′) is split. Moreover, ind A′ = ind A_{E} = 2^{k} divides \frac{1}{2} deg A′, so deg A′ ≡ 0 mod 2^{k+1} and deg Ã′ ≡ 0 mod 2^{k}. On the other hand, since Ã′ is Brauer-equivalent to A′ ⊗_{F} L, we have ind Ã′ = 2^{k−1}, hence ind Ã′ divides \frac{1}{2} deg Ã′. By Theorem 3.2, it follows that σ̃′ is hyperbolic, hence there is an idempotent e ∈ Ã′ such that σ̃′(e) = 1 − e. This idempotent also lies in A′ and satisfies σ′(e) = 1 − e, hence σ′ is hyperbolic. As (A_{E}, σ_{E}) is Witt-equivalent to (A′, σ′), the involution σ_{E} also is hyperbolic. Theorem 1.13 then shows that σ is hyperbolic, since the degree of E/F is odd. □

We next apply Theorem 4.1 to obtain a classification of skew-hermitian forms and symplectic involutions.

Let D be a central division algebra over an arbitrary field F, and let θ be an orthogonal involution on D. Recall from [15, (4.2)] that the adjoint involutions of alternating skew-hermitian forms with respect to θ are symplectic.

Corollary 4.3. Suppose sd_{2} F ≤ 2 and let V be a D-vector space. Any two alternating skew-hermitian forms h, h′ on V with respect to θ are isometric.

Proof. Since Witt cancellation holds for alternating skew-hermitian forms by [21, Corollary 7.9.2], it suffices to prove that h ⊥ −h′ is hyperbolic or, equivalently, that ad_{h⊥−h′} is hyperbolic. This follows from Theorem 4.1 since ind End_{D}(V ⊕ V) = deg D divides \frac{1}{2} deg End_{D}(V ⊕ V) = dim_{D} V deg D. □

Corollary 4.4. Suppose sd_{2} F ≤ 2. Any two symplectic involutions σ, σ′ on a central simple F -algebra A are conjugate.

Proof. Choose a representation A = End_{D} V for some vector space V over a central division F -algebra D. Let also θ be an orthogonal involution on D. Then σ = ad_{h} and σ′ = ad_{h′} for some alternating skew-hermitian forms h, h′ on V by [15, (4.2)]. Corollary 4.3 shows that h ≃ h′, hence σ and σ′ are conjugate. □

4.2. Conjecture II for groups of type C_{n}. Every simply connected absolutely simple group of type C_{n} over a field F is isomorphic to the symplectic group Sp(A, σ) for some central simple F -algebra A of degree 2n and some symplectic involution σ on A. There is an exact sequence

1 → μ_{2} → Sp(A, σ) ⟶^{π} PGSp(A, σ) → 1

and an induced exact sequence in (flat) cohomology:

PGSp(A, σ)(F) ⟶^{δ^{0}} H^{1}(F, μ_{2}) → H^{1}(F, Sp(A, σ)) ⟶^{π^{1}} H^{1}(F, PGSp(A, σ)).

Corollary 4.5. If sd_{2} F ≤ 2, then H^{1}(F, Sp(A, σ)) = 1.

Proof. By a theorem of Gille [13, Théorème 6], the map δ^{0} is surjective, hence ker π^{1} = 1. To complete the proof, we show im π^{1} = 1.

Note that π factors through the inclusion

Sp(A, σ) ↪ GSp(A, σ).

By [15, (29.23)], the set H^{1}(F, GSp(A, σ)) can be identified with the set of conjugacy classes of symplectic involutions on A. Corollary 4.4 then yields

H^{1}(F, GSp(A, σ)) = 1,

hence im π^{1} = 1. □

5. Quadratic pairs

5.1. Classification. For the next theorem, recall from [15, (8.10)] that the discriminant of a quadratic pair (σ, f) on a central simple F -algebra A determines the center of the Clifford algebra C(A, σ, f). In particular, if the discriminant is trivial, then C(A, σ, f) decomposes as a direct product of two central simple F -algebras,

C(A, σ, f) ≃ C^{+} × C^{−}.

Theorem 5.1. Let F be an arbitrary field and let (σ, f) be a quadratic pair on a central simple F -algebra of even degree. If (σ, f) is hyperbolic, then ind A divides \frac{1}{2} deg A, disc(σ, f) is trivial and one (at least) of the components C^{±} of C(A, σ, f) is split. The converse holds if sd_{2} F ≤ 2.

Proof. If (σ, f) is hyperbolic, then ind A divides \frac{1}{2} deg A as was observed in Section 1.4. Moreover, disc(σ, f) is trivial and one of the components of C(A, σ, f) is split by [15, (8.31)].

To prove the converse, we argue by induction on ind A. If ind A = 1, then A is split and the theorem follows from Theorem 2.1. For the rest of the proof, assume ind A = 2^{k} with k ≥ 1, hence deg A is divisible by 2^{k+1}. Arguing as in Lemma 3.3(2) (or by [4, Lemma 3.3.3]), we may find a separable odd-degree extension E/F and a separable quadratic extension L/E such that ind A_{L} = 2^{k−1}. Note that ind A_{E} = 2^{k} since the degree of E/F is odd, hence A_{E} is not split. By the induction hypothesis, the quadratic pair (σ_{L}, f_{L}) is hyperbolic. Therefore, Theorem 1.16 yields an embedding ε: L ↪ A_{E} such that

Trd_{A_{E}}(ε(ℓ)x) = T_{L/E}(ℓ)f_{E}(x) for all x ∈ Sym(σ_{E}) and ℓ ∈ L.

Let Ã = Cent_{A_{E}} L be the centralizer of ε(L) and let σ̃ = σ|Ã, a unitary involution on Ã. Since k ≥ 1 and 2^{k+1} divides deg A, we have deg A ≡ 0 mod 4. It follows from [11, Lemma 4.5]^{5} that, substituting for (A, σ, f) a Witt-equivalent algebra with quadratic pair if necessary, we may assume that D(Ã, σ̃) is split. Since ind Ã = ind A_{L} = 2^{k−1} divides \frac{1}{2} deg Ã = \frac{1}{4} deg A and sd_{2} E ≤ 2, Theorem 3.2 shows that σ̃ is hyperbolic. Therefore (σ_{E}, f_{E}) is hyperbolic by [11, Lemma 4.3], hence (σ, f) is hyperbolic by Theorem 1.14. □

Following the pattern of Sections 3 and 4, we now derive from Theorem 5.1 the classification of generalized quadratic forms.

Let D be a central division algebra over an arbitrary field F, and let V be a D-vector space. Assume deg D dim_{D} V is even, and consider the factor group

B_{D} = Br(F)/{0, [D]}.

For generalized quadratic forms q, q′ on V with disc q = disc q′, the quadratic pair (σ, f) adjoint to q ⊥ −q′ has trivial discriminant, hence

C(End_{D}(V ⊕ V), σ, f) = C^{+} × C^{−}

for some central simple F -algebras C^{+}, C^{−}. Since deg D dim_{D} V is even, it follows that deg End_{D}(V ⊕ V) ≡ 0 mod 4, hence by [15, (9.12)]

2[C] = 2[C′] = 0  and  [C^{+}] + [C^{−}] = [D] in Br(F).

Therefore, [C^{+}] and [C^{−}] have the same image in B_{D}, and we may set

c(q, q′) = image of [C^{+}] or [C^{−}] in B_{D}.

This relative invariant of q and q′ was first defined by Bartels through Galois cohomology under the assumption that char F ≠ 2, see [2, Section 7]. The definition above comes from [11, Section 2].

Corollary 5.2. Suppose sd_{2} F ≤ 2. The forms q, q′ are isometric if and only if disc q = disc q′ and c(q, q′) = 0.

Proof. The forms q, q′ are isometric if and only if the quadratic pair (σ, f) adjoint to q ⊥ −q′ is hyperbolic. By Theorem 5.1, this condition holds if and only if one (at least) of the components C^{±} is split, i.e. c(q, q′) = 0. □

Corollary 5.3. Suppose sd_{2} F ≤ 2. Quadratic pairs (σ, f), (σ′, f′) on a central simple F -algebra A of even degree are conjugate if and only if C(A, σ, f) ≃ C(A, σ′, f′) as F -algebras.

Proof. The “only if” part is clear. We only sketch the arguments for the “if” part and refer to [16, Proposition 6] and [11, Theorem 5.3] for details. Choose a representation A = End_{D} V and quadratic forms q, q′ on V whose adjoint quadratic pairs are (σ, f) and (σ′, f′). For any λ ∈ F^{×}, let (σ_{λ}, f_{λ}) be the quadratic pair on M_{2}(A) = End_{D}(V ⊕ V) adjoint to q ⊥ ⟨−λ⟩q′. Since C(A, σ, f) ≃ C(A, σ′, f′), we have disc q = disc q′, hence there are central simple F -algebras C_{λ}^{+}, C_{λ}^{−} such that

C(M_{2}(A), σ_{λ}, f_{λ}) ≃ C_{λ}^{+} × C_{λ}^{−}.

The hypothesis C(A, σ, f) ≃ C(A, σ′, f′) further implies that the center Z of C(A, σ, f) splits one of C_{λ}^{+}, C_{λ}^{−}. If it splits C_{λ}^{+}, then there exists μ ∈ F^{×} such that

[C_{λ}^{+}] = [(Z, μ)_{F}].

5The arguments in §4 of [11] hold without change in characteristic different from 2.

Then C_{λμ}^{+} splits, hence c(q, ⟨λμ⟩q′) = 0 and it follows from Corollary 5.2 that q ≃ ⟨λμ⟩q′. Since (σ′, f′) is also adjoint to ⟨λμ⟩q′, we conclude that (σ, f) and (σ′, f′) are conjugate. □

5.2. Conjecture II for groups of type D_{n}. The simply connected absolutely simple groups of type D_{n} (trialitarian D_{4} excluded) over a field F are isomorphic to spin groups Spin(A, σ, f) where A is a central simple F -algebra of degree 2n and (σ, f) is a quadratic pair on A. Let μ be the center of Spin(A, σ, f). There is an exact sequence (see [15, §31.A])

(15)  1 → μ → Spin(A, σ, f) ⟶^{χ} PGO^{+}(A, σ, f) → 1

and an induced exact sequence in (flat) cohomology

PGO^{+}(A, σ, f)(F) ⟶^{δ^{0}} H^{1}(F, μ) →

→ H^{1}(F, Spin(A, σ, f)) ⟶^{χ^{1}} H^{1}(F, PGO^{+}(A, σ, f)).

(16)

Let Z be the center of the Clifford algebra C(A, σ, f). Recall from [15, §29.F] that the elements in H^{1}(F, PGO^{+}(A, σ, f)) can be identified with the F -isomorphism classes of 4-tuples (A′, σ′, f′, φ) consisting of a central simple F -algebra A′ of degree 2n, a quadratic pair (σ′, f′) on A′ and an isomorphism φ: Z′ → Z from the center Z′ of C(A′, σ′, f′) to Z.

Lemma 5.4. If (A′, σ′, f′, φ) represents an element in the image of χ^{1}, then A′ ≃ A and φ extends to an F -algebra isomorphism C(A′, σ′, f′) ≃ C(A, σ, f).

Proof. As in the proof of Lemma 2.5, we relate the exact sequence (15) to the exact sequence derived from the Skolem-Noether theorem, using the homomorphism ρ arising from the functorial property of Clifford algebras:

1 ⟶ μ ⟶ Spin(A, σ, f) ⟶^{χ} PGO^{+}(A, σ, f) ⟶ 1
↓  ↓  ↓^{ρ}
1 ⟶ GL_{1}(Z) ⟶ GL_{1}(C(A, σ, f)) ⟶^{Int} Aut_{Z}(C(A, σ, f)) ⟶ 1.

There is an induced commutative diagram

H^{1}(F, PGO^{+}(A, σ, f)) ⟶^{δ^{1}}  H^{2}(F, μ)
ρ^{1}↓  ↓
H^{1}(F, Aut_{Z}(C(A, σ, f))) ⟶^{∂^{1}} H^{2}(F, GL_{1}(Z)).

If (A′, σ′, f′, φ) represents an element in im χ^{1} = ker δ^{1}, then

(17)  ∂^{1} ∘ ρ^{1}(A′, σ′, f′, φ) = 1.

To compute the left side, recall from [15, (29.13)] that H^{1}(F, Aut_{Z}(C(A, σ, f))) can be identified with the set of F -isomorphism classes of pairs (C, ψ) consisting of an F -algebra C and an F -algebra embedding ψ: Z ↪ C which become isomorphic over the separable closure of F to the pair (C(A, σ, f), i) where i: Z ↪ C(A, σ, f) is the inclusion. Under this identification, ρ^{1}(A′, σ′, f′, φ) is represented by the pair (C(A′, σ′, f′), φ^{−1}). On the other hand, GL_{1}(Z) can be viewed as the Weil transfer of the multiplicative group over Z, and we have by Shapiro’s lemma

H^{2}(F, GL_{1}(Z)) = H^{2}(F, R_{Z/F} (G_{m,Z} )) = H^{2}(Z, G_{m,Z} ) = Br(Z).

Under these identifications,

∂ ∘ ρ^{1}(A′, σ′, f′, φ) = [C(A′, σ′, f′) ⊗_{Z′} Z] − [C(A, σ, f)] in Br(Z),

where the tensor product is taken with respect to φ. Therefore, (17) yields

C(A′, σ′, f′) ⊗_{Z′} Z ≃ C(A, σ, f) as Z-algebras,

hence φ extends to an F -algebra isomorphism C(A′, σ′, f′) ≃ C(A, σ, f). To prove A′ ≃ A, observe that the map χ^{1} factors through the canonical map

H^{1}(F, GO^{+}(A, σ, f)) → H^{1}(F, PGO^{+}(A, σ, f)).

From the description of the set on the left in [15, §29.F], it follows that the image consists of F -isomorphism classes of 4-tuples (A′, σ′, f′, φ) where A′ = A. □

Corollary 5.5. If sd_{2} F ≤ 2, then H^{1}(F, Spin(A, σ, f)) = 1.

Proof. Consider the exact sequence (16). By a theorem of Gille [13, Théorème 6] the map δ^{0} is surjective, hence ker χ^{1} = 1. On the other hand, if (A′, σ′, f′, φ) represents an element in im χ^{1}, then by Lemma 5.4 and Corollary 5.3

(A′, σ′, f′, φ) ≃ (A, σ, f, Id_{Z}).

Therefore im χ^{1} = 1 and the proof is complete. □

Appendix A. The Witt kernel of odd-degree and separable quadratic extensions

In this appendix, we give the proofs of the results stated in Section 1.5.

A.1. Metabolic involutions. Besides hyperbolic involutions, we need in Section A.2 the weaker notion of metabolic involution, which we define below after the following easy observation on idempotents in central simple algebras:

Lemma A.1. Let A be a central simple algebra of even degree over a field F and let e, e′ ∈ A be two idempotents. Any two of the following conditions imply the third one:

(a) ee′ = 0,
(b) (1 − e′)(1 − e) = 0,
(c) dim_{F} eA + dim_{F} e′A = dim_{F} A.

In particular, if dim_{F} eA = dim_{F} e′A = \frac{1}{2} dim_{F} A, then ee′ = 0 if and only if (1 − e′)(1 − e) = 0.

Proof. Condition (a) is equivalent to e′ = (1 − e)e′, hence also to e′A ⊂ (1 − e)A. Likewise, condition (b) is equivalent to the reverse inclusion (1 − e)A ⊂ e′A. Since

A = eA ⊕ (1 − e)A,

we have dim_{F} (1 − e)A = dim_{F} A − dim_{F} eA, hence condition (c) is equivalent to

dim_{F} e′A = dim_{F} (1 − e)A.

The lemma is now clear. □

Definition A.2. Let A be a central simple algebra of even degree over an arbitrary field F. An involution μ (of any type) on A is called metabolic if A contains an idempotent e such that μ(e)e = 0 and

(18)  dim_{F} eA = \frac{1}{2} dim_{F} A.

Since

(19)  dim μ(e)A = dim μ(Ae) = dim Ae = dim eA,

it follows by Lemma A.1 that

(20)  (1 − e)(1 − μ(e)) = 0.

Note that if the idempotent e satisfies μ(e)e = 0 and (20), then Lemma A.1 and (19) show that it also satisfies (18). Therefore, we may substitute (20) for (18) in the requirements for e.

If A is represented as in (1) as the algebra of endomorphisms of some hermitian or skew-hermitian space (V, h), the involution μ = ad_{h} is metabolic if and only if V contains a totally isotropic subspace U with dim_{D} U = \frac{1}{2} dim_{D} V (or, equivalently, U^{⊥} = U).

Proposition A.3. A symplectic or unitary involution is metabolic if and only if it is hyperbolic.

Proof. If μ is symplectic or unitary, we may find x ∈ A such that μ(x) = 1 − x, see [15, (2.6), (2.17)]. If e ∈ A is an idempotent such that μ(e)e = 0 and dim eA = \frac{1}{2} dim A, let

e′ = e − exμ(e).

Computation shows that e′ is an idempotent. Moreover,

μ(e′) = μ(e) − eμ(x)μ(e) = μ(e) − eμ(e) + exμ(e).

In view of (20), the right side is 1 − e′, hence μ is hyperbolic.

Conversely, any idempotent e ∈ A such that μ(e) = 1−e clearly satisfies μ(e)e = 0 and (1 − e)(1 − μ(e)) = 0. □

Corollary A.4. Let K be an arbitrary field. Let (A, σ) be a central simple Kalgebra with involution of symplectic or unitary type, and let F ⊂ K be the subfield of σ-invariant elements (so F = K if σ is symplectic). If B is a central simple F -algebra with a metabolic orthogonal involution μ, then the involution σ ⊗ μ on A ⊗_{F} B is hyperbolic.

Proof. To simplify notation, let τ = σ ⊗ μ. Let e ∈ B be an idempotent such that μ(e)e = 0 and (1 − e)(1 − μ(e)) = 0, and let e′ = 1 ⊗ e ∈ A ⊗ B. This is an idempotent such that τ(e′)e′ = 0 and (1 − e′)(1 − τ(e′)) = 0, hence τ is metabolic. Moreover, since μ is orthogonal, τ has the same type as σ, hence the corollary follows from Proposition A.3. □

The next proposition gives a corresponding result for quadratic pairs. Recall from [15, (5.18)] that if (σ, f) is a quadratic pair on a central simple F -algebra A and μ is an orthogonal involution on a central simple F -algebra B, a quadratic pair (σ ⊗ μ, f_{∗}) is defined on A ⊗_{F} B by the condition

f_{∗}(x ⊗ y) = f(x) Trd_{B}(y) for x ∈ Sym(σ) and y ∈ Sym(μ).

Proposition A.5. The quadratic pair (σ ⊗ μ, f_{∗}) is hyperbolic if μ is metabolic.

Proof. Let e ∈ B be an idempotent such that μ(e)e = 0 and (1 − e)(1 − μ(e)) = 0, and let ℓ ∈ A be such that

Trd_{A}(ℓx) = f(x) for x ∈ Sym(σ).

Let e′ = 1 ⊗ e − ℓ ⊗ eμ(e). To prove the proposition, we show that

(21)  f_{∗}(z) = Trd_{A⊗B}(e′z)  for z ∈ Sym(σ ⊗ μ).

By Remark 1.10, we have σ(ℓ) = 1 − ℓ, hence the same computation as in the proof of Proposition A.3 yields

(σ ⊗ μ)(e′) = 1 − e′.

By Remark 1.10, it follows that (21) holds for z ∈ Symd(σ ⊗ μ). Since

Sym(σ ⊗ μ) = Symd(σ ⊗ μ) + Sym(σ) ⊗ Sym(μ)

by [15, (5.17)], it suffices to prove (21) for z ∈ Sym(σ) ⊗ Sym(μ).

For x ∈ Sym(σ) and y ∈ Sym(μ), we have

Trd_{A⊗B}(e′ · x ⊗ y) = Trd_{A}(x) Trd_{B}(ey) − Trd_{A}(ℓx) Trd_{B}(eμ(e)y).

Since Trd_{A}(x) = 2f(x) = 2 Trd_{A}(ℓx), the right side is

(22)  f(x)(2 Trd_{B}(ey) − Trd_{B}(eμ(e)y)).

Now, use Trd_{B}(ey) = Trd_{B}(μ(ey)) = Trd_{B}(μ(e)y) to rewrite (22) in the form

f(x) Trd_{B}((e + μ(e) − eμ(e))y).

Since (1 − e)(1 − μ(e)) = 0 we have e + μ(e) − eμ(e) = 1, hence finally

Trd_{A⊗B}(e′ · x ⊗ y) = f(x) Trd_{B}(y),

proving (21) for z = x ⊗ y. □

A.2. The Witt kernel of an odd-degree extension. We now turn to the proof of Theorems 1.13 and 1.14, and use the notation of these theorems. Arguing by induction on the number of generators, we may restrict to the case where E/F is a simple extension. Let E = F (u), dim_{F} E = n, and define a linear map s: E → F by

s(α_{0} + α_{1}u + ⋯ + α_{n−1}u^{n−1}) = α_{0} for α_{0}, …, α_{n−1} ∈ F.

Consider the symmetric bilinear form b: E × E → F defined by

b(x, y) = s(xy) for x, y ∈ E.

This form is regular, so we may consider the adjoint orthogonal involution ad_{b} on End_{F} E. The regular representation E → End_{F} E is an embedding

(E, Id_{E}) ↪ (End_{F} E, ad_{b}).

Composing s with the embedding F ↪ E, we may view s as an idempotent in End_{F} E. For x, y ∈ E,

b(x, s(y)) = s(x)s(y) = b(s(x), y),

hence ad_{b}(s) = s. Let E_{0} = ker s and s_{0} = 1 − s. Let also b_{0} be the restriction of b to E_{0}.

Lemma A.6. The involution ad_{b_{0}} is metabolic.

Proof. The span of u, …, u^{(n−1)/2} is a totally isotropic subspace of E_{0} of dimension \frac{1}{2} dim E_{0}. Any projection on this subspace is an idempotent e such that ad_{b_{0}}(e)e = 0 and dim_{F} (e End_{F} E_{0}) = \frac{1}{2} dim_{F} End_{F} E. □

Proposition A.7. Let (A, σ) be a central simple algebra with involution as in Theorem 1.13. The algebra with involution (A ⊗_{F} End_{F} E, σ ⊗ ad_{b}) is Witt-equivalent to (A, σ). Similarly, every central simple algebra with quadratic pair (A, σ, f) as in Theorem 1.14 is Witt-equivalent to (A ⊗_{F} End_{F} E, σ ⊗ ad_{b}, f_{∗}), where f_{∗} is defined as in Proposition A.5.

Proof. For every symmetric idempotent e ∈ End_{F} E, we may identify

e(End_{F} E)e = End_{F} im e,

and the restriction of ad_{b} to this algebra is the adjoint involution with respect to the restriction of b to im e. In particular, for e = s and e = s_{0},

s(End_{F} E)s = End_{F} F = F  and  s_{0}(End_{F} E)s_{0} = End_{F} E_{0},

and the restriction of ad_{b} to these algebras is Id_{F}, resp. ad_{b_{0}}. Tensoring with A, we obtain

(1 ⊗ s)(A ⊗_{F} End_{F} E)(1 ⊗ s) = A

and

(1 ⊗ s_{0})(A ⊗_{F} End_{F} E)(1 ⊗ s_{0}) = A ⊗_{F} End_{F} E_{0},

hence an orthogonal sum decomposition

(A ⊗_{F} End_{F} E, σ ⊗ ad_{b}) = (A, σ) ⊞ (A ⊗_{F} End_{F} E_{0}, σ ⊗ ad_{b_{0}}).

The first part follows, since the last summand is hyperbolic by Lemma A.6 and Corollary A.4.

If (σ, f) is a quadratic pair on a central simple F -algebra A, we obtain likewise an orthogonal sum decomposition

(A ⊗_{F} End_{F} E, σ ⊗ ad_{b}, f_{∗}) = (A, σ, f) ⊞ (A ⊗_{F} End_{F} E_{0}, σ ⊗ ad_{b_{0}}, f_{∗}),

and the second part follows by Lemma A.6 and Proposition A.5. □

Proof of Theorem 1.13. The regular representation E → End_{F} E yields an embedding

(E, Id_{E}) ↪ (End_{F} E, ad_{b}).

Tensoring with (A, σ), we obtain

(A_{E}, σ_{E}) ↪ (A ⊗_{F} End_{F} E, σ ⊗ ad_{b}).

If σ_{E} is hyperbolic, we may find an idempotent e ∈ A_{E} such that σ_{E}(e) = 1 − e. Viewing this idempotent in A ⊗ End_{F} E, we also have

(σ ⊗ ad_{b})(e) = 1 − e,

hence σ ⊗ ad_{b} is hyperbolic. It follows that σ is hyperbolic since (A, σ) and (A ⊗_{F} End_{F} E, σ ⊗ ad_{b}) are Witt-equivalent by Proposition A.7. □

Proof of Theorem 1.14. The proof follows the same pattern as the proof of Theorem 1.13. The regular representation of E yields an embedding

(A_{E}, σ_{E}) ↪ (A ⊗_{F} End_{F} E, σ ⊗ ad_{b}).

The main point is to show that (σ ⊗ ad_{b}, f_{∗}) is hyperbolic if (σ_{E}, f_{E}) is hyperbolic. The conclusion then follows from Proposition A.7.

Let e ∈ A_{E} be an idempotent such that

(23)  f_{E}(x) = Trd_{A_{E}}(ex)  for all x ∈ Sym(σ_{E}).

To complete the proof, we show that

(24)  f_{∗}(z) = Trd_{A⊗End E}(ez)  for all z ∈ Sym(σ ⊗ ad_{b}).

Since σ_{E}(e) = 1 − e by Remark 1.10, we have (σ ⊗ ad_{b})(e) = 1 − e. Therefore, as in the proof of Proposition A.5, it suffices to prove (24) for z ∈ Sym(σ) ⊗ Sym(ad_{b}). Thus, we aim to prove

(25) f(x) Tr(y) = Trd_{A⊗End E}(e · x ⊗ y)  for x ∈ Sym(σ) and y ∈ End_{F} E.

Let ρ_{v} ∈ End E denote the image of v ∈ E under the regular representation, i.e. ρ_{v}(w) = vw for v, w ∈ E. Using the basis 1, u, …, u^{n−1} of E, it is easily seen that (ρ_{u^{i}} ∘ s ∘ ρ_{u^{j}})_{0≤i,j≤n−1} is a basis of End E, hence End E is spanned by elements of the form ρ_{v} ∘ s ∘ ρ_{w} with v, w ∈ E. Moreover, computation shows

Tr(ρ_{v} ∘ s) = s(v) for v ∈ E.

Claim. For a ∈ A_{E}, Trd_{A⊗End E}(a · 1 ⊗ s) = s(Trd_{A_{E}}(a)).

Since both sides are linear in a, it suffices to prove the claim for a = a_{0} ⊗ v with a_{0} ∈ A and v ∈ E. Then

Trd(a · 1 ⊗ s) = Trd_{A}(a_{0}) Tr(ρ_{v} ∘ s) = Trd_{A}(a_{0})s(v) = s(Trd_{A_{E}}(a)),

proving the claim.

We may now prove (25). Since both sides are linear in y, it suffices to prove it when y = ρ_{v} ∘ s ∘ ρ_{w} for some v, w ∈ E. We may then decompose

x ⊗ y = (x ⊗ ρ_{v})(1 ⊗ s)(1 ⊗ ρ_{w}).

Since e ∈ A_{E} ⊂ A ⊗ End E commutes with 1 ⊗ ρ_{w}, we have

Trd(e · x ⊗ y) = Trd(e · (x ⊗ ρ_{w} ∘ ρ_{v}) · (1 ⊗ s)).

Using the claim proved above and (23), we may simplify the right side to

s(Trd_{A_{E}}(e · x ⊗ vw)) = s(f(x)vw) = f(x)s(vw).

Equation (25) follows, since

Tr(y) = Tr(ρ_{v} ∘ s ∘ ρ_{w}) = Tr(ρ_{w} ∘ ρ_{v} ∘ s) = s(vw).
□

A.3. The Witt kernel of a separable quadratic extension. We now turn to the proof of Theorems 1.15 and 1.16. The following lemma already yields the “if” part of these results:

Lemma A.8. Let A be an F -algebra, and let σ be an F -linear involution on A. Let also L/F be a separable quadratic field extension with non-trivial automorphism ι. If there is an embedding ε: (L, ι) ↪ (A, σ), then A ⊗_{F} L contains an idempotent e such that σ_{L}(e) = 1 − e. If A is central simple and (σ, f) is a quadratic pair on A, and if there is an embedding ε: L ↪ A such that (3) holds, then A ⊗_{F} L contains an idempotent e such that f_{L}(x) = Trd_{A_{L}}(ex) for all x ∈ Sym(σ_{L}).

Proof. Let ℓ ∈ L \ F and

s = (1 ⊗ ℓ − ℓ ⊗ 1)(1 ⊗ (ℓ − ι(ℓ))^{−1}) ∈ L ⊗_{F} L.

The element s is the separability idempotent of L. It satisfies

(ι ⊗ Id_{L})(s) = 1 − s = (Id_{L} ⊗ι)(s),

and therefore it is mapped under ε ⊗ Id_{L}: L ⊗_{F} L ↪ A ⊗_{F} L to an idempotent e such that σ_{L}(e) = 1 − e.

Now, assume (σ, f) is a quadratic pair on the central simple F -algebra A and ε satisfies (3). For x ∈ Sym(σ),

Trd_{A_{L}}(e(x ⊗ 1)) = Trd_{A}(x)\frac{ℓ}{ℓ − ι(ℓ)} − Trd_{A}(ε(ℓ)x)\frac{1}{ℓ − ι(ℓ)}

= Trd_{A}(x)\frac{ℓ}{ℓ − ι(ℓ)} − T_{L/F} (ℓ)f(x)\frac{1}{ℓ − ι(ℓ)}.

Since T_{L/F} (ℓ) = ℓ + ι(ℓ) and Trd_{A}(x) = 2f(x), the right side is f(x). By linearity, it follows that Trd_{A_{L}}(ex) = f_{L}(x) for all x ∈ Sym(σ_{L}). □

We now consider the “only if” part of Theorem 1.15 in the anisotropic case.

Lemma A.9. Using the same notation as in Theorem 1.15, suppose (A, σ) is anisotropic. If σ_{L} is hyperbolic, then there is an embedding ε: (L, ι) ↪ (A, σ).

Proof. Let e ∈ A_{L} be an idempotent such that σ_{L}(e) = 1 − e. As observed in the proof of Proposition A.3, this condition implies

(26)  dim_{L}(1 − e)A_{L} = \frac{1}{2} dim_{L} A_{L},  hence dim_{F} (1 − e)A_{L} = dim_{F} A.

It also implies σ(x)x = 0 for x ∈ (1 − e)A_{L}, hence

(A ⊗ 1) ∩ (1 − e)A_{L} = {0}

since σ on A is anisotropic. Moreover, we have

dim_{F} (A ⊗ 1) + dim_{F} (1 − e)A_{L} = 2 dim_{F} A = dim_{F} A_{L},

hence

A_{L} = (A ⊗ 1) ⊕ (1 − e)A_{L}.

Therefore, for x ∈ A_{L} there is a unique a ∈ A such that x − (a ⊗ 1) ∈ (1 − e)A_{L}, i.e.

e(x − (a ⊗ 1)) = 0.

We may then define a map ε: L → A as follows: for ℓ ∈ L, ε(ℓ) ∈ A is the unique element such that

e(1 ⊗ ℓ − ε(ℓ) ⊗ 1) = 0.

The map ε clearly is F -linear and injective, and ε(1) = 1. Moreover, for ℓ, ℓ′ ∈ L we have

1 ⊗ ℓ′ℓ − ε(ℓ)ε(ℓ′) ⊗ 1 = (1 ⊗ ℓ′ − ε(ℓ′) ⊗ 1)(1 ⊗ ℓ) + (1 ⊗ ℓ − ε(ℓ) ⊗ 1)(ε(ℓ′) ⊗ 1),

so

e(1 ⊗ ℓℓ′ − ε(ℓ)ε(ℓ′) ⊗ 1) = 0,

and therefore

ε(ℓℓ′) = ε(ℓ)ε(ℓ′).

This proves that ε is an F -algebra embedding L ↪ A. To show that ε ∘ ι = σ ∘ ε, consider ℓ ∈ L \ F and

(27)  e′ = (1 ⊗ ℓ − ε(ℓ) ⊗ 1)(1 ⊗ (ℓ − ι(ℓ))^{−1}) ∈ A_{L}.

This is an idempotent (it is the image under ε ⊗ Id_{L} of the separability idempotent of L), and it satisfies

(Id_{A} ⊗ι)(e′) = 1 − e′.

Since dim_{L}(Id_{A} ⊗ι)(e′)A_{L} = dim_{L} e′A_{L} and A_{L} = e′A_{L} ⊕ (1 − e′)A_{L},

(28)  dim_{L} e′A_{L} = \frac{1}{2} dim_{L} A_{L},  and therefore dim_{F} e′A_{L} = dim_{F} A.

By definition of ε(ℓ), we have ee′ = 0, hence by (26) and (28) Lemma A.1 yields

(1 − e′)(1 − e) = 0.

Applying σ_{L}, we obtain

(29)  e(1 − σ_{L}(e′)) = 0.

On the other hand,

1 − e′ = (Id_{A} ⊗ι)(e′) = (1 ⊗ ι(ℓ) − ε(ℓ) ⊗ 1)(1 ⊗ (ι(ℓ) − ℓ)^{−1}),

hence (29) yields ε ∘ ι(ℓ) = σ ∘ ε(ℓ). □

The corresponding result for quadratic pairs is the following:

Lemma A.10. Using the same notation as in Theorem 1.16, suppose (σ, f) is anisotropic. If (σ_{L}, f_{L}) is hyperbolic, then there is an embedding ε: L ↪ A satisfying (3).

Proof. Let e ∈ A_{L} be an idempotent such that f_{L}(x) = Trd_{A_{L}}(ex) for x ∈ Sym(σ_{L}). By Remark 1.10,

e + σ_{L}(e) = 1.

The ideal (1 − e)A_{L} is isotropic, hence the same arguments as in the proof of Lemma A.9 show that

A_{L} = (A ⊗ 1) ⊕ (1 − e)A_{L}

and yield an F -algebra embedding ε: L ↪ A defined by

e(1 ⊗ ℓ − ε(ℓ) ⊗ 1) = 0 for ℓ ∈ L.

As in the proof of Lemma A.9, we have

σ ∘ ε = ε ∘ ι.

Condition (3) holds for ℓ ∈ F since then

T_{L/F} (ℓ)f(x) = 2ℓf(x) = ℓ Trd_{A}(x) for x ∈ Sym(σ).

Now, let ℓ ∈ L \ F and consider the idempotent e′ defined in (27). To simplify notation, denote = Id_{A} ⊗ι. Since σ ∘ ε = ε ∘ ι, we have

(30)  σ_{L}(e′) = 1 − e′ = \overline{e′}.

For x ∈ Sym(σ),

Trd_{A_{L}}(e(x ⊗ 1)) = f(x) ∈ F,

hence

Trd_{A_{L}}(e(x ⊗ 1)) = Trd_{A_{L}}(\overline{e}(x ⊗ 1)).

By linearity, it follows that

Trd_{A_{L}}((e − \overline{e})x) = 0 for x ∈ Sym(σ_{L}).

In particular,

(31)  Trd_{A_{L}}((e − \overline{e})σ_{L}(e′)xe′) = 0  for x ∈ Sym(σ_{L}).

Applying to ee′ = 0, and taking (30) into account, we obtain

\overline{e}σ_{L}(e′) = 0.

On the other hand, it also follows from ee′ = 0 and (30) that

eσ_{L}(e′) = e.

Therefore, (31) yields

Trd_{A_{L}}(exe′) = 0 for x ∈ Sym(σ_{L}).

Substituting for e′ the right side of (27), we obtain

(32) Trd_{A_{L}}((ε(ℓ) ⊗ 1)ex) = Trd_{A_{L}}(ex)ℓ = f_{L}(x)ℓ  for x ∈ Sym(σ_{L}).

Similarly, ee′ = 0 yields Trd_{A_{L}}(ee′x) = 0, hence

Trd_{A_{L}}(e(ε(ℓ) ⊗ 1)x) = Trd_{A_{L}}(ex)ℓ = f_{L}(x)ℓ for x ∈ Sym(σ_{L}).

Substituting ι(ℓ) for ℓ in the last equation, and using σ ∘ ε = ε ∘ ι, we also have for x ∈ Sym(σ_{L})

(33)  Trd_{A_{L}}((ε(ℓ) ⊗ 1)σ_{L}(e)x) = Trd_{A_{L}}(xe(ε ∘ ι(ℓ) ⊗ 1)) = f_{L}(x)ι(ℓ).

Adding (32) and (33), and using σ_{L}(e) + e = 1, we finally obtain

Trd_{A_{L}}((ε(ℓ) ⊗ 1)x) = T_{L/F} (ℓ)f_{L}(x) for x ∈ Sym(σ_{L}),

proving (3). □

Lemma A.11. Using the same notation as in Theorem 1.15, suppose σ is hyperbolic. Then there is an embedding ε: (L, ι) ↪ (A, σ).

Proof. Since σ is hyperbolic, we have A ≃ M_{2}(A′) ≃ M_{2}(F) ⊗_{F} A′ for some central simple K-algebra A′ which has involutions of the same kind as σ. Let σ′ be an involution on A′, of orthogonal type if σ is symplectic, of unitary type if σ is unitary, with σ′|_{K} = σ|_{K}, and let J be the (unique) symplectic involution on M_{2}(F),

J(x) = Tr(x) − x for x ∈ M_{2}(F).

Since J is hyperbolic, the involution J ⊗σ′ on M_{2}(F)⊗_{F} A′ is hyperbolic. As hyperbolic involutions of a given type on a central simple algebra are all conjugate (since hyperbolic hermitian or skew-hermitian forms of a given dimension are isometric), it follows that

(A, σ) ≃ (M_{2}(F) ⊗_{F} A′, J ⊗ σ′).

Now, let ℓ ∈ L \ F and define

b(x, y) = \frac{ι(x)y − xι(y)}{ℓ − ι(ℓ)} for x, y ∈ L.

The map b: L×L → F is a nonsingular alternating bilinear form, hence its adjoint involution ad_{b} on End_{F} L is symplectic, and

(34)  (End_{F} L, ad_{b}) ≃ (M_{2}(F), J).

Moreover, for u, x, y ∈ L,

b(ux, y) = b(x, ι(u)y),

hence the regular representation L → End_{F} L is an embedding

(35)  (L, ι) ↪ (End_{F} L, ad_{b}).

The lemma follows by composing the maps

(L, ι) ↪ (End_{F} L, ad_{b}) ≃ (M_{2}(F), J) ↪ (M_{2}(F) ⊗_{F} A′, J ⊗ σ′) ≃ (A, σ).
□

Lemma A.12. Using the same notation as in Theorem 1.16, suppose (σ, f) is hyperbolic and deg A ≡ 0 mod 4. Then there is an embedding ε: L ↪ A satisfying (3).

Proof. Since (σ, f) is hyperbolic, we have A ≃ M_{2}(A′) ≃ M_{2}(F) ⊗_{F} A′ for some central simple F -algebra A′ with involutions of the first kind. The degree of A′ is even since deg A is divisible by 4, hence A′ carries an involution σ′ of symplectic type. Let J be the (unique) symplectic involution on M_{2}(F). We may consider the quadratic pair (J ⊗ σ′, f_{⊗}) determined by the condition

f_{⊗}(x_{1} ⊗ x_{2}) = 0 for x_{1} ∈ Skew(J) and x_{2} ∈ Skew(σ′),

see [15, (5.20)].

Claim: The quadratic pair (J ⊗ σ′, f_{⊗}) is hyperbolic.

Consider e = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ⊗ 1 ∈ M_{2}(F) ⊗ A′. To prove the claim, we show that

(36)  f_{⊗}(x) = Trd(ex)  for x ∈ Sym(J ⊗ σ′).

By [15, (5.17)],

(37)  Sym(J ⊗ σ′) = Symd(J ⊗ σ′) + Sym(J) ⊗ Sym(σ′),

hence it suffices to prove (36) for x ∈ Symd(J ⊗ σ′) and for x ∈ Sym(J) ⊗ Sym(σ′). Remark 1.10 shows that (36) holds for x ∈ Symd(J ⊗ σ′).

Suppose next x = x_{1} ⊗ x_{2} with x_{1} ∈ Sym(J) and x_{2} ∈ Sym(σ′). Then

Trd(ex) = Tr(( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} )x_{1}) Trd_{A′}(x_{2}).

If char F = 2, then Trd_{A′}(x_{2}) = 0 since σ′ is symplectic. On the other hand, Sym(J) = Skew(J) and Sym(σ′) = Skew(σ′), so f_{⊗}(x) = 0 by definition.

If char F ≠ 2, then Sym(J) = F, hence

Trd(ex) = x_{1} Trd_{A′}(x_{2}).

On the other hand,

f_{⊗}(x) = \frac{1}{2} Trd(x) = x_{1} Trd_{A′}(x_{2}),

so the claim is proved.

Since all the hyperbolic quadratic pairs on a given central simple algebra are conjugate (because hyperbolic quadratic forms of a given dimension are isometric), we may assume henceforth (A, σ, f) = (M_{2}(F) ⊗_{F} A′, J ⊗ σ′, f_{⊗}).

To complete the proof, we show that the embedding ε: L ↪ A which factors through the regular representation ρ: L → End_{F} L satisfies (3). Again, by (37), it suffices to prove (3) for x ∈ Symd(J ⊗ σ′) and for x ∈ Sym(J) ⊗ Sym(σ′).

If x ∈ Symd(J ⊗ σ′), let x = y + (J ⊗ σ′)(y). For ℓ ∈ L,

Trd_{A}(ε(ℓ)x) = Trd_{A}(ε(ℓ)y) + Trd_{A}(σ ∘ ε(ℓ)y).

Since ε(ℓ) + σ ∘ ε(ℓ) = ρ(ℓ) + J ∘ ρ(ℓ) = T_{L/F} (ℓ), the right side is

T_{L/F} (ℓ) Trd_{A}(y) = T_{L/F} f_{⊗}(x),

and it follows that (3) holds for x ∈ Symd(J ⊗ σ′).

If x = x_{1} ⊗ x_{2} with x_{1} ∈ Sym(J) and x_{2} ∈ Sym(σ′), then

Trd_{A}(ε(ℓ)x) = Tr(ρ(ℓ)x_{1}) Trd_{A′}(x_{2}).

If char F ≠ 2, then Sym(J) = F, hence the right side is x_{1}T_{L/F} (ℓ) Trd_{A′}(x_{2}). On the other hand,

f_{⊗}(x) = \frac{1}{2} Trd_{A}(x) = x_{1} Trd_{A′}(x_{2}),

so (3) holds for x ∈ Sym(J) ⊗ Sym(σ′) if char F ≠ 2.

If char F = 2, then Trd_{A′}(x_{2}) = 0 since σ′ is symplectic. On the other hand, Sym(J) = Skew(J) and Sym(σ′) = Skew(σ′) so f_{⊗}(x) = 0 by definition. Therefore,

(3) holds for x ∈ Sym(J) ⊗ Sym(σ′) if char F = 2. □

Proof of Theorem 1.15. If there is an embedding (L, ι) ↪ (A, σ), then Lemma A.8 shows that σ_{L} is hyperbolic. For the converse, use a Witt decomposition

(A, σ) = (A_{0}, σ_{0}) ⊞ (A_{1}, σ_{1})

where σ_{0} is anisotropic and σ_{1} is hyperbolic, see Proposition 1.8. If σ_{L} is hyperbolic, Witt cancellation implies that σ_{0L} is hyperbolic. By Lemma A.9, it follows that there is an embedding ε_{0} : (L, ι) ↪ (A_{0}, σ_{0}). On the other hand, Lemma A.11 yields an embedding ε_{1} : (L, ι) ↪ (A_{1}, σ_{1}). The direct sum of these embeddings is an embedding

ε_{0} ⊞ ε_{1} : (L, ι) ↪ (A_{0}, σ_{0}) ⊞ (A_{1}, σ_{1}) = (A, σ).
□

The proof of Theorem 1.16 follows the same lines:

Proof of Theorem 1.16. The “if” part readily follows from Lemma A.8. For the converse, use a Witt decomposition as in Proposition 1.11,

(A, σ, f) = (A_{0}, σ_{0}, f_{0}) ⊞ (A_{1}, σ_{1}, f_{1})

where (σ_{0}, f_{0}) is anisotropic and (σ_{1}, f_{1}) is hyperbolic. Then ind A = ind A_{1} divides \frac{1}{2} deg A_{1}. If deg A_{1} ≡ 2 mod 4, then ind A is odd, hence the algebra is split and (σ, f) is adjoint to a quadratic form with odd Witt index. This special case is excluded for the converse, so we assume deg A_{1} is divisible by 4. Lemma A.12 then yields an embedding ε_{1} : L ↪ A_{1} for which (3) holds. If (σ_{L}, f_{L}) is hyperbolic, then Witt cancellation implies (σ_{0L}, f_{0L}) is hyperbolic, hence Lemma A.10 yields an embedding ε_{0} : L ↪ A_{0} for which (3) holds. To complete the proof, we show that the direct sum ε = ε_{0} ⊞ ε_{1} : L ↪ A also satisfies (3).

Let e_{0}, e_{1} ∈ A be the symmetric idempotents such that A_{i} = e_{i}Ae_{i} for i = 0, 1. Since e_{0} + e_{1} = 1, we have

x = e_{0}xe_{0} + e_{0}xe_{1} + e_{1}xe_{0} + e_{1}xe_{1} for x ∈ A.

The reduced trace of the two middle terms vanishes since e_{0}e_{1} = e_{1}e_{0} = 0 implies

Trd_{A}(e_{0}xe_{1}) = Trd_{A}(e_{1}e_{0}x) = 0  and  Trd_{A}(e_{1}xe_{0}) = Trd_{A}(e_{0}e_{1}x) = 0.

Therefore,

Trd_{A}(x) = Trd_{A_{0}}(e_{0}xe_{0}) + Trd_{A_{1}}(e_{1}xe_{1}).

For ℓ ∈ L, we have ε(ℓ) = ε_{0}(ℓ)e_{0} + ε_{1}(ℓ)e_{1}, hence

Trd_{A}(ε(ℓ)x) = Trd_{A_{0}}(ε_{0}(ℓ)e_{0}xe_{0}) + Trd_{A_{1}}(ε_{1}(ℓ)e_{1}xe_{1}) for x ∈ Sym(σ).

Since ε_{0} and ε_{1} satisfy (3), it follows that

Trd_{A}(ε(ℓ)x) = T_{L/F} (ℓ)(f_{0}(e_{0}xe_{0}) + f_{1}(e_{1}xe_{1})).

Since, by definition of the orthogonal sum,

f(x) = f_{0}(e_{0}xe_{0}) + f_{1}(e_{1}xe_{1}),

the proof is complete. □

Appendix B. Flat cohomology

Let G be an affine algebraic group scheme (not necessarily smooth) defined over a field F and let \overline{F} be a fixed algebraic closure of F. For any integers n ≥ 1 and i with n ≥ i ≥ 0, define

ε^{i}: ⊗_{F}^{n} \overline{F} → ⊗_{F}^{n+1}\overline{F} , x_{1} ⊗ ⋯ ⊗ x_{n} ↦ x_{1} ⊗ ⋯ ⊗ x_{i−1} ⊗ 1 ⊗ x_{i} ⊗ ⋯ ⊗ x_{n},

and let

d^{i}: G(⊗_{F}^{n} \overline{F}) → G(⊗_{F}^{n+1}\overline{F})

be the induced map.

An element g ∈ G(\overline{F} ⊗ \overline{F}) is a 1-cocycle if

d^{1}g = (d^{0}g)(d^{2}g) in G(\overline{F} ⊗ \overline{F} ⊗ \overline{F}).

Two cocycles g, g′ ∈ G(\overline{F} ⊗ \overline{F}) are cohomologous if there exists h ∈ G(\overline{F} ⊗ \overline{F}) such that

g′ = (d^{0}h)g(d^{1}h)^{−1}.

We thus obtain an equivalence relation, and we denote by H^{1}(F, G) the quotient set. This a pointed set, the distinguished element being the class of the neutral element. If G is abelian, this set has a natural group structure, and one can define higher cohomology groups. We recall only the definition of H^{2}(F, G). A 2-cocycle is an element g ∈ G(\overline{F} ⊗ \overline{F} ⊗ \overline{F}) satisfying

(d^{0}g)(d^{1}g)^{−1}(d^{2}g)(d^{3}g)^{−1} = 1.

A 2-coboundary is a 2-cocycle g ∈ G(\overline{F} ⊗ \overline{F} ⊗ \overline{F}) of the form (d^{0}h)(d^{1}h)^{−1}(d^{2}h) for some h ∈ G(\overline{F} ⊗ \overline{F}). We define H^{2}(F, G) to be the quotient group of the group of 2-cocycles by the group of 2-coboundaries.

If 1 → N → G → H → 1 is an exact sequence of algebraic groups, we have an exact sequence in flat cohomology

1 → N(F) → G(F) → H(F) → H^{1}(F, N) → H^{1}(F, G) → H^{1}(F, H),

see [27, §18.1]. If N, G, H are abelian, all the morphisms are group morphisms. If N is a central subgroup of G, the exact sequence extends to

1 → N(F) → G(F) → H(F) → H^{1}(F, N) → H^{1}(F, G) → H^{1}(F, H) → H^{2}(F, N).

If G is smooth, the flat cohomology sets can be identified with the Galois cohomology sets

H^{1}(F, G) = H^{1}(Gal(F_{s}/F), G(F_{s}))

where F_{s} is a separable closure of F, see [27, §18.5]. Moreover, connecting maps induced by exact sequences agree under this isomorphism. In view of these properties, the standard identifications valid in Galois cohomology under assumptions on the characteristic of F extend to arbitrary fields; for example H^{1}(F, μ_{n}) ≃ F^{×}/F^{×n} and H^{2}(F, μ_{n}) ≃ _{n} Br(F).

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School of Mathematics, University of Southampton, Highfield, SO17 1BJ, United Kingdom

E-mail address: G.W.Berhuy@soton.ac.uk
URL: http://www.maths.soton.ac.uk/staff/Berhuy

Institut de Mathématique Pure et Appliquée, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

E-mail address: tignol@math.ucl.ac.be
URL: http://www.math.ucl.ac.be/membres/tignol