The chain equivalence of totally decomposable orthogonal involutions in characteristic two

A.-H. Nokhodkar

February 21, 2016

Abstract

It is shown that two totally decomposable algebras with involution of orthogonal type over a field of characteristic two are isomorphic if and only if they are chain equivalent.

1 Introduction

The chain equivalence theorem for bilinear Pfister forms describes the isometry class of n-fold Pfister forms in terms of the isometry class of 2-fold Pfister forms (see [6, (3.2)] and [1, (A.1)]). There exist some related results in the literature for certain classes of central simple algebras over a field. In [11], the chain equivalence theorem for biquaternion algebras over a field of characteristic not two was proved (see [2] for the corresponding result in characteristic two). Also, the chain equivalence theorem for tensor products of quaternion algebras over a field of arbitrary characteristic was recently obtained in [3].

Let F be a field of characteristic 2. An algebra with involution (A, σ) over F is called totally decomposable if it decomposes as tensor products of quaternion F -algebras with involution. In [4], a bilinear Pfister form 𝔓𝔣(A, σ), called the Pfister invariant, was associated to every totally decomposable algebra with orthogonal involution (A, σ) over F. In [9, (6.5)], it was shown that the Pfister invariant can be used to classify totally decomposable algebras with orthogonal involution over F. Regarding this result, an analogue chain equivalence for these algebras was defined in [9, (6.7)]. A relevant problem then is whether the isomorphism of such algebras with involution implies that they are chain equivalent (see [9, (6.8)]). In this work we present a solution to this problem.

2 Preliminaries

In this paper, F is a field of characteristic 2. Let V be a finite dimensional vector space over F. A bilinear form 𝔟 : V ×V → F is called anisotropic if 𝔟(v, v) ≠ 0 for every nonzero vector v ∈ V. The form 𝔟 is called metabolic if V has a subspace W with dim W = \frac{1}{2} dim V and 𝔟|_{W ×W} = 0. For λ_{1}, ⋯, λ_{n} ∈ F^{×}, the form ⟨⟨λ_{1}, ⋯, λ_{n}⟩⟩ := ⨂_{i=1}^{n}⟨1, λ_{i}⟩ is called a bilinear Pfister form, where ⟨1, λ_{i}⟩ is the diagonal form 𝔟((x_{1}, x_{2}), (y_{1}, y_{2})) = x_{1}y_{1} + λ_{i}x_{2}y_{2}. By [5, (6.3)], a bilinear Pfister form is either metabolic or anisotropic. We say that 𝔟 = ⟨⟨α_{1}, ⋯, α_{n}⟩⟩ and 𝔟′ = ⟨⟨β_{1}, ⋯, β_{n}⟩⟩ are simply P-equivalent, if either n = 1 and α_{1}F^{×2} = β_{1}F^{×2} or n ⩾ 2 and there exist 1 ⩽ i < j ⩽ n such that ⟨⟨α_{i}, α_{j}⟩⟩ ≃ ⟨⟨β_{i}, β_{j}⟩⟩ and α_{k} = β_{k} for all other k. We say that 𝔟 and 𝔟′ are chain P-equivalent, if there exist bilinear Pfister forms 𝔟_{0}, ⋯, 𝔟_{m} such that 𝔟_{0} = 𝔟, 𝔟_{m} = 𝔟′ and every 𝔟_{i} is simply P-equivalent to 𝔟_{i−1}.

A quaternion algebra over F is a central simple F -algebra of degree 2. Every quaternion algebra Q has a quaternion basis, i.e., a basis {1, u, v, w} satisfying u^{2} + u ∈ F, v^{2} ∈ F^{×} and uv = w = vu + v. It is easily seen that every element v ∈ Q \ F with v^{2} ∈ F^{×} extends to a quaternion basis {1, u, v, uv} of Q. A tensor product of two quaternion algebras is called a biquaternion algebra.

An involution on a central simple F -algebra A is an antiautomorphism of A of period 2. Involutions which restrict to the identity on F are said to be of the first kind. An involution of the first kind is either symplectic or orthogonal (see [7, (2.5)]). The discriminant of an orthogonal involution σ is denoted by disc σ (see [7, (7.2)]). If K/F is a field extension, the scalar extension of (A, σ) to K is denoted by (A, σ)_{K}. We also use the notation Alt(A, σ) = {a − σ(a) | a ∈ A}.

Let (A, σ) be a totally decomposable algebra of degree 2^{n} with orthogonal involution over F. In [9], it was shown that there exists a unique, up to isomorphism, subalgebra S ⊆ F + Alt(A, σ) such that (i) x^{2} ∈ F for x ∈ S; (ii) dim_{F} S = deg_{F} A = 2^{n}; (iii) S is self-centralizing; (iv) S is generated as an F -algebra by n elements. Also, S has a set of alternating generators, i.e., a set {u_{1}, ⋯, u_{n}} consisting of units such that S ≃ F [u_{1}, ⋯, u_{n}] and u_{i_{1}} ⋯ u_{i_{l}} ∈ Alt(A, σ) for every 1 ≤ l ≤ n and 1 ≤ i_{1} < ⋯ < i_{l} ≤ n. We denote the isomorphism class of the subalgebra S by Φ(A, σ). Note that Φ(A, σ) is commutative by [9, (3.2 (i))]. Also, if deg_{F} A ⩽ 4, then Φ(A, σ) is unique as a set. In fact if A is a quaternion algebra, then Φ(A, σ) = F + Alt(A, σ) by dimension count. If A is a biquaternion algebra, then Φ(A, σ) = F + Alt(A, σ)^{+}, where Alt(A, σ)^{+} is the set of square-central elements in Alt(A, σ) (see [10, (4.4)] and [10, (3.9)]).

Let (A, σ) = ⨂_{i=1}^{n}(Q_{i}, σ_{i}) be a decomposition of (A, σ) and choose α_{i} ∈ F^{×} such that disc σ_{i} = α_{i}F^{×2}, i = 1, ⋯, n. As in [4], we call the form ⟨⟨α_{1}, ⋯, α_{n}⟩⟩ the Pfister invariant of (A, σ) and we denote it by 𝔓𝔣(A, σ). Note that by [4, (7.2)], the Pfister invariant is independent of the decomposition of (A, σ).

With the above notations, we have the following results.

Theorem 2.1. ([9, (5.7)]) For a totally decomposable algebra of degree 2^{n} with orthogonal involution (A, σ) over F, the following conditions are equivalent:

(i) (A, σ) ≃ (M_{2^{n}}(F), t), where t is the transpose involution. (ii) 𝔓𝔣(A, σ) ≃ ⟨⟨1, ⋯, 1⟩⟩. (iii) x^{2} ∈ F^{2} for every x ∈ Φ(A, σ).

Theorem 2.2. ([9, (6.5)]) Let (A, σ) and (A′, σ′) be two totally decomposable algebras with orthogonal involution over F. If A ≃ A′ and 𝔓𝔣(A, σ) ≃ 𝔓𝔣(A′, σ′), then (A, σ) ≃ (A′, σ′).

3 The chain lemma

Our first result, which strengthens [9, (5.6)], gives a natural description of the Pfister invariant.

Lemma 3.1. Let (A, σ) be a totally decomposable algebra of degree 2^{n} with orthogonal involution over F. If 𝔓𝔣(A, σ) ≃ ⟨⟨α_{1}, ⋯, α_{n}⟩⟩ for some α_{1}, ⋯, α_{n} ∈ F^{×}, then there exists a decomposition (A, σ) ≃ ⨂_{i=1}^{n}(Q_{i}, σ_{i}) into quaternion F - algebras with involution such that disc σ_{i} = α_{i}F^{×2}, i = 1, ⋯, n.

Proof. By [9, (5.5)] and [9, (5.6)] there exists a set of alternating generators {u_{1}, ⋯, u_{n}} of Φ(A, σ) such that u_{i}^{2} = α_{i}, i = 1, ⋯, n. If α_{i} ∈ F^{2} for every i, the result follows from (2.1). Thus (by re-indexing if necessary) we may assume that α_{n} ∉ F^{2}. It is enough to prove that there exists a decomposition (A, σ) ≃ ⨂_{i=1}^{n}(Q_{i}, σ_{i}) such that u_{i} ∈ Alt(Q_{i}, σ_{i}), i = 1, ⋯, n. We use induction on n. The case n = 1 is evident, so suppose that n > 1. Let B = C_{A}(u_{n}) be the centralizer of u_{n} in A and set K = F [u_{n}] = F (\sqrt{α}_{n}). By [9, (6.3)] and [9, (6.4)], (B, σ|_{B}) is a totally decomposable algebra with orthogonal involution over K and {u_{1}, ⋯, u_{n−1}} is a set of alternating generators of Φ(B, σ|_{B}). By induction hypothesis there exists a decomposition

(B, σ|_{B}) ≃_{K} (Q′_{1}, σ′_{1}) ⊗_{K} ⋯ ⊗_{K} (Q′_{n−1}, σ′_{n−1}),

into quaternion K-algebras with involution such that u_{i} ∈ Alt(Q′_{i}, σ′_{i}) for i = 1, ⋯, n − 1. By dimension count we have Φ(Q′_{i}, σ′_{i}) = K + Ku_{i}. Since K^{2} ⊆ F and u_{i}^{2} ∈ F, we get x^{2} ∈ F for every x ∈ Φ(Q′_{i}, σ′_{i}). By [9, (6.1)] there exists a quaternion F -algebra Q_{i} ⊆ Q′_{i} such that (Q′_{i}, σ′_{i}) ≃_{K} (Q_{i}, σ|_{Q_{i}}) ⊗ (K, id) and u_{i} ∈ Alt(Q_{i}, σ|_{Q_{i}}), i = 1, ⋯, n − 1. Set Q_{n} = C_{A}(Q_{1} ⊗ ⋯ ⊗ Q_{n−1}). Then Q_{n} is a quaternion F -algebra and (A, σ) ≃ (Q_{1}, σ|_{Q_{1}}) ⊗ ⋯ ⊗ (Q_{n}, σ|_{Q_{n}}). Since u_{n} ∈ K = Z(B) ⊆ C_{A}(Q_{1} ⊗ ⋯ ⊗ Q_{n−1}) = Q_{n}, we obtain u_{n} ∈ Q_{n}. Finally [8, (3.5)] implies that u_{n} ∈ Alt(Q_{n}, σ|_{Q_{n}}). This completes the proof.

Lemma 3.2. Let K/F be a field extension satisfying K^{2} ⊆ F. Let Q and Q′ be quaternion algebras over F and let v′ ∈ Q′ \ F with v^{′2} ∈ F^{×}. If there exists an isomorphism of K-algebras f : Q′_{K} ≃ Q_{K} such that f(v′⊗1) ∈ Q⊗F, then there exists η ∈ K such that f(Q′ ⊗ F) ⊆ Q ⊗ F [η]. In addition, if {1, u, v, uv} and {1, u′, v′, u′v′} are respective quaternion bases of Q and Q′ and f(v′ ⊗1) = v⊗1, then f(u′ ⊗ 1) = 1 ⊗ λ + u ⊗ 1 + v ⊗ η for some λ ∈ F.

Proof. The first statement follows from the second, since f(u′ ⊗1) and f(v′ ⊗1) generate f(Q′ ⊗ F) as an F -algebra. To prove the second statement write f(u′ ⊗ 1) = 1 ⊗ η_{1} + u ⊗ η_{2} + v ⊗ η_{3} + uv ⊗ η_{4} for some η_{1}, ⋯, η_{4} ∈ K. Since

v ⊗ 1 = f(v′ ⊗ 1) = f((u′v′ + v′u′) ⊗ 1)
= f(u′ ⊗ 1)(v ⊗ 1) + (v ⊗ 1)f(u′ ⊗ 1) = v ⊗ η_{2} + v^{2} ⊗ η_{4},

we get η_{4} = 0 and η_{2} = 1, i.e., f(u′ ⊗ 1) = 1 ⊗ η_{1} + u ⊗ 1 + v ⊗ η_{3}. Hence

f((u^{′2} + u′) ⊗ 1) = f(u′ ⊗ 1)^{2} + f(u′ ⊗ 1)
= 1 ⊗ η_{1}^{2} + u^{2} ⊗ 1 + v^{2} ⊗ η_{3}^{2} + (uv + vu) ⊗ η_{3}
+ 1 ⊗ η_{1} + u ⊗ 1 + v ⊗ η_{3}
= 1 ⊗ η_{1}^{2} + (u^{2} + u) ⊗ 1 + v^{2} ⊗ η_{3}^{2} + 1 ⊗ η_{1}.

As f((u^{′2} + u′) ⊗ 1) ∈ F and K^{2} ⊆ F, the above relations imply that η_{1} ∈ F, proving the result.

The following definition was given in [9, (6.7)].

Definition 3.3. Let (A, σ) = ⨂_{i=1}^{n}(Q_{i}, σ_{i}) and (A′, σ′) = ⨂_{i=1}^{n}(Q′_{i}, σ′_{i}) be two totally decomposable algebras with orthogonal involution over F. We say that (A, σ) and (A′, σ′) are simply equivalent if either n = 1 and (Q_{1}, σ_{1}) ≃ (Q′_{1}, σ′_{1}) or n ⩾ 2 and there exist 1 ⩽ i < j ⩽ n such that (Q_{i}, σ_{i}) ⊗ (Q_{j}, σ_{j}) ≃ (Q′_{i}, σ′_{i}) ⊗ (Q′_{j}, σ′_{j}) and (Q_{k}, σ_{k}) ≃ (Q′_{k}, σ′_{k}) for k ≠ i, j. We say that (A, σ) and (A′, σ′) are chain equivalent if there exist totally decomposable algebras with involution (A_{0}, τ_{0}), ⋯, (A_{m}, τ_{m}) such that (A, σ) = (A_{0}, τ_{0}), (A′, σ′) = (A_{m}, τ_{m}) and for every i = 0, ⋯, m−1, (A_{i}, τ_{i}) and (A_{i+1}, τ_{i+1}) are simply equivalent. We write (A, σ) ≈ (A′, σ′) if (A, σ) and (A′, σ′) are chain equivalent.

Since the symmetric group is generated by transpositions, for every isometry ρ of {1, ⋯, n} we have

⨂_{i=1}^{n}(Q_{i}, σ_{i}) ≈ ⨂_{i=1}^{n}(Q_{ρ(i)}, σ_{ρ(i)}).

Lemma 3.4. Let (A, σ) = ⨂_{i=1}^{3}(Q_{i}, σ_{i}) and (A′, σ′) = ⨂_{i=1}^{3}(Q′_{i}, σ′_{i}) be totally decomposable algebras with orthogonal involution over F. Let α_{i} ∈ F^{×} (resp. α′_{i} ∈ F^{×}) be a representative of the class disc σ_{i} (resp. disc σ′_{i}), i = 1, 2, 3. Suppose that A ≃ A′, ⟨⟨α_{1}, α_{2}⟩⟩ ≃ ⟨⟨α′_{1}, α′_{2}⟩⟩ and α_{3} = α′_{3}. If ⟨⟨α_{1}, α_{2}⟩⟩ is metabolic, then (A, σ) and (A′, σ′) are chain equivalent.

Proof. By [1, (A.5)] there exists β ∈ F such that ⟨⟨α_{1}, α_{2}⟩⟩ ≃ ⟨⟨1, β⟩⟩. Thus, according to (3.1) and (2.1), one can write

(Q_{1}, σ_{1}) ⊗ (Q_{2}, σ_{2}) ≃ (M_{2}(F), t) ⊗ (Q_{0}, σ_{0}),
(Q′_{1}, σ′_{1}) ⊗ (Q′_{2}, σ′_{2}) ≃ (M_{2}(F), t) ⊗ (Q′_{0}, σ′_{0}),

where (Q_{0}, σ_{0}) and (Q′_{0}, σ′_{0}) are quaternion algebras with orthogonal involution over F and disc σ_{0} = disc σ′_{0} = βF^{×2}. It follows that A ≃ M_{2}(F) ⊗ Q_{0} ⊗ Q_{3} and A′ ≃ M_{2}(F) ⊗ Q′_{0} ⊗ Q′_{3}. Since A ≃ A′, we get Q_{0} ⊗ Q_{3} ≃ Q′_{0} ⊗ Q′_{3}. We also have

𝔓𝔣(Q_{0} ⊗ Q_{3}, σ_{0} ⊗ σ_{3}) ≃ ⟨⟨β, α_{3}⟩⟩ ≃ 𝔓𝔣(Q′_{0} ⊗ Q′_{3}, σ′_{0} ⊗ σ′_{3}),

which implies that (Q_{0}, σ_{0}) ⊗ (Q_{3}, σ_{3}) ≃ (Q′_{0}, , σ′_{0}) ⊗ (Q′_{3}, σ′_{3}) by (2.2). Thus,

(A, σ) = ⨂_{i=1}^{3}(Q_{i}, σ_{i}) ≈ (M_{2}(F), t) ⊗ (Q_{0}, σ_{0}) ⊗ (Q_{3}, σ_{3})
≈ (M_{2}(F), t) ⊗ (Q′_{0}, σ′_{0}) ⊗ (Q′_{3}, σ′_{3}) ≈ ⨂_{i=1}^{3}(Q′_{i}, σ′_{i}) = (A′, σ′).

Lemma 3.5. ([9, (6.3)]) Let (A, σ) be a totally decomposable algebra with orthogonal involution over F. For every v ∈ Φ(A, σ) with v^{2} ∈ F^{×} \F^{×2}, there exists a σ-invariant quaternion F -algebra Q ⊆ A such that v ∈ Φ(Q, σ|_{Q}).

Proof. By [9, (6.3)], there exists a σ-invariant quaternion F -algebra Q ⊆ A containing v. Write v = λ + w for some λ ∈ F and w ∈ Alt(A, σ). Then w ∈ Alt(Q, σ|_{Q}) by [8, (3.5)], hence v = λ + w ∈ F + Alt(Q, σ|_{Q}) = Φ(Q, σ|_{Q}).

Lemma 3.6. Let (A, σ) = ⨂_{i=1}^{3}(Q_{i}, σ_{i}) and (A′, σ′) = ⨂_{i=1}^{3}(Q′_{i}, σ′_{i}) be two totally decomposable algebras with orthogonal involution over F. If 𝔓𝔣(A, σ) and 𝔓𝔣(A′, σ′) are simply P-equivalent and A ≃ A′, then (A, σ) and (A′, σ′) are chain equivalent.

Proof. Choose invertible elements v_{i} ∈ Alt(Q_{i}, σ_{i}) and v′_{i} ∈ Alt(Q′_{i}, σ′_{i}), i = 1, 2, 3. Set α_{i} = v_{i}^{2} ∈ F^{×} and α′_{i} = v_{i}^{′2} ∈ F^{×}. Then 𝔓𝔣(A, σ) ≃ ⟨⟨α_{1}, α_{2}, α_{3}⟩⟩ and 𝔓𝔣(A′, σ′) ≃ ⟨⟨α′_{1}, α′_{2}, α′_{3}⟩⟩. By re-indexing if necessary, we may assume that α_{3} = α′_{3} and ⟨⟨α_{1}, α_{2}⟩⟩ ≃ ⟨⟨α′_{1}, α′_{2}⟩⟩. In view of (3.4), it suffices to consider the case where ⟨⟨α_{1}, α_{2}⟩⟩ is anisotropic. Set K = F [v_{1}, v_{2}] and K′ = F [v′_{1}, v′_{2}]. Then K ≃ K′ ≃ F (\sqrt{α}_{1}, \sqrt{α}_{2}), because ⟨⟨α_{1}, α_{2}⟩⟩ is anisotropic. Consider C_{A}(K) ≃_{K} Q_{3} ⊗_{F} K and C_{A′}(K′) ≃_{K′} Q′_{3} ⊗_{F} K′. As K ≃ K′, one may consider Q′_{3} ⊗_{F} K′ as a quaternion algebra over K, which is isomorphic to Q_{3} ⊗_{F} K. Since disc(σ_{3} ⊗ id) = disc(σ′_{3} ⊗ id) = α_{3}K^{×2}, by [7, (7.4)] there exists an isomorphism of K-algebras with involution

f : (Q′_{3} ⊗ K′, σ′_{3} ⊗ id) → (Q_{3} ⊗ K, σ_{3} ⊗ id).  (1)

Dimension count shows that Alt(Q_{3} ⊗ K, σ_{3} ⊗ id) = v_{3} ⊗ K and Alt(Q′_{3} ⊗ K′, σ′_{3} ⊗ id) = v′_{3} ⊗ K′, hence f(v′_{3} ⊗ 1) = v_{3} ⊗ β for some β ∈ K. The relations v_{3}^{2} = v_{3}^{′2} = α_{3} then imply that β = 1, i.e., f(v′_{3} ⊗ 1) = v_{3} ⊗ 1 ∈ Q_{3} ⊗ F. By (3.2) there exists η ∈ K such that f(Q′_{3} ⊗ F) ⊆ Q_{3} ⊗ F [η].

If η ∈ F, then Q′_{3} ≃ Q_{3}. Hence (Q′_{3}, σ′_{3}) ≃ (Q_{3}, σ_{3}) by [7, (7.4)]. The isomorphism A ≃ A′ then implies that Q_{1} ⊗Q_{2} ≃ Q′_{1} ⊗Q′_{2}. Thus, (Q_{1}, σ_{1})⊗(Q_{2}, σ_{2}) ≃ (Q′_{1}, σ′_{1}) ⊗ (Q′_{2}, σ′_{2}) by (2.2), i.e., (A, σ) and (A′, σ′) are simply equivalent. Suppose that η ∉ F, hence η^{2} ∈ F^{×} \ F^{×2}. As η ∈ K ≃ Φ(Q_{1} ⊗ Q_{2}, σ_{1} ⊗ σ_{2}), by (3.5) there exists a σ-invariant quaternion algebra Q_{4} ⊆ Q_{1} ⊗ Q_{2} such that η ∈ Φ(Q_{4}, σ|_{Q_{4}}). Let Q_{5} be the centralizer of Q_{4} in Q_{1} ⊗ Q_{2}. Then

(Q_{1}, σ_{1}) ⊗ (Q_{2}, σ_{2}) ≃ (Q_{4}, σ|_{Q_{4}}) ⊗ (Q_{5}, σ|_{Q_{5}}),  (2)

which implies that

𝔓𝔣(Q_{4} ⊗ Q_{5}, σ|_{Q_{4}} ⊗ σ|_{Q_{5}}) ≃ ⟨⟨α_{1}, α_{2}⟩⟩,  (3)

by [4, (7.2)]. Since f(Q′_{3} ⊗ F) ⊆ Q_{3} ⊗ F [η] and η ∈ Q_{4}, we get f(Q′_{3} ⊗ F) ⊆ Q_{3} ⊗ Q_{4}. Let Q_{6} be the centralizer of f(Q′_{3} ⊗ F) in Q_{3} ⊗ Q_{4}. Then

(Q_{3}, σ_{3}) ⊗ (Q_{4}, σ|_{Q_{4}}) ≃ (Q_{6}, σ|_{Q_{6}}) ⊗ f(Q′_{3} ⊗ F, σ′_{3} ⊗ id).  (4)

By (2) and (4) we have

(A, σ) = (Q_{1}, σ_{1}) ⊗ (Q_{2}, σ_{2}) ⊗ (Q_{3}, σ_{3})
≈ (Q_{4}, σ|_{Q_{4}}) ⊗ (Q_{5}, σ|_{Q_{5}}) ⊗ (Q_{3}, σ_{3})
≈ (Q_{5}, σ|_{Q_{5}}) ⊗ (Q_{3}, σ_{3}) ⊗ (Q_{4}, σ|_{Q_{4}})
≈ (Q_{5}, σ|_{Q_{5}}) ⊗ (Q_{6}, σ|_{Q_{6}}) ⊗ (Q′_{3}, σ′_{3}).  (5)

We claim that disc σ|_{Q_{6}} = disc σ|_{Q_{4}}. If this is true, then

𝔓𝔣(Q_{5} ⊗ Q_{6}, σ|_{Q_{5}} ⊗ σ|_{Q_{6}}) ≃ 𝔓𝔣(Q_{5} ⊗ Q_{4}, σ|_{Q_{5}} ⊗ σ|_{Q_{4}}).

Thus, using (3) we obtain

𝔓𝔣(Q_{5} ⊗ Q_{6}, σ|_{Q_{5}} ⊗ σ|_{Q_{6}}) ≃ ⟨⟨α_{1}, α_{2}⟩⟩ ≃ 𝔓𝔣(Q′_{1} ⊗ Q′_{2}, σ′_{1} ⊗ σ′_{2}).  (6)

The chain equivalence (5) together with A ≃ A′ yields A′ ≃ Q_{5} ⊗ Q_{6} ⊗ Q′_{3}, hence Q_{5} ⊗ Q_{6} ≃ Q′_{1} ⊗ Q′_{2}. By (6) and (2.2) we have (Q_{5}, σ|_{Q_{5}}) ⊗ (Q_{6}, σ|_{Q_{6}}) ≃ (Q′_{1}, σ′_{1})⊗(Q′_{2}, σ′_{2}). This, together with (5) yields the desired chain equivalence:

(A, σ) ≈ (Q_{5}, σ|_{Q_{5}}) ⊗ (Q_{6}, σ|_{Q_{6}}) ⊗ (Q′_{3}, σ′_{3})
≈ (Q′_{1}, σ′_{1}) ⊗ (Q′_{2}, σ′_{2}) ⊗ (Q′_{3}, σ′_{3}) = (A′, σ′).

We now proceed to prove the claim. Let v_{6} ∈ Alt(Q_{6}, σ|_{Q_{6}}) ⊆ Q_{3} ⊗ Q_{4} and v_{4} ∈ Alt(Q_{4}, σ|_{Q_{4}}) be two units. It is enough to show that v_{6} = μ(1 ⊗ v_{4}) for some μ ∈ F. The element

v_{6} ∈ Alt(Q_{6}, σ|_{Q_{6}}) ⊆ Alt(Q_{3} ⊗ Q_{4}, σ_{3} ⊗ σ|_{Q_{4}})

is square-central. Hence v_{6} ∈ Φ(Q_{3} ⊗ Q_{4}, σ_{3} ⊗ σ|_{Q_{4}}) = F [v_{3} ⊗ 1, 1 ⊗ v_{4}] by [10, (4.4)], i.e., there exist a, b, c, d ∈ F such that

v_{6} = a(v_{3} ⊗ 1) + b(1 ⊗ v_{4}) + c(v_{3} ⊗ v_{4}) + d.

Since σ_{3} ⊗ σ|_{Q_{4}} is orthogonal, by [7, (2.6)] we have 1 ∉ Alt(Q_{3} ⊗ Q_{4}, σ_{3} ⊗ σ|_{Q_{4}}), hence d = 0. On the other hand by extending {v_{3}} and {v′_{3}} to quaternion bases {u_{3}, v_{3}, u_{3}v_{3}} of Q_{3} and {u′_{3}, v′_{3}, u′_{3}v′_{3}} of Q′_{3} and using (3.2) for the map f in (1), we get

f(u′_{3} ⊗ 1) = 1 ⊗ λ + u_{3} ⊗ 1 + v_{3} ⊗ η ∈ Q_{3} ⊗ F [η] ⊆ Q_{3} ⊗ Q_{4},

for some λ ∈ F. Thus,

v_{6}f(u′_{3}⊗ 1) + f(u′_{3}⊗ 1)v_{6} = a(v_{3}u_{3} + u_{3}v_{3}) ⊗ 1 + bv_{3} ⊗ (v_{4}η + ηv_{4})
+ c(v_{3}u_{3} + u_{3}v_{3}) ⊗ v_{4} + cα_{3} ⊗ (v_{4}η + ηv_{4})
= av_{3}⊗ 1 + (bv_{3} + cα_{3}) ⊗ (v_{4}η + ηv_{4}) + cv_{3}⊗ v_{4}. (7)

As v_{4}, η ∈ Φ(Q_{4}, σ|_{Q_{4}}), we have ηv_{4} = v_{4}η. Also, v_{6} ∈ Q_{6} commutes with f(u′_{3} ⊗ 1) ∈ f(Q′_{3} ⊗ F), i.e., v_{6}f(u′_{3} ⊗ 1) + f(u′_{3} ⊗ 1)v_{6} = 0. Therefore, (7) leads to a = c = 0, hence v_{6} = b(1 ⊗ v_{4}), proving the claim.

Remark 3.7. Let K = F (\sqrt{α}) be a quadratic field extension. Consider a bilinear Pfister form 𝔟 over F and let 𝔟_{K} be the scalar extension of 𝔟 to K. If 𝔟_{K} is metabolic then 𝔟 ⊗ ⟨⟨α⟩⟩ is also metabolic. In fact if 𝔟 is itself metabolic, the conclusion is evident. Otherwise, 𝔟 is anisotropic and the result follows form [5, (34.29 (2))] and [5, (34.7)]. Note that this implies that if 𝔟_{K} ≃ 𝔟′_{K} for some bilinear Pfister form 𝔟′ over F, then 𝔟 ⊗ ⟨⟨α⟩⟩ ≃ 𝔟′ ⊗ ⟨⟨α⟩⟩.

The following result gives a solution to [9, (6.8)].

Theorem 3.8. Let (A, σ) ≃ ⨂_{i=1}^{n}(Q_{i}, σ_{i}) and (A′, σ′) ≃ ⨂_{i=1}^{n}(Q′_{i}, σ′_{i}) be two totally decomposable algebras with orthogonal involution over F. Then (A, σ) ≃ (A′, σ′) if and only if (A, σ) and (A′, σ′) are chain equivalent.

Proof. The “if” part is evident. To prove the converse, let deg_{F} A = 2^{n}. The case n ⩽ 2 is trivial, so suppose that n ⩾ 3. By [4, (7.2)] we have 𝔓𝔣(A, σ) ≃ 𝔓𝔣(A′, σ′). Hence by [1, (A. 1)], there exist bilinear Pfister forms 𝔟_{0}, ⋯, 𝔟_{m} such that 𝔟_{0} = 𝔓𝔣(A, σ), 𝔟_{m} = 𝔓𝔣(A′, σ′) and for i = 0, ⋯, m − 1, 𝔟_{i} is simply P-equivalent to 𝔟_{i+1}. Set (A_{0}, τ_{0}) = (A, σ) and (A_{m}, τ_{m}) = (A′, σ′). By (3.1) every 𝔟_{i}, i = 1, ⋯, m − 1, can be realised as the Pfister invariant of a totally decomposable algebra with orthogonal involution (A_{i}, τ_{i}) over F with A_{i} ≃ A. We show that for i = 0, ⋯, m−1, (A_{i}, τ_{i}) and (A_{i+1}, τ_{i+1}) are chain equivalent. By induction, it suffices to consider the case where m = 1 (i.e., we may assume that 𝔓𝔣(A, σ) and 𝔓𝔣(A′, σ′) are simply P-equivalent). If n = 3 the result follows from (3.6). So suppose that n ⩾ 4.

For i = 1, ⋯, n, choose invertible elements v_{i} ∈ Alt(Q_{i}, σ_{i}) and v′_{i} ∈ Alt(Q′_{i}, σ′_{i}) and set α_{i} = v_{i}^{2} ∈ F^{×} and α′_{i} = v_{i}^{′2} ∈ F^{×}. Then 𝔓𝔣(A, σ) ≃ ⟨⟨α_{1}, ⋯, α_{n}⟩⟩ and 𝔓𝔣(A′, σ′) ≃ ⟨⟨α′_{1}, ⋯, α′_{n}⟩⟩. By re-indexing if necessary, we may assume that

⟨⟨α_{1}, α_{2}⟩⟩ ≃ ⟨⟨α′_{1}, α′_{2}⟩⟩ and α_{i} = α′_{i} for i = 3, ⋯, n.  (8)

Suppose first that α_{n} ∈ F^{×2}. Then (Q_{n}, σ_{n}) ≃ (Q′_{n}, σ′_{n}) ≃ (M_{2}(F), t) by (2.1). Set C = ⨂_{i=1}^{n−1} Q_{i} and C′ = ⨂_{i=1}^{n−1} Q′_{i}, so that C ≃ C_{A}(Q_{n}) ≃ C_{A′}(Q′_{n}) ≃ C′. Using (8) we have

𝔓𝔣(C, σ|_{C}) ≃ 𝔓𝔣(C′, σ′|_{C′}) ≃ ⟨⟨α_{1}, ⋯, α_{n−1}⟩⟩,

hence (C, σ|_{C}) ≃ (C′, σ′|_{C′}) by (2.2). By induction hypothesis, (C, σ|_{C}) and (C′, σ′|_{C′}) are chain equivalent. Thus, (A, σ) ≈ (C, σ|_{C})⊗(M_{2}(F), t) and (A′, σ′) ≈ (C′, σ′|_{C′}) ⊗ (M_{2}(F), t) are also chain equivalent.

Suppose now that α_{n} ∉ F^{×2}. Set B = C_{A}(v_{n}), B′ = C_{A′}(v′_{n}), K = F [v_{n}] and K′ = F [v′_{n}] ≃ F (\sqrt{α}_{n}) ≃ K. Then

(B, σ|_{B}) ≃_{K} ⨂_{i=1}^{n−1}(Q_{i}, σ_{i})_{K} and (B′, σ′|_{B′}) ≃_{K} ⨂_{i=1}^{n−1}(Q′_{i}, σ′_{i})_{K′},

are totally decomposable algebras with orthogonal involution over K and K′ respectively. Since A ≃_{F} A′, we have B ≃_{K} B′. Also, using (8) we get

𝔓𝔣(B, σ|_{B}) ≃ ⟨⟨α_{1}, ⋯, α_{n−1}⟩⟩_{K} ≃ ⟨⟨α′_{1}, ⋯, α′_{n−1}⟩⟩_{K} ≃ 𝔓𝔣(B′, σ′|_{B′}).

Thus, (B, σ|_{B}) ≃_{K} (B′, σ′|_{B′}) by (2.2). By induction hypothesis we get (B, σ|_{B}) ≈ (B′, σ′|_{B′}). Again, using induction, it suffices to consider the case where (B, σ|_{B}) and (B′, σ′|_{B′}) are simply equivalent (note that every totally decomposable algebra with involution (B′′, σ′′) over K with (B′′, σ′′) ≃ (B, σ|_{B}) has a decomposition of the form ⨂_{i=1}^{n−1}(Q′′_{i}, σ′′_{i})_{K}, where every (Q′′_{i}, σ′′_{i}) is a quaternion algebra with orthogonal involution over F). By re-indexing, we may assume that (Q_{n−2}, σ_{n−2})_{K} ⊗ (Q_{n−1}, σ_{n−1})_{K} ≃_{K} (Q′_{n−2}, σ′_{n−2})_{K} ⊗ (Q′_{n−1}, σ′_{n−1})_{K} and

(Q_{i}, σ_{i})_{K} ≃_{K} (Q′_{i}, σ′_{i})_{K′}, for i = 1, ⋯, n − 3,  (9)

In particular, ⨂_{i=2}^{n−1}(Q_{i}, σ_{i})_{K} ≈ ⨂_{i=2}^{n−1}(Q′_{i}, σ′_{i})_{K′}, which implies that

⟨⟨α_{2}, ⋯, α_{n−1}⟩⟩_{K} ≃ ⟨⟨α′_{2}, ⋯, α′_{n−1}⟩⟩_{K}.  (10)

Since n − 3 ⩾ 1, (9) gives an isomorphism of K-algebras with involution

f : (Q_{1}, σ_{1}) ⊗ (K, id) ≃ (Q′_{1}, σ′_{1}) ⊗ (K′, id).

The element 1 ⊗ v_{n} lies in the center of Q_{1} ⊗ K. Thus, there exist a, b ∈ F such that f(1⊗v_{n}) = 1⊗(a+bv′_{n}). Squaring both sides implies that α_{n} = a^{2} +b^{2}α_{n}. The assumption α_{n} ∉ F^{2} then yields a = 0 and b = 1, i.e., f(1⊗v_{n}) = 1⊗v′_{n}. As (K′, id) ⊆ (Q′_{n}, σ′_{n}), the isomorphism f induces a monomorphism of F -algebras with involution

g : (Q_{1}, σ_{1}) ⊗ (K, id) ↪ (Q′_{1}, σ′_{1}) ⊗ (Q′_{n}, σ′_{n}),

with g(1 ⊗ v_{n}) = 1 ⊗ v′_{n}. Let Q′_{0} be the centralizer of g(Q_{1} ⊗ F) in Q′_{1} ⊗ Q′_{n}. Then

(Q′_{1}, σ′_{1}) ⊗ (Q′_{n}, σ′_{n}) ≃_{F} (Q′_{0}, σ′|_{Q′_{0}}) ⊗ (Q_{1}, σ_{1}).  (11)

Since the element 1 ⊗ v_{n} ∈ Q_{1} ⊗ K commutes with Q_{1} ⊗ F, we get 1 ⊗ v′_{n} = g(1 ⊗ v_{n}) ∈ Q′_{0}, which implies that 1 ⊗ v′_{n} ∈ Alt(Q′_{0}, σ′|_{Q′_{0}}) by [8, (3.5)]. Thus,

disc σ′|_{Q′_{0}} = α_{n}F^{×2} ∈ F^{×}/F^{×2}.  (12)

Using (11) we have

(A′, σ′) ≈ (Q′_{2}, σ′_{2}) ⊗ ⋯ ⊗ (Q′_{n−1}, σ′_{n−1}) ⊗ (Q′_{0}, σ′|_{Q′_{0}}) ⊗ (Q_{1}, σ_{1}).  (13)

This, together with A ≃ A′ implies that

Q_{2} ⊗ ⋯ ⊗ Q_{n} ≃ Q′_{2} ⊗ ⋯ ⊗ Q′_{n−1} ⊗ Q′_{0}.  (14)

The isometry (10) and (3.7) show that

⟨⟨α_{2}, ⋯, α_{n−1}, α_{n}⟩⟩ ≃ ⟨⟨α′_{2}, ⋯, α′_{n−1}, α_{n}⟩⟩,

hence, thanks to (12), the Pfister invariants of (D, τ) := (Q_{2}, σ_{2})⊗⋯⊗(Q_{n}, σ_{n}) and (D′, τ′) := (Q′_{2}, σ′_{2}) ⊗ ⋯ ⊗ (Q′_{n−1}, σ′_{n−1}) ⊗ (Q′_{0}, σ′|_{Q′_{0}}) are isometric. Using (14) and (2.2) we obtain (D, τ) ≃ (D′, τ′). By induction hypothesis, (D, τ) and (D′, τ′) are chain equivalent. Thus, by (13) we have

(A′, σ′) ≈ (Q′_{2}, σ′_{2}) ⊗ ⋯ ⊗ (Q′_{n−1}, σ′_{n−1}) ⊗ (Q′_{0}, σ′|_{Q′_{0}}) ⊗ (Q_{1}, σ_{1})
= (D′, τ′) ⊗ (Q_{1}, σ_{1}) ≈ (D, τ) ⊗ (Q_{1}, σ_{1})
= (Q_{2}, σ_{2}) ⊗ ⋯ ⊗ (Q_{n}, σ_{n}) ⊗ (Q_{1}, σ_{1}) ≈ (A, σ).

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A.-H. Nokhodkar, a.nokhodkar@kashanu.ac.ir

Department of Pure Mathematics, Faculty of Science, University of Kashan, P. O. Box 87317-53153, Kashan, Iran.