INVOLUTIONS ON TENSOR PRODUCTS OF QUATERNION ALGEBRAS

DEMBA BARRY

Abstract. We study possible decompositions of totally decomposable algebras with involution, that is, tensor products of quaternion algebras with involution. In particular, we are interested in decompositions in which one or several factors are the split quaternion algebra M_{2}(F), endowed with an orthogonal involution. Using the theory of gauges, developed by Tignol-Wadsworth, we construct examples of algebras isomorphic to a tensor product of quaternion algebras with k split factors, endowed with an involution which is totally decomposable, but does not admit any decomposition with k factors M_{2}(F) with involution. This extends an earlier result of Sivatski where the algebra considered is of degree 8 and index 4, and endowed with some orthogonal involution.

1. Introduction

Let F be a field of characteristic different from 2. In [Mer81], Merkurjev showed that a central simple F -algebra of exponent 2 is always Brauer-equivalent to a tensor product of quaternion algebras. In degree 8, this result was first proved by Tignol (see for instance [Jac96, Thm. 5.6.38]). More precisely, let A be a central simple F -algebra of degree 8 and exponent 2. Tignol showed that M_{2}(A) is isomorphic to a tensor product of four quaternion algebras. Moreover, if A is indecomposable (i.e, A does not contain any quaternion subalgebra, see [ART79]), one may check that M_{2}(A) does not admit any decomposition as a tensor product of quaternion algebras in which one the quarternion factors is split, that is isomorphic to M_{2}(F). In this paper, we are interested in the analogue question for algebras with involution.

A central simple algebra A of exponent 2 is called totally decomposable if A ≃ Q_{1} ⊗⋯⊗Q_{m}, for some quaternion algebras Q_{i}. The set {Q_{1}, …, Q_{m}} is called a set of factors of A. Moreover, we say that A is totally decomposable with k split factors if it admits a set of factors containing k split factors M_{2}(F). This implies that the co-index of A, which is the ratio deg A/ ind A is a multiple of 2^{k}; but as explained before, this co-index condition is not sufficient for totally decomposable algebra to admit a decomposition with k split factors. Assume now that A is endowed with an F -linear involution σ. We say that (A, σ) is totally decomposable if (A, σ) ≃

2010 Mathematics Subject Classification. Primary 16W10; Secondary 16K20, 16W60, 16W70, 11E39.

Key words and phrases. Central simple algebra, Quaternion algebra, Involution, Valuation, Gauge, Armature.

2 D. BARRY

(Q_{1}, σ_{1}) ⊗ ⋯ ⊗ (Q_{m}, σ_{m}) and totally decomposable with k split factors if it admits such a decomposition with k factors isomorphic to M_{2}(F) endowed with some F - linear involution. The main question we are interested in is the following: let A be a central simple algebra of co-index 2^{k} such that A is totally decomposable with k split factors. Let σ be an anisotropic involution on A. If (A, σ) is totally decomposable, is it totally decomposable with k split factors?

As we now proceed to explain, a complete answer is known is the orthogonal case. Let (A, σ) be a totally decomposable algebra with orthogonal involution of degree 2^{m}. If A is split, Becher showed in [Bec08] that (A, σ) ≃ (M_{2^{m}}(F), ad_{π}) where π is an m-fold Pfister form over F and ad_{π} is the involution on M_{2^{m}}(F) that is adjoint to the polar form of π. By definition of Pfister forms, it follows that (A, σ) is totally decomposable with m split factors. This result is known in the literature as the Pfister Factor Conjecture (see [Sha00, Chap. 9]). Assume that (A, σ) is of co-index 2^{m−1}, i.e, the algebra A is Brauer-equivalent to a quaternion algebra H. Again a positive answer is given in [Bec08]: (A, σ) ≃ (H, σ_{H}) ⊗ (M_{2^{m−1}}(F), ad_{π′}) for some orthogonal involution σ_{H} on H and some (m − 1)-fold Pfister form π′ over F. In higher index, a counter example is provided by Sivatski in [Siv05, Prop. 5]. He produces an algebra with orthogonal involution of degree 8 and co-index 2 which is totally decomposable, but not totally decomposable with a split factor. Observe that the algebra in this example is isomorphic to M_{2}(D) for some biquaternion algebra D. Hence, it is totally decomposable with a split factor. Starting from Sivatski’s example, we construct examples of algebras with orthogonal involution, of index 2^{r} and co-index 2^{k}, where r ≥ 2 and k ≥ 1 are arbitrary, that are not totally decomposable with k split factors as algebras with involution, even though the underlying algebra does admit a total decomposition with k split factors (see Remark 5.5). Those algebras have center some iterated Laurent power series fields, and the theory of gauges play a crucial role in the proofs.

Assume now that the involution is of symplectic type. We give a partial answer to our main question in this case. Since we assume σ is anisotropic, the index of A is at least 2. In index 2, Becher showed in [Bec08] that (A, σ) ≃ (H′, γ) ⊗ (M_{2^{m−1}}(F), ad_{π′′}) where H′ is some quaternion algebra, γ is the canonical involution on H′ and π′′ is some (m − 1)-fold Pfister form over F. In degree 8 and index 4, we prove that (A, σ) is totally decomposable with 1 split factor (see Prop. 2.2). Finally, in index 2^{r} and co-index 2^{k} for arbitrary r ≥ 3 and k ≥ 1, we give examples of (A, σ) such that A is totally decomposable with k split factors but (A, σ) is not totally decomposable with k split factors (see Remark 5.5).

The paper is organized as follows: in Section 2 we deal with the degree 8 algebras with involution. In particular, we show that if (A, σ) is a central simple algebra of degree 8 and co-index 2^{k} (where k ≤ 2) with a symplectic involution then (A, σ) is totally decomposable with k split factors (see Prop. 2.2). Section 3 collects preliminary results on gauges, which are the main tool in the remaining part of the paper. We give in Sections 4 and 5 the proof of our main results and the examples

INVOLUTIONS ON TENSOR PRODUCTS OF QUATERNION ALGEBRAS 3

that extend Sivatski’s result to algebras with involution of arbitrarily large co-index (see Cor. 4.4 and Remark 5.5) and to algebras with involution of arbitrarily large index (see Cor. 5.4 and Remark 5.5). Corollaries 4.4 and 5.4 are the immediate consequences of the main Theorems 1.1 and 1.2 below.

1.1. Statement of main results. Throughout this paper the characteristic of the base field F is assumed to be different from 2 and all algebras are associative and finite-dimensional over their centers. We will use freely the standard terminology and notation from the theory of finite-dimensional algebras, the theory of valuations on division algebras and the theory of involutions on central simple algebras. For these, as well as background information, we refer the reader to [Pie82] and [KMRT98].

Let us fix some notations: let F be a field, consider the fields

K = F ((t)) and L = F ((t_{1}))((t_{2})),

where t, t_{1}, t_{2} are independent indeterminates over F, and the quaternion algebra Q = (t_{1}, t_{2}) over L. For an n-dimensional quadratic form q, following [Bec08], we denote by Ad_{q} the split algebra with orthogonal involution (M_{n}(F), ad_{q}) where ad_{q} is the involution on M_{n}(F) that is adjoint to the polar form of q. Let S be a central simple algebra over F and let ρ be an anisotropic involution on S. Let ρ′ be any involution of orthogonal or symplectic type on Q and let

(S_{1}, σ) = (S, ρ) ⊗_{F} Ad_{⟨⟨t⟩⟩}, (S_{2}, τ) = (S, ρ) ⊗_{F} (Q, ρ′).

Theorem 1.1. If (S_{1}, σ) is totally decomposable with (n+1) split factors then (S, ρ) is totally decomposable with n split factors.

Theorem 1.2. If (S_{2}, τ) is totally decomposable with n split factors then (S, ρ) is totally decomposable with n split factors.

Notice that if (S, ρ) is totally decomposable then the algebras with involution (S_{1}, σ) and (S_{2}, τ) are totally decomposable. But we have:

Remark 1.3. Decompositions of the algebra (S_{1}, σ) of Thm. 1.1 do not necessarily arise from a decomposition of (S, ρ) by multiplication with the factor Ad_{⟨⟨t⟩⟩}. For instance, the arguments in the proof of Lemma 4.2 show that if (S, ρ) is a quaternion algebra (a, b) with its canonical involution, then

(S, ρ) ⊗ Ad_{⟨⟨t⟩⟩} ≃ ((at, b), ν) ⊗ ((t, b), γ)

where γ is the canonical involution and ν an orthogonal involution with discriminant at. Likewise, decompositions of the algebra (S_{2}, τ) of Thm. 1.2 do not necessarily arise from a decomposition of (S, ρ).

2. Degree 8

Let A be a central simple algebra of degree 8 and co-index 2 and let σ be an anisotropic involution on A. If σ is orthogonal, it follows from [MQT09, Thm. 1.1] that (A, σ) ≃ (D, θ) ⊗ (M_{2}(F), ad_{φ}) where (D, θ) is some biquaternion algebra

4 D. BARRY

with orthogonal involution and ad_{φ} is the adjoint involution with respect to some binary quadratic form φ. Notice that (D, θ) need not to be decomposable in general. Indeed, the following result is due to Sivatski.

Proposition 2.1 ([Siv05, Prop. 5]). There exists a totally decomposable F -algebra with orthogonal involution (A′, σ′) of degree 8 and co-index 2 such that (A′, σ′) is not totally decomposable with 1 split factor.

Now assume that σ is symplectic. It is shown in [BMT03] that (A, σ) ≃ (Q, γ)⊗_{F} (A_{0}, σ_{0}) where (Q, γ) is a quaternion subalgebra of A with the canonical involution and σ_{0} is an orthogonal involution on the subalgebra A_{0} of A. Contrary to the orthogonal case, we show in the following proposition that there exists no analogue of the above result of Sivatski when σ is symplectic. First, let us recall that a relative invariant for symplectic involutions on central simple algebras of degree divisible by 4 is defined in [BMT03] and the authors show that the relative invariant is trivial for an algebra of degree 8 and co-index 2 with symplectic involution (A, σ) if and only if (A, σ) is totally decomposable with 1 split factor ([BMT03, Thm. 8]). In [GPT09], this invariant leads to an absolute invariant Δ, with value in H^{3}(F, μ_{2}), for symplectic involutions. For a central simple algebra A of degree 8 with a symplectic involution σ, the element Δ(A, σ) is zero if and only if (A, σ) is totally decomposable (see [GPT09, Thm. B]).

Proposition 2.2. Let (A, σ) be a totally decomposable algebra of degree 8 with an anisotropic symplectic involution, of co-index 2^{m} (where m ≤ 2). Then (A, σ) is totally decomposable with m split factors.

Proof. If the co-index of A is 4, it follows from Becher’s result [Bec08, Cor] that (A, σ) is totally decomposable with 2 split factors. Assume that the co-index of A is 2. Since (A, σ) is totally decomposable, the absolute invariant Δ(A, σ), defined in [GPT09], is zero. Therefore it readily follows by [BMT03, Thm. 8] that (A, σ) is totally decomposable with 1 split factors. □

3. Gauges on algebras with involution

Gauges on algebras with involution play a major role in the next sections. In this section we recall the notions of value functions and gauges introduced in [RTW07], [TW10], [TW11] and [TW15] and gather some results for the sequel.

We fix a divisible totally ordered abelian group Γ, which will contain the value of all the valuations and the degree of all the gradings we consider. Thus, a valued division algebra is a pair (D, v) where D is a division algebra and v : D ⟶ Γ∪{∞} is a valuation. The group v(D^{×}) of values of D is denoted by Γ_{D}, and the residue division algebra by \overline{D}. The valuation v defines a filtration on D: for γ ∈ Γ, set

D_{≥γ} = {d ∈ D | v(d) ≥ γ}, D_{>γ} = {d ∈ D | v(d) > γ}

and

D_{γ} = D_{≥γ}/D_{>γ}.

The associated graded ring (of v on D) is

gr_{v}(D) = ⨁_{γ∈Γ} D_{γ}

with the multiplication induced by the multiplication in D. The homogeneous elements of gr_{v}(D) are those in ⋃_{γ∈Γ} D_{γ}. For d ∈ D^{×}, we write d̃for the image d+D_{>v(d)} for d ∈ D_{v(d)}, and ˜0_{D} = 0 in gr_{v}(D). It follows from the fact v(cd) = v(c) + v(d), for c, d ∈ D^{×}, that c̃d = c̃d̃. So, in particular, we have d̃{ \atop (d^{−1})} = ˜1 for any d ∈ D^{×}. This shows that gr_{v}(D) is graded division ring. The grade group of gr_{v}(D), denoted Γ_{gr_{v}(D)}, is {γ ∈ Γ | D_{γ} ≠ 0}. Notice that Γ_{gr_{v}(D)} = Γ_{D} and the degree 0 component of gr_{v}(D) is D_{0} = D_{≥0}/D_{>0} = \overline{D}.

Let (F, v) be a valued field, and let M be a (right) F -vector space. A v-value function on M is a map α : M ⟶ Γ ∪ {∞} such that

(i) α(x) = ∞ if and only if x = 0;

(ii) α(x + y) ≥ min(α(x), α(y)) for x, y ∈ M;

(iii) α(xc) = α(x) + v(c) for all x ∈ M and c ∈ F. If α is a v-value function on M, in the same way as for the construction of the graded division algebra, we associate to M a graded gr_{v}(F)-module that we denote by gr_{α}(M). In addition, if M is finite-dimensional and [M : F] = [gr_{α}(M) : gr_{v}(F)], we say that α is a norm (or a v-norm). A v-value function g on an F -algebra A is surmultiplicative if

g(1) = 0 and g(xy) ≥ g(x) + g(y) for all x, y ∈ A.

If g is a surmultiplicative v-value function on an F -algebra A, then gr_{g}(A) is an algebra over gr_{v}(F), in which multiplication is defined by

ãb̃ = ab + A_{>g(a)+g(b)} = \{ \begin{matrix}ãb & if g(ab) = g(a) + g(b), \\ 0 & if g(ab) > g(a) + g(b), \end{matrix} for a, b ∈ A.

Suppose A is a finite-dimensional simple F -algebra. A surmultiplicative v-value function g on A is called an F -gauge (or a v-gauge) if:

(i) g is a v-norm, i.e, [A : F] = [gr_{g}(A) : gr_{v}(F)]; (ii) gr_{g}(A) is a graded semisimple gr_{v}(F)-algebra, i.e, gr_{g}(A) does not contain

any non-zero nilpotent homogeneous two-sided ideal. We give the following examples that will be used in the proof of our main results.

Examples 3.1. The algebras S_{1} and S_{2} are as in Theorems 1.1 and 1.2. (1) Let

i = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) , j = (\begin{matrix}0 & t \\ 1 & 0 \end{matrix}) ∈ M_{2}(K),

so ad_{⟨⟨t⟩⟩}(i) = i and ad_{⟨⟨t⟩⟩}(j) = −j. Let (s_{ℓ})_{1≤ℓ≤n} be an F -basis of S, so (s_{ℓ} ⊗ 1, s_{ℓ} ⊗ i, s_{ℓ} ⊗ j, s_{ℓ} ⊗ ij)_{1≤ℓ≤n} is a K-basis of S_{1}. The t-adic valuation v on K extends to a

6 D. BARRY

map

g_{1} : S_{1} ⟶ (\frac{1}{2}ℤ) ∪ \{∞\}

defined by

g_{1}( ∑_{1≤ℓ≤n} s_{ℓ} ⊗ (α_{ℓ} + β_{ℓ}i + γ_{ℓ} j + δ_{ℓ}ij)) = min_{1≤ℓ≤n} (v(α_{ℓ}), v(β_{ℓ}), v(γ_{ℓ}) + \frac{1}{2}, v(δ_{ℓ}) + \frac{1}{2}).

We readily check that g_{1}(a_{1}a_{2}) ≥ g_{1}(a_{1}) + g_{1}(a_{2}) for a_{1}, a_{2} ∈ S_{1}. It then follows by [TW15, Lemma 3.23] that g_{1} defines a filtration on S_{1}. The associated graded ring is noted gr_{g_{1}}(S_{1}). Let t̃ be the image of t in gr_{v}(K). We have

gr_{g_{1}}(S_{1}) = S ⊗_{F} M_{2}(F [t̃, t̃^{−1}])

with the grading defined by

gr_{g_{1}}(S_{1})_{γ} = (\begin{matrix}St̃^{γ} & 0 \\ 0 & St̃^{γ} \end{matrix})  for γ ∈ ℤ

gr_{g_{1}}(S_{1})_{γ} = (\begin{matrix}0 & St̃^{γ+ \frac{1}{2}} \\ St̃^{γ− \frac{1}{2}} & 0 \end{matrix} )  for γ ∈ \frac{1}{2}ℤ\ℤ.

Let x be a nonzero homogeneous element in gr_{g_{1}}(S_{1}). Since t̃is invertible in gr_{g_{1}}(S_{1}), the two-sided ideal I(x) ⊂ gr_{g_{1}}(S_{1}) generated by x contains a nonzero element of M_{2}(S). But M_{2}(S) is simple and contained in gr_{g_{1}}(S_{1}), hence I(x) contains 1. Therefore, gr_{g_{1}}(S_{1}) is graded simple algebra, and g_{1} is a gauge on S_{1} (see [TW15, Def. 3.31]).

(2) The (t_{1}, t_{2})-adic valuation v on L being Henselian, it extends to a valuation w on Q defined by

w(q) = \frac{1}{2}v( Nrd(q)) ∈ (\frac{1}{2}ℤ)^{2} ∪ \{∞\} for q ∈ Q,

where Nrd is the reduced norm. One further extends w to a map

g_{2} : S_{2} ⟶ (\frac{1}{2}ℤ)^{2} ∪ \{∞\}

as follows: we consider the F -base (s_{ℓ})_{1≤ℓ≤n} of S as above. Every element in S_{2} has a unique representation of the form ∑_{ℓ} s_{ℓ} ⊗ q_{ℓ} for some q_{ℓ} ∈ Q = (t_{1}, t_{2}). We define

g_{2}(∑_{ℓ} s_{ℓ} ⊗ q_{ℓ}) = min{w(q_{ℓ}) | 1 ≤ ℓ ≤ n}.

It is easy to see that this definition does not depend on the choice of the base (s_{ℓ})_{1≤ℓ≤n}. We easily check that g_{2} is a gauge. The associated graded ring is

gr_{g_{2}}(S_{2}) = S ⊗_{F} (t̃_{1}, t̃_{2})_{gr_{v}(L)}

where t̃_{1} and t̃_{2} are respectively the images of t_{1} and t_{2} in gr_{v}(L) = F [t̃_{1}, t̃_{1} ^{−1}, t̃_{2}, t̃_{2} ^{−1}].

INVOLUTIONS ON TENSOR PRODUCTS OF QUATERNION ALGEBRAS 7

Now, let θ be an F -linear involution on a central simple algebra A over F. Assume that v is a valuation on F and A carries a v-gauge g. The gauge g is said to be invariant under θ (or θ-invariant) if

g(θ(x)) = g(x) for all x ∈ A.

In this case, the involution θ preserves the filtration on A defined by g. Thus θ induces an involution θ̃on gr_{g}(A) such that θ̃(x̃) = θ̃(x) for all x ∈ A. The involution θ is said to be anisotropic if there is no nonzero elements x ∈ A such that θ(x)x = 0 (see [KMRT98, §6.A]). Likewise, θ̃ is said to be anisotropic if there is no nonzero homogeneous element ε ∈ gr_{g}(A) such that θ̃(ε)ε = 0. Clearly, if θ̃ is anisotropic then θ is anisotropic. If g is a v-gauge on A that is invariant under θ, we denote by θ_{0} the 0 component of θ̃. Thus, θ_{0} is an involution on the \overline{F} -algebra A_{0} = A_{≥0}/A_{>0}, which may be viewed as the residue involution on the residue algebra A_{0}.

Let (F, v) be a Henselian valued field with char(\overline{F}) ≠ 2 and let (A, θ) be a central simple F -algebra with (F -linear) involution. If the involution θ is anisotropic, it is shown in [TW11, Thm. 2.2] that there exists a unique v-gauge g on A such that g is invariant under θ and g(θ(a)a) = 2g(a) for all a ∈ A. Following Tignol-Wadsworth’s terminology ([TW11]) such a gauge g is called θ-special.

The following proposition gives an explicit description of the special gauges on (S_{1}, σ) and (S_{2}, τ).

Proposition 3.2. We consider the gauges g_{1} and g_{2} constructed in Examples 3.1 and the algebras with involution (S_{1}, σ) and (S_{2}, τ) are as in Theorems 1.1 and 1.2.

(1) The gauge g_{1} is the unique gauge on S_{1} that is invariant under σ, and moreover it satisfies g_{1}(σ(a)a) = 2g_{1}(a) for all a ∈ S_{1}.

(2) The gauge g_{2} is the unique gauge on S_{2} that is invariant under τ, and more-

over it satisfies g_{2}(τ(b)b) = 2g_{2}(b) for all b ∈ S_{2}. (That is, g_{1} and g_{2} are σ-special and τ-special respectively.)

Proof. (1) It follows from the definition of g_{1} that it is invariant under σ, that is, g_{1}(σ(a)) = g_{1}(a) for all a ∈ S_{1}. On gr_{g_{1}}(S_{1})_{0} = S × S, the induced involution is ρ × ρ. Therefore, since ρ is anisotropic, σ also is by [TW11, Cor. 2.3], and the statement follows by [TW11, Thm. 2.2 and Cor. 2.3].

(2) One may check that g_{2}(τ(b)) = g_{2}(b) for all b ∈ S_{2} (that is, g_{2} is invariant under τ), and g_{2}(τ(b)b) = 2g_{2}(b). We may therefore consider the residue algebra with involution gr_{g_{2}}(S_{2}, τ)_{0}. Since the residue of Q is F and S is defined over F, we have gr_{g_{2}}(S_{2}, τ)_{0} ≃ (S, ρ). The involution ρ being anisotropic, the assertion follows by the same arguments as above. □

4. Decomposition with arbitrarily large co-index

The goal of this section is to give examples of a totally decomposable algebras with orthogonal involution of degree 2^{m} and co-index 2^{m−2} (for m ≥ 3) which are not totally decomposable with (m − 2) split factors. We start out by proving Thm. 1.1. We first prove the following:

Lemma 4.1. Let (H, ν) be a quaternion algebra with anisotropic involution over K = F ((t)) and let γ denote the canonical involution on H.

(1) Suppose the algebra H is not defined over F. Then H ≃ (at, b) for some a, b ∈ F^{×}.

(2) Suppose the algebra with involution (H, ν) is not defined over F and ν is orthogonal and let disc(ν) denote the discriminant of ν. Then either disc(ν) = at · K^{×2} for some a ∈ F^{×} or disc(ν) = b · K^{×2} for some b ∈ F^{×}.

– If disc(ν) = at·K^{×2} (resp. b·K^{×2}) then there exist quaternion generators i, j of H such that i^{2} = at, j^{2} = b and ν = Int(i)∘γ (resp. ν = Int(j)∘γ).

– If H is split, then (H, ν) = Ad_{⟨⟨at⟩⟩}.

Proof. We first recall that (see [Lam05, Cor. VI.1.3])

K^{×}/K^{×2} = {α · K^{×2} | α ∈ F } ∪ {αt · K^{×2} | α ∈ F }.

(1) Let q be a pure quaternion in H; we have q^{2} ∈ K^{×}/K^{×2}. Up to scaling, we may assume that either q^{2} = at for some a ∈ F^{×} or q^{2} = b for some b ∈ F^{×}. Since we assume that H is not defined over F, the algebra H is either isomorphic to H ≃ (at, b) or to H ≃ (at, bt) ≃ (at, −ab).

(2) Now assume that ν is orthogonal. By [KMRT98, (2.7)], the involution ν has the form Int(s)∘γ for some invertible s ∈ H with γ(s) = −s. [KMRT98, (7.3)] shows that disc(ν) = − Nrd(s) · K^{×2} ∈ K^{×}/K^{×2}, where Nrd denotes the reduced norm. Therefore, either disc(ν) = at · K^{×2} for some a ∈ F^{×} or disc(ν) = b · K^{×2} for some b ∈ F^{×}. If disc(ν) = at · K^{×2}, we choose i = s. One has ν = Int(i) ∘ γ. Let j ∈ H^{×} be such that j^{2} = y ∈ K^{×}/K^{×2} and ij = −ji (recall that such a j always exists). Up to scaling we may assume y ∈ F^{×} or y ∈ F^{×} · t. We have the isomorphism (H, ν) ≃ ((at, b), Int(i) ∘ γ) if y = b ∈ F^{×} and (H, ν) ≃ ((at, bt), Int(i) ∘ γ) if y has the form y = bt. In the latter case, we may substitute ij^{−1} for j and reduce to the case where j^{2} ∈ F^{×}.

Likewise, if disc(ν) = b · K^{×2} for some b ∈ F^{×}, we choose j = s. So ν = Int(j) ∘ γ. Let i ∈ H^{×} be such that i^{2} ∈ K^{×}/K^{×2} and ij = −ji. Since (H, ν) is not defined over F, up to scaling we have necessarily i^{2} = at for some a ∈ F^{×}. This yields that (H, ν) ≃ ((at, b), Int(j) ∘ γ).

Now, suppose H is split. If so, (H, ν) ≃ Ad_{⟨1,− disc(ν)⟩}. Since it is not defined over F, we get disc(ν) = at · K^{×2} for some a ∈ F^{×}, that is, (H, ν) ≃ Ad_{⟨⟨at⟩⟩}. □

Lemma 4.2. Let (H_{1}, ν_{1}), (H_{2}, ν_{2}) be quaternion algebras with anisotropic involutions over K = F ((t)). There exist quaternion algebras with involution (H′_{1}, ν′_{1}), (H′_{2}, ν′_{2}) with (H′_{1}, ν′_{1}) defined over F such that

(H_{1}, ν_{1}) ⊗ (H_{2}, ν_{2}) ≃ (H′_{1}, ν′_{1}) ⊗ (H′_{2}, ν′_{2}).

Moreover,

INVOLUTIONS ON TENSOR PRODUCTS OF QUATERNION ALGEBRAS 9

– if (H_{1}, ν_{1}) and (H_{2}, ν_{2}) are split, we may find (H′_{1}, ν′_{1}) and (H′_{2}, ν′_{2}) such that both are split;

– if (H_{1}, ν_{1}) is split and (H_{2}, ν_{2}) is not split, we may find (H′_{1}, ν′_{1}) and (H′_{2}, ν′_{2}) such that (H′_{1}, ν′_{1}) is split and (H′_{2}, ν′_{2}) is not split.

Proof. If (H_{1}, ν_{1}) or (H_{2}, ν_{2}) is defined over F, there is nothing to show. Otherwise, write H_{1} = (a_{1}t, b_{1}) and H_{2} = (a_{2}t, b_{2}) for some a_{1}, a_{2}, b_{1}, b_{2} ∈ F. By Lemma 4.1, there exist quaternion generators i_{1}, j_{1} and i_{2}, j_{2} of H_{1} and H_{2} respectively such that i_{ℓ}^{2} = a_{ℓ}t, j_{ℓ}^{2} = b_{ℓ}, and ν_{ℓ}(i_{ℓ}) = ±i_{ℓ} and ν_{ℓ}(j_{ℓ}) = ±j_{ℓ} for ℓ = 1, 2. Let H′_{1}, H′_{2} ⊂ H_{1} ⊗ H_{2} be quaternion subalgebras generated by t^{−1}i_{1} ⊗ i_{2}, j_{1} ⊗ 1 and 1 ⊗ i_{2}, j_{1} ⊗ j_{2} respectively. In fact, H′_{1} = (a_{1}a_{2}, b_{1}) and H′_{2} = (a_{2}t, b_{1}b_{2}). We easily check that H′_{1} and H′_{2} are stable under ν_{1} ⊗ ν_{2}. Denote by ν′_{ℓ} (for ℓ = 1, 2) the restriction of ν_{1} ⊗ ν_{2} to H′_{ℓ}. We have the isomorphism

(H_{1}, ν_{1}) ⊗ (H_{2}, ν_{2}) ≃ (H′_{1}, ν′_{1}) ⊗ (H′_{2}, ν′_{2})

with (H′_{1}, ν′_{1}) defined over F.

Now, if (H_{1}, ν_{1}) and (H_{2}, ν_{2}) are split, then the quadratic forms ⟨1, −b_{ℓ}⟩ ⊥ ⟨−a_{ℓ}t⟩⟨1, −b_{ℓ}⟩ are isotropic (for ℓ = 1, 2), hence b_{1} and b_{2} are squares. Thus, the algebras H′_{1} = (a_{1}a_{2}, b_{1}) and H′_{2} = (a_{2}t, b_{1}b_{2}) are split. Finally, if (H_{1}, ν_{1}) is split and (H_{2}, ν_{2}) is not split, then b_{1} ∈ F^{×2} and b_{2} ∉ F^{×2}. So H′_{1} is split and H′_{2} is not split. This concludes the proof. □

Proposition 4.3. The algebra with involution (S_{1}, σ) is as Thm. 1.1. Assume (S_{1}, σ) is totally decomposable with (n+1) split factors. Then (S_{1}, σ) ≃ Ad_{⟨⟨at⟩⟩} ⊗(S′, σ′) for some a ∈ F and some totally decomposable algebra with involution (S′, σ′) with n split factors. Moreover, each factor of (S′, σ′) is defined over F.

Proof. Write (S_{1}, σ) ≃ ⨂_{k=1}^{m}(Q_{k}, σ_{k}). It follows by part (1) of Lemma 4.1 that each Q_{k} is either defined over F or isomorphic to a quaternion algebra of the form (at, b) for some a, b ∈ F. Applying repeatedly Lemma 4.2, we get a total decomposition of (S_{1}, σ) in which each factor is defined over F except only one factor.

We show that the number of split factors in the decomposition of (S_{1}, σ) does not change by applying Lemma 4.2. Without loss of generality we may assume that (S_{1}, σ) is not totally decomposable with more than (n + 1) split factors. Let (Q_{i}, σ_{i}) and (Q_{j}, σ_{j}) be two factors of (S_{1}, σ). If (Q_{i}, σ_{i}) and (Q_{j}, σ_{j}) both are split, Lemma 4.2 shows that the tensor product (Q_{i}, σ_{i}) ⊗ (Q_{j}, σ_{j}) is exchanged for another one in which each factor is split. If (Q_{i}, σ_{i}) is split and (Q_{j}, σ_{j}) is not split, again Lemma 4.2 shows that the product (Q_{i}, σ_{i}) ⊗ (Q_{j}, σ_{j}) is exchanged for a product in which only one factor is split. Finally, suppose (Q_{i}, σ_{i}) and (Q_{j}, σ_{j}) both are division algebras with involution. Since we assume that (S_{1}, σ) is not totally decomposable with more than (n + 1) split factors, the product (Q_{i}, σ_{i}) ⊗ (Q_{j}, σ_{j}) is necessarily exchanged for another one in which each factor is a division algebra with involution. It then follows that the number of split factors in the decomposition of (S_{1}, σ) does not change by applying Lemma 4.2.

10 D. BARRY

Let us denote by ((at, b), σ_{r}) the factor of (S_{1}, σ) which is not defined over F. We claim that ((at, b), σ_{r}) ≃ Ad_{⟨⟨a′t⟩⟩} for some a′ ∈ F^{×}. To see this, we first show that (at, b) is split. Assume the contrary, that is, (at, b) is a division quaternion algebra over K. On the one hand, according to Prop. 3.2, there exists a unique gauge g_{1} on S_{1} that is invariant under σ and its residue algebra is gr_{g_{1}}(S_{1})_{0} = S × S with the induced involution ρ × ρ. On the other hand, write (S_{1}, σ) ≃ ((at, b), σ_{r}) ⊗ (S′, σ′) where S′ is the centralizer of (at, b) in S_{1}. As we showed above, (S′, σ′) is totally decomposable with n split factors and each factor of (S′, σ′) is defined over F. Now, we construct the same gauge on S_{1} by taking into account the fact that (at, b) is a division algebra: the field K being Henselian, the t-adic valuation v of K extends to a valuation w on (at, b). Let (a_{i})_{1≤i≤s} be an F -basis of S′. So every element in S_{1} has a unique representation ∑_{i} q_{i} ⊗ a_{i} for some q_{i} ∈ (at, b). It is easy to verify that the map g_{3} : S_{1} ⟶ \frac{1}{2}ℤ ∪ \{∞\} defined by

g_{3}(∑_{i} q_{i} ⊗ a_{i}) = min{w(q_{i}) | 1 ≤ i ≤ s}

is a v-gauge on S_{1}. Moreover, it follows from the definition that g_{3} is invariant under σ and g_{3}(σ(a)a) = 2g_{3}(a) for all a ∈ S_{1}. Hence, g_{3} is the unique v-gauge on S_{1} that is preserved by σ by the same arguments as above. Therefore g_{3} = g_{1} by uniqueness arguments. This leads to a contradiction, since the residue of S_{1} with respect to g_{3} is gr_{g_{3}}(S_{1})_{0} ≃ A′ ⊗ F (\sqrt{b}), which is simple, as opposed to the residue of S_{1} with respect to g_{1}, which is not simple. Thus the quaternion algebra (at, b) is split. Since the involution σ is anisotropic, σ_{r} is necessarily orthogonal. It Then follows by part (2) of Lemma 4.1 that ((at, b), σ_{r}) ≃ Ad_{⟨⟨a′t⟩⟩} for some a′ ∈ F^{×}. Therefore (S_{1}, σ) ≃ Ad_{⟨⟨a′t⟩⟩} ⊗(S′, σ′). We have the stated description of (S_{1}, σ). □

Proof of Theorem 1.1. Assume that (S_{1}, σ) is totally decomposable with (n + 1) split factors. By Prop. 4.3, (S_{1}, σ) ≃ Ad_{⟨⟨at⟩⟩} ⊗(S′, σ′), where (S′, σ′) is totally decomposable with n split factors, and each factor of (S′, σ′) is defined over F. We write (S′, σ′) ≃ ⊗_{ℓ=2}^{m}(Q_{ℓ}, σ_{ℓ}) ⊗ K. Consider the gauge g_{1} on S_{1} from Prop. 3.2, and recall that the residue algebra with involution of (S_{1}, σ) is gr_{g_{1}}(S_{1}, σ)_{0} ≃ (S×S, ρ×ρ). For ℓ = 2, …, m, let i_{ℓ}, j_{ℓ} ∈ S′ be such that

(4.1)  i_{ℓ}^{2} = a_{ℓ}, j_{ℓ}^{2} = b_{ℓ}, i_{ℓ}j_{ℓ} = −j_{ℓ}i_{ℓ}, σ′(i_{ℓ}) = ±i_{ℓ}, σ′(j_{ℓ}) = ±j_{ℓ},

so Q_{ℓ} = (a_{ℓ}, b_{ℓ}) for all ℓ. Since g_{1}(σ(a)a) = 2g_{1}(a) for all a ∈ S_{1}, we have g_{1}(i_{ℓ}) = \frac{1}{2}g_{1}(±a_{ℓ}) = 0 for all ℓ. Similarly, g_{1}(j_{ℓ}) = \frac{1}{2}g_{1}(±b_{ℓ}) = 0 for all ℓ. The images ĩ_{ℓ}, j̃_{ℓ} of i_{ℓ}, j_{ℓ} in gr_{g_{1}}(S_{1})_{0} ≃ S × S satisfy conditions similar to (4.1). The involution σ being preserved by g_{1}, it induces an involution σ̃ on gr_{g_{1}}(S_{1}). Consider a projection gr_{g_{1}}(S_{1})_{0} → S. This projection is a homomorphism of algebras with involution p : (gr_{g_{1}}(S_{1}), σ̃)_{0} → (S, ρ). The images p(ĩ_{ℓ}), p(j̃_{ℓ}) generate a copy of ⊗_{ℓ=2}^{m}(Q_{ℓ}, σ_{ℓ}) in (S, ρ). Thus, we have the isomorphism ⊗_{ℓ=2}^{m}(Q_{ℓ}, σ_{ℓ}) ≃ (S, ρ) by dimension count. Therefore (S, ρ) is totally decomposable with n split factors. □

INVOLUTIONS ON TENSOR PRODUCTS OF QUATERNION ALGEBRAS 11

Corollary 4.4. Let (A′, σ′) be the algebra with orthogonal involution of Prop. 2.1. The algebra with involution Ad_{⟨⟨t⟩⟩} ⊗(A′, σ′) is not totally decomposable with 2 split factors.

Proof. Assume the contrary, that is, Ad_{⟨⟨t⟩⟩} ⊗(A′, σ′) is totally decomposable with 2 split factors. It follows by Thm. 1.1 that (A′, σ′) is totally decomposable with 1 split factor; that is impossible. Hence the corollary is proved. □

Remark 4.5. The construction in Cor. 4.4 can be iterated to obtain examples of algebras with orthogonal involution of degree 2^{m} (for m ≥ 4) and co-index 2^{m−2} that are not totally decomposable with (m−2) split factors although the underlying algebras are totally decomposable with (m − 2) split factors.

5. Decomposition with arbitrarily large index

The main object of this section is to prove Thm. 1.2. This theorem leads to examples of totally decomposable algebras with involution of degree 2^{m} (for m ≥ 3) and co-index 2 which are not totally decomposable with 1 split factor. For the proof of Thm. 1.2 we need some preliminary results. Let E be an arbitrary central simple algebra of exponent 2 over F. As in part 1.1 of Section 1, Q denotes the quaternion algebra (t_{1}, t_{2}) over the field L = F ((t_{1}))((t_{2})). We set

C = E ⊗_{F} Q.

Consider the gauge as constructed in part (2) of Example 3.1, where we take E instead of S. Here, we denote this gauge by φ. Let θ be an L-linear anisotropic involution on C such that φ is invariant under θ. Let us denote by θ_{0} the induced involution on gr_{φ}(C)_{0} = E. We have the following:

Lemma 5.1. The algebras with involution (C, θ) and (E, θ_{0}) are as above. Denote by γ the canonical involution on Q and let 1, i, j, ij be a quaternion base of Q such that i^{2} = t_{1} and j^{2} = t_{2}. We have the decomposition

(C, θ) ≃ (E, θ_{0}) ⊗_{F} (Q, Int(u) ∘ γ),

where u is one of the elements 1, i, j, ij.

Proof. As in Example 3.1, recall that

gr_{φ}(C) = E ⊗_{F} (t̃_{1}, t̃_{2})_{gr_{v}(L)}

where v denotes the (t_{1}, t_{2})-adic valuation on L. Since θ̃induces θ_{0} on E, it suffices to evaluate the restriction θ̃|_{gr_{φ}(Q)} of θ̃ to gr_{φ}(Q) = (t̃_{1}, t̃_{2})_{gr_{v}(L)}. Notice that the degrees (\frac{1}{2}, 0) and (0, \frac{1}{2}) components of gr_{φ}(C) are respectively E ⊗ ĩ and E ⊗ j̃. Since θ preserves the filtration on gr_{φ}(C) defined by the gauge φ, one has θ̃(1 ⊗ ĩ) = ζ_{1} ⊗ ĩ for some ζ_{1} ∈ E. Moreover, it is easy to see that ζ_{1} ⊗ ĩ centralizes E since 1 ⊗ ĩ centralizes E ⊗ 1 and θ̃(1 ⊗ ĩ) = ζ_{1} ⊗ ĩ. Therefore ζ_{1} ∈ F. On the other hand, we have

1 ⊗ ĩ = θ̃^{2}(1 ⊗ ĩ) = θ̃(ζ_{1} ⊗ ĩ) = (θ_{0}(ζ_{1})ζ_{1}) ⊗ ĩ.

This shows that θ_{0}(ζ_{1})ζ_{1} = 1, and so ζ_{1} = ±1. Likewise, θ̃(1 ⊗ j̃) = ζ_{2} ⊗ j̃ with ζ_{2} = ±1. Thus, we have:

θ̃|_{gr_{φ}(Q)} = \{ \begin{matrix}γ & if ζ_{1} = ζ_{2} = −1, \\ Int(j̃) ∘ γ & if ζ_{1} = 1, ζ_{2} = −1, \\ Int(ĩ) ∘ γ & if ζ_{1} = −1, ζ_{2} = 1, \\ Int(ĩj) ∘ γ & if ζ_{1} = 1, ζ_{2} = −1. \end{matrix}

Hence, our arguments show that θ̃ = θ_{0}⊗(Int(ũ)∘γ) where u is one of the quaternion base elements 1, i, j, ij of Q. It then follows by [TW11, Thm. 2.6] that the algebras with involution (E ⊗_{F} Q, θ) and (E ⊗_{F} Q, θ_{0} ⊗ Int(u) ∘ γ) are isomorphic. This concludes the proof. □

From now on, we assume that (C, θ) is totally decomposable. We want to show that (E, θ_{0}) is totally decomposable. We first introduce the notion of an armature, which originated in [Tig82] and [TW87]. However, we shall refer frequently to the recent book [TW15] that gives a more extensive treatment of armatures. Let A be a finite-dimensional algebra over a field F. Recall from [TW15, Def. 7.27] that a subgroup A ⊂ A^{×}/F^{×} is an armature of A if A is abelian, A has cardinality |A| = [A : F], and {a ∈ A | aF^{×} ∈ A} spans A as an F -vector space. For a ∈ A^{×}/F^{×}, we fix a representative x_{a} in A whose image in A^{×}/F^{×} is a, that is, a = x_{a}F^{×}. In what follows, we always suppose A is a central simple algebra of exponent 2 over F, so the exponent of the group A is 2, that is, x_{a}^{2} ∈ F^{×} for all a ∈ A.

For example, let A be a quaternion algebra over F. The image in A^{×}/F^{×} of the standard generators i, j generate an armature of A. More generally, in a tensor product of quaternion algebras the images of the product of standard generators generate an armature. Actually, there exists a characterization of tensor products of quaternion algebras in terms of the existence of armatures: a central simple algebra A over F of exponent 2 has an armature if and only if A is isomorphic to a tensor product of quaternion algebras over F (see [TW15, Cor. 7.34]).

For an armature A of a finite-dimensional F -algebra A of exponent 2, there is an associated armature pairing

⟨ , ⟩ : A × A → μ_{2}(F) = {±1} defined by ⟨a, b⟩ = x_{a}x_{b}x_{a}^{−1}x_{b}^{−1}.

It is shown in [TW15, Prop. 7.26] that ⟨ , ⟩ is a well-defined alternating bimultiplicative pairing. The set {a_{1}, …, a_{r}} is called a base of A if A is the internal direct product

A = (a_{1}) × ⋯ × (a_{r})

where (a_{i}) denotes the cyclic subgroup of A generated by a_{i}. If ⟨ , ⟩ is nondegenerate then (A, ⟨ , ⟩) is a symplectic module and has a symplectic base with respect to ⟨ , ⟩, i.e, a base {a_{1}, b_{1}, …, a_{n}, b_{n}} such that for all i, j

⟨a_{i}, b_{i}⟩ = −1, where ord(a_{i}) = ord(b_{i}) = 2
⟨a_{i}, a_{j}⟩ = ⟨b_{i}, b_{j}⟩ = 1 and, if i ≠ j, ⟨a_{i}, b_{j}⟩ = 1

INVOLUTIONS ON TENSOR PRODUCTS OF QUATERNION ALGEBRAS 13

(see [TW15, Thm. 7.2] or [Tig82, 1.8]). Given any subgroup B of A, let F [B] denote the F -subspace (and subalgebra) of A generated by the representatives of the elements of B. It is easy to check that B is an armature of F [B].

Let v be a valuation on F. Assume that A has an armature A. Using the armature A, an F -gauge on A is constructed in [TW15, §7.2.3]: a map g_{A} : A ⟶ Γ is defined as follows. We choose an F -basis (x_{a})_{a∈A} of A and set

g_{A}(∑_{a∈A} λ_{a}x_{a}) = min_{a∈A}(v(λ_{a}) + \frac{1}{2}v(x_{a}^{2})) ∈ Γ ∪ \{∞\} for λ_{a} ∈ F,

where we recall that x_{a}^{2} ∈ F^{×} for all a ∈ A. One also defines a map \overline{g}_{A} : A ⟶ Γ/Γ_{F} by

\overline{g}_{A}(a) = g_{A}(x_{a}) + Γ_{F} .

The map \overline{g}_{A} is well-defined since x_{a} is uniquely determined by a up to a factor in F. The map g_{A} is a v-gauge on A and \overline{g}_{A} is a surjective group homomorphism, see [TW15, Lemma 7.46, Cor. 7.48 and Thm. 7.49]. Following Tignol-Wadsworth’s terminology in [TW15], the gauge g_{A} is called the armature gauge associated to A.

We now return to the central simple L-algebra with involution (C, θ) defined above. We fix the setting we will consider in the next proposition. Assume that C has an armature C and θ(x_{a}) = ±x_{a} for all a ∈ C. Let g_{C} be the armature gauge associated to C. It is clear from the definition of the armature gauge g_{C} that g_{C}(θ(c)) = g_{A}(c) and g_{C}(θ(c)c) = 2g_{C}(c) for all c ∈ C. The (t_{1}, t_{2})-adic valuation of L being Henselian, and θ being anisotropic, g_{C} is the unique gauge on C invariant under θ (see [TW11, Thm. 2.2]), that is, g_{C} = φ (where φ is the gauge used in the proof of Lemma 5.1) . Set C_{0} = ker \overline{g}_{C}. Since Γ_{C}/Γ_{F} = (\frac{1}{2}ℤ/ℤ)^{2}, it follows from the surjectivity of \overline{g}_{C} that

|C_{0}| = |C|/|(\frac{1}{2}ℤ/ℤ)^{2}| = \frac{1}{4}|C| = dim_{F} E.

We have the following result.

Proposition 5.2. The algebras with involution (C, θ) and (E, θ_{0}) are as in Lemma 5.1. Assume that (C, θ) is totally decomposable. Then (E, θ_{0}) is totally decomposable.

Proof. Write (C, θ) = ⨂_{m=1}^{k}(Q_{m}, θ_{m}) for some quaternion algebras Q_{m} and some involutions θ_{m} on Q_{m} (for m = 1, …, k). We choose the quaternion generators i_{m} and j_{m} of Q_{m} such a way that θ(i_{m}) = ±i_{m} and θ(j_{m}) = ±j_{m}. Let C be the armature of C associated to the generators i_{m}, j_{m} for m = 1, …, k, that is, C is the subgroup of C^{×}/L^{×} generated by the image of {i_{1}, j_{1}, …, i_{k}, j_{k}}. It is shown in [TW10, Prop. 4.8] that the degree 0 component of gr_{φ}(C), which is gr_{φ}(C)_{0} = E, has an armature canonically isomorphic to C_{0} with armature pairing isometric to the restriction to C_{0} of the armature pairing of C. We recall that the gauge φ is nothing but the armature gauge associated to C by uniqueness arguments. Since the center F of E is a field, [TW87, Cor. 2.8] indicates that E is a tensor product of quaternion algebras. In other words, let {a_{1}, b_{1}, …, a_{k−1}, b_{k−1}} be a symplectic base of C_{0} and let denote

14 D. BARRY

by F [a_{i}, b_{i}] the quaternion subalgebra of E generated by the representatives x_{a_{i}}, x_{b_{i}} of a_{i}, b_{i}. The algebra E is the tensor product E = F [a_{1}, b_{1}] ⊗ ⋯ ⊗ F [a_{k−1}, b_{k−1}]. Since θ(x_{a}) = ±x_{a} for all a ∈ C, the algebra with involution (E, θ_{0}) is totally decomposable. The proof is complete. □

Lemma 5.3. Let θ_{1} and θ_{2} be L-linear anisotropic involutions on M_{2}(L) and Q respectively. Then there exist λ_{0} ∈ F^{×} such that

(M_{2}(L), θ_{1}) ⊗ (Q, θ_{2}) ≃ (M_{2}(F), ad_{⟨⟨λ_{0}⟩⟩}) ⊗ (Q, θ_{2}).

Proof. Setting V = Q × Q, there is λ ∈ L^{×} such that θ_{1} ⊗ θ_{2} is the adjoint of the binary Hermitian form h = ⟨1, −λ⟩ : V × V → Q with respect to θ_{2} on Q (see for instance [KMRT98, Thm. (4.2)]). We claim that we may always suppose w(λ) = (0, 0). Indeed, multiplying λ by a square if it is necessary, the valuation w(λ) is either (0, 0) or (1, 0) or (0, 1) or (1, 1). For instance, assume w(λ) = (1, 0). We set θ′ = θ_{1} ⊗ θ_{2} for simplicity. Choose a quaternion base 1, i, j, ij of Q such a way that θ′(1 ⊗ i) = ±1 ⊗ i, θ′(1 ⊗ j) = ±1 ⊗ j and i^{2} = t_{1}. Let e_{1}, e_{2} be the base of V corresponding to the diagonalization h = ⟨1, −λ⟩. Replacing the base e_{1}, e_{2} by e_{1}, i^{−1}e_{2}, we get the isometry h = ⟨1, −λ⟩ ≃ ⟨1, −θ′(1 ⊗ i)^{−1}λ(1 ⊗ i)^{−1}⟩. Moreover, we have w(θ′(1 ⊗ i)^{−1}λ(1 ⊗ i)^{−1}) = 0. Hence, we may always suppose w(λ) = (0, 0) as claimed.

Since w(λ) = (0, 0), one has λ̄ = λ_{0} for some λ_{0} ∈ F^{×}. Thus, \overline{λλ_{0}^{−1}} = 1. By Hensel’s lemma it follows that there exists l ∈ L^{×} such that λ_{0}l^{2} = λ. Hence, ⟨1, −λ⟩ = ⟨1, −λ_{0}l^{2}⟩ ≃ ⟨1, −λ_{0}⟩ for some λ_{0} ∈ F^{×}. So

(M_{2}(L) ⊗ Q, θ′) ≃ (M_{2}(F), ad_{⟨⟨λ_{0}⟩⟩}) ⊗ (Q, θ_{2}).

The lemma is proved. □

Proof of Theorem 1.2. Assume that (S_{2}, τ) is totally decomposable with n split factors, say

(S_{2}, τ) ≃ ⨂_{k=1}^{n}(M_{2}(L), τ_{k}) ⊗ (B, η),

for some involutions τ_{k} on M_{2}(L) and some totally decomposable involution η on the centralizer B of M_{2^{n}}(L) in S_{2}.

We first show that B = B_{0} ⊗ Q for some subalgebra B_{0} of B defined over F. For this, let D be the division algebra Brauer equivalent to S. The algebra S_{2} = S ⊗ Q is Brauer equivalent to B ∼ D ⊗ Q. Hence B ≃ M_{r}(D) ⊗ Q for some r ∈ ℕ. It then follows that B ≃ B_{0} ⊗ Q with B_{0} ≃ M_{r}(D) defined over F.

As in Lemma 5.1, notice that there exists a η-invariant gauge on B and

(B, η) ≃ (B_{0}, η_{0}) ⊗ (Q, Int(u) ∘ γ)

for some u ∈ Q^{×}, where η_{0} is the induced involution on B_{0} by this gauge and γ is the canonical involution on Q. Thus, we have

(S_{2}, τ) ≃ ⨂_{k=1}^{n}(M_{2}(L), τ_{k}) ⊗ (B_{0}, η_{0}) ⊗ (Q, Int(u) ∘ γ).

Moreover the involution η_{0} is totally decomposable by Prop. 5.2 since η is totally decomposable. Applying repeatedly Lemma 5.3, one finds λ_{1}, … λ_{n} ∈ F^{×} such that

(S_{2}, τ) ≃ ⨂_{k=1}^{n}(M_{2}(F), ad_{⟨⟨λ_{k}⟩⟩}) ⊗ (B_{0}, η_{0}) ⊗ (Q, Int(u) ∘ γ).

Consider the gauge g_{2} on S_{2} constructed in Example 3.1. The residue algebra with involution gr_{g_{2}}(S_{2}, τ)_{0} is

gr_{g_{2}}(S_{2}, τ)_{0} ≃ (S, ρ) ≃ ⨂_{k=1}^{n}(M_{2}(F), ad_{⟨⟨λ_{k}⟩⟩}) ⊗ (B_{0}, η_{0}).

Since (B_{0}, η_{0}) is totally decomposable, (S, ρ) is totally decomposable with n split factors. That concludes the proof. □

Now, let (A′, σ′) be the algebra with orthogonal involution of Prop. 2.1. Notice that A′ ⊗ Q is of degree 16 and index 8, and it is totally decomposable with 1 split factor since A′ is totally decomposable with 1 split factor. It is also clear that (A′, σ′) ⊗ (Q, ρ) is totally decomposable. But, we have the following:

Corollary 5.4. The algebra with involution (A′, σ′) ⊗ (Q, ρ) is not totally decomposable with 1 split factor.

Proof. Assume that (A′, σ′) ⊗ (Q, ρ) is totally decomposable with 1 split factor. It follows from Thm. 1.2 that (A′, σ′) is totally decomposable with 1 split factor; that is impossible. Therefore (A′, σ′) ⊗ (Q, ρ) is not totally decomposable with 1 split factor. □

Remark 5.5. Combining Corollaries 4.4 and 5.4, one can construct examples of totally decomposable algebras with involution of index 2^{r}, where r ≥ 2 for orthogonal involutions and r ≥ 3 for symplectic involutions, and co-index 2^{k} with k ≥ 1 that are not totally decomposable with k split factors.

Acknowledgements. I am very grateful to Jean-Pierre Tignol for triggering the idea presented in this paper and for his availability. I would like to thank Anne Quéguiner-Mathieu for inspiring discussion and for the financial support provided by the French Agence Nationale de la Recherche (ANR) under reference ANR-12-BL01-0005. I further gratefully acknowledge the financial support provided by the French Embassy in Bamako (Grant N^{o} 830592G (2014)).

References

[ART79] S. A. Amitsur, L. H. Rowen, J.-P. Tignol, Division algebras of degree 4 and 8 with involution, Isreal J. Math., 33, 133–148, (1979).

[Bec08] K. J. Becher, A proof of the Pfister factor conjecture, Invent. Math. 173 (2008), 1–6.

[BMT03] G. Berhuy, M. Monsurrò, J.-P. Tignol, The discriminant of a symplectic involution, Pacific J. Math. 209 (2003), 201–218.

[GPT09] S. Garibaldi, R. Parimala, J.-P. Tignol, Discriminant of symplectic involutions, Pure App. Math. Quart., 5 (2009), 349 - 374.

[Jac96] N. Jacobson, Finite-dimensional division algebras over fields, Springer-Verlag, Berlin, (1996).

[KMRT98] M.-A. Knus, S. A. Merkurjev, M. Rost, J.-P. Tignol, The book of involutions, Colloquium Publ., vol. 44, AMS, Providence, RI, (1998).

[Lam05] T. Y. Lam, Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67, AMS, Providence, (2005).

[MQT09] A. Masquelein, A. Quéguiner-Mathieu, J.-P. Tignol, Quadratic forms of dimension 8 with trivial discriminant and Clifford algebra of index 4, Arch. Math., 93 (2009), 129–138.

[Mer81] A. S. Merkurjev, On the norm residue symbol of degree 2, Sov. Math. Dokl., 24, 546– 551, (1981).

[Pie82] R. S. Pierce, Associative Algebras, Graduate Texts in Mathematics, vol. 88. Springer-Verlag, New York-Heidelberg-Berlin, (1982).

[RTW07] J.-F. Renard, J.-P. Tignol, A. R. Wadsworth, Graded Hermitian forms and Springer’s theorem, Indag. Math. N. S, 18 (2007), 97–134.

[Sha00] D. B. Shapiro, Compositions of quadratic forms, Expositions in Mathematics, vol. 33, De Gruyter, (2000).

[Siv05] A. S. Sivatski, Applications of Clifford algebras to involutions and quadratic forms, Com. in Algebra, 33 (2005), 937–951

[Tig82] J.-P. Tignol, Sur les décompositions des algèbres à division en produit tensoriel d’algèbres cycliques, Brauer Groups in Ring Theory and Algebraic Geometry (F. Van Oystaeyen and A. Verschoren, eds.), Lecture notes in Math., Springer-Verlag, Berlin and New York, 917, 126–145, (1982).

[TW87] J.-P. Tignol, A. R. Wadsworth, Totally ramified valuations on finite-dimensional division algebras, Trans. Amer. Math. Soc., 302, No. 1, 223–250 (1987).

[TW10] J.-P. Tignol, A. R. Wadsworth, Value functions and associated graded rings for semi simple algebras, Trans. Amer. Math. Soc., 362, No. 2 (2010), 687–726.

[TW11] J.-P. Tignol, A. R. Wadsworth, Valuations on algebras with involution, Math. Ann., 351 (2011), 109–148.

[TW15] J.-P. Tignol, A. R. Wadsworth, Value functions on simple algebras, and associated graded rings, Springer Monographs in Mathematics. Springer, Cham, 2015. DOI 10.1007/978-3-319-16360-4.

DER de Mathématiques et Informatique, Université des Sciences des Techniques et des Technologies de Bamako, BP: E3206 Bamako, Mali

E-mail address: barry.demba@gmail.com