Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 40 (1999), No. 1, 97-112.

Compositions of Quadratic Forms with Quadratic, Bilinear or Hermitian Forms

Remo Züger

Mathematisches Institut, Universität Bern

1. Introduction

In this paper we construct compositions of quadratic spaces with quadratic, symplectic or hermitian spaces by using properties of the Clifford-Algebra. Over fields of characteristic not equal to 2, we know by the Radon-Hurwitz formula that there exists a composition (M, q) × (E, p) → (E, p) for some quadratic spaces M of dimension m and E of dimension l if and only if m ≤ ρ(m) where ρ is the Radon-Hurwitz function (see e.g. [3], p.133ff). We want to generalize this result by considering arbitrary rings (we do not suppose \frac{1}{2} ∈ R), but also by replacing the quadratic space (E, p) by a symplectic space or by a hermitian space over a quadratic or quaternion algebra.

Throughout this paper we fix a commutative ring R. In order to simplify notations, we assume that R is connected, so the rank of a projective module is well-defined (in the general case we decompose R into connected components). All bilinear and hermitian forms are supposed to be nonsingular.

A quadratic space of rank m is a pair (M, q), where M is a projective R-module of rank m and q is a regular quadratic form on M if m is even, and a semiregular quadratic form (in the sense of Kneser ([4]; see also [1], p.207ff) if m is odd. Let C(M, q) (resp. C_{0}(M, q)) denote the Clifford algebra (resp. the even Clifford algebra) of the quadratic space (M, q). We call the involution on C(M, q) which extends the identity on M the canonical involution and the involution which extends the map x ↦ −x on M the standard involution.

If b is a bilinear form on a projective module E and g ∈ End_{R}(E), there is a homomorphism \overline{g} such that b(g(x), y) = b(x, \overline{g}(y)) holds for all x, y in E. If b is symmetric or skew-symmetric, the map g ↦ \overline{g} is an involution, which we call the adjoint of b. Conversely, let σ be an involution on End_{R}(E) . By the Skolem-Noether Theorem, σ is the adjoint of a bilinear form b_{σ} on E with values in an invertible module. We say that σ is of orthogonal (resp. symplectic) type if b_{σ} is symmetric (resp. skew-symmetric). An orthogonal (resp. symplectic) involution σ is even if there is a bilinear form on S^{m} such that b_{σ}(x, y) = c(x, y) + c(y, x)

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1999 Heldermann Verlag

(resp. b_{σ}(x, y) = c(x, y) − c(y, x)). If τ is an involution on an arbitrary Azumaya algebra A, and if S is a faithfully flat R-algebra such that A ⊗ S ≃ M_{m}(R), then we call τ orthogonal (resp. symplectic, resp. even) if σ ⊗ 1_{S} is orthogonal (resp. symplectic, resp. even).

2. Hyperbolic Spaces

Let P be a finitely generated projective module of rank m, and let P^{∗} = Hom_{R}(P, R). We call (M, q) = (P ⊕ P^{∗}, ((x, f) ↦ f(x)) a hyperbolic space. The Clifford-Algebra C(M, q) of a hyperbolic space is given by the following

Theorem 2.1. (a) There is an isomorphism

α : C(q) := C(P ⊕ P^{∗}, q) → End_{R}(⋀ P),

where ⋀ P denotes the exterior algebra of P.

(b) Let τ be the canonical involution of C(M, q). Then we have

ατα^{−1}(g) = u^{−1}(g^{∗} ⊗ 1_{I})u,

where I = ⋀^{m}(P) is invertible and u : ⋀ P → Hom((⋀ P), I) is the adjoint of the bilinear form b given by

b(x, y) = \{ \begin{matrix}σ(x)y & (r + s = m) \\ 0 & (r + s ≠ m) \end{matrix} ,

where x ∈ ⋀^{r} P, y ∈ ⋀^{s} P and σ is the involution of ⋀ P which extends the identity of P.

(c) Let τ′ be the standard involution of C(q). Then we have

ατα^{−1}(g) = u^{′−1}(g^{∗} ⊗ 1)u′,

where u′ : ⋀ P → Hom((⋀ P), I), is the adjoint of the bilinear form given by

b′(x, y) = \{ \begin{matrix}σ′(x)y & (r + s = m) \\ 0 & (r + s ≠ m) \end{matrix} ,

where x ∈ ⋀^{r} P, y ∈ ⋀^{s} P and σ′ denotes the involution of ⋀ P which extends the map P → P, x ↦ −x.

(d)  b(x, y) = \{ \begin{matrix}b(y, x) & (m ≡ 0, 1 (mod 4)) \\ −b(y, x) & (m ≡ 2, 3 (mod 4)) \end{matrix} .

(e)  b′(x, y) = \{ \begin{matrix}b′(y, x) & (m ≡ 0, 3 (mod 4)) \\ −b′(y, x) & (m ≡ 1, 2 (mod 4)) \end{matrix} .

Proof. See [5]. □

We will show now that the form b or b′ in the above theorem is not only symmetric (resp. skew-symmetric), but that b is the polar of a quadratic form on ⋀ P (resp. is an alternating form) even if 2 is not a unit in R.

Lemma 2.2. Let ⋀ P be generated by e_{1}, …, e_{l} where e_{i} = x_{i_{1}} ∧ ⋯ ∧ x_{i_{r_{i}}}. Let b be the bilinear form b or b′ in the above theorem and μ_{ij} := b(e_{i}, e_{j}). For x = λ_{1}e_{1} + ⋯ + λ_{l}e_{l} we consider b(x, x) as a polynomial in Z[λ_{1}, …, λ_{l}, μ_{12}, …].

(a) If b is symmetric, there exists g ∈ Z[λ_{1}, …, λ_{l}, μ_{12}, …] such that b(x, x) = 2g(λ_{1}, …, λ_{l}, μ_{12}, …).

(b) If b is skew-symmetric, then we have b(x, x) = 0 ∀x ∈ ⋀ P.

Proof. We have b(e_{i}, e_{i}) = 0. Thus b(x, x) = ∑_{i,j} λ_{i}λ_{j}b(e_{i}, e_{j}) = ∑_{i} λ_{i}^{2}b(e_{i}, e_{i})+ ∑_{i≠j} λ_{i}λ_{j}b(e_{i}, e_{j}) =

2 ∑_{i<j} λ_{i}λ_{j}μ_{ij}. Now (a) follows if we set

g(λ_{1}, …, μ_{12}, …) := ∑_{i<j} λ_{i}λ_{j}μ_{ij},

and (b) follows immediately from b(e_{i}, e_{j}) = −b(e_{j}, e_{i}). □

Theorem 2.3. Let (M, q) be a quadratic space of rank 2m such that there is an isomorphism α : C(M, q) → End_{R}(E) . Let τ denote the canonical or standard involution of C(M, q). Then ατα^{−1} = τ_{c} where τ_{c} is the adjoint of a bilinear form on E which is either alternating or the polar of a quadratic form p. Moreover, in the second case, if a linear map g : E → E satisfies c(x, y) = c(g(x), g(y)) · a for some a ∈ R, we have p(x) = p(g(x)) · a.

Proof. For (M, q) hyperbolic, the assertions are in 2.1 and 2.2. In the general case, there exists a splitting

β : (M, q) ⊗ S ≃ H(P)

(S a commutative, faithfully flat R-algebra) and an isometry

(E ⊗ S, c ⊗ S) → (⋀ P, b).

where b denotes the form b or b′ of 2.1. The form b is even. It follows by descent that c is even, which proves the first statement. The second follows from the same reasons and the fact that polynomial rings over Z are integral domains. □

A variation of this proof shows that the theorem remains correct if there is an isomorphism α : C(M, q) ⊗ Q → End_{R}(E) where Q is a quaternion algebra (i.e. an Azumaya-algebra of rank 4) and the involution is τ ⊗ σ where σ is the standard involution of Q).

We consider now quadratic spaces of odd rank. We begin with a special case:

Lemma 2.4. Let P a projective module of rank m. Let q′ be the quadratic form q′(v, f, μ) = f(v) + aμ^{2} on M = P ⊕ P^{∗} ⊕ R (a ∈ R^{•}). Let τ be the standard involution auf C_{0}(M, q). Then the following holds:

(a) There is an isomorphism α′ : C_{0}(q) → End_{R}(⋀ P),

(b) α′τα^{′−1}(g) = u^{′−1}(x^{∗} ⊗ 1)u′, where u′ is the homomorphism defined in 2.1 (c).

Proof. The lemma follows immediately from 2.1 and the fact that C(U, p) and C_{0}(U ⊕ R, p⊥⟨a⟩) are isomorphic as algebras with involutions for every quadratic space (U, p) if C(U, p) carries the standard involution. □

In the same way as 2.3 we can prove:

Theorem 2.5. Let (M, q) be a quadratic space of rank 2m + 1 such that there is an isomorphism α : C_{0}(M, q) → End_{R}(E). Let τ denote the standard involution of C_{0}(M, q). Then ατα^{−1} = τ_{c}, where τ_{c} is the adjoint of a bilinear form on E which is either alternating or the polar of a quadratic form p. Moreover, in the second case, if a linear map g : E → E satisfies c(x, y) = c(g(x), g(y)) · a for some a ∈ R, we have p(x) = p(g(x)) · a.

3. Compositions with quadratic forms

Let (M, q) and (E, p) be quadratic spaces. A composition of (M, q) with (E, p) is a linear map ϕ : M × E → E such that the following holds:

p(ϕ(y)) = p(y)q(x) ∀x ∈ M, y ∈ E.

We shall write ϕ : (M, q) × (E, p) → (E, p) for such a composition. Of course, more general definitions are possible, e.g. compositions (M, q) × (E, p) → (E′, p′). We allow (E, p) to take values in some invertible module I (i.e. p is a map E → I with the usual properties of a quadratic form), but we always suppose (M, q) to be R-valued. By the type of a composition we mean the triple (rank(M), rank(E), rank(E)). We often shall write ϕ_{x} for the map y ↦ ϕ(x, y).

Let m be a positive integer. We will denote by π_{+}(m) the following function: let m = 8a + b where a, b are integers with 0 ≤ b < 8. Then we have

\begin{matrix}b & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ π_{+}(m) & 2^{4a−1} & 2^{4a} & 2^{4a+1} & 2^{4a+2} & 2^{4a+2} & 2^{4a+3} & 2^{4a+3} & 2^{4a+3} \end{matrix}

If R is a field of characteristic not equal to 2, it is known that π_{+}(m) is the smallest integer l such that there exists a composition of type (m, l, l) (see [3], p.133ff). We will show in the following that over an arbitrary ring this result is still true except for obvious counterexamples.

We will consider the algebra C(M, q) resp. C_{0}(M, q) as an element of the Brauer group of R. We will write [A] = 1 if the Azumaya algebra A is Brauer-equivalent to R, i.e. if A ≃ End_{R}(E) for some projective R-module E, and [A] = Q will mean that A is equivalent to a quaternion algebra Q. Moreover, Z(B) will denote the center of the algebra B.

Theorem 3.1. Let (M, q) be a quadratic space of rank m. We suppose that (M, q) is pointed, i.e. that there exists an element z ∈ M with q(z) = 1, and that the following conditions are satisfied:

(a) If m ≡ 0 (mod 8) then [C(M, q)] = 1 in Br(R) and Z(C_{0}(M, q)) ≃ R × R.

(b) If m ≡ ±1 (mod 8) then [C_{0}(M, q)] = 1 in Br(R).

(c) If m ≡ ±2 (mod 8) then [C_{0}(M, q)] = 1 in Br(Z) where Z = Z(C_{0}(M, q)).

(d) If m ≡ ±3 (mod 8) then [C(M, q)] = Q in Br(R) for a quaternion algebra Q.

(e) If m ≡ 4 (mod 8) then [C(M, q)] = Q in Br(R) for a quaternion algebra Q, and Z(C_{0}(M, q)) ≃ R × R.

Then (M, q) admits a composition of type (m, π_{+}(m), π_{+}(m)).

Proof. (a) Let α : C(M, q) → End_{R}(E) be an isomorphism, and let e be a nontrivial idempotent of Z(C_{0}(M, q)). We define

E_{1} = (α(e))(E), E_{2} = (α(1 − e))(E).

It is easy to show that E = E_{1} ⊕ E_{2}, α(C_{0}(M, q)) = End_{R}(E_{1}) × End_{R}(E_{2}) and rank(E_{1}) = rank(E_{2}) = 2^{\frac{m}{2} −1} = π_{+}(m). Since m ≡ 0 (mod 8), the standard involution of C(M, q) is of orthogonal type. Thus we have

ατα^{−1}(g) = u^{−1}(g^{∗} ⊗ 1)u ∀g ∈ End_{R}(E)

for an isomorphism u : E → Hom_{R}(E, I), where I is an invertible R-module, and we have u = u^{∗} ⊗ 1_{I}. Thus u is the adjoint of a symmetric bilinear form on E which is, by 2.3, polar of a quadratic form p. We can show that the decomposition E = E_{1} → E_{2} is orthogonal with respect to p if we use the fact that the standard involution of C_{0}(M, q) induces the identity on Z(C_{0}(M, q)). Now set

ϕ(x, y) = α(zx(y)) (x ∈ M, y ∈ E_{1}).

We show that ϕ is a composition:

(u(ϕ(x, y)))(ϕ(x, y′)) = (uα(zx)y)(α(zx)y′)
= (α(zx)^{∗} ⊗ 1_{I})(uα(zx)y)(y′)
= (uα(τα^{−1}(α(zx)))α(zx)y)(y′)
= (uα(τ(x)τ(z)zx)y)(y′)
= (uα(xzzx)y)(y′)
= (u(y))(y′)q(z)q(x)

By 2.3, ϕ is a composition (M, q) × (E_{1}, p) → (E_{1}, p). The type of the composition is (m, π_{+}(m), π_{+}(m)).

(b) Let α : C_{0}(M, q) → End_{R}(E) be an isomorphism, and let u : E → Hom(E, I) be as in (a). Let p denote the polar of the adjoint bilinear form of u. Then we can prove as in (a) that ϕ : M × E → E, ϕ(x, y) = α(zx(y)) is a composition of the wanted type.

(c) Let β : C_{0}(M, q) → End_{Z}(E) be an isomorphism. The homomorphism γ : M → End_{R}(E⊕E), γ(x)(y, y′) = (β(xz)(y′), β(zx)(y)) can (by the universal property of the Clifford algebra) be extended to a homomorphism of algebras

γ : C(M, q) → End_{R}(E ⊕ E).

Now we define

Q = {g ∈ End_{R}(E ⊕ E′); ag = ga ∀a ∈ γ(C(M, q))}.

By well-known results on Azumaya algebras, Q is a quaternion algebra and there exists a homomorphism

α : C(M, q) ⊗ Q → End_{R}(E ⊕ E′).

It is defined by α(c ⊗ g) = γ(c)g. The checker-board grading of End_{R}(E ⊕ E) induces, by an easy calculation, via α a grading on C(M, q) which coincides with the usual one, and a nontrivial grading on Q. Therefore we have an isomorphism

α : C_{0}(M, q) ⊗ Q_{0} ⊕ C_{1}(M, q) ⊗ Q_{1} → End_{R}(E) ⊕ End_{R}(E′).

Let τ be the standard involution of C(M, q) and σ the standard involution of Q. Since both involutions are even and of symplectic type, α(τ ⊗ σ)α^{−1} is an even orthogonal involution on E ⊕ E. Thus the corresponding bilinear form is the polar of a quadratic form p with values in some invertible module I (as in (a)). We will show now that the decomposition E ⊕ E is orthogonal with respect to p. Let e be the inverse image under α of the first projection of E ⊕ E. Then we have e ∈ Z(C_{0}(M, q) ⊗ Q_{0} ⊕ C_{1}(M, q) ⊗ Q_{1}) ⊂ C_{0}(M, q) ⊗ Q_{0}. Using the fact that the restriction of τ on Z(C_{0}(M, q)) is the standard involution of Z(C_{0}(M, q)), we get (τ ⊗ σ)(e) = e. Now let be y_{i} in the ith component of (E ⊕ E). Then we have

u(y_{1})(y_{2}) = u((α(e))(t))(α(1 − e))(t′)) (t, t′ ∈ E ⊕ E)
= (α(1 − e))^{∗}u((α(e))(t))(t′)
= u(α(τ ⊗ σ)(1 − e))(α(e))(t)(t′)
= uα((1 − e)e)(t)(t′) = 0.

Thus p is a nonsingular quadratic form on E. As in (a), we have a composition (M, q) × (E, p) → (E, p) defined by ϕ(x, y) = β(zx)(y).

(d) Let α : C_{0}(M, q) ⊗ Q → End_{R}(E) be an isomorphism. Let τ be the standard involution of C_{0}(M, q) and σ the standard involution of Q. Then τ ⊗σ is an even orthogonal involution. Now we can continue as in (a).

(e) is proved by the same arguments as in (a) and (d). □

We now want to prove the converse of the above theorem. We use the following construction: If ϕ : (M, q) × (E, p) → (E, p) is a composition, the map

α : M → End_{R}(E ⊕ E), α(x)(y, y′) = (u^{−1}ϕ_{x}^{∗}u(y′), ϕ_{x}(y)),

where ϕ_{x}(y) = ϕ(x, y) and u is the adjoint of the polar of p, can be extended by the universal property of Clifford algebras to an algebra homomorphism

α : C(M, q) → End_{R}(E ⊕ E)

whose restriction yields a homomorphism

α : C_{0}(M, q) → End_{R}(E).

This homomorphism preserves the involution if C(M, q) carries the canonical (resp. standard)

involution and End_{R}(E ⊕ E) the involution induced by ( \begin{matrix}u & 0 \\ 0 & u \end{matrix} ) (resp. ( \begin{matrix}u & 0 \\ 0 & −u \end{matrix} ).

Theorem 3.2. For a quadratic space (M, q) of rank m, the conditions of Theorem 3.1 (without the existence of z with q(z) = 1) are necessary for the existence of a composition of type (m, π_{+}(m), π_{+}(m)) except if 2 is a zero divisor in R and m ≡ 4 (mod 8).

Proof. (a) The map α defined above is a homomorphism of Azumaya algebras of the same rank. Therefore it is an isomorphism. We have α(C_{0}(M, q)) = End_{R}(E) × End_{R}(E). The restriction on the center yields Z(C_{0}(M, q)) ≃ R × R.

(b) The map α : C_{0}(M, q) → End_{R}(E) defined above is a homomorphism of Azumaya algebras of the same rank and therefore an isomorphism.

(c) The map α : C_{0}(M, q) → End_{R}(E) is a homomorphism of Azumaya algebras. We have a Z-module structure on E via α and therefore a Z-algebra homomorphism α : C_{0}(M, q) → End_{Z}(E). Since both algebras have the same rank and C_{0}(M, q) is Azumaya over Z, it suffices to show that the Z-module E is faithful. Now suppose that R is local, and let {1, z} be a basis of Z which satisfies z^{2} = z + r, 1 + 4r ∈ R^{•} and \overline{z} = 1 − z (z ↦ \overline{z} denotes the standard involution of Z) (see [1]). Since the standard involution of C_{0}(M, q) induces the standard involution on Z, we have β(z) = 1−z and therefore β(\overline{sz + t}) = β(−sz +s+t) = 0. This implies β(s + 2t) = 0 and s + 2t = 0.

Thus we get β(sz + t) = β(t(−2z + 1)) = 0 and β(t(−2z + 1)^{2}) = β(t)β(4r + 1) = 0. This equality implies t = 0, sz + t = 0. If R is arbitrary, the statement is true since it’s true in every localization.

(d) The map α : C_{0}(M, q) → End_{R}(E) defined above is a homomorphism of Azumaya algebras. We define

Q = {g ∈ End_{R}(E); gc = cg ∀c ∈ β(C_{0}(M, q))}.

Then Q is a quaternion algebra and there is a homomorphism

β : C_{0}(M, q) ⊗ Q → End_{R}(E).

The Azumaya algebras C_{0}(M, q) ⊗ Q and End_{R}(E) are of the same rank and therefore β is an isomorphism.

(e) The existence of an isomorphism α : C(M, q) ⊗ Q → End_{R}(E ⊕ E)) is proved as in (d). The checker-board grading on End_{R}(E ⊕ E) induces via α a grading of C(M, q) ⊗ Q, and we can show that it induces also a grading of Q. It is not difficult to show that the involution ρ on End_{R}(E ⊕ E) induces the standard involution τ on C(M, q), which is of symplectic type, and an involution σ on Q which is symplectic since ρ is orthogonal and ρ = α(τ ⊗ σ)α^{−1}. Thus σ is the standard involution of Q, and it induces the standard involution on Z(Q_{0}). We have

Z(C_{0}(M, q) ⊗ Q_{0} ⊕ C_{1}(M, q) ⊗ Q_{1}) ≃ R × R.

Using the fact that τ induces the identity on Z = Z(C_{0}(M, q)) we get

Z(C_{0}(M, q) ⊗ Q_{0} ⊕ C_{1}(M, q) ⊗ Q_{1}) ⊂ {z ∈ Z ⊗ Z(Q_{0}); (τ ⊗ σ)(z) = z}
= {z ∈ Z ⊗ Z(Q_{0}); (1 ⊗ σ)(z) = z}
= Z

(The last equality holds because 2 is not a zero divisor). Since rank(Z(C_{0}(M, q))) = 2 we conclude Z(C_{0}(M, q)) ≃ R × R. □

Note that in the case m ≡ 4 (mod 8) the condition that 2 is not a zero divisor cannot be dropped. However the statement of Theorem 3.2 holds for m = 4, see [1]. Now we show that the compositions constructed in 3.1 are the best possible:

Theorem 3.3. Let m be a positive integer. The smallest number l such that there exists a composition with quadratic spaces of type (m, l, l) is exactly π_{+}(m) except in the cases m ≡ 3, 4, 5 (mod 8) and 2 = 0 in R. In these cases we have l = \frac{π_{+}(m)}{2}.

Proof. If m ≡ 0 (resp. 1,7, resp. 2,6) (mod 8), the theorem follows by considering the injective homomorphism of Azumaya algebras (see 3.2) C(M, q) → End_{R}(E ⊕ E) (resp. C_{0}(M, q) → End_{R}(E), resp. C_{0}(M, q) → End_{Z}(E)) and comparing the ranks. If m ≡ 4 (resp. 3,5) and 2 ≠ 0 this homomorphism cannot be surjective because the standard involution of C(M, q) (resp. C_{0}(M, q)) and the involution on End_{R}(E ⊕ E) (resp. End_{R}(E)) are not of the same type. For the exceptional cases, see Theorem 4.1. □

4. Compositions of quadratic forms with alternating forms

Let (M, q) be a quadratic space and (E, b) a symplectic space. A composition of (M, q) with (E, b) is a linear map ϕ : M × E → E such that the following holds:

b(ϕ(x, y), ϕ(x, y′)) = b(y, y′)q(x) ∀x ∈ M, y, y′ ∈ E.

We will write ϕ : (M, q) × (E, b) → (E, b). The bilinear form b may have values in some invertible module I.

Let m be a positive integer. We will denote by π_{−}(m) the following function: let m = 8a + b where a, b are integers with 0 ≤ b < 8. Then we have

\begin{matrix}b & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ π_{−}(m) & 2^{4a} & 2^{4a+1} & 2^{4a+1} & 2^{4a+1} & 2^{4a+1} & 2^{4a+2} & 2^{4a+3} & 2^{4a+4} \end{matrix}

If R is a field of characteristic not 2, it is known that π_{−}(m) is the smallest integer l such that there exists a composition with an alternating form of type (m, l, l). This result is still true over an arbitrary ring with the same exceptions as in 3.3. The following results are analogous to those for composition with quadratic forms and have nearly the same proofs as 3.1, 3.2 and 3.3.

Theorem 4.1. Let (M, q) be a quadratic space of rank m. We suppose that (M, q) ist pointed, i.e. that there exists an element z ∈ M with q(z) = 1, and that the following conditions are satisfied:

(a) If m ≡ 4 (mod 8) then [C(M, q)] = 1 in Br(R) and Z(C_{0}(M, q)) ≃ R × R.

(b) If m ≡ ±3 (mod 8) then [C_{0}(M, q)] = 1 in Br(R).

(c) If m ≡ ±2 (mod 8) then [C_{0}(M, q)] = 1 in Br(Z) where Z = Z(C_{0}(M, q)).

(d) If m ≡ ±1 (mod 8) then [C(M, q)] = Q in Br(R) for a quaternion algebra Q.

(e) If m ≡ 0 (mod 8) then [C(M, q)] = Q in Br(R) for a quaternion algebra Q, and Z(C_{0}(M, q)) ≃ R × R.

Then (M, q) admits a composition of type (m, π_{−}(m), π_{−}(m)) with an alternating form.

Theorem 4.2. For a quadratic space (M, q) of rank m, the conditions of Theorem 4.1 (without the existence of z with q(z) = 1) are necessary for the existence of a composition of type (m, π_{−}(m), π_{−}(m)) except if 2 is a zero divisor in R and m ≡ 0 (mod 8).

Theorem 4.3. Let m be a positive integer. The smallest number l such that there exists a composition of a quadratic space with a symplectic space of type (m, l, l) is exactly π_{−}(m) except in the case m ≡ 7, 0, 1 (mod 8) and 2 = 0 in R. In that case we have l = \frac{π_{−}(m)}{2}.

5. Composition with hermitian forms

We consider hermitian spaces (E, b) over a quadratic R-algebra S. Let (M, q) be a quadratic space. Compositions (M, q) × (E, b) → (E, b) are defined as in the preceding section. We suppose that a composition is S-linear in the second variable, i.e. the equality ϕ(x, sy) = sϕ(x, y) holds for x ∈ M, y ∈ E and s ∈ S. The hermitian form may have values in an invertible S-module.

For a positive integer m, let π_{h}(m) be defined as follows: if m is even we set π_{h}(m) = 2^{(m−2)/2}, and if m is odd we set π_{h}(m) = 2^{(m−1)/2}. We want to show that π_{h}(m) is the smallest S-rank of a hermitian space (E, b) such that there exist a composition (M, q)×(E, b) → (E, b) for some quadratic space (M, q) of rank m.

Theorem 5.1. Let (M, q) be a quadratic space of rank m. We suppose that (M, q) is pointed. Then the following holds:

(a) If m ≡ 0 (mod 4), [C(M, q)] = Q in Br(R), where Q is a quaternion algebra with a nontrivial grading Q = Q_{0} ⊕ Q_{1}, and Z(C_{0}(M, q)) ≃ R × R, then (M, q) admits a composition with a hermitian space of rank π_{h}(m) over Q_{0}.

(b) If m ≡ ±1 (mod 4) and [C(M, q)] = Q in Br(R), where Q is a quaternion algebra with a nontrivial grading Q = Q_{0} ⊕ Q_{1}, then (M, q) admits a composition with a hermitian space of rank π_{h}(m) over Q_{0}.

(c) If m ≡ ±2 (mod 8) [C_{0}(M, q)] = Z in Br(Z), where Z = Z(C_{0}(M, q), then (M, q) admits a composition with a hermitian space of rank π_{h}(m) over Z.

Proof. (a) As in Theorem 3.1 (e), we have an isomorphism

α : C_{0}(M, q) ⊗ Q → End_{R}(E_{1}) × End_{R}(E_{2}).

Now let

S = Z(C_{0}(M, q) ⊗ Q_{0}) ⊕ (C_{1}(M, q) ⊗ Q_{1})).

Let ρ be the standard involution of Z(C_{0}(M, q)) and σ that of Q. We have S = {s ∈ Z(C_{0}(M, q) ⊗ Q_{0}); (ρ ⊗ σ)(s) = s}. The map α induces a S-module-structure on E_{1} ⊕ E_{2}. For c ∈ (C_{0}(M, q) ⊗ Q_{0}) ⊕ (C_{1}(M, q) ⊗ Q_{1}) we get immediately

α(c) ∈ End_{S′}(E).

By the theory of quadratic algebras with involution (see e.g. [1], p.129), and using Z(C_{0}(M, q)) ≃ R × R, we conclude S ≃ Q_{0} and therefore

C_{0}(M, q) ⊗ Q_{0} ≃ End_{Q_{0}}(E_{1}) × End_{Q_{0}}(E_{2}).

The standard involution τ on C_{0}(M, q) is of the first kind, and σ induces the standard involution of Q_{0}. Thus τ ⊗ σ is of the second kind. Therefore τ ⊗ σ is the adjoint of a nonsingular hermitian form on E_{1} ⊕ E_{2} or, as in Theorem 3.1 (a), on E_{1}. Now we define the composition ϕ : M × E_{1} → E_{1} by ϕ(x, y) = (α(zx ⊗ 1))(y). As in the preceding sections we can prove that ϕ is a composition of the wanted type.

(b) Let α : C_{0}(M, q) ⊗ Q → End_{R}(E) be an isomorphism. Via α, E becomes a Q_{0}-module of rank π_{h}(m). The standard involution of C_{0}(M, q) and the standard involution of Q_{0} induce an involution of the second kind on End_{R}(E) and therefore a nonsingular hermitian form b on E. As in the preceding sections we can show that ϕ : M × E → E, ϕ(x, y) = (α(zx) ⊗ 1)(y) is a composition of (M, q) with (E, b).

(c) Let α : C_{0}(M, q) → End_{Z}(E) be an isomorphism. The standard involution of C_{0}(M, q) is of the second kind and therefore induces a hermitian form on E. Now we continue as in (b).

□

Theorem 5.2. For a quadratic space (M, q) of rank m, the sufficient conditions (a), (b) and (c) of Theorem 5.1 are necessary for the existence of a composition M × E → E with a hermitian space over a quadratic algebra S, where rank(M) = m and rank_{S}(E) = π_{h}(m). Proof. (a) As in the bilinear case, the composition induces an isomorphism α : C(M, q)⊗Q → End_{R}(E ⊗ E) for a quaternion algebra Q. Since the composition is S-linear, multiplication with s ∈ S of E defines an element of C(M, q) ⊗ Q which lies in R ⊗ Q because of the S-linearity of α. Thus Q contains a copy of S which is denoted by Q_{0}. Now α induces an isomorphism

α : C_{0}(M, q) ⊗ Q_{0} → End_{S}(E) × End_{S}(E)

Let ρ be the involution on End_{R}(E ⊕E) induced by the hermitian form on E. Then α^{−1}ρα = τ ⊗ σ, where τ is the standard involution of C_{0}(M, q). Since ρ is of the second and τ of the first kind, σ is the standard involution of Q_{0}. Now we set π : E ⊕E → E ⊕E, π(y, y′) = (y, 0) and e = α^{−1}. We have ρ(π) = π and therefore (1 ⊗ σ)(e) = e. Thus e ∈ Z(C_{0}(M, q)), and e is idempotent because π is. Therefore we have Z(C(M, q)) ≃ R × R.

(b) The composition induces an injective homomorphism α : C_{0}(M, q) → End_{R}(E). Now, if Q = {g ∈ End_{R}(E); gc = cg ∀c ∈ α(C_{0}(M, q))}, then Q is a quaternion algebra and the first statement is proved. As above one shows that Q contains a copy of S.

(c) The composition induces an injective homomorphism

α : C_{0}(M, q) → End_{R}(E)

whose image is End_{Z}(E). It is easy to prove that S ≃ Z. □

By similar arguments as in the bilinear case, we can prove: Theorem 5.3. Let m be a positive integer. The smallest number l such that there exists a composition of a quadratic space of rank m with a hermitian space of S-rank l over a quadratic algebra S is exactly π_{h}(m).

6. Composition of quadratic forms with hermitian forms over quaternion algebras

In this section we consider compositions with hermitian spaces (E, b) over a quaternion algebra Q. We suppose that E is a left Q-module and the involution on Q is always the standard involution.

In the bilinear case, we used the fact that an involution on End_{R}(E) induces a bilinear form on E with values in an invertible module (this is a consequence of a generalized version of the Skolem-Noether theorem). In order to generalize our results for composition with hermitian forms over quaternion algebras (or, more generally, over Azumaya algebras with involution) we need the following

Lemma 6.1. Let A be an Azumaya algebra with involution σ, and let E be a faithfully projective A-module. If τ is an involution on End_{A}(E). Then τ(g) = u^{−1}g^{∗}u for a semilinear map u : E → Hom_{A}(E, A ⊗ I) where I is an invertible R-module. Moreover u is the adjoint of a hermitian oder skew-hermitian form.

Proof. See [2]. □

Let (E, b) be a symplectic space. The form

B(( \begin{matrix}x \\ y \end{matrix} ) , ( \begin{matrix}x′ \\ y′ \end{matrix} )) = ( \begin{matrix}b(y, x′) & −b(x, x′) \\ b(y, y′) & −b(x, y′) \end{matrix} )

on E ⊕ E is a hermitian form over M_{2}(R). Conversely we can prove by Morita-theory that if (F, B) is a hermitian space over M_{2}(R) then F ≃ E ⊕ E and B can be constructed from an alternating form b as above. Thus there is a 1-1-correspondence between hermitian spaces over M_{2}(R) and symplectic spaces over R or between compositions with hermitian spaces over M_{2}(R) and compositions with symplectic spaces over R. If a composition of a quadratic space with a hermitian space over a quaternion algebra Q is given, we choose a faithfully flat commutative R-algebra S such that Q ⊗ S ≃ M_{2}(S). Therefore the properties of such a composition can be taken over from the theory of compositions with alternating forms:

Theorem 6.2. Let Q be a quaternion algebra. Let (M, q) a pointed quadratic space of rank m which statisfies the following conditions:

(a) If m ≡ 4 (mod 8) then [C(M, q)] = Q in Br(R) and Z(C_{0}(M, q)) ≃ R × R.

(b) If m ≡ ±3 (mod 8) then [C_{0}(M, q)] = Q in Br(R).

(c) If m ≡ ±2 (mod 8) then [C_{0}(M, q)] = Q ⊗ Z in Br(Z) if we denote by Z the algebra Z(C_{0}(M, q)).

(d) If m ≡ ±1 (mod 8) then [C(M, q)] = Q ⊗ Q′ in Br(R) for a quaternion algebra Q′.

(e) If m ≡ 0 (mod 8) then [C(M, q)] = Q ⊗ Q′ in Br(R) for a quaternion algebra Q, and Z(C_{0}(M, q)) ≃ R × R.

Then (M, q) admits a composition with a hermitian space (E, B) over Q, and the rank of E as R-module is 2π_{−}(m).

Theorem 6.3. For a quadratic space (M, q) of rank m, the conditions of Theorem 6.2 (without the existence of z with q(z) = 1) are necessary for the existence of a composition of (M, q) with a hermitian space of R-rank 2π_{−}(m) over a quaternion algebra except if 2 is a zero divisor in R and m ≡ 0 (mod 8).

Theorem 6.4. Let m be a positive integer. The smallest number l such that there exists a composition of a quadratic space of rank m with a hermitian space of rank l (as an R-module) over a quaternion algebra Q is exactly 2π_{−}(m) excepted in the cases m ≡ 7, 0, 1 (mod 8) and 2 = 0 in R. In these cases we have l = π_{−}(m).

Note that if we take a symmetric bilinear form instead of an alternating one, the construction above yields compositions with skew-hermitian forms. We get analogous results if we use the theorems of Section 3.

These results can be easily generalized for compositions of quadratic spaces with hermitian or skew-hermitian spaces over an Azumaya algebra with involution.

If B is an even hermitian form over a quaternion algebra, then p(x) = B(x, x) is a quadratic form over R. Let (M, q) × (E, B) → (E, B) be a composition with a hermitian form over a quaternion algebra. Then we get a composition of quadratic spaces (M, q) × (E, p) → (E, p). We want to compare this composition with those constructed in Section 3, which will be denoted (M, q) × (E_{0}, p_{0}) → (E_{0}, p_{0}). Thus we have rank(E_{0}) = π_{+}(m) where m =rank(M). As usual we suppose that (E, p) and (E_{0}, p_{0}) have values in an invertible R-module.

Theorem 6.5. Let ϕ : (M, q)×(E, B) → (E, B) be a composition of a quadratic space (M, q) of rank m, with a hermitian space (E, B) over a fixed quaternion algebra Q such that the rank of E is minimal. Then

(a) If m ≡ 0, (resp. 1,7) (mod 8), and if [C(M, q)] = 1 (resp. [C_{0}(M, q)] = 1) in Br(R), we have (E, p) ≃ (E_{0}, p_{0}) ⊗ (Q, r), where r is a symmetric bilinear form on Q with values in an invertible R-module.

(b) If m ≡ 3, 4, 5 (mod 8), then (E, p) is similar to (E_{0}, p_{0}).

Proof. (a) We only give the proof for m ≡ ±1 (mod 8). The other case is similar. We have isomorphisms

α : C_{0}(M, q) →≃ End_{End_{R}Q}E

and, by Theorem 3.2,

β : C_{0}(M, q) → End_{R}(E).

By Morita-theory we conclude that E = E_{0} ⊗ Q. Let u be the adjoint of B and \overline{u} = u + u^{∗} : E → Hom_{R}(E, Q ⊗ I). We have (as in the bilinear case): u^{−1}(β(c ⊗ d))^{∗}u = (τ ⊗ σ′)(c ⊗ d), and therefore u^{∗−1}(β(c ⊗ d))^{∗}u^{∗} = (τ ⊗ σ′)(c ⊗ d). For d = 1 we conclude from these formulas

(α(c))^{∗}\overline{u} = \overline{u}τ(c).

As in the proof of Theorem 3.1, this formula can be used to construct a composition (M, q)× (E, \overline{B}) → (E, \overline{B}) with a bilinear form \overline{B} such that \overline{u} is adjoint of \overline{B}. We obviously have \overline{B}(x, y) = B(x, y) + B(y, x), and \overline{B} is the polar of p. We will show below (see Theorem 7.2) that

(E, \overline{B}) = (E_{0}, b_{0}) ⊗ (Q, r)

where b_{0} is the polar of p_{0} and r is a symmetric bilinear form on Q.

(b) We only give the proof for m ≡ ±3 (mod 8). The other case is similar. We have an isomorphism

α : C_{0}(M, q) →≃ End_{End_{R}Q}E.

If u is the adjoint of B and \overline{u} = u + u^{∗} : E → Hom_{R}(E, Q ⊗ I), we prove as in (a) that

(α(c ⊗ 1))^{∗}\overline{u} = \overline{u}(τ ⊗ σ)(c ⊗ 1).

By an easy calculation, using the fact that the trace tr on Q satisfies tr(ed) = tr(de), the formula holds also for c ⊗ d (d ∈ Q). We now conclude as in (a). □

Note that there is no analogous statement in (a) if [C(M, q)] ≠ 1 in Br(R), and there is no similar result if m ≡ 2 (mod 4). It can be proved that the symmetric bilinear space (Q, r) in (a) is a tensor product of two symplectic spaces of rank 2.

7. Tensor products of compositions

We fix a quadratic space of rank m which admits a composition with a bilinear space of minimal rank (symmetric or alternating). In this section we compare such a “minimal composition” with a composition of the same quadratic space with an arbitrary bilinear space. In the following, let

ϕ_{0} : (M, q) × (E_{0}, b_{0}) → (E_{0}, b_{0})

denote this minimal composition. Thus b_{0} is symmetric and rank(E_{0}) = π_{+}(m) if m ≡ 0, 1, 7 (mod 8), and b_{0} is skew-symmetric and rank(E_{0}) = π_{−}(m) if m ≡ 3, 4, 5 (mod 8). In the case m = 4k + 2 the type of b_{0} is not determined, but we should exclude this case because there is no analogous results to the following theorem.

Lemma 7.1. Let ϕ : (M, q) × (E, b) → (E, b) be a composition. Let m be the rank of M and suppose m ≠ 4k + 2. Let (E_{0}, b_{0}) be as above. Then we have E ≃ E_{0} ⊗ F for an R-module F.

Proof. The composition ϕ induces a homomorphism

C_{0}(M, q) → End_{R}(E)

Thus E is a C_{0}(M, q)-module. Since (E_{0}, b_{0}) admits a composition and the rang of E_{0} is minimal, we have by Theorem 3.2 or 4.2

C_{0}(M, q) ≃ \{ \begin{matrix}End_{R}(E_{0}) & m = 2k + 1 \\ End_{R}(E_{0}) ⊕ End_{R}(E′_{0}) & m = 4k \end{matrix}

Therefore E is an End_{R}(E_{0})-module. The statement follows by Morita-theory. □

In the following, for invertible modules I and J, let B_{I,J} denote the bilinear form I × J → I ⊗ J, (i, j) ↦ i ⊗ j.

Theorem 7.2. Let (M, q) a quadratic space of rank m, which satisfies one of the following conditions:

(1) rank(M) = 2k + 1 and α : C_{0}(M, q) → End_{R}(E_{0}) is an isomorphism.

(2) rank(M) = 4k and α : C(M, q) → End_{R}(E_{0}) is an isomorphism, and Z((C_{0}(M, q)) ≃ R × R.

Let ϕ_{0} : (M, q) × (E_{0}, b_{0}) → (E_{0}, b_{0}) be a composition where the rank of E_{0} is minimal (see above).

(a) Suppose that E = E_{0} ⊗ F and ϕ = ϕ_{0} ⊗ 1_{F} : M × E → E. Let b′ be an arbitrary nonsingular bilinear form on E with values in an invertible module I′. Then the following holds: the map ϕ is a composition (M, q) × (E, b′) → (E, b′) if and only if b′ = b_{0} ⊗ c where c is a bilinear form on F (with values in an invertible module.)

(b) Let (E, b′) be a bilinear space. Then we have the following: There exists a composition ϕ′ : (M, q) × (E, b′) → (E, b′) if and only if b′ is isometric with b = b_{0} ⊗ c where c is a bilinear form on some R-module F.

Proof. (a): The “if”-part is obvious. For the proof of the “only if”, let c̃ be a nonsingular bilinear form on F with values in an arbitrary invertible module. We put b = b_{0} ⊗ c̃. The form b has values in an invertible module, say I. Let u : E → Hom(E, I) (resp. u′ : E → Hom(E, I′)) be the adjoint of b (resp. b′.) Since ϕ is a composition, we have u(ϕ_{x})(y)((ϕ_{x})(y′)) = (ϕ_{x} ⊗ 1_{I})^{∗}(u(ϕ_{x})(y))(y′) = q(x)u(y)(y′) for all x ∈ M and y, y′ ∈ E, therefore

(ϕ_{x} ⊗ 1_{I})^{∗}uϕ_{x} = q(x) · u

and in the same way

(ϕ_{x} ⊗ 1_{I′})^{∗}u′ϕ_{x} = q(x) · u′.

We define the invertible R-module L by L = I^{∗} ⊗ I′. We get

1_{L} ⊗ ((ϕ_{x} ⊗ 1_{I})^{∗}uϕ_{x}u^{−1}) = (ϕ_{x} ⊗ 1_{I′})^{∗}u′ϕ_{x}u^{′−1}

and after some calculations

(ϕ_{x} ⊗ 1_{L})((u^{−1} ⊗ 1_{L})u′) = ((u^{−1} ⊗ 1_{L})u′)ϕ_{x} ∀x ∈ M.

Now the set {ϕ_{x}; x ∈ M} generates the algebra α(C_{0}(M, q)). Thus (u^{−1} ⊗ 1_{L})u′ ∈ End_{R}(E) commutes with α(C_{0}(M, q)). In the case m = 2k + 1 we have α(C_{0}(M, q)) = End_{R}(E_{0}), therefore (u^{−1}⊗1_{L})u′ commutes with End_{R}(E_{0}). Because of Hom(E, E⊗L) ≃ End_{R}(E)⊗L ≃ End_{R}(E_{0}) ⊗ End_{R}(F) ⊗ L ≃ End_{R}(E_{0}) ⊗ Hom(F, F ⊗ L) we conclude

(u^{−1} ⊗ 1_{L})u′ ∈ Hom(F, F ⊗ L).

In other words, there is a f ∈ Hom(F, F ⊗ L) such that u′ = uf. Thus, for y = y_{0} ⊗ z, y′ = y′_{0} ⊗ z′ ∈ E = E_{0} ⊗ F we have

b′(y_{0} ⊗ z, y′_{0} ⊗ z′) = (b ⊗ B_{L,R})(y_{0} ⊗ f(z), y′_{0} ⊗ z′) = b_{0}(y_{0}, y′_{0}) ⊗ (c̃ ⊗ B_{L,R})(f(z), z′).

Therefore, for c we can take

c(z, z′) = (c̃ ⊗ B_{L,R})(f(z), z) (z, z′ ∈ F).

The case m = 4k is similar (we use α(C_{0}(M, q)) = End_{R}(E_{0}) ⊕ End_{R}(E′_{0})).

(b): The “if”-part is clear. Let us prove “only if” in the case m = 2k + 1. Let

β′ : C_{0}(M, q) → End_{R}(E)

be the homomorphism of algebras with involution induced by ϕ′ (see Section 3). We set D = {g ∈ End_{R}(E), gc = cg ∀g ∈ β′(C_{0}(M, q))}. We get an isomorphism

γ : β′(C_{0}(M, q)) ⊗ D ≃ End_{R}(E),

and, using the theory of Azumaya algebras, we can prove

D ≃ End_{R}(F′)

where E_{0} ⊗ F′ ⊗ U ≃ E for an invertible module U. We define F = F′ ⊗ U. Then there is an isomorphism

δ : D → End_{R}(F)

and an isomorphism

ϵ : α(C_{0}(M, q)) ⊗ End_{R}(F) → End_{R}(E)

which is induced by E_{0} ⊗ F ≃ E. Now let c̃ be a nonsingular bilinear form on F with values in an invertible module. We define b = b_{0} ⊗ c̃. The form b has values in an invertible module, say I. Let u : E → Hom(E, I) (resp. u′ : E → Hom(E, I′)) be the adjont of b (resp. b′). Set

β : C_{0}(M, q) → End_{R}(E) ≃ End_{R}(E_{0}) ⊗ End_{R}(F), β(a) = ϵ(α(a) ⊗ 1)

for a ∈ C_{0}(M, q). We consider C_{0}(M, q) to be equipped with the standard involution τ and End_{R}(E) with the involution induced by u. Then β is a homomorphism of algebras with involution, therefore

β(τ(a)) = u^{−1}((β(a))^{∗} ⊗ 1_{I})u, β′(τ(a)) = u^{′−1}((β′(a))^{∗} ⊗ 1′_{I})u′.

Now we define

ψ : End_{R}(E) → End_{R}(E), ψ = γ(β′α^{−1} ⊗ δ^{−1})ϵ^{−1}.

Then, for a ∈ C_{0}(M, q), we have

ψβ(a) = γ(β′α^{−1} ⊗ δ^{−1})ϵ^{−1}β(a) = γ(β′α^{−1} ⊗ δ^{−1})(α(a) ⊗ 1_{F}) = γ(β′(a) ⊗ 1_{F}) = β′(a).

Therefore, by a generalized version of the Skolem-Noether theorem, there exists an invertible module J and an isomorphism h : E ⊗ J → E such that

ψ(g) = h(g ⊗ 1_{J})h^{−1}.

Because β′ = ψβ = h(β ⊗ 1_{J})h^{−1}, ∀x ∈ M, we have

h(u^{−1}(β(a)^{∗} ⊗ 1_{I})u ⊗ 1_{J})h^{−1} = u^{′−1}(h^{∗−1}(β(a)^{∗} ⊗ 1_{J}^{∗})h^{∗} ⊗ 1_{J})u′,

and by an easy calculation

u^{−1}(β(a)^{∗} ⊗ 1_{I})u = (h^{−1}u^{′−1}(h^{∗−1} ⊗ 1_{I′})) ⊗ 1_{J^{∗}}(β(a)^{∗} ⊗ 1_{J^{∗}⊗I′})((h^{∗} ⊗ 1_{I′})u′h ⊗ 1_{J}.

Set

u′′ = (h^{∗} ⊗ 1_{I′})u′h ⊗ 1_{J^{∗}} : E → Hom_{R}(E, I′ ⊗ J^{∗} ⊗ J^{∗}).

We have

u^{−1}(β(a)^{∗} ⊗ 1_{I})u = u^{′′−1}(β(a)^{∗} ⊗ 1_{J^{∗}⊗J^{∗}⊗I′})u′′

and conclude, using that u and u′′ are symmetric oder skew-symmetric,

(β(a)(u^{−1}(u′′ ⊗ 1_{L}))) = (u^{−1}(u′′ ⊗ 1_{L}))(β(a) ⊗ 1_{L}),

where L is the invertible module L = J ⊗ J ⊗ I^{′∗} ⊗ I. Therefore the homomorphism u^{−1}(u′′ ⊗ 1_{L}) ∈ Hom(E ⊗ L, E) commutes with β(C_{0}(M, q)) = End_{R}(E_{0}). From E ≃ E_{0} ⊗ F we get that f = u^{−1}(u′′ ⊗ 1_{L}) lies in Hom(F ⊗ L, F). Now u′′ is the adjoint of a bilinear form b′′. For y = y_{0} ⊗ z, y′ = y′_{0} ⊗ z′ ∈ E = E_{0} ⊗ F we have

b′′(y_{0} ⊗ z, y′_{0} ⊗ z′) = (b ⊗ B_{L,R})(y_{0} ⊗ f(z), y′_{0} ⊗ z′) = b_{0}(y_{0}, y′_{0}) ⊗ (c̃ ⊗ B_{L,R})(f(z), z′).

We now can take for c the bilinear form

c(z, z′) = (c̃ ⊗ B_{L,R}(f(z), z) (z, z′ ∈ F)

since b′′ = b_{0} ⊗ c and b′ isometric with b′′. This concludes the proof. The case m = 4k is similar (we replace End_{R}(E) by End_{R}(E_{0}) ⊕ End_{R}(E_{0}) in the first part of the proof). □

Corollary 7.3. Let (M, q) be a quadratic space of rank m ≠ 4k + 2. Bilinear spaces (E_{0}, b_{0}) of minimal rank which admit compositions ϕ_{0} : (M, q) × (E_{0}, b_{0}) → (E_{0}, b_{0}) are determined up to isometry by (M, q).

References

[1] Knus, M. A.: Quadratic and Hermitian Forms over Rings. Springer 1991.

[2] Knus, M. A.: Algèbres d’Azumaya et modules projectifs, Comm. Math. Helv. 45 (1970), 372–383.

[3] Lam, T. Y.: Algebraic theory of quadratic forms. Benjamin, New York 1973.

[4] Kneser, M.: Quadratische Formen. Ausarbeitung einer Vorlesung, Math. Institut, Göttingen 1974.

[5] Parimala, R.; Srinivas, V.: Analogue of the Brauer group for algebras with involution. Duke J. Math. 66 (1992), 207–232.

Received December 1, 1997