© Hindawi Publishing Corp. QUADRATIC PAIRS IN CHARACTERISTIC 2 AND THE WITT CANCELLATION THEOREM

We define the orthogonal sum of quadratic pairs and we show that there is no Witt cancellation theorem for this operation in characteristic 2.


Introduction.
Quadratic pairs on central simple algebras were defined in [5].They play the same role for quadratic forms as involutions for symmetric or skew-symmetric bilinear forms.In particular, they can be used to define twisted orthogonal groups in characteristic 2. In this paper, a notion of orthogonal sum of quadratic pairs is introduced on the model of Dejaiffe's orthogonal sum of involutions [2].Moreover, an example is given to show that there is no cancellation for this operation.

Orthogonal sum of quadratic pairs
Definition 2.1.Let A be a central simple algebra of degree n over a field F of characteristic 2. A quadratic pair on A is a pair (σ , f ), where σ is a symplectic involution on A and f : Sym(A, σ ) → F is a linear map satisfying the following condition: for all ∈ A. In this case, n is always even.
We recall from [2] that a Morita equivalence ((A, σ ), (B, τ), M, N, f , g, ν) between two algebras with involutions of the first kind (A, σ ) and (B, τ) is given by ( for all a ∈ A, m ∈ M, b ∈ B, and ν −1 is its inverse. If ((A, σ ), (A ,σ ), M, N, g, h, ν) is a Morita equivalence of two algebras with symplectic involutions and (σ , f ) and (σ ,f ) are quadratic pairs, respectively, on A and A , then we define the orthogonal sum of (A,σ ,f ) and (A ,σ ,f ) as follows: where We have and f : Sym(S, * ) → F defined by Proposition 2.2.The orthogonal sum (S, * ,f ) is an algebra with quadratic pair.
Proof.It is clear that the involution * is symplectic, and we have, for all (2.7) A particular case of this definition is the situation where M = N = A = A .If A is a central simple algebra over a field of characteristic 2, we consider the two algebras with quadratic pairs (A,σ ,f ) and (A, σ ,f ), where σ and σ are symplectic involutions on A. Then we have σ = Int(s) The map ν is defined by ν(x) = λsσ (x), for all x ∈ A, and we define the map (2.9) It is clear that (M 2 (A), θ λ ,g) is an algebra with quadratic pair.We recall that

Generalized quadratic forms.
Let V be a right vector space on a central division F -algebra with involution (D, * ).A generalized quadratic form on V is a pair (ψ, Q) consisting of a hermitian form ψ and a map Let D be a central division algebra over F with an involution * of any kind, V a D-vector space, and (ψ, Q) a generalized quadratic form.Then we have an for all v, w ∈ V and α ∈ D.
In fact, ϕ ψ is one-to-one, by [5, page 54, Theorem 5.1].If σ is the adjoint involution on End D (V ) with respect to ψ, then we have In [3], we established a relation between quadratic pairs and generalized quadratic forms.
Theorem 3.1.To every generalized quadratic form (V ,ψ,Q), a quadratic pair (σ , f ) can be associated on End D (V ), where σ is the adjoint involution to ψ and f is defined by ψ(x, y)) for all x, y ∈ V .The pair (σ , f ) is called the adjoint quadratic pair.
From [3], we recall the following result.Theorem 3.2.Every quadratic pair on End D (V ) is associated to a unique generalized quadratic form up to a scalar.
We now have the following theorem.
Theorem 3.3.The quadratic pair associated to the orthogonal sum of two generalized quadratic forms is the orthogonal sum of the associated quadratic pairs.
Proof.Let (V ,ψ,Q) and (W , ψ ,Q ) be two generalized quadratic forms.We can construct two algebras with quadratic pairs: be defined by the condition We can easily verify that End D (V ), σ ψ , End D (W ), σ ψ , Hom(W , V ), Hom D (V , W ), g, h, ν (3.7) is a Morita equivalence (with the same notation as in Section 2), and Using this isomorphism, we deduce that the quadratic pair (σ ψ⊕ψ ,f Q⊕Q ) corresponds to the orthogonal sum of quadratic pairs (σ ψ ,f Q ) and (σ ψ ,f Q ).In fact, for we want to show that then we have to show that (3.12) We have (3.13) On the other hand, for all h ∈ Hom D (W , V).Since ν is bijective, h=ν −1 ( ) for some ∈ Hom D (V , W ), and we have that for all ∈ Hom D (V , W ), which implies that and this proves (3.12).
Observe that Sym(End D (V ⊕W ),σ ψ⊕ψ ) is linearly generated by elements of the two following types.
Type 1.The first type of generators is Type 2. The second type is ( For two symmetric elements f and g, we have, by definition, So it suffices to show the following equality: where is a generator of Type 1 or Type 2.
We have the identification The definition of ϕ ψ⊕ψ implies that for all x 1 ,x 2 ∈ V , and it follows that Now take x ∈ V , y ∈ V , and d ∈ Sym(D, * ).Then by the definition of the associated quadratic pair.
On the other hand, (3.28) On the other hand, Counterexample 3.5.Let k be a field of characteristic 2 and F = k(x, y, z, t).We consider the quadratic forms q = 1, 1,x,y [1, t], q = 1, 1,x,z [1,t], q = 1,x,y,yz [1,t] (3.31) (see [1, page 5] for notation).Then q ⊥ q and q ⊥ q are isometric up to a scalar factor, but q and q are not isometric up to a scalar factor since the first form is isotropic whereas the second is anisotropic.We conclude that, in general, there is no Witt cancellation theorem for quadratic pairs.