i=1 i=1

Since q(v) = f —¦ •q (v — v), the ¬rst assertion is proved. Identifying EndF (V ) with

Mn (F ) by means of the basis (e1 , . . . , en ) maps to the matrix

«

0

1

1 q(e2i )

¬ ·

..

= where i = q(e

.

2i’1 ) 0

0 m

The characteristic polynomial of is the product of the characteristic polynomials

of 1 , . . . , m . This implies the second assertion.

(5.13) Example. Suppose that char F = 2 and let (1, u, v, w) be a quaternion

basis of a quaternion F -algebra Q = [a, b)F . In every quadratic pair (σ, f ) on Q,

the involution σ is symplectic. It is therefore the canonical involution γ. The space

Sym(Q, γ) is the span of 1, v, w, and Alt(Q, γ) = F . Since 1 = u + γ(u) and

TrdQ (u) = 1, the map f may be any linear form on Sym(Q, γ) such that f (1) = 1.

An element corresponding to f as in (??) is

= u + f (w)b’1 v + f (v)b’1 w.

(For a given f , the element is uniquely determined up to the addition of an element

in F .)

Quadratic pairs on tensor products. Let A1 , A2 be central simple F -

algebras. Given a quadratic pair (σ1 , f1 ) on A1 and an involution σ2 on A2 , we

aim to de¬ne a quadratic pair on the tensor product A1 —F A2 . If char F = 2, this

amounts to de¬ning an orthogonal involution on A1 —F A2 , and it su¬ces to take

σ1 — σ2 , assuming that σ2 is orthogonal, see (??). For the rest of this section, we

may thus focus on the case where char F = 2.

(5.14) Lemma. Let (A1 , σ1 ) and (A2 , σ2 ) be central simple algebras with involu-

tion of the ¬rst kind over a ¬eld F of characteristic 2.

(5.15) Symd(A1 , σ1 ) — Symd(A2 , σ2 ) =

Symd(A1 , σ1 ) — Sym(A2 , σ2 ) © Sym(A1 , σ1 ) — Symd(A2 , σ2 );

(5.16) Symd(A1 — A2 , σ1 — σ2 ) © Sym(A1 , σ1 ) — Sym(A2 , σ2 ) =

Symd(A1 , σ1 ) — Sym(A2 , σ2 ) + Sym(A1 , σ1 ) — Symd(A2 , σ2 );

§5. QUADRATIC FORMS 61

(5.17) Sym(A1 — A2 , σ1 — σ2 ) =

Symd(A1 — A2 , σ1 — σ2 ) + Sym(A1 , σ1 ) — Sym(A2 , σ2 ) .

Proof : Equation (??) is clear. For x1 ∈ A1 and s2 ∈ Sym(A2 , σ2 ),

x1 + σ1 (x1 ) — s2 = x1 — s2 + (σ1 — σ2 )(x1 — s2 ),

hence Symd(A1 , σ1 ) — Sym(A2 , σ2 ) ‚ Symd(A1 — A2 , σ1 — σ2 ). Similarly,

Sym(A1 , σ1 ) — Symd(A2 , σ2 ) ‚ Symd(A1 — A2 , σ1 — σ2 ),

hence the left side of (??) contains the right side. To prove the reverse inclusion,

consider x ∈ A1 — A2 . If x + (σ1 — σ2 )(x) ∈ Sym(A1 , σ1 ) — Sym(A2 , σ2 ), then

x + (σ1 — σ2 )(x) is invariant under σ1 — IdA2 , hence

x + (σ1 — IdA2 )(x) + (IdA1 — σ2 )(x) + (σ1 — σ2 )(x) = 0.

Therefore, the element u = x + (σ1 — IdA2 )(x) is invariant under IdA1 — σ2 , hence it

lies in A1 — Sym(A2 , σ2 ). Similarly, the element v = (σ1 — IdA2 )(x) + (σ1 — σ2 )(x)

is in Sym(A1 , σ1 ) — A2 . On the other hand, it is clear by de¬nition that u ∈

Symd(A1 , σ1 ) — A2 and v ∈ A1 — Symd(A2 , σ2 ), hence

u ∈ Symd(A1 , σ1 ) — Sym(A2 , σ2 ) and v ∈ Sym(A1 , σ1 ) — Symd(A2 , σ2 ).

Since x + (σ1 — σ2 )(x) = u + v, the proof of (??) is complete.

Since the left side of equation (??) obviously contains the right side, it suf-

¬ces to prove that both sides have the same dimension. Let ni = deg Ai , so that

dimF Sym(Ai , σi ) = 2 ni (ni + 1) and dimF Symd(Ai , σi ) = 1 ni (ni ’ 1) for i = 1, 2.

1

2

From (??), it follows that

dimF Symd(A1 , σ1 ) — Sym(A2 , σ2 ) + Sym(A1 , σ1 ) — Symd(A2 , σ2 ) =

1

’ 1)(n2 + 1) + 1 n1 n2 (n1 + 1)(n2 ’ 1) ’ 1 n1 n2 (n1 ’ 1)(n2 ’ 1)

4 n1 n2 (n1 4 4

= 1 n1 n2 (n1 n2 + n1 + n2 ’ 3).

4

Therefore, (??) yields

dimF Symd(A1 — A2 , σ1 — σ2 ) + Sym(A1 , σ1 ) — Sym(A2 , σ2 ) =

1

’ 1) + 1 n1 n2 (n1 + 1)(n2 + 1) ’ 1 n1 n2 (n1 n2 + n1 + n2 ’ 3)

2 n1 n2 (n1 n2 4 4

= 1 n1 n2 (n1 n2 + 1) = dimF Sym(A1 — A2 , σ1 — σ2 ).

2

(5.18) Proposition. Suppose that char F = 2. Let (σ1 , f1 ) be a quadratic pair

on a central simple F -algebra A1 and let (A2 , σ2 ) be a central simple F -algebra

with involution of the ¬rst kind. There is a unique quadratic pair (σ1 — σ2 , f1— ) on

A1 —F A2 such that

f1— (s1 — s2 ) = f1 (s1 ) TrdA2 (s2 )

for s1 ∈ Sym(A1 , σ1 ) and s2 ∈ Sym(A2 , σ2 ).

Proof : Since σ1 is symplectic, (??) shows that σ1 — σ2 is symplectic. To prove the

existence of a quadratic pair (σ1 —σ2 , f1— ) as above, we have to show that the values

that f1— is required to take on Symd(A1 — A2 , σ1 — σ2 ) because of the quadratic

pair conditions agree with the prescribed values on Sym(A1 , σ1 ) — Sym(A2 , σ2 ). In

view of the description of Symd(A1 — A2 , σ1 — σ2 ) © Sym(A1 , σ1 ) — Sym(A2 , σ2 )

in the preceding lemma, it su¬ces to consider the values of f1— on elements of the

62 I. INVOLUTIONS AND HERMITIAN FORMS

form x1 + σ1 (x1 ) — s2 = x1 — s2 + (σ1 — σ2 )(x1 — s2 ) or s1 — x2 + σ2 (x2 ) =

s1 — x2 + (σ1 — σ2 )(s1 — x2 ) with xi ∈ Ai and si ∈ Sym(Ai , σi ) for i = 1, 2. Since

(σ1 , f1 ) is a quadratic pair on A1 , we have

f1 x1 + σ1 (x1 ) TrdA2 (s2 ) = TrdA1 (x1 ) TrdA2 (s2 ) = TrdA1 —A2 (x1 — s2 ),

as required. For the second type of element we have

f1 (s1 ) TrdA2 x2 + σ2 (x2 ) = 0.

On the other hand, since σ1 is symplectic we have TrdA1 (s1 ) = 0, hence

TrdA1 —A2 (s1 — x2 ) = TrdA1 (s1 ) TrdA2 (x2 ) = 0.

Therefore,

f1 (s1 ) TrdA2 x2 + σ2 (x2 ) = TrdA1 —A2 (s1 — x2 )

for s1 ∈ Sym(A1 , σ1 ) and x2 ∈ A2 , and the existence of the quadratic pair (σ1 —

σ2 , f1— ) is proved.

Uniqueness is clear, since the values of the linear map f1— are determined on

the set Symd(A1 — A2 , σ1 — σ2 ) and on Sym(A1 , σ1 ) — Sym(A2 , σ2 ), and (??) shows

that these subspaces span Sym(A1 — A2 , σ1 — σ2 ).

(5.19) Example. Let (V1 , q1 ) be a nonsingular quadratic space of even dimen-

sion and let (V2 , b2 ) be a nonsingular symmetric bilinear space over a ¬eld F of

characteristic 2. Let (σ1 , f1 ) be the quadratic pair on A1 = EndF (V1 ) associ-

ated with q1 (see (??)) and let σ2 = σb2 denote the adjoint involution with re-

spect to b2 on A2 = EndF (V2 ). We claim that, under the canonical isomorphism

A1 — A2 = EndF (V1 — V2 ), the quadratic pair (σ1 — σ2 , f1— ) is associated with the

quadratic form q1 — b2 on V1 — V2 whose polar form is bq1 — b2 and such that

(q1 — b2 )(v1 — v2 ) = q1 (v1 )b2 (v2 , v2 ) for v1 ∈ V1 and v2 ∈ V2 .

∼

Indeed, letting •1 , •2 and • denote the standard identi¬cations V1 — V1 ’ ’

∼ ∼

EndF (V1 ), V2 — V2 ’ EndF (V2 ) and (V1 — V2 ) — (V1 — V2 ) ’ EndF (V1 — V2 )

’ ’

associated with the bilinear forms bq1 , b2 and bq1 — b2 , we have

•(v1 — v2 — v1 — v2 ) = •1 (v1 — v1 ) — •2 (v2 — v2 )

and

f1 •1 (v1 — v1 ) TrdA2 •2 (v2 — v2 ) = q1 (v1 )b2 (v2 , v2 ),

hence

f1— •(v1 — v2 — v1 — v2 ) = q1 — b2 (v1 — v2 ).

(5.20) Corollary. Let (A1 , σ1 ), (A2 , σ2 ) be central simple algebras with symplectic

involutions over a ¬eld F of arbitrary characteristic. There is a unique quadratic

pair (σ1 — σ2 , f— ) on A1 — A2 such that f— (s1 — s2 ) = 0 for all s1 ∈ Skew(A1 , σ1 ),

s2 ∈ Skew(A2 , σ2 ).

1

Proof : If char F = 2, the linear form f— which is the restriction of TrdA1 —A2 to

2

Sym(A1 — A2 , σ1 — σ2 ) satis¬es

1

f— (s1 — s2 ) = TrdA1 (s1 ) TrdA2 (s2 ) = 0

2

EXERCISES 63

for all s1 ∈ Skew(A1 , σ1 ), s2 ∈ Skew(A2 , σ2 ). Suppose next that char F = 2.

For any linear form f1 on Sym(A1 , σ1 ) we have f1 (s1 ) TrdA2 (s2 ) = 0 for all s1 ∈

Sym(A1 , σ1 ), s2 ∈ Sym(A2 , σ2 ), since σ2 is symplectic. Therefore, we may set

(σ1 — σ2 , f— ) = (σ1 — σ2 , f1— )

for any quadratic pair (σ1 , f1 ) on A1 . Uniqueness of f— follows from (??).

(5.21) De¬nition. The quadratic pair (σ1 — σ2 , f— ) of (??) is called the canonical

quadratic pair on A1 — A2 .

Exercises

1. Let A be a central simple algebra over a ¬eld F and ¬x a ∈ A. Show that there

is a canonical F -algebra isomorphism EndA (aA) EndA (Aa) which takes f ∈

EndA (aA) to the endomorphism f ∈ EndA (Aa) de¬ned by (xa)f = xf (a) for

x ∈ A, and the inverse takes g ∈ EndA (Aa) to the endomorphism g ∈ EndA (aA)

de¬ned by g(ax) = ag x for x ∈ A.

∼

Show that there is a canonical F -algebra isomorphism EndA (Aa)op ’ ’

EndAop (aop Aop ) which, for f ∈ EndA (Aa), maps f op to the endomorphism

˜ ˜

f de¬ned by f (mop ) = (mf )op . Therefore, there is a canonical isomorphism

op

EndAop (Aop aop ). Use it to identify (»k A)op = »k (Aop ), for

EndA (Aa)

k = 1, . . . , deg A.

2. Let Q be a quaternion algebra over a ¬eld F of arbitrary characteristic. Show

that the conjugation involution is the only linear map σ : Q ’ Q such that

σ(1) = 1 and σ(x)x ∈ F for all x ∈ F .

3. (Rowen-Saltman [?]) Let V be a vector space of dimension n over a ¬eld F and

let „ be an involution of the ¬rst kind on EndF (V ). Prove that „ is orthogonal

if and only if there exist n symmetric orthogonal10 idempotents in EndF (V ).

Find a similar characterization of the symplectic involutions on EndF (V ).

4. Let A be a central simple algebra with involution σ of the ¬rst kind. Show

that σ is orthogonal if and only if it restricts to the identity on a maximal ´tale

e

(commutative) subalgebra of A.

Hint: Extend scalars and use the preceding exercise.

5. Show that in a central simple algebra with involution, every left or right ideal is

generated by a symmetric element, unless the algebra is split and the involution

is symplectic.

6. (Albert) Let b be a symmetric, nonalternating bilinear form on a vector space V

over a ¬eld of characteristic 2. Show that V contains an orthogonal basis for b.

7. Let (ai )i=1,...,n2 be an arbitrary basis of a central simple algebra A, and let

(bi )i=1,...,n2 be the dual basis for the bilinear form TA , which means that

TrdA (ai bj ) = δij for i, j = 1, . . . , n2 . Show that the Goldman element of

n2

A is ai — b i .

i=1

Hint: Reduce by scalar extension to the split case and show that it su¬ces

to prove the assertion for the standard basis of Mn (F ).

10 Two idempotents e, f are called orthogonal if ef = f e = 0.

64 I. INVOLUTIONS AND HERMITIAN FORMS

8. Let (1, i, j, k) be a quaternion basis in a quaternion algebra Q of characteristic

di¬erent from 2. Show that the Goldman element in Q — Q is g = 1 (1 — 1 + i —

2

i’1 + j — j ’1 + k — k ’1 ). Let (1, u, v, w) be a quaternion basis in a quaternion

algebra Q of characteristic 2. Show that the Goldman element in Q — Q is

g = 1 — 1 + u — 1 + 1 — u + w — v ’1 + v ’1 — w.

9. Let K/F be a quadratic extension of ¬elds of characteristic di¬erent from 2,

and let a ∈ F — , b ∈ K — . Prove the “projection formula” for the norm of the

quaternion algebra (a, b)K :

NK/F (a, b)K ∼ a, NK/F (b) F

(where ∼ denotes Brauer-equivalence). Prove corresponding statements in

characteristic 2: if K/F is a separable quadratic extension of ¬elds and a ∈ F ,

b ∈ K — , c ∈ K, d ∈ F — ,

NK/F [a, b)K ∼ a, NK/F (b) and NK/F [c, d)K ∼ TK/F (c), d .

F F

Hint (when char F = 2): Let ι be the non-trivial automorphism of K/F

and let (1, i1 , j1 , k1 ) (resp. (1, i2 , j2 , k2 )) denote the usual quaternion basis of

a, ι(b) K = ι (a, b)K (resp. (a, b)K ). Let s be the switch map on ι (a, b)K —K

1

(a, b)K . Let u = 2 1 + a’1i1 — i2 + b’1 j1 — j2 ’ (ab)’1 k1 — k2 ∈ a, ι(b) K —K

(a, b)K . Show that s(u)u = 1, and that s = Int(u) —¦ s is a semi-linear auto-

morphism of order 2 of ι (a, b)K —K (a, b)K which leaves invariant i1 — 1, 1 — i2

and j1 — j2 . Conclude that the F -subalgebra of elements invariant under s is

Brauer-equivalent to a, NK/F (b) F . If b ∈ F , let v = 1 + u; if b ∈ F , pick

c ∈ K such that c2 ∈ F and let v = c + uι(c). Show that u = s (v)’1 v and

that Int(v) maps the subalgebra of s-invariant elements onto the subalgebra of

s -invariant elements.

10. (Knus-Parimala-Srinivas [?, Theorem 4.1]) Let A be a central simple algebra

over a ¬eld F , let V be an F -vector space and let

ρ : A —F A ’ EndF (V )

be an isomorphism of F -algebras. We consider V as a left A-module via av =

ρ(a — 1)(v) and identify EndA (A) = A, HomA (A, V ) = V by mapping every

homomorphism f to 1f . Moreover, we identify EndA (V ) = Aop by setting

op

va = ρ(1 — a)(v) for v ∈ V , a ∈ A

and HomA (V, A) = V — = HomF (V, F ) by mapping h ∈ HomA (V, A) to the

linear form

v ’ TrdA (v h ) for v ∈ V .

Let

EndA (A) HomA (A, V ) A V

B = EndA (A • V ) = = ;

V— Aop

HomA (V, A) EndA (V )

this is a central simple F -algebra which is Brauer-equivalent to A, by Proposi-

tion (??). Let γ = ρ(g) ∈ EndF (V ), where g ∈ A—F A is the Goldman element.

Show that

av b γ(v)

’

op

γ (f ) aop

t

fb

is an involution of orthogonal type of B.

EXERCISES 65

11. (Knus-Parimala-Srinivas [?, Theorem 4.2]) Let K/F be a separable quadratic

extension with nontrivial automorphism ι, let A be a central simple K-algebra,

let V be an F -vector space and let

ρ : NK/F (A) ’ EndF (V )

be an isomorphism of F -algebras. Set W = V —F K and write

ρ : A —K ιA ’ EndK (W )

for the isomorphism induced from ρ by extension of scalars. We consider W as

a left A-module via av = ρ(a — 1)(v). Let

EndA (A) HomA (A, W ) A W

B = EndA (A • W ) = =

W— ι op

HomA (W, A) EndA (W ) A

with identi¬cations similar to those in Exercise ??. The K-algebra B is Brauer

equivalent to A by (??). Show that

a w b ι(w)

’

ι op ι op

f b ι(f ) a

is an involution of the second kind of B.

12. Let A be a central simple F -algebra with involution σ of the ¬rst kind. Recall

the F -algebra isomorphism

σ— : A —F A ’ EndF A

de¬ned in the proof of Corollary (??) by

σ— (a — b)(x) = axσ(b) for a, b, x ∈ A.

Show that the image of the Goldman element g ∈ A—A under this isomorphism

is δσ where δ = +1 if σ is orthogonal and δ = ’1 if σ is symplectic. Use this

result to de¬ne canonical F -algebra isomorphisms

op

EndF A/ Alt(A, σ) EndF Sym(A, σ) if δ = +1,

2

sA op

EndF A/ Symd(A, σ) EndF Skew(A, σ) if δ = ’1,

EndF Alt(A, σ) if δ = +1,

»2 A

EndF Symd(A, σ) if δ = ’1.

13. (Saltman [?, Proposition 5]) Let A, B be central simple algebras of degrees m, n

over a ¬eld F . For every F -algebra homomorphism f : A ’ B, de¬ne a map

f : Aop —F B ’ B by f (aop — b) = f (a)b. Show that f is a homomorphism of

right Aop —F B-modules if B is endowed with the following Aop —F B-module

structure:

x —f (aop — b) = f (a)xb for a ∈ A and b, x ∈ B.

Show that ker f ‚ Aop —F B is a right ideal of reduced dimension mn ’ (n/m)

generated by the elements aop — 1 ’ 1 — f (a) for a ∈ A and that

Aop —F B = (1 — B) • ker f .

Conversely, show that every right ideal I ‚ Aop —F B of reduced dimension

mn ’ (n/m) such that

Aop —F B = (1 — B) • I

de¬nes an F -algebra homomorphism f : A ’ B such that I = ker f .

66 I. INVOLUTIONS AND HERMITIAN FORMS

Deduce from the results above that there is a natural one-to-one correspon-

dence between the set of F -algebra homomorphisms A ’ B and the rational

points in an open subset of the Severi-Brauer variety SBd (Aop —F B) where

d = mn ’ (n/m).

14. Let (A, σ) be a central simple algebra with involution of the ¬rst kind over

a ¬eld F of characteristic di¬erent from 2. Let a ∈ A be an element whose

minimal polynomial over F is separable. Show that a is symmetric for some

symplectic involution on A if and only if its reduced characteristic polynomial

is a square.

15. Let (A, σ) be a central simple algebra with involution of orthogonal type. Show

that every element in A is the product of two symmetric elements, one of which

is invertible.

Hint: Use (??).

16. Let V be a ¬nite dimensional vector space over a ¬eld F of arbitrary character-

istic and let a ∈ EndF (V ). Extend the notion of involution trace by using the

structure of V as an F [a]-module to de¬ne a nonsingular symmetric bilinear

form b : V — V ’ F such that a is invariant under the adjoint involution σb .

17. Let K/F be a separable quadratic extension of ¬elds with nontrivial automor-

phism ι. Let V be a vector space of dimension n over K and let b ∈ EndK (V )

be an endomorphism whose minimal polynomial has degree n and coe¬cients

in F . Show that V is a free F [b]-module of rank 1 and de¬ne a nonsingular her-

mitian form h : V —V ’ K such that b is invariant under the adjoint involution

σh .

Show that a matrix m ∈ Mn (K) is symmetric under some involution of the

second kind whose restriction to K is ι if and only if all the invariant factors

of m have coe¬cients in F .

18. Let q be a nonsingular quadratic form of dimension n over a ¬eld F , with n even

if char F = 2, and let a ∈ Mn (F ) be a matrix representing q, in the sense that

q(X) = X · a · X t . After identifying Mn (F ) with EndF (F n ) by mapping every

matrix m ∈ Mn (F ) to the endomorphism x ’ m · x, show that the quadratic

pair (σq , fq ) on EndF (F n ) associated to the quadratic map q : F n ’ F is the

same as the quadratic pair (σa , fa ) associated to a.

19. Let Q1 , Q2 be quaternion algebras with canonical involutions γ1 , γ2 over a ¬eld

F of arbitrary characteristic. Show that Alt(Q1 —Q2 , γ1 —γ2 ) = { q1 —1’1—q2 |

TrdQ1 (q1 ) = TrdQ2 (q2 ) }. If char F = 2, show that f (q1 — 1 + 1 — q2 ) =

TrdQ1 (q1 ) = TrdQ2 (q2 ) for all q1 — 1 + 1 — q2 ∈ Symd(Q1 — Q2 , γ1 — γ2 ) and for

all quadratic pairs (γ1 — γ2 , f ) on Q1 — Q2 .

20. The aim of this exercise is to give a description of the variety of quadratic

pairs on a central simple algebra in the spirit of (??). Let σ be a symplectic

involution on a central simple algebra A over a ¬eld F of characteristic 2 and

let σ— : A — A ’ EndF (A) be the isomorphism of Exercise ??. Let Iσ ‚ A — A

denote the right ideal corresponding to σ by (??) and let J ‚ A — A be the left

ideal generated by 1 ’ g, where g is the Goldman element. Denote by A0 ‚ A

the kernel of the reduced trace: A0 = { a ∈ A | TrdA (a) = 0 }. Show that

σ— (Iσ ) = Hom(A, A0 ), σ— (J ) = Hom A/ Sym(A, σ), A

and

σ— (J 0 ) = Hom A, Sym(A, σ) .

NOTES 67

Let now I ‚ A — A be a left ideal containing J , so that

σ— (I) = Hom(A/U, A)

for some vector space U ‚ Sym(A, σ). Show that σ— I·(1+g) = Hom(A/W, A),

where W = { a ∈ A | a + σ(a) ∈ U }, and deduce that σ— [I · (1 + g)]0 = Iσ if

and only if W = A0 , if and only if U © Symd(A, σ) = ker Trpσ .

Observe now that the set of rational points in SB s2 (Aop ) is in canonical

one-to-one correspondence with the set of left ideals I ‚ A — A containing J 0

and such that rdim I ’ rdim J = 1. Consider the subset U of such ideals which

satisfy [I · (1 + g)]0 • (1 — A) = A — A. Show that the map which carries every

’1

quadratic pair (σ, f ) on A to the left ideal σ— Hom(A/ ker f, A) de¬nes a

bijection from the set of quadratic pairs on A onto U.

Hint: For I ∈ U, the right ideal [I · (1 + g)]0 corresponds by (??) to

some symplectic involution σ. If U ‚ A is the subspace such that σ— (I) =

Hom(A/U, A), there is a unique quadratic pair (σ, f ) such that U = ker f .

21. Let (V1 , b1 ) and (V2 , b2 ) be vector spaces with nonsingular alternating forms

over an arbitrary ¬eld F . Show that there is a unique quadratic form q on

V1 — V2 whose polar form is b1 — b2 and such that q(v1 — v2 ) = 0 for all

v1 ∈ V1 , v2 ∈ V2 . Show that the canonical quadratic pair (σb1 — σb2 , f— ) on

EndF (V1 ) — EndF (V2 ) = EndF (V1 — V2 ) is associated with the quadratic form

q.

22. Let (A, σ) be a central simple algebra with involution of the ¬rst kind over an

arbitrary ¬eld F . Assume σ is symplectic if char F = 2. By (??), there exists

an element ∈ A such that +σ( ) = 1. De¬ne a quadratic form q(A,σ) : A ’ F

by

q(A,σ) (x) = TrdA σ(x) x for x ∈ A.

Show that this quadratic form does not depend on the choice of such that

+ σ( ) = 1. Show that the associated quadratic pair on EndF (A) corresponds

to the canonical quadratic pair (σ — σ, f— ) on A — A under the isomorphism

∼

σ— : A — A ’ EndF (A) such that σ— (a — b)(x) = axσ(b) for a, b, x ∈ A.

’

Notes

§??. Additional references for the material in this section include the classical

books of Albert [?], Deuring [?] and Reiner [?]. For Severi-Brauer varieties, see

Artin™s notes [?]. A self-contained exposition of Severi-Brauer varieties can be

found in Jacobson™s book [?, Chapter 3].

§??. The ¬rst systematic investigations of involutions of central simple alge-

bras are due to Albert. His motivation came from the theory of Riemann matrices:

on a Riemann surface of genus g, choose a basis (ω± )1¤±¤g of the space of holo-

morphic di¬erentials and a system of closed curves (γβ )1¤β¤2g which form a basis

of the ¬rst homology group, and consider the matrix of periods P = ( γβ ω± ). This

is a complex g — 2g matrix which satis¬es Riemann™s period relations: there exists