(8.27) Proposition. Let deg A = n = 2m. For all ∈ Λ, we have c( ) ∈

Z(A, σ, f ), ι c( ) = c( ) + 1 and

m(m’1)

c( )2 + c( ) = SrdA ( ) + .

2

Proof : It su¬ces to check the equations above after a scalar extension. We may

therefore assume that A is split. Moreover, it su¬ces to consider a particular choice

of ; indeed, if 0 , ∈ Λ, then = 0 + x + σ(x) for some x ∈ A, hence

c( ) = c( 0 ) + c x + σ(x) .

Since c and f have the same restriction to Sym(A, σ), the last term on the right

side is f x + σ(x) = TrdA (x), hence

c( ) = c( 0 ) + TrdA (x).

Therefore, we have c( ) ∈ Z(A, σ, f ) and ι c( ) = c( ) + 1 if and only if c( 0 ) satis-

¬es the same conditions. Moreover, (??) yields SrdA ( ) = SrdA ( 0 ) + „˜ TrdA (x) ,

hence „˜ c( ) = SrdA ( ) + m(m’1) if and only if the same equation holds for 0 .

2

We may thus assume A = EndF (V ) and (σ, f ) = (σq , fq ) for some nonsingular

quadratic space (V, q), and consider only the case of

m

= •q ( e2i’1 — e2i’1 q(e2i ) + e2i — e2i q(e2i’1 ) + e2i’1 — e2i )

0 i=1

where (e1 , . . . , en ) is a symplectic basis of V for the polar form bq :

bq (e2i’1 , e2i ) = 1, bq (e2i , e2i+1 ) = 0 and bq (ei , ej ) = 0 if |i ’ j| > 1.

Using the standard identi¬cation C(A, σ, f ) = C0 (V, q) of (??), we then have

m m

e2 q(e2i ) + e2 q(e2i’1 ) + e2i’1 · e2i =

c( 0 ) = e2i’1 · e2i

2i’1 2i

i=1 i=1

and the required equations follow by computation (compare with (??)).

(8.28) Corollary. For any ∈ Λ,

m(m’1)

F [X]/ X 2 + X + SrdA ( ) +

Z(A, σ, f ) .

2

Proof : The proposition above shows that Z(A, σ, f ) = F c( ) and c( )2 + c( ) +

SrdA ( ) + m(m’1) = 0.

2

106 II. INVARIANTS OF INVOLUTIONS

8.E. The Cli¬ord algebra of a hyperbolic quadratic pair. Let (σ, f ) be

a hyperbolic quadratic pair on a central simple algebra A over an arbitrary ¬eld F

and let deg A = n = 2m. By (??), the discriminant of (σ, f ) is trivial, hence (??)

shows that the Cli¬ord algebra C(A, σ, f ) decomposes as a direct product of two

central simple algebras of degree 2m’1 :

C(A, σ, f ) = C + (A, σ, f ) — C ’ (A, σ, f ).

Our aim is to show that one of the factors C± (A, σ, f ) is split if m is even.

We start with some observations on isotropic ideals in a central simple algebra

with an arbitrary quadratic pair: suppose I ‚ A is a right ideal of even reduced

dimension rdim I = r = 2s with respect to a quadratic pair (σ, f ). Consider the

image c Iσ(I) ‚ C(A, σ, f ) of Iσ(I) under the canonical map c : A ’ C(A, σ, f )

and let

s

ρ(I) = c Iσ(I) = x1 · · · x s x1 , . . . , xs ∈ c Iσ(I) .

(8.29) Lemma. The elements in c Iσ(I) commute. The F -vector space ρ(I) ‚

C(A, σ, f ) is 1-dimensional; it satis¬es σ ρ(I) · ρ(I) = {0} and

dimF ρ(I) · C(A, σ, f ) = dimF C(A, σ, f ) · ρ(I) = 2n’r’1 .

Proof : It su¬ces to check the lemma over a scalar extension. We may therefore

assume that A is split and identify A = EndF (V ), C(A, σ, f ) = C0 (V, q) for some

nonsingular quadratic space (V, q) of dimension n, by (??). The ideal I then has the

form I = HomF (V, U ) for some m-dimensional totally isotropic subspace U ‚ V .

Let (u1 , . . . , ur ) be a basis of U . Under the identi¬cation A = V — V described

in (??), the vector space Iσ(I) is spanned by the elements ui — uj for i, j = 1,

. . . , r, hence c Iσ(I) is spanned by the elements ui · uj in C0 (V, q). Since U is

totally isotropic, we have u2 = 0 and ui · uj + uj · ui = 0 for all i, j = 1, . . . , r, hence

i

the elements ui · uj commute. Moreover, the space ρ(I) is spanned by u1 · · · ur .

The dimensions of ρ(I) · C0 (V, q) and C0 (V, q) · ρ(I) are then easily computed, and

since σ(u1 · · · ur ) = ur · · · u1 we have σ(x)x = 0 for all x ∈ ρ(I).

(8.30) Corollary. Let deg A = n = 2m and suppose the center Z = Z(A, σ, f ) of

C(A, σ, f ) is a ¬eld. For any even integer r, the relation r ∈ ind(A, σ, f ) implies

2m’r’1 ∈ ind C(A, σ, f ), σ .

Proof : If r is an even integer in ind(A, σ, f ), then A contains an isotropic right

ideal I of even reduced dimension r. The lemma shows that ρ(I) · C(A, σ, f ) is an

isotropic ideal for the involution σ. Its reduced dimension is

1

dimF ρ(I) · C(A, σ, f )

dimZ ρ(I) · C(A, σ, f )

= 2m’r’1 .

2

=

2m’1

deg C(A, σ, f )

We next turn to the case of hyperbolic quadratic pairs:

(8.31) Proposition. Let (σ, f ) be a hyperbolic quadratic pair on a central simple

algebra A of degree 2m over an arbitrary ¬eld F . If m is even, then one of the

factors C± (A, σ, f ) of the Cli¬ord algebra C(A, σ, f ) is split.

§9. THE CLIFFORD BIMODULE 107

Proof : Let I ‚ A be a right ideal of reduced dimension m which is isotropic with

respect to (σ, f ), and consider the 1-dimensional vector space ρ(I) ‚ C(A, σ, f )

de¬ned above. Multiplication on the left de¬nes an F -algebra homomorphism

» : C(A, σ, f ) ’ EndF C(A, σ, f ) · ρ(I) .

Dimension count shows that this homomorphism is not injective, hence the kernel

is one of the nontrivial ideals C + (A, σ, f ) — {0} or {0} — C ’ (A, σ, f ). Assuming

for instance ker » = C + (A, σ, f ) — {0}, the homomorphism » factors through an

injective F -algebra homomorphism C ’ (A, σ, f ) ’ EndF C(A, σ, f ) · ρ(I) . This

homomorphism is surjective by dimension count.

§9. The Cli¬ord Bimodule

Although the odd part C1 (V, q) of the Cli¬ord algebra of a quadratic space (V, q)

is not invariant under similarities, it turns out that the tensor product V — C1 (V, q)

is invariant, and therefore an analogue can be de¬ned for a central simple algebra

with quadratic pair (A, σ, f ). The aim of this section is to de¬ne such an analogue.

This construction will be used at the end of this section to obtain fundamental

relations between the Cli¬ord algebra C(A, σ, f ) and the algebra A (see (??)); it

will also be an indispensable tool in the de¬nition of spin groups in the next chapter.

We ¬rst review the basic properties of the vector space V — C1 (V, q) that we

want to generalize.

9.A. The split case. Let (V, q) be a quadratic space over a ¬eld F (of arbi-

trary characteristic). Let C1 (V, q) be the odd part of the Cli¬ord algebra C(V, q).

Multiplication in C(V, q) endows C1 (V, q) with a C0 (V, q)-bimodule structure. Since

V is in a natural way a left End(V )-module, the tensor product V — C1 (V, q) is

at the same time a left End(V )-module and a C0 (V, q)-bimodule: for f ∈ End(V ),

v ∈ V , c0 ∈ C0 (V, q) and c1 ∈ C1 (V, q) we set

f · (v — c1 ) = f (v) — c1 , c0 — (v — c1 ) = v — c0 c1 , (v — c1 ) · c0 = v — c1 c0 .

These various actions clearly commute.

We summarize the basic properties of V — C1 (V, q) in the following proposition:

(9.1) Proposition. Let dim V = n.

(1) The vector space V — C1 (V, q) carries natural structures of left End(V )-module

and C0 (V, q)-bimodule, and the various actions commute.

(2) The standard identi¬cation End(V ) = V — V induced by the quadratic form q

(see (??)) and the embedding V ’ C1 (V, q) de¬ne a canonical map

b : End(V ) ’ V — C1 (V, q)

which is an injective homomorphism of left End(V )-modules.

(3) dimF V — C1 (V, q) = 2n’1 n.

The proof follows by straightforward veri¬cation.

Until the end of this subsection we assume that the dimension of V is even:

dim V = n = 2m. This is the main case of interest for generalization to central

simple algebras with involution, since central simple algebras of odd degree with

involution of the ¬rst kind are split (see (??)). Since dim V is even, the center of

C0 (V, q) is an ´tale quadratic F -algebra which we denote Z. Let ι be the nontrivial

e

automorphism of Z/F . In the Cli¬ord algebra C(V, q) we have

v · ζ = ι(ζ) · v for v ∈ V , ζ ∈ Z,

108 II. INVARIANTS OF INVOLUTIONS

hence

(9.2) ι(ζ) — (v — c1 ) = (v — c1 ) · ζ for v ∈ V , c1 ∈ C1 (V, q), ζ ∈ Z.

In view of this equation, we may consider V — C1 (V, q) as a right module over the

Z-algebra ι C0 (V, q)op —Z C0 (V, q): for v ∈ V , c1 ∈ C1 (V, q) and c0 , c0 ∈ C0 (V, q)

we set

(v — c1 ) · (ιcop — c0 ) = c0 — (v — c1 ) · c0 = v — c0 c1 c0 .

0

On the other hand V — C1 (V, q) also is a left module over End(V ); since the actions

of End(V ) and C0 (V, Q) commute, there is a natural homomorphism of F -algebras:

ν : ι C0 (V, q)op —Z C0 (V, q) ’ EndEnd(V ) V — C1 (V, q) = EndF C1 (V, q) .

This homomorphism is easily seen to be injective: this is obvious if Z is a ¬eld,

because then the tensor product on the left is a simple algebra. If Z F — F,

the only nontrivial ideals in the tensor product are generated by elements in Z.

However the restriction of ν to Z is injective, since the condition v · ζ = 0 for all

v ∈ V implies ζ = 0. Therefore, ν is injective.

The image of ν is determined as follows: through ν, the center Z of C0 (V, q) acts

on V — C1 (V, q) by End(V )-linear homomorphisms; the set V — C1 (V, q) therefore

has a structure of left End(V ) — Z-module (where the action of Z is through the

right action of C0 (V, q)): for f ∈ End(V ), ζ ∈ Z, v ∈ V and c1 ∈ C1 (V, q),

(f — ζ) · (v — c1 ) = f (v) — c1 ζ.

The map ν may then be considered as an isomorphism of Z-algebras:

∼

ν : ι C0 (V, q)op —Z C0 (V, q) ’ EndEnd(V )—Z V — C1 (V, q) = EndZ C1 (V, q) .

’

Equivalently, ν identi¬es ι C0 (V, q)op —Z C0 (V, q) with the centralizer of Z (= ν(Z))

in EndEnd(V ) V — C1 (V, q) .

9.B. De¬nition of the Cli¬ord bimodule. In order to de¬ne an analogue

of V — C1 (V, q) for a central simple algebra with quadratic pair (A, σ, f ), we ¬rst

de¬ne a canonical representation of the symmetric group S2n on A—n .

Representation of the symmetric group. As in §??, we write A for the

underlying vector space of the F -algebra A. For any integer n ≥ 2, we de¬ne a

generalized sandwich map

Sandn : A—n ’ HomF (A—n’1 , A)

by the condition:

Sandn (a1 — · · · — an )(b1 — · · · — bn’1 ) = a1 b1 a2 b2 · · · bn’1 an .

(Thus, Sand2 is the map denoted simply Sand in §??).

(9.3) Lemma. For any central simple F -algebra A, the map Sandn is an isomor-

phism of vector spaces.

Proof : Since it su¬ces to prove Sandn is an isomorphism after scalar extension, we

may assume that A = Mn (F ). It su¬ces to prove injectivity of Sandn , since A—n

and HomF (A—n’1 , A) have the same dimension. Let eij (i, j = 1, . . . , n) be the

matrix units of Mn (F ). Take any nonzero ± ∈ A—n and write

n n n n

±= ··· ci1 j1 ...in jn ei1 j1 — · · · — ein jn

i1 =1 j1 =1 in =1 jn =1

§9. THE CLIFFORD BIMODULE 109

with the ci1 j1 ...in jn ∈ F . Some coe¬cient of ±, say cp1 q1 ...pn qn is nonzero. Then,

Sandn (±)(eq1 p2 — eq2 p3 — · · · — eqi pi+1 — · · · — eqn’1 pn ) =

n n

ci1 q1 p2 q2 ...pn jn ei1 jn

i1 =1 jn =1

which is not zero, since its p1 qn -entry is not zero.

(9.4) Proposition. Let (A, σ) be a central simple F -algebra with involution of the

¬rst kind. If char F = 2, suppose further that σ is orthogonal. For all n ≥ 1 there is

a canonical representation ρn : S2n ’ GL(A—n ) of the symmetric group S2n which

is described in the split case as follows: for every nonsingular symmetric bilinear

space (V, b) and v1 , . . . , v2n ∈ V ,

ρn (π) •b (v1 — v2 ) — · · · — •b (v2n’1 — v2n ) =

•b vπ’1 (1) — vπ’1 (2) — · · · — •b (vπ’1 (2n’1) — vπ’1 (2n) )

∼

for all π ∈ S2n where •b : V — V ’ EndF (V ) is the standard identi¬cation (??).

’

Proof : We ¬rst de¬ne the image of the transpositions „ (i) = (i, i + 1) for i = 1,

. . . , 2n ’ 1.

If i is odd, i = 2 ’ 1, let

ρn „ (i) = IdA — · · · — IdA — σ — IdA — · · · — IdA ,

where σ lies in -th position. In the split case, σ corresponds to the twist under the

standard identi¬cation A = V — V ; therefore,

ρn „ (2 ’ 1) (v1 — · · · — v2 — v2 — · · · — v2n ) =

’1

v1 — · · · — v 2 — v 2 — · · · — v2n .

’1

If i is even, i = 2 , we de¬ne ρn „ (i) by the condition:

Sandn ρn „ (i) (u) (x) = Sandn (u) IdA — · · · — IdA — σ — IdA — · · · — IdA (x)

for u ∈ A—n and x ∈ A—n’1 where σ lies in -th position. The same computation

as in (??) shows that ρn „ (2 ) satis¬es the required condition in the split case.

In order to de¬ne ρn (π) for arbitrary π ∈ S2n , we use the fact that „ (1), . . . ,

„ (2n ’ 1) generate S2n : we ¬x some factorization

π = „1 —¦ · · · —¦ „ s where „1 , . . . , „s ∈ {„ (1), . . . , „ (2n ’ 1)}

and de¬ne ρn (π) = ρn („1 ) —¦ · · · —¦ ρn („s ). The map ρn (π) thus de¬ned meets the

requirement in the split case, hence ρn is a homomorphism in the split case. By

extending scalars to a splitting ¬eld, we see that ρn also is a homomorphism in the

general case. Therefore, the de¬nition of ρn (π) does not actually depend on the

factorization of π.

The de¬nition. Let (σ, f ) be a quadratic pair on a central simple F -algebra A.

For all n ≥ 1, let γn = ρn (1, 2, . . . , 2n)’1 ∈ GL(A—n ) where ρn is as in (??),

and let γ = •γn : T (A) ’ T (A) be the induced linear map. Thus, in the split case

(A, σ, f ) = EndF (V ), σq , fq , we have, under the standard identi¬cation A = V —V

of (??):

γ(v1 — · · · — v2n ) = γn (v1 — · · · — v2n ) = v2 — · · · — v2n — v1

110 II. INVARIANTS OF INVOLUTIONS

for v1 , . . . , v2n ∈ V .

Let also T+ (A) = •n≥1 A—n . The vector space T+ (A) carries a natural structure

of left and right module over the tensor algebra T (A). We de¬ne a new left module

structure — as follows: for u ∈ T (A) and v ∈ T+ (A) we set

u — v = γ ’1 u — γ(v) .

Thus, in the split case A = V — V , the product — avoids the ¬rst factor:

(u1 — · · · — u2i ) — (v1 — · · · — v2j ) = v1 — u1 — · · · — u2i — v2 — · · · — v2j

for u1 , . . . , u2i , v1 , . . . , v2j ∈ V . (Compare with the de¬nition of — in §??).

(9.5) De¬nition. The Cli¬ord bimodule of (A, σ, f ) is de¬ned as

T+ (A)

B(A, σ, f ) =

J1 (σ, f ) — T+ (A) + T+ (A) · J1 (σ, f )

where J1 (σ, f ) is the two-sided ideal of T (A) which appears in the de¬nition of the

Cli¬ord algebra C(A, σ, f ) (see (??)).

The map a ∈ A ’ a ∈ T+ (A) induces a canonical F -linear map

(9.6) b : A ’ B(A, σ, f ).

(9.7) Theorem. Let (A, σ, f ) be a central simple F -algebra with a quadratic pair.

(1) The F -vector space B(A, σ, f ) carries a natural C(A, σ, f )-bimodule structure

where action on the left is through —, and a natural left A-module structure.

(2) In the split case (A, σ, f ) = EndF (V ), σq , fq , the standard identi¬cation

∼

•q : V — V ’ EndF (V )

’

induces a standard identi¬cation of Cli¬ord bimodules

∼

V —F C1 (V, q) ’ B(A, σ, f ).

’

(3) The canonical map b : A ’ B(A, σ, f ) is an injective homomorphism of left 13

A-modules.

(4) dimF B(A, σ, f ) = 2(deg A)’1 deg A.

Proof : By extending scalars to split A, it is easy to verify that

J2 (σ, f ) — T+ (A) ⊆ T+ (A) · J1 (σ, f ) and T+ (A) · J2 (σ, f ) ⊆ J1 (σ, f ) — T+ (A).

Therefore, the actions of T (A) on T+ (A) on the left through — and on the right

through the usual product induce a C(A, σ, f )-bimodule structure on B(A, σ, f ).

We de¬ne on T+ (A) a left A-module structure by using the multiplication map

A ’ A which carries a—b to ab. Explicitly, for a ∈ A and u = u1 —· · ·—ui ∈ A—i ,

—2

we set

a · u = au1 — u2 — · · · — ui .

Thus, in the split case (A, σ, f ) = (EndF (V ), σq , fq ), we have, under the standard

identi¬cation A = V — V :

a · (v1 — · · · — v2i ) = a(v1 ) — v2 — · · · — v2i .

It is then clear that the left action of A on T+ (A) commutes with the left and right

actions of T (A). Therefore, the subspace J1 (σ, f ) — T+ (A) + T+ (A) · J1 (σ, f ) is

13 Therefore, the image of a ∈ A under b will be written ab .

§9. THE CLIFFORD BIMODULE 111

preserved under the action of A, and it follows that B(A, σ, f ) inherits this action

from T+ (A).

∼

In the split case, the standard identi¬cation •’1 : A ’ V — V induces a sur-

’

q

jective linear map from B(A, σ, f ) onto V — C1 (V, q). Using an orthogonal decom-

position of (V, q) into 1- or 2-dimensional subspaces, one can show that

dimF B(A, σ, f ) ¤ dimF V dimF C1 (V, q).

Therefore, the induced map is an isomorphism. This proves (??) and (??), and (??)

follows by dimension count. Statement (??) is clear in the split case (see (??)), and

the theorem follows.

As was observed in the preceding section, there is no signi¬cant loss if we

restrict our attention to the case where the degree of A is even, since A is split if

its degree is odd. Until the end of this subsection, we assume deg A = n = 2m.

According to (??), the center Z of C(A, σ, f ) is then a quadratic ´tale F -algebra.

e

Let ι be the non-trivial automorphism of Z/F . By extending scalars to split the

algebra A, we derive from (??):

(9.8) x · ζ = ι(ζ) — x for x ∈ B(A, σ, f ), ζ ∈ Z.

Therefore, we may consider B(A, σ, f ) as a right module over ι C(A, σ, f )op —Z

C(A, σ, f ): for c, c ∈ C(A, σ, f ) and x ∈ B(A, σ, f ), we set

x · (ιcop — c ) = c — x · c .

Thus, B(A, σ, f ) is an A-ι C(A, σ, f )op —Z C(A, σ, f )-bimodule, and there is a natural

homomorphism of F -algebras:

ν : ι C(A, σ, f )op —Z C(A, σ, f ) ’ EndA B(A, σ, f ).

By comparing with the split case, we see that the map ν is injective, and that its

image is the centralizer of Z (= ν(Z)) in EndA B(A, σ, f ). Endowing B(A, σ, f )

with a left A —F Z-module structure (where the action of Z is through ν), we may

thus view ν as an isomorphism

∼

ν : ι C(A, σ, f )op —Z C(A, σ, f ) ’ EndA—Z B(A, σ, f );

(9.9) ’

ι op

it is de¬ned by xν( c —c )

= c — x · c for c, c ∈ C(A, σ, f ) and x ∈ B(A, σ, f ).

The canonical involution. We now use the involution σ on A to de¬ne an

involutorial A-module endomorphism ω of B(A, σ, f ). As in §??, σ denotes the

involution of C(A, σ, f ) induced by σ, and „ is the involution on C(V, q) which is

the identity on V .

(9.10) Proposition. The A-module B(A, σ, f ) is endowed with a canonical endo-

morphism14 ω such that for c1 , c2 ∈ C(A, σ, f ), x ∈ B(A, σ, f ) and a ∈ A:

(c1 — x · c2 )ω = σ(c2 ) — xω · σ(c1 ) (ab )ω = ab ,

and

where b : A ’ B(A, σ, f ) is the canonical map. Moreover, in the split case

(A, σ, f ) = (EndF (V ), σq , fq )

we have ω = IdV — „ under the standard identi¬cations A = V — V , B(A, σ, f ) =

V — C1 (V, q).

14 Since B(A, σ, f ) is a left A-module, ω will be written to the right of its arguments.

112 II. INVARIANTS OF INVOLUTIONS

Proof : Let ω = γ ’1 —¦σ : T+ (A) ’ T+ (A) where σ is the involution on T (A) induced

by σ. Thus, in the split case A = V — V :

ω(v1 — · · · — v2n ) = v1 — v2n — v2n’1 — · · · — v3 — v2 .

By extending scalars to a splitting ¬eld of A, it is easy to check that for a ∈ A, u 1 ,

u2 ∈ T (A) and v ∈ T+ (A),

ω(u1 — v · u2 ) = σ(u2 ) — ω(v) · σ(u1 ), ω(a · v) = a · ω(v) and ω(a) = a.

It follows from the ¬rst equation that

ω J1 (σ, f ) — T+ (A) = T+ (A) · σ J1 (σ, f ) ⊆ T+ (A) · J1 (σ, f )

and

ω T+ (A) · J1 (σ, f ) = σ J1 (σ, f ) — T+ (A) ⊆ J1 (σ, f ) — T+ (A),

hence ω induces an involutorial F -linear operator ω on B(A, σ, f ) which satis¬es

the required conditions.

We thus have ω ∈ EndA B(A, σ, f ). Moreover, it follows from the ¬rst property

of ω in the proposition above and from (??) that for x ∈ B(A, σ, f ) and ζ ∈ Z,

(x · ζ)ω = σ(ζ) — xω = xω · ι —¦ σ(ζ) .

The restriction of σ to Z is determined in (??): σ is of the ¬rst kind if m is even

and of the second kind if m is odd. Therefore, ω is Z-linear if m is odd, hence it

belongs to the image of ι C(A, σ, f )op —Z C(A, σ, f ) in EndA B(A, σ, f ) under the

natural monomorphism ν. By contrast, when m is even, ω is only ι-semilinear. In

this case, we de¬ne an F -algebra

ι

C(A, σ, f ) —Z C(A, σ, f ) • ι C(A, σ, f ) —Z C(A, σ, f ) · z

E(A, σ, f ) =

where multiplication is de¬ned by the following equations:

z(ιc — c ) = (ιc — c)z z 2 = 1.

for c, c ∈ C(A, σ, f ),

We also de¬ne a map ν : E(A, σ, f ) ’ EndA B(A, σ, f ) by

ω

(ιc1 —c2 +ιc3 —c4 ·z)

xν = σ(c1 ) — x · c2 + σ(c3 ) — x · c4

for x ∈ B(A, σ, f ) and c1 , c2 , c3 , c4 ∈ C(A, σ, f ). The fact that ν is a well-de¬ned

F -algebra homomorphism follows from the properties of ω in (??), and from the

hypothesis that deg A is divisible by 4 which ensures that σ is an involution of the

¬rst kind: see (??).

(9.11) Proposition. If deg A ≡ 0 mod 4, the map ν is an isomorphism of F -

algebras.

Proof : Suppose ν (u + vz) = 0 for some u, v ∈ ι C(A, σ, f ) —Z C(A, σ, f ). Then

ν (u) = ’ν (vz); but ν (u) is Z-linear while ν (vz) is ι-semilinear. Therefore,

ν (u) = ν (vz) = 0. It then follows that u = v = 0 since the natural map ν is

injective. This proves injectivity of ν . Surjectivity follows by dimension count:

since dimF B(A, σ, f ) = 2deg A’1 deg A, we have rdimA B(A, σ, f ) = 22m’1 hence

deg EndA B(A, σ, f ) = 22m’1 by (??). On the other hand,

dimF E(A, σ, f ) = 2 dimF ι C(A, σ, f ) —Z C(A, σ, f )

2

= 22(2m’1) .

= dimF C(A, σ, f )

§9. THE CLIFFORD BIMODULE 113

9.C. The fundamental relations. In this section, A denotes a central simple

F -algebra of even degree n = 2m with a quadratic pair (σ, f ). The fundamental

relations between the Brauer class [A] of A and the Brauer class of the Cli¬ord

algebra C(A, σ, f ) are the following:

(9.12) Theorem. Let Z be the center of the Cli¬ord algebra C(A, σ, f ).

(1) If deg A ≡ 0 mod 4 (i.e., if m is even), then

2

(9.13) C(A, σ, f ) =1 in Br(Z).

(9.14) NZ/F C(A, σ, f ) = [A] in Br(F ).

(2) If deg A ≡ 2 mod 4 (i.e., if m is odd ), then

2

(9.15) C(A, σ, f ) = [AZ ] in Br(Z).

(9.16) NZ/F C(A, σ, f ) = 1 in Br(F ).

(If Z = F — F , the norm NZ/F is de¬ned by NF —F/F (C1 — C2 ) = C1 —F C2 : see

the end of §??).

Proof : Equations (??) and (??) follow by (??) from the fact that the canonical

involution σ on C(A, σ, f ) is of the ¬rst kind when deg A ≡ 0 mod 4 and of the

second kind when deg A ≡ 2 mod 4.

To prove equations (??) and (??), recall the natural isomorphism (??):

∼

ν : ι C(A, σ, f )op —Z C(A, σ, f ) ’ EndA—Z B(A, σ, f ).

’

By (??), it follows that AZ = A —F Z is Brauer-equivalent to ι C(A, σ, f )op —Z

C(A, σ, f ). If m is odd, the canonical involution σ is of the second kind; it therefore

de¬nes an isomorphism of Z-algebras:

ι

C(A, σ, f )op C(A, σ, f ),

and (??) follows. Note that the arguments above apply also in the case where

Z F —F ; then C(A, σ, f ) = C + —C ’ for some central simple F -algebras C + , C ’ ,

and there is a corresponding decomposition of B(A, σ, f ) which follows from its Z-

module structure:

B(A, σ) = B+ — B’ .

Then EndA—Z B(A, σ, f ) = EndA B+ —EndA B’ and ι C(A, σ, f )op = C ’op —C +op ,

and the isomorphism ν can be considered as

∼

ν : (C + —F C ’op ) — (C ’ —F C +op ) ’ (EndA B+ ) — (EndA B’ ).

’

Therefore, A is Brauer-equivalent to C + —F C ’op and to C ’ — C +op . Since σ is

of the second kind when m is odd, we have in this case C ’op C + , hence A is

Brauer-equivalent to C +—2 and to C ’—2 , proving (??).

Similarly, (??) is a consequence of (??), as we proceed to show. Let

B (A, σ) = B(A, σ)ω

denote the F -subspace of ω-invariant elements in B(A, σ). For the rest of this

section, we assume deg A ≡ 0 mod 4. As observed before (??), the involution ω is

ι-semilinear, hence the multiplication map

B (A, σ) —F Z ’ B(A, σ)

is an isomorphism of Z-modules. Relation (??) follows from (??) and the following

claim:

114 II. INVARIANTS OF INVOLUTIONS

Claim. The isomorphism ν of (??) restricts to an F -algebra isomorphism

∼

NZ/F C(A, σ, f ) ’ EndA B (A, σ, f ).

’

To prove the claim, observe that the subalgebra

EndA B (A, σ, f ) ‚ EndA B(A, σ, f )

is the centralizer of Z and ω, hence it is also the F -subalgebra of elements which

commute with ω in EndA—Z B(A, σ, f ). The isomorphism ν identi¬es this algebra

with the algebra of switch-invariant elements in ι C(A, σ, f ) —Z C(A, σ, f ), i.e., with

NZ/F C(A, σ, f ).

§10. The Discriminant Algebra

A notion of discriminant may be de¬ned for hermitian spaces on the same model

as for symmetric bilinear spaces. If (V, h) is a hermitian space over a ¬eld K, with

respect to a nontrivial automorphism ι of K (of order 2), the determinant of the

Gram matrix of h with respect to an arbitrary basis (e1 , . . . , en ) lies in the sub¬eld

F ‚ K elementwise ¬xed under ι and is an invariant of h modulo the norms of K/F .

We may therefore de¬ne the determinant by

· N (K/F ) ∈ F — /N (K/F )

det h = det h(ei , ej ) 1¤i,j¤n

where N (K/F ) = NK/F (K — ) is the group of norms of K/F . The discriminant is

the signed determinant:

disc h = (’1)n(n’1)/2 det h(ei , ej ) ∈ F — /N (K/F ).

1¤i,j¤n

If δ ∈ F — is a representative of disc h, the quaternion algebra K • Kz where

multiplication is de¬ned by zx = ι(x)z for x ∈ K and z 2 = δ does not depend on

the choice of the representative δ. We denote it by (K, disc h)F ; thus

√

(±, δ)F if K = F ( ±) (char F = 2),

(K, disc h)F =

[±, δ)F if K = F „˜’1 (±) (char F = 2).

Our aim in this section is to generalize this construction, associating a central

simple algebra D(B, „ ) to every central simple algebra with involution of unitary

type (B, „ ) of even degree, in such a way that D EndK (V ), σh is Brauer-equivalent

to (K, disc h)F (see (??)). In view of this relation with the discriminant, the algebra

D(B, „ ) is called the discriminant algebra, a term suggested by A. Wadsworth. This

algebra is endowed with a canonical involution of the ¬rst kind.

As preparation for the de¬nition of the discriminant algebra, we introduce in the

¬rst four sections various constructions related to exterior powers of vector spaces.

For every central simple algebra A over an arbitrary ¬eld F , and for every positive

integer k ¤ deg A, we de¬ne a central simple F -algebra »k A which is Brauer-

k

equivalent to A—k and such that in the split case »k EndF (V ) = EndF ( V ). In

the second section, we show that when the algebra A has even degree n = 2m, the

algebra »m A carries a canonical involution γ of the ¬rst kind. In the split case

A = EndF (V ), the involution γ on »m A = EndF ( m V ) is the adjoint involution

m m n

with respect to the exterior product § : V— V’ V F . The third

section is more speci¬cally concerned with the case where char F = 2: in this case,

we extend the canonical involution γ on »m A into a canonical quadratic pair (γ, f ),

when m ≥ 2. Finally, in §??, we show how an involution on A induces an involution

on »k A for all k ¤ deg A.

§10. THE DISCRIMINANT ALGEBRA 115

10.A. The »-powers of a central simple algebra. Let A be a central sim-

ple algebra of degree n over an arbitrary ¬eld F . Just as for the Cli¬ord bimodule,

the de¬nition of »k A uses a canonical representation of the symmetric group, based

on Goldman elements.

(10.1) Proposition. For all k ≥ 1, there is a canonical homomorphism g k : Sk ’

—

A—k from the symmetric group Sk to the group of invertible elements in A—k ,

such that in the split case A = EndF (V ) we have under the identi¬cation A—k =

EndF (V —k ):

gk (π)(v1 — · · · — vk ) = vπ’1 (1) — · · · — vπ’1 (k)

for all π ∈ Sk and v1 , . . . , vk ∈ V .

Proof : We ¬rst de¬ne the image of the transpositions „ (i) = (i, i + 1) for i = 1,

. . . , k ’ 1, by setting

gk „ (i) = 1 — · · · — 1 — g — 1 — · · · — 1

i’1 k’i’1

where g ∈ A — A is the Goldman element de¬ned in (??). From (??), it follows

that in the split case

gk „ (i) (v1 — · · · — vk ) = v1 — · · · — vi+1 — vi — · · · — vk ,

as required.

In order to de¬ne gk (π) for arbitrary π ∈ Sk , we ¬x some factorization

π = „1 —¦ · · · —¦ „ s where „1 , . . . , „s ∈ {„ (1), . . . , „ (k ’ 1)}

and set gk (π) = gk („1 ) · · · gk („s ). Then, in the split case

gk (π)(v1 — · · · — vk ) = vπ’1 (1) — · · · — vπ’1 (k) ,

and it follows that gk is a homomorphism in the split case. It then follows by scalar

extension to a splitting ¬eld that gk is also a homomorphism in the general case.

Therefore, the de¬nition of gk does not depend on the factorization of π.

(10.2) Corollary. For all π ∈ Sk and a1 , . . . , ak ∈ A,

gk (π) · (a1 — · · · — ak ) = (aπ’1 (1) — · · · — aπ’1 (k) ) · gk (π).

Proof : The equation follows by scalar extension to a splitting ¬eld of A from the

description of gk (π) in the split case.

For all k ≥ 2, de¬ne

sgn(π)gk (π) ∈ A—k ,

sk =

π∈Sk

where sgn(π) = ±1 is the sign of π.

(10.3) Lemma. The reduced dimension of the left ideal A—k sk is given by

n

for 2 ¤ k ¤ n = deg A,

—k k

rdim(A sk ) =

0 for k > n = deg A.

If A is split: A = EndF (V ), A—k = EndF (V —k ), then there is a natural isomor-

phism of A—k -modules:

k

A—k sk = HomF ( V , V —k ).

116 II. INVARIANTS OF INVOLUTIONS

Proof : Since the reduced dimension does not change under scalar extension, it

su¬ces to prove the second part. Under the correspondence between left ideals in

A—k = EndF (V —k ) and subspaces of V —k (see §??), we have

A—k sk = HomF (V —k / ker sk , V —k ).

From the description of gk (π) in the split case, it follows that ker sk contains the

subspace of V —k spanned by the products v1 — · · · — vk where vi = vj for some

k

V ’ V —k / ker sk .

indices i = j. Therefore, there is a natural epimorphism

To prove that this epimorphism is injective, pick a basis (e1 , . . . , en ) of V . For

the various choices of indices i1 , . . . , ik such that 1 ¤ i1 < i2 < · · · < ik ¤ n, the

images sk (ei1 — · · · — eik ) are linearly independent, since they involve di¬erent basis

vectors in V —k . Therefore,

n

dim(V —k / ker sk ) = dim im sk ≥ ,

k

and the epimorphism above is an isomorphism.

(10.4) De¬nition. Let A be a central simple algebra of degree n over a ¬eld F .

For every integer k = 2, . . . , n we de¬ne the k-th »-power of A as

»k A = EndA—k (A—k sk ).

We extend this de¬nition by setting »1 A = A. Note that for k = 2 we recover the

de¬nition of »2 A given in §?? (see (??)).

The following properties follow from the de¬nition, in view of (??), (??) and (??):

(a) »k A is a central simple F -algebra Brauer-equivalent to A—k , of degree

n

deg »k A = .

k

(b) There is a natural isomorphism:

k

»k EndF (V ) = EndF ( V ).

(10.5) Corollary. If k divides the index ind A, then ind A—k divides (ind A)/k.

Proof : By replacing A by a Brauer-equivalent algebra, we may assume that A is

a division algebra. Let n = deg A = ind A. Arguing by induction on the number

of prime factors of k, it su¬ces to prove the corollary when k = p is a prime

number. If K is a splitting ¬eld of A of degree n, then K also splits »k A, hence

ind »p A divides n. On the other hand, ind »p A divides deg »p A = n , and the

p

n p

greatest common divisor of n and p is n/p, hence ind » A divides n/p. Since

»p A is Brauer-equivalent to A—p , we have ind »p A = ind A—p , and the proof is

complete.

10.B. The canonical involution. Let V be a vector space of even dimen-

n

sion n = 2m over a ¬eld F of arbitrary characteristic. Since dim V = 1, the

composition of the exterior product

m m n

§: V— V’ V

n m

∼

with a vector-space isomorphism V ’ F is a bilinear form on

’ V , which is

uniquely determined up to a scalar factor.

§10. THE DISCRIMINANT ALGEBRA 117

(10.6) Lemma. The bilinear map § is nonsingular. It is symmetric if m is even

and skew-symmetric if m is odd. If char F = 2, the map § is alternating for

all m. Moreover, the discriminant of every symmetric bilinear form induced from

n

§ through any isomorphism V F is trivial.

Proof : Let (e1 , . . . , en ) be a basis of V . For every subset of m indices

S = {i1 , . . . , im } ‚ {1, . . . , n} with i1 < · · · < im

m

we set eS = ei1 § · · · § eim ∈ V . As S runs over all the subsets of m indices,

m

the elements eS form a basis of V.

Since eS § eT = 0 when S and T are not disjoint, we have for x = xS e S

x § eT = ±xT e1 § · · · § en ,

where T is the complementary subset of T in {1, . . . , n}. Therefore, if x § eT = 0

for all T , then x = 0. This shows that the map § is nonsingular. Moreover, for all

subsets S, T of m indices we have

2

eS § eT = (’1)m eT § eS ,

hence § is symmetric if m is even and skew-symmetric if m is odd. Since eS §eS = 0

for all S, the form § is alternating if char F = 2.

n

Suppose m is even and ¬x an isomorphism V F to obtain from § a sym-

m m

metric bilinear form b on V . The space V decomposes into an orthogonal

direct sum:

⊥

m

V= ES

S∈R

where ES is the subspace spanned by the basis vectors eS , eS where S is the

complement of S and R is a set of representatives of the equivalence classes of

subsets of m indices under the relation S ≡ T if and only if S = T or S = T . (For

instance, one can take R = { S | 1 ∈ S }.)

On basis elements eS , eS the matrix of b has the form ± ± for some ± ∈ F — .

0

0

m d/2 —2

Therefore, if d = dim V we have det b = (’1) · F , hence

disc b = 1.

m

Since § is nonsingular, there is an adjoint involution γ on EndF ( V ) de¬ned

by

γ(f )(x) § y = x § f (y)

m m

for all f ∈ EndF ( V ), x, y ∈ V . The involution γ is of the ¬rst kind and its

discriminant is trivial; it is of orthogonal type if m is even and char F = 2, and it

is of symplectic type if m is odd or char F = 2. We call γ the canonical involution

m

on EndF ( V ).

Until the end of this subsection, A is a central simple F -algebra of degree

n = 2m. Our purpose is to de¬ne a canonical involution on »m A in such a way as

to recover the de¬nition above in the split case.

—

We ¬rst prove a technical result concerning the elements sk ∈ A—k de¬ned

in the preceding section:

118 II. INVARIANTS OF INVOLUTIONS

(10.7) Lemma. Let A be a central simple F -algebra of degree n = 2m. Since

sm ∈ A—m , we may consider sm — sm ∈ A—n . Then

sn ∈ A—n (sm — sm ).

Proof : In the symmetric group Sn , consider the subgroup Sm,m Sm — Sm con-

sisting of the permutations which preserve {1, . . . , m} (and therefore also the set

{m + 1, . . . , n}). The split case shows that gn (π1 , π2 ) = gm (π1 ) — gm (π2 ) for π1 ,

π2 ∈ Sm . Therefore,

sm — s m = sgn(π)gn (π).

π∈Sm,m

Let R be a set of representatives of the left cosets of Sm,m in Sn , so that each

π ∈ Sn can be written in a unique way as a product π = ρ —¦ π for some ρ ∈ R and

some π ∈ Sm,m . Since gn is a homomorphism, it follows that

sgn(π)gn (π) = sgn(ρ)gn (ρ) · sgn(π )gn (π ),

hence, summing over π ∈ Sn :

sn = sgn(ρ)gn (ρ) sm — sm .

ρ∈R

Recall from (??) that »m A = EndA—m (A—m sm ). There is therefore a natural

isomorphism:

»m A —F »m A = EndA—n A—n (sm — sm ) .

Since (??) shows that sn ∈ A—n (sm — sm ), we may consider

I = { f ∈ EndA—n A—n (sm — sm ) | sf = {0} }.

(10.8) n

This is a right ideal in EndA—n A—n (sm — sm ) = »m A —F »m A.

(10.9) Lemma. If A = EndF (V ), then under the natural isomorphisms

m m m m

»m A —F »m A = EndF ( V ) —F EndF ( V ) = EndF ( V— V)

the ideal I de¬ned above is

m m

I = { • ∈ EndF ( V— V ) | § —¦• = 0}

m

where § is the canonical bilinear form on V , viewed as a linear map

m m n

§: V— V’ V.

m

Proof : As observed in (??), we have A—m sm = HomF ( V, V —m ), hence

m m

A—n (sm — sm ) = HomF ( V , V —n ).

V—

n

Moreover, we may view sn as a map sn : V —n ’ V —n which factors through V:

there is a commutative diagram:

s

V —n ’ ’ ’ V —n

’n’

¦

¦ ¦

¦

sn

§

m m n

V— V ’’’

’’ V

m m

The image of sn ∈ A—n (sm — sm ) in HomF ( V, V —n ) under the

V—

identi¬cation above is then the induced map sn .

By (??) every endomorphism f of the A—n -module A—n (sm — sm ) has the form

xf = x —¦ •

§10. THE DISCRIMINANT ALGEBRA 119

m m

for some uniquely determined • ∈ EndF ( V— V ). The correspondence

f ” • yields the natural isomorphism

m m

»m A — »m A = EndA—n A—n (sm — sm ) = EndF ( V— V ).

Under this correspondence, the elements f ∈ EndA—n A—n (sm — sm ) which vanish

m m

on sn correspond to endomorphisms • ∈ EndF ( V— V ) such that sn —¦ • =

0. It is clear from the diagram above that ker sn = ker §, hence the conditions

sn —¦ • = 0 and § —¦ • = 0 are equivalent.

(10.10) Corollary. The right ideal I ‚ »m A —F »m A de¬ned in (??) above sat-

is¬es the following conditions:

(1) »m A —F »m A = I • (1 — »m A).

0

(2) I contains the annihilator (»m A — »m A) · (1 ’ g) , where g is the Goldman

element of »m A —F »m A, if m is odd or char F = 2; it contains 1 ’ g but not

0

(»m A — »m A) · (1 ’ g) if m is even and char F = 2.

Proof : It su¬ces to check these properties after scalar extension to a splitting ¬eld

of A. We may thus assume A = EndF (V ). The description of I in (??) then shows

that I is the right ideal corresponding to the canonical involution γ on EndF ( m V )

under the correspondence of (??).

(10.11) De¬nition. Let A be a central simple F -algebra of degree n = 2m. The

canonical involution γ on »m A is the involution of the ¬rst kind corresponding to

the ideal I de¬ned in (??) under the correspondence of (??).

The following properties follow from the de¬nition by (??) and (??), and by

scalar extension to a splitting ¬eld in which F is algebraically closed:

m

(a) If A = EndF (V ), the canonical involution γ on »m A = EndF ( V ) is the

m

adjoint involution with respect to the canonical bilinear map § : V—

m n

V’ V.

(b) γ is of symplectic type if m is odd or char F = 2; it is of orthogonal type if

m is even and char F = 2; in this last case we have disc(γ) = 1.

In particular, if A has degree 2 (i.e., A is a quaternion algebra), then the

canonical involution on A = »1 A has symplectic type, hence it is the quaternion

conjugation.

10.C. The canonical quadratic pair. Let A be a central simple F -algebra

of even degree n = 2m. As observed in (??), the canonical involution γ on »m A

is symplectic for all m if char F = 2. We show in this section that the canonical

involution is actually part of a canonical pair (γ, f ) on »m A for all m ≥ 2 if

char F = 2. (If char F = 2, a quadratic pair is uniquely determined by its involution;

thus »m A carries a canonical quadratic pair if and only if γ is orthogonal, i.e., if

and only if m is even).

We ¬rst examine the split case.

(10.12) Proposition. Assume char F = 2 and let V be an F -vector space of di-

mension n = 2m ≥ 4. There is a unique quadratic map

m n

q: V’ V

which satis¬es the following conditions:

(1) q(v1 § · · · § vm ) = 0 for all v1 , . . . , vm ∈ V ;

120 II. INVARIANTS OF INVOLUTIONS

m m n

(2) the polar form bq : V— V’ V is the canonical pairing §. In

particular, the quadratic map q is nonsingular.

Moreover, the discriminant of q is trivial.

Proof : Uniqueness of q is clear, since decomposable elements v1 § · · · § vm span

m

V . To prove the existence of q, we use the same notation as in the proof

m

of (??): we pick a basis (e1 , . . . , en ) of V and get a basis eS of V , where S runs

over the subsets of m indices in {1, . . . , n}. Fix a partition of these subsets into two

classes C, C such that the complement S of every S ∈ C lies in C and conversely.

(For instance, one can take C = { S | 1 ∈ S }, C = { S | 1 ∈ S }.) We may then

/

m

de¬ne a quadratic form q on V by

q xS e S = xS xS e 1 § · · · § e n .

S S∈C

The polar form bq satis¬es

bq xS e S , yT eT = xS y S + x S y S e 1 § · · · § e n

S T S∈C

= xS y S e 1 § · · · § e n .

S

Since the right side is also equal to ( S xS eS ) § ( T yT eT ), the second condition

is satis¬ed.

It remains to prove that q vanishes on decomposable elements. We show that q

m’1

actually vanishes on all the elements of the type v§·, where v ∈ V and · ∈ V.

n

Let v = i=1 vi ei and · = I ·I eI , where I runs over the subsets of m ’ 1

indices in {1, . . . , n}, so that

v§· = vi ·I e{i}∪I = vi ·S eS .

{i}

i,I,i∈I S i∈S

We thus get

q(v § ·) = vi ·S vj ·S .

{i} {j}

S∈C i∈S j∈S

The right side is a sum of terms of the form vi vj ·I ·J where I, J are subsets of

m ’ 1 indices such that {i, j} ∪ I ∪ J = {1, . . . , n}. Each of these terms appears

twice: vi vj ·I ·J appears in the term corresponding to S = {i} ∪ I or S = {j} ∪ J

(depending on which one of these two sets lies in C) and in the term corresponding

to S = {i} ∪ J or {j} ∪ I. Therefore, q(v § ·) = 0.

To complete the proof, we compute the discriminant of q. From the de¬nition,

it is clear that q decomposes into an orthogonal sum of 2-dimensional subspaces:

q = ⊥S∈C qS ,

where qS (xS eS + xS eS ) = xS xS e1 § · · · § en . It is therefore easily calculated that

disc q = 0.

(10.13) Remark. The quadratic map q may be de¬ned alternately by representing

m

V as the quotient space F V m /W , where F V m is the vector space of formal

linear combinations of m-tuples of vectors in V , and W is the subspace generated

by all the elements of the form

(v1 , . . . , vm )

where v1 , . . . , vm ∈ V are not all distinct, and

(v1 , . . . , vi ± + vi ± , . . . , vm ) ’ (v1 , . . . , vi , . . . , vm )± ’ (v1 , . . . , vi , . . . , vm )±

where i = 1, . . . , m and v1 , . . . , vi , vi , . . . , vm ∈ V , ±, ± ∈ F . Since m-

tuples of vectors in V form a basis of F V m , there is a unique quadratic map

§10. THE DISCRIMINANT ALGEBRA 121

n

q: F V m ’ V whose polar form bq factors through the canonical pairing §

and such that q(v) = 0 for all v ∈ V m . It is easy to show that this map q factors

through the quadratic map q.

By composing q with a vector-space isomorphism n V ’ F , we obtain a

∼

’

m

quadratic form on V which is uniquely determined up to a scalar factor. There-

m

fore, the corresponding quadratic pair (σq , fq ) on EndF ( V ) is unique. In this

pair, the involution σq is the canonical involution γ, since the polar form bq is the

canonical pairing.

Given an arbitrary central simple F -algebra A of degree n = 2m, we will

construct on »m A a quadratic pair (γ, f ) which coincides with the pair (σq , fq ) in

the case where A = EndF (V ). The ¬rst step is to distinguish the right ideals in

»m A which correspond in the split case to the subspaces spanned by decomposable

elements.

The following construction applies to any central simple algebra A over an

arbitrary ¬eld F : if I ‚ A is a right ideal of reduced dimension k, we de¬ne

ψk (I) = { f ∈ »k A = EndA—k (A—k sk ) | sf ∈ I —k · sk } ‚ »k A.

k

This set clearly is a right ideal in »k A.

(10.14) Lemma. If A = EndF (V ) and I = Hom(V, U ) for some k-dimensional

k k

subspace U ‚ V , then we may identify ψk (I) = Hom( V, U ). In particular,

rdim ψk (I) = 1.

Proof : If I = Hom(V, U ), then I —k = Hom(V —k , U —k ) and

k

I —k · sk = Hom( V, U —k ).

k

V ), we have sf ∈ I —k · sk if and only if the

Therefore, for f ∈ »k A = EndF ( k

image of the composite map

f s

k k

V ’’ V —k

k

V ’’

is contained in U —k . Since s’1 (U —k ) = k U , this condition is ful¬lled if and only

k

k

if im f ‚ U . Therefore, we may identify

k k

ψk Hom(V, U ) = Hom( V, U ).

k

Since dim U = 1 if dim U = k, we have rdim ψk (I) = 1 for all right ideals I of

reduced dimension k.

In view of the lemma, we have

ψk : SBk (A) ’ SB(»k A);

if A = EndF (V ), this map is the Pl¨cker embedding

u

k

ψk : Grk (V ) ’ P( A)

which maps every k-dimensional subspace U ‚ V to the 1-dimensional subspace

k k

U‚ V (see §??).

Suppose now that σ is an involution of the ¬rst kind on the central simple

F -algebra A. To every right ideal I ‚ A, we may associate the set I · σ(I) ‚ A.

(10.15) Lemma. Suppose σ is orthogonal or char F = 2. If rdim I = 1, then

I · σ(I) is a 1-dimensional subspace in Sym(A, σ).

122 II. INVARIANTS OF INVOLUTIONS

Proof : It su¬ces to prove the lemma in the split case. Suppose therefore that

A = EndF (V ) and σ is the adjoint involution with respect to some nonsingular

bilinear form b on V . Under the standard identi¬cation •b , we have A = V — V

and I = v — V for some nonzero vector v ∈ V , and σ corresponds to the switch

map. Therefore, I · σ(I) = v — v · F , proving the lemma.

Under the hypothesis of the lemma, we thus get a map

• : SB(A) ’ P Sym(A, σ)

which carries every right ideal I ‚ A of reduced dimension 1 to I · σ(I). If A =

EndF (V ), we may identify SB(A) = P(V ) and P Sym(A, σ) = P(W ), where W ‚

V — V is the subspace of symmetric tensors; the proof above shows that • : P(V ) ’

P(W ) maps v · F to v — v · F .

The relevance of this construction to quadratic pairs appears through the fol-

lowing lemma:

(10.16) Lemma. Suppose char F = 2 and σ is symplectic. The map (σ, f ) ’

ker f de¬nes a one-to-one correspondence between quadratic pairs (σ, f ) on A and

hyperplanes in Sym(A, σ) whose intersection with Symd(A, σ) is ker Trpσ .

Proof : For every quadratic pair (σ, f ), the map f extends Trpσ (this is just condi-

tion (??) of the de¬nition of a quadratic pair, see (??)); therefore

ker f © Symd(A, σ) = ker Trpσ .

If U ‚ Sym(A, σ) is a hyperplane whose intersection with Symd(A, σ) is ker Trpσ ,

then Sym(A, σ) = U +Symd(A, σ), hence there is only one linear form on Sym(A, σ)

with kernel U which extends Trpσ .

Suppose now that A is a central simple algebra of degree n = 2m over a ¬eld

F of characteristic 2. We consider the composite map

ψm •

SBm (A) ’’ SB(»m A) ’’ P Sym(»m A, γ) ,

where γ is the canonical involution on »m A.

(10.17) Proposition. If m ≥ 2, there is a unique hyperplane in Sym(»m A, γ)

which contains the image of • —¦ ψm and whose intersection with Symd(»m A, γ) is

ker Trpσ .

Proof : The proposition can be restated as follows: the subspace of Sym(»m A, γ)

spanned by the image of • —¦ ψm and ker Trpσ is a hyperplane which does not

contain Symd(»m A, γ). Again, it su¬ces to prove the result in the split case.

We may thus assume A = EndF (V ). From the description of ψm and • in this

case, it follows that the image of • —¦ ψm consists of the 1-dimensional spaces in

m m

V— V spanned by elements of the form (v1 § · · · § vm ) — (v1 § · · · § vm ),

with v1 , . . . , vm ∈ V . By (??), hyperplanes whose intersection with Symd(»m A, γ)

coincides with ker Trpγ correspond to quadratic pairs on »m A with involution γ,

m

hence to quadratic forms on V whose polar is the canonical pairing, up to a

scalar factor. Those hyperplanes which contain the image of • —¦ ψm correspond to

nonsingular quadratic forms which vanish on decomposable elements v1 § · · · § vm ,

and Proposition (??) shows that there is one and only one such quadratic form up

to a scalar factor.

§10. THE DISCRIMINANT ALGEBRA 123

(10.18) De¬nition. Let A be a central simple algebra of degree n = 2m over a

¬eld F of characteristic 2. By (??), Proposition (??) de¬nes a unique quadratic

pair (γ, f ) on »m A, which we call the canonical quadratic pair. The proof of (??)

shows that in the case where A = EndF (V ) this quadratic pair is associated with

m

the canonical map q on V de¬ned in (??). Since A may be split by a scalar

extension in which F is algebraically closed, and since the discriminant of the

canonical map q is trivial, by (??), it follows that disc(γ, f ) = 0.

If char F = 2, the canonical involution γ on »m A is orthogonal if and only if m

1

is even. Letting f be the restriction of 2 TrdA to Sym(»m A, γ), we also call (γ, f )

the canonical quadratic pair in this case. Its discriminant is trivial, as observed

in (??).

10.D. Induced involutions on »-powers. In this section, ι is an automor-

phism of the base ¬eld F such that ι2 = IdF (possibly ι = IdF ). Let V be a (¬nite

dimensional) vector space over F . Every hermitian15 form h on V with respect to ι

induces for every integer k a hermitian form h—k on V —k such that

h—k (x1 — · · · — xk , y1 — · · · — yk ) = h(x1 , y1 ) · · · h(xk , yk )

for x1 , . . . , xk , y1 , . . . , yk ∈ V . The corresponding linear map

h—k : V —k ’ ι (V —k )—

ˆ

(see (??)) is (h)—k under the canonical identi¬cation ι (V —k )— = (ι V — )—k , hence

h—k is nonsingular if h is nonsingular. Moreover, the adjoint involution σh—k on

EndF (V —k ) = EndF (V )—k is the tensor product of k copies of σh :

σh—k = (σh )—k .

k

The hermitian form h also induces a hermitian form h§k on V such that

h§k (x1 § · · · § xk , y1 § · · · § yk ) = det h(xi , yj ) 1¤i,j¤k

for x1 , . . . , xk , y1 , . . . , yk ∈ V . The corresponding linear map

k k

V ’ ι( V )—

h§k :

is k h under the canonical isomorphism k (ι V — ) ’ ι ( k

ˆ ∼

V )— which maps ι •1 §

’

· · · § ι •k to ι ψ where ψ ∈ ( k V )— is de¬ned by

ψ(x1 § · · · § xk ) = det •i (xj ) ,

1¤i,j¤k

for •1 , . . . , •k ∈ V — and x1 , . . . , xk ∈ V . Therefore, h§k is nonsingular if h is

nonsingular.

k

We will describe the adjoint involution σh§k on EndF ( V ) in a way which

generalizes to the »k -th power of an arbitrary central simple F -algebra with invo-

lution.

k

We ¬rst observe that if : V —k ’ V is the canonical epimorphism and

—k —k

sk : V ’V is the endomorphism considered in §??:

sk (v1 — · · · — vk ) = sgn(π)vπ’1 (1) — · · · — vπ’1 (k) ,

π∈Sk

then for all u, v ∈ V —k we have

h—k sk (u), v = h§k (u), (v) = h—k u, sk (v) .

(10.19)

15 By convention, a hermitian form with respect to IdF is a symmetric bilinear form.

124 II. INVARIANTS OF INVOLUTIONS

—k

In particular, it follows that σh (sk ) = sk .

(10.20) De¬nition. Let A be a central simple F -algebra with an involution σ

such that σ(x) = ι(x) for all x ∈ F . Recall from (??) that for k = 2, . . . , deg A,

»k A = EndA—k (A—k sk ).

According to (??), every f ∈ »k A has the form f = ρ(usk ) for some u ∈ A—k , i.e.,

there exists u ∈ A—k such that

xf = xusk for x ∈ A—k sk .

We then de¬ne σ §k (f ) = ρ σ —k (u)sk , i.e.,

§k

xσ (f )

= xσ —k (u)sk for x ∈ A—k sk .

To check that the de¬nition of σ §k (f ) does not depend on the choice of u, observe

¬rst that if f = ρ(usk ) = ρ(u sk ), then

sf = sk usk = sk u sk .

k

By applying σ —k to both sides of this equation, and taking into account the fact

that sk is symmetric under σ —k (see (??)), we obtain

sk σ —k (u)sk = sk σ —k (u )sk .

Since every x ∈ A—k sk has the form x = ysk for some y ∈ A—k , it follows that

xσ —k (u)sk = ysk σ —k (u)sk = ysk σ —k (u )sk = xσ —k (u )sk .

This shows that σ §k (f ) is well-de¬ned. Since σ(x) = ι(x) for all x ∈ F , it is easily

veri¬ed that σ §k also restricts to ι on F .

1

A = A and we set σ §1 = σ.

For k = 1, we have

(10.21) Proposition. If A = EndF (V ) and σ = σh is the adjoint involution

with respect to some nonsingular hermitian form h on V , then under the canonical

isomorphism »k A = EndF ( k V ), the involution σ §k is the adjoint involution with

k

respect to the hermitian form h§k on V.

Proof : Recall the canonical isomorphism of (??):

k k

A—k sk = HomF ( V, V —k ), hence »k A = EndF ( V ).

k

For f = ρ(usk ) ∈ EndA—k (A—k sk ), the corresponding endomorphism • of V is

de¬ned by

•(x1 § · · · § xk ) = —¦ u —¦ sk (x1 — · · · — xk )

or

• —¦ = —¦ u —¦ sk

k

where : V —k ’ V is the canonical epimorphism. In order to prove the propo-

sition, it su¬ces, therefore, to show:

h§k (x), —¦ u —¦ sk (y) = h§k —¦ σ —k (u) —¦ sk (x), (y)

for all x, y ∈ V —k . From (??) we have

h§k (x), —¦ u —¦ sk (y) = h—k sk (x), u —¦ sk (y)

and

h§k —¦ σ —k (u) —¦ sk (x), (y) = h—k σ —k (u) —¦ sk (x), sk (y) .

§10. THE DISCRIMINANT ALGEBRA 125

The proposition then follows from the fact that σ —k is the adjoint involution with

respect to h—k .

The next proposition is more speci¬cally concerned with symmetric bilinear

§m

forms b. In the case where dim V = n = 2m, we compare the involution σb with

m

the canonical involution γ on EndF ( V ).

(10.22) Proposition. Let b be a nonsingular symmetric, nonalternating, bilinear

form on an F -vector space V of dimension n = 2m. Let (e1 , . . . , en ) be an orthog-

n

onal basis of V and let e = e1 § · · · § en ∈ V ; let also

n

m

δ = (’1) b(ei , ei ),

i=1

m

so that disc b = δ · F —2 . There is a map u ∈ EndF ( V ) such that

b§m u(x), y e = x § y = (’1)m b§m x, u(y) e

(10.23)

m

for all x, y ∈ V , and

u2 = δ ’1 · Id§m V .

(10.24)

If σ = σb is the adjoint involution with respect to b, then the involution σ §m on

m

EndF ( V ) is related to the canonical involution γ by

σ §m = Int(u) —¦ γ.

In particular, the involutions σ §m and γ commute.

Moreover, if char F = 2 and m ≥ 2, the map u is a similitude of the canonical

m n

V of (??) with multiplier δ ’1 , i.e.,

quadratic map q : V’

q u(x) = δ ’1 q(x)

m

for all x ∈ V.

Proof : Let ai = b(ei , ei ) ∈ F — for i = 1, . . . , n. As in (??), we set

m

e S = e i1 § · · · § e im ∈ V and let a S = a i1 · · · a im

for S = {i1 , . . . , im } ‚ {1, . . . , n} with i1 < · · · < im . If S = T , the matrix

b(ei , ej ) (i,j)∈S—T has at least one row and one column of 0™s, namely the row corre-

sponding to any index in S T and the column corresponding to any index in T S.

Therefore, b§m (eS , eT ) = 0. On the other hand, the matrix b(ei , ej ) (i,j)∈S—S is

diagonal, and b§m (eS , eS ) = aS . Therefore, as S runs over all the subsets of m

m

indices, the elements eS are anisotropic and form an orthogonal basis of V with

§m

respect to the bilinear form b .

On the other hand, if S = {1, . . . , n} S is the complement of S, we have

eS § e S = µ S e

for some µS = ±1. Since § is symmetric when m is even and skew-symmetric when

m is odd (see (??)), it follows that

µS µS = (’1)m .

(10.25)

De¬ne u on the basis elements eS by

u(eS ) = µS a’1 eS

(10.26) S

126 II. INVARIANTS OF INVOLUTIONS

m

and extend u to V by linearity. We then have

b§m u(eS ), eS e = µS e = eS § eS

and

b§m eS , u(eS ) = µS e = (’1)m eS § eS .

Moreover, if T = S, then

b u(eS ), eT = 0 = eS § eT = b eS , u(eT ) .

The equations (??) thus hold when x, y run over the basis (eS ); therefore they hold

for all x, y ∈ V by bilinearity.

n

For all S we have aS aS = i=1 b(ei , ei ), hence (??) follows from (??) and (??).

From (??), it follows that for all f ∈ EndF (V ) and all x, y ∈ V ,

b§m u(x), f (y) e = x § f (y),

hence

b§m σ §m (f ) —¦ u(x), y e = γ(f )(x) § y.

The left side also equals

b§m u —¦ u’1 —¦ σ §m (f ) —¦ u(x), y e = u’1 —¦ σ §m (f ) —¦ u (x) § y,

hence

u’1 —¦ σ §m (f ) —¦ u = γ(f ) for f ∈ EndF (V ).

Therefore, σ §m = Int(u) —¦ γ.

We next show that σ §m and γ commute. By (??), we have σ §m (u) = (’1)m u,

hence γ(u) = (’1)m u. Therefore, γ —¦ σ §m = Int(u’1 ), while σ §m —¦ γ = Int(u).

Since u2 ∈ F — , we have Int(u) = Int(u’1 ), and the claim is proved.

Finally, assume char F = 2 and m ≥ 2. The proof of (??) shows that the quad-

ratic map q may be de¬ned by partitioning the subsets of m indices in {1, . . . , n}

into two classes C, C such that the complement of every subset in C lies in C and

vice versa, and letting

q(x) = xS xS e

S∈C

xS a’1 eS , hence

for x = xS eS . By de¬nition of u, we have u(x) =

S S S

xS a’1 xS a’1 e.

q u(x) = S S

S∈C

Since aS aS = δ for all S, it follows that q u(x) = δ ’1 q(x).

10.E. De¬nition of the discriminant algebra. Let (B, „ ) be a central

simple algebra with involution of the second kind over a ¬eld F . We assume that

the degree of (B, „ ) is even: deg(B, „ ) = n = 2m. The center of B is denoted K;

it is a quadratic ´tale F -algebra with nontrivial automorphism ι. We ¬rst consider

e

the case where K is a ¬eld, postponing to the end of the section the case where

F — F . The K-algebra B is thus central simple. The K-algebra »m B has a

K

canonical involution γ, which is of the ¬rst kind, and also has the involution „ §m

induced by „ , which is of the second kind. The de¬nition of the discriminant algebra

D(B, „ ) is based on the following crucial result:

§10. THE DISCRIMINANT ALGEBRA 127

(10.27) Lemma. The involutions γ and „ §m on »m B commute. Moreover, if

char F = 2 and m ≥ 2, the canonical quadratic pair (γ, f ) on »m B satis¬es

f „ §m (x) = ι f (x)

for all x ∈ Sym(»m B, γ).

Proof : We reduce to the split case by a scalar extension. To construct a ¬eld

extension L of F such that K —F L is a ¬eld and B —F L is split, consider the

division K-algebra D which is Brauer-equivalent to B. By (??), this algebra has

an involution of the second kind θ. We may take for L a maximal sub¬eld of

Sym(D, θ).

We may thus assume B = EndK (V ) for some n-dimensional vector space V

over K and „ = σh for some nonsingular hermitian form h on V . Consider an

orthogonal basis (ei )1¤i¤n of V and let V0 ‚ V be the F -subspace of V spanned

by e1 , . . . , en . Since h(ei , ei ) ∈ F — for i = 1, . . . , n, the restriction h0 of h to

V0 is a nonsingular symmetric bilinear form which is not alternating. We have

V = V0 —F K, hence

B = EndF (V0 ) —F K.

Moreover, since „ is the adjoint involution with respect to h,

„ = „0 — ι

where „0 is the adjoint involution with respect to h0 on EndF (V0 ). Therefore, there

is a canonical isomorphism

m

»m B = EndF ( V0 ) — F K

and

„ §m = „0 — ι.

§m

On the other hand, the canonical bilinear map

m m n

§: V— V’ V

is derived by scalar extension to K from the canonical bilinear map § on m V0 ,

m

hence γ = γ0 — IdK where γ0 is the canonical involution on EndF ( V0 ). By

§m §m

Proposition (??), „0 and γ0 commute, hence „ and γ also commute.

Suppose now that char F = 2 and m ≥ 2. Let z ∈ K F . In view of the canon-

m

ical isomorphism »m B = EndF ( V0 ) —F K, every element x ∈ Sym(»m B, γ)

m

may be written in the form x = x0 — 1 + x1 — z for some x0 , x1 ∈ EndF ( V0 )

m

symmetric under γ0 . Proposition (??) yields an element u ∈ EndF ( V0 ) such

§m

that „0 = Int(u) —¦ γ0 , hence

„ §m (x) = „0 (x0 ) — 1 + „0 (x1 ) — ι(z) = (ux0 u’1 ) — 1 + (ux1 u’1 ) — ι(z).

§m §m

To prove f „ §m (x) = ι f (x) , it now su¬ces to show that f (uyu’1 ) = f (y) for

m

all y ∈ Sym EndF ( V0 ), γ0 .

m

Let q : V0 ’ F be the canonical quadratic form uniquely de¬ned (up to a

scalar multiple) by (??). Under the associated standard identi¬cation, the elements

m m m

in Sym EndF ( V0 ), γ0 correspond to symmetric tensors in V0 — V0 , and

m

we have f (v — v) = q(v) for all v ∈ V0 . Since symmetric tensors are spanned

by elements of the form v — v, it su¬ces to prove

m

f u —¦ (v — v) —¦ u’1 = f (v — v) for all v ∈ V0 .

128 II. INVARIANTS OF INVOLUTIONS

The proof of (??) shows that γ0 (u) = u and u2 = δ ’1 ∈ F — , hence

u —¦ (v — v) —¦ u’1 = δu —¦ (v — v) —¦ γ0 (u) = δu(v) — u(v);

therefore, by (??),

f u —¦ (v — v) —¦ u’1 = δq u(v) = q(v) = f (v — v),

and the proof is complete.

The lemma shows that the composite map θ = „ §m —¦ γ is an automorphism of

order 2 on the F -algebra B. Note that θ(x) = ι(x) for all x ∈ K, since „ §m is an

involution of the second kind while γ is of the ¬rst kind.

(10.28) De¬nition. The discriminant algebra D(B, „ ) of (B, „ ) is the F -subal-

gebra of θ-invariant elements in »m B. It is thus a central simple F -algebra of

degree

n

deg D(B, „ ) = deg »m B = .

m

The involutions γ and „ §m restrict to the same involution of the ¬rst kind „ on

D(B, „ ):

„ = γ|D(B,„ ) = „ §m |D(B,„ ) .

Moreover, if char F = 2 and m ≥ 2, the canonical quadratic pair (γ, f ) on »m B

restricts to a canonical quadratic pair („ , fD ) on D(B, „ ); indeed, for an element

x ∈ Sym D(B, „ ), „ we have „ §m (x) = x, hence, by (??), f (x) = ι f (x) , and

therefore f (x) ∈ F .

The following number-theoretic observation on deg D(B, „ ) is useful:

2m

(10.29) Lemma. Let m be an integer, m ≥ 1. The binomial coe¬cient m

satis¬es

2 mod 4 if m is a power of 2;

2m

≡

m

0 mod 4 if m is not a power of 2.

Proof : For every integer m ≥ 1, let v(m) ∈ N be the exponent of the highest

power of 2 which divides m, i.e., v(m) is the 2-adic valuation of m. The equation

(m + 1) 2(m+1) = 2(2m + 1) 2m yields

m+1 m

2(m+1) 2m

v =v + 1 ’ v(m + 1) for m ≥ 1.

m+1 m

On the other hand, let (m) = m0 + · · · + mk where the 2-adic expansion of m

is m = m0 + 2m1 + 22 m2 + · · · + 2k mk with m0 , . . . , mk = 0 or 1. It is easily

seen that the function (m) satis¬es the same recurrence relation as v 2m and

m

2 2m 2m

(1) = 1 = v 1 , hence (m) = v m for all m ≥ 1. In particular, v m = 1 if m

is a power of 2, and v 2m ≥ 2 otherwise.

m

(10.30) Proposition. Multiplication in »m B yields a canonical isomorphism

D(B, „ ) —F K = »m B

such that „ — IdK = γ and „ — ι = „ §m . The index ind D(B, „ ) divides 4, and

ind D(B, „ ) = 1 or 2 if m is a power of 2.

The involution „ is of symplectic type if m is odd or char F = 2; it is of

orthogonal type if m is even and char F = 2.

§10. THE DISCRIMINANT ALGEBRA 129

Proof : The ¬rst part follows from the de¬nition of D(B, „ ) and its involution „ . By

(??) we have ind »m B = 1 or 2 since »m B and B —m are Brauer-equivalent, hence

ind D(B, „ ) divides 2[K : F ] = 4. However, if m is a power of 2, then ind D(B, „ )

cannot be 4 since the preceding lemma shows that deg D(B, „ ) ≡ 2 mod 4.

Since γ = „ — IdK , the involutions γ and „ have the same type, hence „ is

orthogonal if and only if m is even and char F = 2.

For example, if B is a quaternion algebra, i.e., n = 2, then m = 1 hence »m B =

B and „ §m = „ . The algebra D(B, „ ) is the unique quaternion F -subalgebra of B

such that B = D(B, „ ) —F K and „ = γ0 — ι where γ0 is the canonical (conjugation)

involution on D(B, „ ): see (??).

To conclude this section, we examine the case where K = F — F . We may then

assume B = E — E op for some central simple F -algebra E of degree n = 2m and

„ = µ is the exchange involution. Note that there is a canonical isomorphism

∼

(»m E)op = EndE —m (E —m sm )op ’ End(E op )—m (E op )—m sop = »m (E op )

’ m

which identi¬es f op ∈ (»m E)op with the endomorphism of (E op )—m sop which maps

m

op

sop to (sf )op (thus, (sop )f = (sf )op ) (see Exercise ?? of Chapter I). Therefore,

m m m m

m op

the notation » E is not ambiguous. We may then set

»m B = »m E — »m E op

and de¬ne the canonical involution γ on »m B by means of the canonical involution

γE on »m E:

γ(x, y op ) = γE (x), γE (y)op for x, y ∈ »m E.

Similarly, if char F = 2 and m ≥ 2, the canonical pair (γE , fE ) on »m E (see (??))

induces a canonical quadratic pair (γ, f ) on »m B by

f (x, y op ) = fE (x), fE (y) ∈ F — F for x, y ∈ Sym(»m E, γ).

We also de¬ne the involution µ§m on »m B as the exchange involution on »m E —

»m E op :

µ§m (x, y op ) = (y, xop ) for x, y ∈ »m E.

The involutions µ§m and γ thus commute, hence their composite θ = µ§m —¦ γ is an

F -automorphism of order 2 on »m B. The invariant elements form the F -subalgebra

D(B, µ) = { x, γ(x)op | x ∈ »m E } »m E.

(10.31)

The involutions µ§m and γ coincide on this subalgebra and induce an involution

which we denote µ.

The following proposition shows that these de¬nitions are compatible with the

notions de¬ned previously in the case where K is a ¬eld:

(10.32) Proposition. Let (B, „ ) be a central simple algebra with involution of

the second kind over a ¬eld F . Suppose the center K of B is a ¬eld. The K-

∼

algebra isomorphism • : (BK , „K ) ’ (B — B op , µ) of (??) which maps x — k to

’

op

xk, „ (x)k induces a K-algebra isomorphism

∼

»m • : (»m B)K ’ »m B — »m B op

’

op

mapping x — k to xk, „ §m (x)k . This isomorphism is compatible with the

canonical involution and the canonical quadratic pair (if char F = 2 and m ≥ 2),

130 II. INVARIANTS OF INVOLUTIONS

and satis¬es »m • —¦ „K = µ§m —¦ »m •. Therefore, »m • induces an isomorphism of

§m

K-algebras with involution

∼

D(B, „ )K , „ K ’ D(B — B op , µ), µ

’

and also, if char F = 2 and m ≥ 2, an isomorphism of K-algebras with quadratic

pair

∼

D(B, „ )K , „ K , (fD )K ’ D(B — B op , µ), µ, f .

’

Proof : The fact that »m • is compatible with the canonical involution and the

canonical pair follows from (??); the equation »m • —¦ „K = µ§m —¦ »m • is clear from

§m

the de¬nition of »m •.

10.F. The Brauer class of the discriminant algebra. An explicit descrip-

tion of the discriminant algebra of a central simple algebra with involution of the

second kind is known only in very few cases: quaternion algebras are discussed after

(??) above, and algebras of degree 4 are considered in §??. Some general results

on the Brauer class of a discriminant algebra are easily obtained however, as we

proceed to show. In particular, we establish the relation between the discriminant

algebra and the discriminant of hermitian forms mentioned in the introduction.

Notation is as in the preceding subsection. Thus, let (B, „ ) be a central simple

algebra with involution of the second kind of even degree n = 2m over an arbi-

trary ¬eld F , and let K be the center of B. For any element d = δ · N (K/F ) ∈

F — /N (K/F ), we denote by (K, d)F (or (K, δ)F ) the quaternion algebra K • Kz

where multiplication is de¬ned by zx = ι(x)z for x ∈ K and z 2 = δ. Thus,

F [X]/(X 2 ’ ±),

(±, δ)F if char F = 2 and K

(K, d)F =

F [X]/(X 2 + X + ±).

[±, δ)F if char F = 2 and K

(In particular, (K, d)F splits if K F — F ). We write ∼ for Brauer-equivalence.

(10.33) Proposition. Suppose B = B0 —F K and „ = „0 — ι for some central

simple F -algebra B0 with involution „0 of the ¬rst kind of orthogonal type; then

D(B, „ ) ∼ »m B0 —F (K, disc „0 )F .

Proof : We have »m B = »m B0 —F K, „ §m = „0 — ι and γ = γ0 — IdK where γ0

§m

is the canonical involution on »m B0 , hence also θ = θ0 — ι where θ0 = „0 —¦ γ0 .

§m

2

Since θ0 leaves F elementwise invariant and θ0 = Id, we have θ0 = Int(t) for some

t ∈ (»m B0 )— such that t2 ∈ F — . After scalar extension to a splitting ¬eld L of B0

in which F is algebraically closed, (??) yields t = uξ for some ξ ∈ L— and some

u ∈ (»m B0 — L)— such that u2 · L—2 = disc „0 . Therefore, letting δ = t2 ∈ F — , we

have δ · L—2 = disc „0 , hence

δ · F —2 = disc „0 ,

since F is algebraically closed in L. The proposition then follows from the following

general result:

(10.34) Lemma. Let A = A0 —F K be a central simple K-algebra and let t ∈ A— 0

be such that t2 = δ ∈ F — . The F -subalgebra A ‚ A of elements invariant under

Int(t) — ι is Brauer-equivalent to A0 —F (K, δ)F .

§10. THE DISCRIMINANT ALGEBRA 131

Proof : Let (K, δ)F = K • Kz where zx = ι(x)z for x ∈ K and z 2 = δ, and let

A1 = A0 — (K, δ)F . The centralizer of K ‚ (K, δ)F , viewed as a subalgebra in A1 ,

is

CA1 K = A0 —F K,

which may be identi¬ed with A. The algebra A is then identi¬ed with the central-

izer of K and tz.

Claim. The subalgebra M ‚ A1 generated by K and tz is a split quaternion

algebra.

Since t ∈ A— , the elements t and z commute, and tzx = ι(x)tz for x ∈ K.

0

Moreover, t = δ = z 2 , hence (tz)2 = δ 2 ∈ F —2 . Therefore, M (K, δ 2 )F , proving

2

the claim.

Since A is the centralizer of M in A1 , Theorem (??) yields

A1 A —F M.

The lemma then follows from the claim.

The split case B = EndK (V ) is a particular case of (??):

(10.35) Corollary. For every nonsingular hermitian space (V, h) of even dimen-

sion over K,

D EndK (V ), σh ∼ (K, disc h)F

where disc h is de¬ned in the introduction to this section.

Proof : Let V0 ‚ V be the F -subspace spanned by an orthogonal K-basis of V .

The hermitian form h restricts to a nonsingular symmetric bilinear form h0 on V0

and we have

EndK (V ) = EndF (V0 ) —F K, σh = σ 0 — ι

where σ0 = σh0 is the adjoint involution with respect to h0 . By (??),

D EndK (V ), σh ∼ (K, disc h0 )F .

The corollary follows, since disc h = disc h0 · N (K/F ) ∈ F — /N (K/F ).

(10.36) Corollary. For all u ∈ Sym(B, „ ) © B — ,

D B, Int(u) —¦ „ ∼ D(B, „ ) —F K, NrdB (u) .

F

Proof : If K F — F , each side is split. We may thus assume K is a ¬eld. Suppose

¬rst B = EndK (V ) for some vector space V , and let h be a nonsingular hermitian

form on V such that „ = σh . The involution Int(u) —¦ „ is then adjoint to the

hermitian form h de¬ned by

h (x, y) = h x, u’1 (y) for x, y ∈ V .

Since the Gram matrix of h is the product of the Gram matrix of h by the matrix

of u’1 , it follows that

disc h = disc h det u’1 = disc h det u ∈ F — /N (K/F ).

The corollary then follows from (??) by multiplicativity of the quaternion symbol.

132 II. INVARIANTS OF INVOLUTIONS

The general case is reduced to the split case by a suitable scalar extension. Let

X = RK/F SB(B) be the Weil transfer (or restriction of scalars) of the Severi-

Brauer variety of B (see Scheiderer [?, §4] for a discussion of the Weil transfer) and

let L = F (X) be the function ¬eld of X. We have