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