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

B —F L = B —K K SB(B) —K SB(ι B) ,

hence BL is split. Therefore, the split case considered above shows that the F -

algebra

op

A = D(B, „ ) —F K, NrdB (u) —F D B, Int(u) —¦ „

F

is split by L. However, the kernel of the scalar extension map Br(F ) ’ Br(L) is

the image under the norm map of the kernel of the scalar extension map Br(K) ’

Br K SB(B) (see Merkurjev-Tignol [?, Corollary 2.12]). The latter is known

to be generated by the Brauer class of B (see for instance Merkurjev-Tignol [?,

Corollary 2.7]), and NK/F (B) splits since B has an involution of the second kind

(see (??)). Therefore, the map Br(F ) ’ Br(L) is injective, hence A is split.

§11. Trace Form Invariants

The invariants of involutions de¬ned in this section are symmetric bilinear forms

derived from the reduced trace. Let A be a central simple algebra over an arbitrary

¬eld F and let σ be an involution of any kind on A. Our basic object of study is

the form

T(A,σ) : A — A ’ F

de¬ned by

T(A,σ) (x, y) = TrdA σ(x)y for x, y ∈ A.

Since σ TrdA σ(y)x = TrdA σ(x)y , by (??) and (??), the form T(A,σ) is sym-

metric bilinear if σ is of the ¬rst kind and hermitian with respect to σ|F if σ is

of the second kind. It is nonsingular in each case, since the bilinear trace form

TA (x, y) = TrdA (xy) is nonsingular, as is easily seen after scalar extension to a

splitting ¬eld of A.

More generally, for any u ∈ Sym(A, σ) we set

T(A,σ,u) (x, y) = TrdA σ(x)uy for x, y ∈ A.

The form T(A,σ,u) also is symmetric bilinear if σ is of the ¬rst kind and hermitian

if σ is of the second kind, and it is nonsingular if and only if u is invertible.

How much information on σ can be derived from the form T(A,σ) is suggested

by the following proposition, which shows that T(A,σ) determines σ — σ if σ is of

the ¬rst kind. To formulate a more general statement, we denote by ι = σ|F the

restriction of σ to F , by ιA = { ιa | a ∈ A } the conjugate algebra of A (see §??)

and by ισ the involution on ιA de¬ned by

ι

σ(ιa) = ι σ(a) for a ∈ A.

∼

(11.1) Proposition. Under the isomorphism σ— : A — ι A ’ EndF (A) such that

’

σ— (a — ι b)(x) = axσ(b),

the involution σ — ισ corresponds to the adjoint involution with respect to the form

T(A,σ) . More generally, for all u ∈ Sym(A, σ)©A— , the involution Int(u’1 )—¦σ — ισ

§11. TRACE FORM INVARIANTS 133

corresponds to the adjoint involution with respect to the form T(A,σ,u) under the

isomorphism σ— .

Proof : The proposition follows by a straightforward computation: for a, b, x, y ∈ A,

T(A,σ,u) σ— (a — b)(x), y = TrdA bσ(x)σ(a)uy

and

T(A,σ,u) x, σ— Int(u’1 ) —¦ σ(a) — σ(b) (y) = TrdA σ(x)u u’1 σ(a)u yb .

The equality of these expressions proves the proposition. (Note that the ¬rst part

(i.e., the case where u = 1) was already shown in (??)).

On the basis of this proposition, we de¬ne below the signature of an involution

σ as the square root of the signature of T(A,σ) . We also show how the form T(A,σ)

can be used to determine the discriminant of σ (if σ is of orthogonal type and

char F = 2) or the Brauer class of the discriminant algebra of (A, σ) (if σ is of the

second kind).

11.A. Involutions of the ¬rst kind. In this section, σ denotes an involution

+ ’

of the ¬rst kind on a central simple algebra A over a ¬eld F . We set Tσ and Tσ

for the restrictions of the bilinear trace form T(A,σ) to Sym(A, σ) and Skew(A, σ)

respectively; thus

+

Tσ (x, y) = TrdA σ(x)y = TrdA (xy) for x, y ∈ Sym(A, σ)

’

Tσ (x, y) = TrdA σ(x)y = ’ TrdA (xy) for x, y ∈ Skew(A, σ).

Also let TA denote the symmetric bilinear trace form on A:

TA (x, y) = TrdA (xy) for x, y ∈ A,

+ ’

so that Tσ (x, y) = TA (x, y) for x, y ∈ Sym(A, σ) and Tσ (x, y) = ’TA (x, y) for x,

y ∈ Skew(A, σ).

(11.2) Lemma. Alt(A, σ) is the orthogonal space of Sym(A, σ) in A for each of

the bilinear forms T(A,σ) and TA . Consequently,

+ ’

(1) if char F = 2, the form Tσ = Tσ is singular;

+ ’

(2) if char F = 2, the forms Tσ and Tσ are nonsingular and there are orthogonal

sum decompositions

⊥

+ ’

A, T(A,σ) = Sym(A, σ), Tσ • Skew(A, σ), Tσ ,

⊥

+ ’

A, TA = Sym(A, σ), Tσ • Skew(A, σ), ’Tσ .

Proof : For x ∈ A and y ∈ Sym(A, σ), we have TrdA σ(x)y = TrdA σ(yx) =

TrdA (xy), hence

TA x ’ σ(x), y = TrdA (xy) ’ TrdA σ(x)y = 0.

This shows Alt(A, σ) ‚ Sym(A, σ)⊥ (the orthogonal space for the form TA ); the

equality Alt(A, σ) = Sym(A, σ)⊥ follows by dimension count. Since

T(A,σ) x ’ σ(x), y = ’TA x ’ σ(x), y for x ∈ A, y ∈ Sym(A, σ),

the same arguments show that Alt(A, σ) is the orthogonal space of Sym(A, σ) for

the form T(A,σ) .

134 II. INVARIANTS OF INVOLUTIONS

(11.3) Examples. (1) Quaternion algebras. Let Q = (a, b)F be a quaternion al-

gebra with canonical involution γ over a ¬eld F of characteristic di¬erent from 2.

Let Q0 denote the vector space of pure quaternions, so that Q0 = Skew(Q, γ). A

direct computation shows that the elements i, j, k of the usual quaternion basis

+ ’

are orthogonal for T(Q,γ) , hence Tγ and Tγ have the following diagonalizations:

+ ’

Tγ = 2 and Tγ = 2 · ’a, ’b, ab .

Now, let σ = Int(i) —¦ γ. Then Skew(Q, σ) = i · F , and Sym(Q, σ) has 1, j, k as

orthogonal basis. Therefore,

+ ’

Tσ = 2 · 1, b, ’ab and Tσ = ’2a .

(2) Biquaternion algebras. Let A = (a1 , b1 )F —F (a2 , b2 )F be a tensor product of

two quaternion F -algebras and σ = γ1 — γ2 , the tensor product of the canonical

involutions. Let (1, i1 , j1 , k1 ) and (1, i2 , j2 , k2 ) denote the usual quaternion bases of

(a1 , b1 )F and (a2 , b2 )F respectively. The element 1 and the products ξ — · where

ξ and · independently range over i1 , j1 , k1 , and i2 , j2 , k2 , respectively, form an

+

orthogonal basis of Sym(A, σ) for the form Tσ . Similarly, i1 — 1, j1 — 1, k1 — 1,

1 — i2 , 1 — j2 , 1 — k2 form an orthogonal basis of Skew(A, σ). Therefore,

+

Tσ = 1 ⊥ a1 , b1 , ’a1 b1 · a2 , b2 , ’a2 b2

and

’

Tσ = ’a1 , ’b1 , a1 b1 , ’a2 , ’b2 , a2 b2 .

(Note Tσ is not an Albert form of A as discussed in §??, unless ’1 ∈ F —2 ).

’

As a further example, consider the split orthogonal case in characteristic dif-

ferent from 2. If b is a nonsingular symmetric bilinear form on a vector space V ,

we consider the forms bS2 and b§2 de¬ned on the symmetric square S 2 V and the

2

exterior square V respectively by

bS2 (x1 · x2 , y1 · y2 ) = b(x1 , y1 )b(x2 , y2 ) + b(x1 , y2 )b(x2 , y1 ),

b§2 (x1 § x2 , y1 § y2 ) = b(x1 , y1 )b(x2 , y2 ) ’ b(x1 , y2 )b(x2 , y1 )

for x1 , x2 , y1 , y2 ∈ V . (The form b§2 has already been considered in §??). Assuming

2

V in V — V by mapping x1 · x2 to 1 (x1 — x2 +

char F = 2, we embed S 2 V and 2

1

x2 — x1 ) and x1 § x2 to 2 (x1 — x2 ’ x2 — x1 ) for x1 , x2 ∈ V .

(11.4) Proposition. Suppose char F = 2 and let (A, σ) = EndF (V ), σb . The

∼

standard identi¬cation •b : V — V ’ A of (??) induces isometries of bilinear

’

spaces

∼

V — V, b — b ’ A, T(A,σ) ,

’

∼

S 2 V, 1 bS2 ’ Sym(A, σ), Tσ ,

+

’

2

2 ∼ ’

1

V, 2 b§2 ’ Skew(A, σ), Tσ .

’

Proof : As observed in (??), we have σ •b (x1 — x2 ) = •b (x2 — x1 ) and

TrdA •b (x1 — x2 ) = b(x2 , x1 ) = b(x1 , x2 ) for x1 , x2 ∈ V .

Therefore,

T(A,σ) •b (x1 — x2 ), •b (y1 — y2 ) = b(x1 , y1 )b(x2 , y2 ) for x1 , x2 , y1 , y2 ∈ V ,

proving the ¬rst isometry. The other isometries follow by similar computations.

§11. TRACE FORM INVARIANTS 135

Diagonalizations of bS2 and b§2 are easily obtained from a diagonalization of b:

if b = ±1 , . . . , ±n , then

bS2 = n 2 ⊥ (⊥1¤i<j¤n ±i ±j ) and b§2 = ⊥1¤i<j¤n ±i ±j .

Therefore, det bS2 = 2n (det b)n’1 and det b§2 = (det b)n’1 .

(11.5) Proposition. Let (A, σ) be a central simple algebra with involution of or-

thogonal type over a ¬eld F of characteristic di¬erent from 2. If deg A is even,

then

det Tσ = det Tσ = 2deg A/2 det σ.

+ ’

Proof : By extending scalars to a splitting ¬eld L in which F is algebraically closed

(so that the induced map F — /F —2 ’ L— /L—2 is injective), we reduce to considering

the case where A is split. If (A, σ) = EndF (V ), σb , then det σ = det b by (??),

and the computations above, together with (??), show that

det Tσ = det Tσ = 2’n(n’1)/2 (det b)n’1 = 2n/2 det b in F — /F —2 ,

+ ’

where n = deg A.

As a ¬nal example, we compute the form T(A,σ,u) for a quaternion algebra with

orthogonal involution. This example is used in §?? (see (??)). In the following

statement, we denote by W F the Witt ring of nonsingular bilinear forms over F .

(11.6) Proposition. Let Q be a quaternion algebra over a ¬eld F of arbitrary

characteristic, let σ be an orthogonal involution on Q and v ∈ Sym(Q, σ) © Q — .

For all s ∈ Q— such that σ(s) = s = ’γ(s),

T(Q,σ,v) TrdQ (v) · NrdQ (vs), disc σ if TrdQ (v) = 0;

T(Q,σ,v) = NrdQ (vs), disc σ = 0 in W F if TrdQ (v) = 0.

Proof : Let γ be the canonical (symplectic) involution on Q and let u ∈ Skew(Q, γ)

F be such that σ = Int(u) —¦ γ. The discriminant disc σ is therefore represented in

F — /F —2 by ’ NrdQ (u) = u2 . Since σ(v) = v, we have v = uγ(v)u’1 , hence

TrdQ (vu) = 0. A computation shows that 1, u are orthogonal for the form T(Q,σ,v) .

Since further T(Q,σ,v) (1, 1) = TrdQ (v) and T(Q,σ,v) (u, u) = TrdQ (v) NrdQ (u), the

subspace spanned by 1, u is totally isotropic if TrdQ (v) = 0, hence T(Q,σ,v) is

metabolic in this case. If TrdQ (v) = 0, a direct calculation shows that for all

s ∈ Q— as above, 1, u, γ(v)s, γ(v)su is an orthogonal basis of Q which yields the

diagonalization

T(Q,σ,v) TrdQ (v) · 1, NrdQ (u), ’ NrdQ (vs), ’ NrdQ (vsu) .

To complete the proof, we observe that if TrdQ (v) = 0, then v and s both anti-

commute with u, hence vs ∈ F [u] and therefore NrdQ (vs) is a norm from F [u]; it

follows that

NrdQ (vs), u2 = 0 in W F.

136 II. INVARIANTS OF INVOLUTIONS

The signature of involutions of the ¬rst kind. Assume now that the base

¬eld F has an ordering P , so char(F ) = 0. (See Scharlau [?, §2.7] for background

information on ordered ¬elds.) To every nonsingular symmetric bilinear form b

there is classically associated an integer sgnP b called the signature of b at P (or

with respect to P ): it is the di¬erence m+ ’m’ where m+ (resp. m’ ) is the number

of positive (resp. negative) entries in any diagonalization of b.

Our goal is to de¬ne the signature of an involution in such a way that in the

split case A = EndF (V ), the signature of the adjoint involution with respect to a

symmetric bilinear form b is the absolute value of the signature of b:

sgnP σb = |sgnP b| .

(Note that σb = σ’b and sgnP (’b) = ’ sgnP b, so sgnP b is not an invariant of σb ).

(11.7) Proposition. For any involution σ of the ¬rst kind on A, the signature of

the bilinear form T(A,σ) at P is a square in Z. If A is split: A = EndF (V ) and

σ = σb is the adjoint involution with respect to some nonsingular bilinear form b

on V , then

(sgnP b)2 if σ is orthogonal,

sgnP T(A,σ) =

0 if σ is symplectic.

Proof : When A is split and σ is orthogonal, (??) yields an isometry T(A,σ) b — b

from which the formula for sgnP T(A,σ) follows. When A is split and σ is symplectic,

we may ¬nd an isomorphism (A, σ) EndF (V ), σb for some vector space V and

some nonsingular skew-symmetric form b on V . The same argument as in (??)

yields an isometry (A, T(A,σ) ) (V — V, b — b). In this case, b — b is hyperbolic.

Indeed, if U ‚ V is a maximal isotropic subspace for b, then dim U = 1 dim V and

2

1

U — V is an isotropic subspace of V — V of dimension 2 dim(V — V ). Therefore,

sgnP T(A,σ) = 0.

In the general case, consider a real closure FP of F for the ordering P . Since

the signature at P of a symmetric bilinear form over F does not change under scalar

extension to FP , we may assume F = FP . The Brauer group of F then has order 2,

the nontrivial element being represented by the quaternion algebra Q = (’1, ’1) F .

Since the case where A is split has already been considered, we may assume for the

rest of the proof that A is Brauer-equivalent to Q. According to (??), we then have

(A, σ) EndQ (V ), σh

for some (right) vector space V over Q, and some nonsingular form h on V , which

is hermitian with respect to the canonical involution γ on Q if σ is symplectic, and

skew-hermitian with respect to γ if σ is orthogonal.

Let (ei )1¤i¤n be an orthogonal basis of V with respect to h, and let

h(ei , ei ) = qi ∈ Q— for i = 1, . . . , n.

Thus qi ∈ F if σ is symplectic and qi is a pure quaternion if σ is orthogonal. For

i, j = 1, . . . , n, write Eij ∈ EndQ (V ) for the endomorphism which maps ej to ei

and maps ek to 0 if k = j. Thus Eij corresponds to the matrix unit eij under the

isomorphism EndQ (V ) Mn (Q) induced by the choice of the basis (ei )1¤i¤n .

A direct veri¬cation shows that for i, j = 1, . . . , n and q ∈ Q,

’1

σ(Eij q) = Eji qj γ(q)qi .

§11. TRACE FORM INVARIANTS 137

Therefore, for i, j, k, = 1, . . . , n and q, q ∈ Q,

0 if i = k or j = ,

T(A,σ) (Eij q, Ek q ) = ’1

TrdQ qj γ(q)qi q if i = k and j = .

We thus have an orthogonal decomposition of EndQ (V ) with respect to the form

T(A,σ) :

(11.8) EndQ (V ) = ⊥1¤i,j¤n Eij · Q.

Suppose ¬rst that σ is orthogonal, so h is skew-hermitian and qi is a pure quaternion

for i = 1, . . . , n. Fix a pair of indices (i, j). If qi qj is a pure quaternion, then

Eij and Eij qi span an isotropic subspace of Eij · Q, so Eij · Q is hyperbolic. If

qi qj is not pure, pick a nonzero pure quaternion h ∈ Q which anticommutes with

’1

qj qi qj . Since Q = (’1, ’1)F and F is real-closed, the square of every nonzero

pure quaternion lies in ’F —2 . For i = 1, . . . , n, let qi = ’±2 for some ±i ∈ F — ;

2

i