—

hermitian forms on V . For s ∈ Sym EndK (V ), σh , we have

disc hs = disc h · det s’1 in F — /N (K/F ),

hence there exists z ∈ K — such that det s = NK/F (z) if and only if disc hs = disc h.

Therefore, we have a canonical bijection of pointed sets

isometry classes of nonsingular

H 1 F, SU(V, h) .

hermitian forms h on V ←’

with disc h = disc h

(29.20) Example. Consider B = Mn (F ) — Mn (F )op , with µ the exchange invo-

lution (a, bop ) ’ (b, aop ). We have

U(B, µ) = { u, (u’1 )op | u ∈ GLn (F ) },

hence SU(B, „ ) = SLn (F ) and PGU(B, „ ) = PGLn (F ). Therefore, by Hilbert™s

Theorem 90 (??) and (??),

H 1 F, U(B, µ) = H 1 F, SU(B, µ) = 1.

The map (a, bop ) ’ bt , (at )op is an outer automorphism of order 2 of (B, µ), and

we may identify

Aut Mn (F ) — Mn (F )op , µ = PGLn S2

where the nontrivial element of S2 acts on PGLn by mapping a · F — to (at )’1 · F — .

The exact sequence

1 ’ PGLn ’ PGLn S2 ’ S2 ’ 1

induces the following exact sequence in cohomology:

H 1 (F, PGLn ) ’ H 1 (F, PGLn S2 ) ’ H 1 (F, S2 ).

404 VII. GALOIS COHOMOLOGY

This cohomology sequence corresponds to

central simple central simple F -algebras quadratic

F -algebras of with unitary involution ´tale

e

’ ’

degree n of degree n F -algebras

A ’ (A — Aop , µ) B ’ Z(B)

where µ is the exchange involution. Observe that S2 acts on H 1 (F, PGLn ) by send-

ing a central simple algebra A to the opposite algebra Aop , and that the algebras

with involution (A — Aop , µ) and (Aop — A, µ) are isomorphic over F .

(29.21) Remark. Let Z be a quadratic ´tale F -algebra. The cohomology set

e

1

H F, (PGLn )[Z] , where the action of “ is twisted through the cocycle de¬n-

ing [Z], classi¬es triples (B , „ , φ) where (B , „ ) is a central simple F -algebra with

∼

unitary involution of degree n and φ is an isomorphism Z(B ) ’ Z. ’

Symplectic involutions. Let A be a central simple F -algebra of degree 2n

with a symplectic involution σ. The group Aut(A, σ) coincides with PGSp(A, σ).

Moreover, since all the nonsingular alternating bilinear forms of dimension 2n are

isometric, all the symplectic involutions on a split algebra of degree 2n are conju-

gate, hence (??) and (??) yield bijections of pointed sets

(29.22)

F -isomorphism classes of

H 1 F, PGSp(A, σ)

central simple F -algebras of degree 2n ←’

with symplectic involution

conjugacy classes of

H 1 F, GSp(A, σ) .

symplectic involutions

(29.23) ←’

on A

The exact sequence

1 ’ Gm ’ GSp(A, σ) ’ PGSp(A, σ) ’ 1

yields a connecting map in cohomology

δ 1 : H 1 F, PGSp(A, σ) ’ H 2 (F, Gm ) = Br(F ).

The commutative diagram

1 ’’ Gm ’’ GSp(A, σ) ’’ PGSp(A, σ) ’’ 1

¦ ¦

¦ ¦

1 ’’ Gm ’’ GL1 (A) ’’ PGL1 (A) ’’ 1

and Proposition (??) show that δ 1 maps the class of (A , σ ) to the Brauer class

[A ] · [A]’1 .

We now consider the group of isometries Sp(A, σ). Our ¬rst goal is to describe

the set Sym(A, σ) . By identifying Asep = M2n (Fsep ), we have σsep = Int(u) —¦ t for

some unit u ∈ Alt M2n (Fsep ), t , where t is the transpose involution. For x ∈ Asep ,

we have

x + σ(x) = xu ’ (xu)t u’1 .

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 405

If x + σ(x) is invertible, then xu ’ (xu)t is an invertible alternating matrix. Since

all the nonsingular alternating forms of dimension 2n are isometric, we may ¬nd

a ∈ GL2n (Fsep ) such that xu ’ (xu)t = auat . Then

x + σ(x) = a · σ(a),

proving that every unit in Symd(A, σ) is in Sym(A, σ) . On the other hand, since

σ is symplectic we have 1 = y + σ(y) for some y ∈ A, hence for all a ∈ Asep

a · σ(a) = a y + σ(y) σ(a) = ay + σ(ay).

Therefore, Sym(A, σ) is the set of all symmetrized units in A, i.e.,

Sym(A, σ) = Symd(A, σ)— ,

and (??) yields a bijection of pointed sets

Symd(A, σ)— /∼ H 1 F, Sp(A, σ) .

(29.24) ←’

(29.25) Example. Let a be a nonsingular alternating bilinear form on an F -

vector space V . To simplify notation, write GSp(V, a) for GSp EndF (V ), σa

and Sp(V, a) for Sp EndF (V ), σa . Since all the nonsingular alternating bilinear

forms on V are isometric to a, we have

H 1 F, GSp(V, a) = H 1 F, Sp(V, a) = 1.

Orthogonal involutions. Let A be a central simple F -algebra of degree n

with an orthogonal involution σ. We have Aut(A, σ) = PGO(A, σ). Assume

that char F = 2 or that F is perfect of characteristic 2. Then Fsep is quadrati-

cally closed, hence all the nonsingular symmetric nonalternating bilinear forms of

dimension n over Fsep are isometric. Therefore, all the orthogonal involutions on

Asep Mn (Fsep ) are conjugate, and the following bijections of pointed sets readily

follow from (??) and (??):

F -isomorphism classes of

H 1 F, PGO(A, σ)

central simple F -algebras of degree n ←’

with orthogonal involution

conjugacy classes of

H 1 F, GO(A, σ) .

orthogonal involutions ←’

on A

The same arguments as in the case of symplectic involutions show that the

connecting map

δ 1 : H 1 F, PGO(A, σ) ’ H 2 (F, Gm ) = Br(F )

in the cohomology sequence arising from the exact sequence

1 ’ Gm ’ GO(A, σ) ’ PGO(A, σ) ’ 1

takes the class of (A , σ ) to the Brauer class [A ] · [A]’1 .

In order to give a description of H 1 F, O(A, σ) , we next determine the set

Sym(A, σ) . We still assume that char F = 2 or that F is perfect. By identifying

Asep = Mn (Fsep ), we have σsep = Int(u) —¦ t for some symmetric nonalternating

matrix u ∈ GLn (Fsep ). For s ∈ Sym(A, σ), we have su ∈ Sym Mn (Fsep ), t . If

su = x ’ xt for some x ∈ Mn (Fsep ), then s = xu’1 ’ σ(xu’1 ). Therefore, su is not

alternating if s ∈ Alt(A, σ). Since all the nonsingular symmetric nonalternating

/

406 VII. GALOIS COHOMOLOGY

bilinear forms of dimension n over Fsep are isometric, we then have su = vuv t for

some v ∈ GLn (Fsep ), hence s = vσ(v). This proves

Sym(A, σ) ‚ Sym(A, σ)— Alt(A, σ).

To prove the reverse inclusion, observe that if aσ(a) = x ’ σ(x) for some a ∈

GLn (Fsep ), then

1 = a’1 xσ(a)’1 ’ σ a’1 xσ(a)’1 ∈ Alt(Asep , σ).

This is impossible since σ is orthogonal (see (??)).

By (??), we have a bijection of pointed sets

Sym(A, σ)— H 1 F, O(A, σ)

Alt(A, σ) /∼ ←’

where the base point in the left set is the equivalence class of 1. Of course, if

char F = 2, then Sym(A, σ) © Alt(A, σ) = {0} hence the bijection above takes the

form

Sym(A, σ)— /∼ H 1 F, O(A, σ) .

(29.26) ←’

Assuming char F = 2, let

SSym(A, σ)— = { (s, z) ∈ Sym(A, σ)— — F — | NrdA (s) = z 2 }

and de¬ne an equivalence relation ≈ on this set by

(s, z) ≈ (s , z ) if and only if s = a · s · σ(a) and z = NrdA (a)z for some a ∈ A— .

The same arguments as in the proof of (??) yield a canonical bijection of pointed

sets

H 1 F, O+ (A, σ) .

SSym(A, σ)— /≈

(29.27) ←’

29.E. Quadratic spaces. Let (V, q) be a nonsingular quadratic space of di-

mension n over an arbitrary ¬eld F . Let W = S 2 (V — ), the second symmetric power

of the dual space of V . Consider the representation

ρ : G = GL(V ) ’ GL(W )

de¬ned by

ρ(±)(f )(x) = f ±’1 (x)

for ± ∈ G, f ∈ W and x ∈ V (viewing S 2 (V — ) as a space of polynomial maps on

V ). The group scheme AutG (q) is the orthogonal group O(V, q).

We postpone until the end of this subsection the discussion of the case where

n is odd and char F = 2. Assume thus that n is even or that char F = 2. Then,

all the nonsingular quadratic spaces of dimension n over Fsep are isometric, hence

Proposition (??) yields a canonical bijection

isometry classes of

H 1 F, O(V, q) .

n-dimensional nonsingular

(29.28) ←’

quadratic spaces over F

To describe the pointed set H 1 F, O+ (V, q) , we ¬rst give another description

of H 1 F, O(V, q) . Consider the representation

ρ : G = GL(V ) — GL C(V, q) ’ GL(W )

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 407

where

W = S 2 (V — ) • HomF V, C(V, q) • HomF C(V, q) — C(V, q), C(V, q)

and

ρ(±, β)(f, g, h) = f ±’1 , βg±’1 , ±h(±’1 — ±’1 ) .

Set w = (q, i, m), where i : V ’ C(V, q) is the canonical map and m is the mul-

tiplication of C(V, q). Then, we obviously also have Aut G (w) = O(V, q) since

automorphisms of (C, q) which map V to V are in O(V, q). If n is even, let Z be

the center of the even Cli¬ord algebra C0 (V, q); if n is odd, let Z be the center of

the full Cli¬ord algebra C(V, q). The group G also acts on

W + = W • EndF (Z)

where the action of G on EndF (Z) is given by ρ(±, β)(j) = jβ|’1 . If we set

Z

w+ = (w, IdZ ), with w as above, we obtain AutG (w+ ) = O+ (V, q). By Proposi-

∼

tion (??) the set H 1 F, O+ (V, q) classi¬es triples (V , q , •) where • : Z ’ Z is

’

an isomorphism from the center of C(V , q ) to Z. We claim that in fact we have a

bijection

isometry classes of

n-dimensional nonsingular

H 1 F, O+ (V, q) .

(29.29) ←’

quadratic spaces (V , q ) over F

such that disc q = disc q

Since the F -algebra Z is determined up to isomorphism by disc q (see (??) when

n is even), the set on the left corresponds to the image of H 1 F, O+ (V, q) in

H 1 F, O(V, q) . Hence we have to show that the canonical map

H 1 F, O+ (V, q) ’ H 1 F, O(V, q)

is injective. If char F = 2 (and n is even), consider the exact sequence

∆

1 ’ O+ (V, q) ’ O(V, q) ’ Z/2Z ’ 0

(29.30) ’

where ∆ is the Dickson invariant, and the induced cohomology sequence

O(V, q) ’ Z/2Z ’ H 1 F, O+ (V, q) ’ H 1 F, O(V, q) .

Since ∆ : O(V, q) ’ Z/2Z is surjective, we have the needed injectivity at the base

point. To get injectivity at a class [x], we twist the sequence (??) by a cocycle x

representing [x] = [(V , q )]; then [x] is the new base point and the claim follows

from O(V, q)x = O(V , q ).

If char F = 2 (regardless of the parity of n), the arguments are the same,

substituting for (??) the exact sequence

det

1 ’ O+ (V, q) ’ O(V, q) ’’ µ2 ’ 1.

’

We now turn to the case where char F = 2 and n is odd, which was put

aside for the preceding discussion. Nonsingular quadratic spaces of dimension n

become isometric over Fsep if and only if they have the same discriminant, hence

408 VII. GALOIS COHOMOLOGY

Proposition (??) yields a bijection

isometry classes of

n-dimensional nonsingular

H 1 F, O(V, q) .

←’

quadratic spaces (V , q ) over F

such that disc q = disc q

The description of H 1 F, O+ (V, q) by triples (V , q , •) where • : Z ’ Z is an

isomorphism of the centers of the full Cli¬ord algebras C(V , q ), C(V, q) still holds,

√ √

but in this case Z = F ( disc q ), Z = F ( disc q) are purely inseparable quadratic

F -algebras, hence the isomorphism • : Z ’ Z is unique when it exists, i.e., when

disc q = disc q. Therefore, we have

H 1 F, O+ (V, q) = H 1 F, O(V, q) .

This equality also follows from the fact that O+ (V, q) is the smooth algebraic group

associated to O(V, q), hence the groups of points of O+ (V, q) and of O(V, q) over

Fsep (as over any reduced F -algebra) coincide.

29.F. Quadratic pairs. Let A be a central simple F -algebra of degree 2n

with a quadratic pair (σ, f ). Consider the representation already used in the proof

of (??):

ρ : G = GL(A) — GL Sym(A, σ) ’ GL(W )

where

W = HomF Sym(A, σ), A • HomF (A —F A, A) • EndF (A) • Sym(A, σ)— ,

with ρ given by

ρ(g, h)(», ψ, •, p) = g —¦ » —¦ h’1 , g(ψ), g —¦ • —¦ g ’1 , p —¦ h

where g(ψ) arises from the natural action of GL(A) on HomF (A — A, A). Consider

also the element w = (i, m, σ, f ) ∈ W where i : Sym(A, σ) ’ A is the inclusion.

The group AutG (w) coincides with PGO(A, σ, f ) (see §??).

Every twisted ρ-form (», ψ, •, p) of w de¬nes a central simple F -algebra with

quadratic pair (A , σ , f ) as follows: on the set A = { x | x ∈ A }, we de¬ne

the multiplication by x y = ψ(x — y) and the involution by σ (x ) = •(x) .

Then Sym(A , σ ) = { »(s) | s ∈ Sym(A, σ) }, and we de¬ne f by the condition

f »(s) = p(s) for s ∈ Sym(A, σ).

Conversely, to every central simple F -algebra with quadratic pair (A , σ , f ) of

degree 2n, we associate an element (», ψ, •, p) ∈ W as follows: we choose arbitrary

∼ ∼

bijective F -linear maps ν : A ’ A and » : Sym(A, σ) ’ ν ’1 Sym(A , σ ) , and

’ ’

de¬ne ψ, •, p by

ψ(x — y) = ν ’1 ν(x)ν(y) for x, y ∈ A,

• = ν ’1 —¦ σ —¦ ν and p = f —¦ ν —¦ ».

Over Fsep , all the algebras with quadratic pairs of degree 2n become isomorphic

to the split algebra with the quadratic pair associated to the hyperbolic quadratic

form. If

∼

θ : (Asep , σsep , fsep ) ’ (Asep , σsep , fsep )

’

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 409

is an isomorphism, then we let g = ν ’1 —¦ θ ∈ GL(Asep ) and de¬ne

h : Sym(Asep , σsep ) ’ Asep by ν —¦ » —¦ h = θ —¦ i.

Then ρ(g, h)(i, m, σ, f ) = (», ψ, •, p), proving that (», ψ, •, p) is a twisted ρ-form of

w. Thus, twisted ρ-forms of w are in one-to-one correspondence with isomorphism

classes of central simple F -algebras with quadratic pair of degree 2n. By (??) there

is a canonical bijection

F -isomorphism classes of

H 1 F, PGO(A, σ, f ) .

central simple F -algebras with ←’

quadratic pair of degree 2n

The center Z of the Cli¬ord algebra C(A, σ, f ) is a quadratic ´tale F -algebra

e

which we call the discriminant quadratic extension. The class [Z] of Z in H 1 (F, S2 )

is the discriminant class of (σ, f ). We have an exact sequence of group schemes

d

1 ’ PGO+ (A, σ, f ) ’ PGO(A, σ, f ) ’ S2 ’ 1

’

where d is the natural homomorphism

PGO(A, σ, f ) ’ Autalg (Z) S2 .

Thus the map

H 1 F, PGO(A, σ, f ) ’ H 1 (F, S2 )

takes (A , σ , f ) to [Z ] ’ [Z] where Z is the center of C(A , σ , f ). As in (??), we

obtain a natural bijection

F -isomorphism classes of 4-tuples (A , σ , f , •)

with a central simple F -algebra A

←’ H 1 F, PGO+ (A, σ, f ) .

of degree 2n, a quadratic pair (σ , f )

and an F -algebra isomorphism • : Z ’ Z of

the centers of the Cli¬ord algebras

(29.31) Remark. In particular, if Z F — F , the choice of • amounts to a

designation of the two components C+ (A, σ, f ), C’ (A, σ, f ) of C(A, σ, f ).

In order to obtain similar descriptions for the cohomology sets of GO(A, σ, f )

and GO+ (A, σ, f ), it su¬ces to let GL1 (A) — GL Sym(A, σ) act on W via ρ and

the map GL1 (A) ’ GL(A) which takes x ∈ A— to Int(x). As in (??), we obtain

bijections

conjugacy classes of

H 1 F, GO(A, σ, f ) ,

←’

quadratic pairs on A

conjugacy classes of triples (σ , f , •)

where (σ , f ) is a quadratic pair on A

H 1 F, GO+ (A, σ, f ) .

←’

and • : Z ’ Z is an isomorphism of

the centers of the Cli¬ord algebras

If char F = 2, the quadratic pair (σ, f ) is completely determined by the orthog-

onal involution σ, hence O(A, σ, f ) = O(A, σ) and we refer to §?? for a description

of H 1 F, O(A, σ) and H 1 F, O+ (A, σ) . For the rest of this subsection, we assume

410 VII. GALOIS COHOMOLOGY

char F = 2. As a preparation for the description of H 1 F, O(A, σ, f ) , we make an

observation on involutions on algebras over separably closed ¬elds.

(29.32) Lemma. Suppose that char F = 2. Let σ be an involution of the ¬rst kind

on A = M2n (Fsep ). For all a, b ∈ A such that a + σ(a) and b + σ(b) are invertible,

there exists g ∈ A— and x ∈ A such that

b = gaσ(g) + x + σ(x).

’1 ’1

Moreover, if Srd a + σ(a) a = Srd b + σ(b) b , then we may assume

’1

Trd b + σ(b) x = 0.

Proof : Let u ∈ A— satisfy σ = Int(u) —¦ t, where t is the transpose involution. Then

u = ut and a + σ(a) = u u’1 a + (u’1 a)t , hence u’1 a + (u’1 a)t is invertible.

Therefore, the quadratic form q(X) = Xu’1 aX t , where X = (x1 , . . . , x2n ), is

nonsingular. Similarly, the quadratic form Xu’1 bX t is nonsingular. Since all

the nonsingular quadratic forms of dimension 2n over a separably closed ¬eld are

isometric, we may ¬nd g0 ∈ A— such that

u’1 b ≡ g0 u’1 ag0

t

mod Alt(A, σ)

hence

b ≡ g1 aσ(g1 ) mod Alt(A, σ)

for g1 = ug0 u’1 . This proves the ¬rst part.

To prove the second part, choose g1 ∈ A— as above and x1 ∈ A such that

b = g1 aσ(g1 ) + x1 + σ(x1 ).

Let g0 = u’1 g1 u and x0 = u’1 x1 , so that

u’1 b = g0 u’1 ag0 + x0 + xt .

t

(29.33) 0

Let also v = u’1 b + (u’1 b)t . We have b + σ(b) = uv and, by the preceding equation,

v = g0 u’1 a + (u’1 a)t g0 .

t

From (??), we derive

s2 (v ’1 u’1 b) = s2 (v ’1 g0 u’1 ag0 ) + „˜ tr(v ’1 x0 ) .

t

(29.34)

’1

b . On the other hand, v ’1 g0 u’1 ag0 is conjugate

t

The left side is Srd b + σ(b)

to

’1

u’1 a + (u’1 a)t u’1 a = a + σ(a) a.

’1 ’1

a , equation (??) yields „˜ tr(v ’1 x0 ) =

Therefore, if Srd b+σ(b) b = Srd a+σ(a)

0, hence tr(v ’1 x0 ) = 0 or 1. In the former case, we are ¬nished since v ’1 x0 =

’1

b + σ(b) x1 . In the latter case, let g2 be an improper isometry of the quadratic

form Xg0 u ag0 X t (for instance a hyperplane re¬‚ection, see (??)). We have

’1 t

g0 u’1 ag0 = g2 g0 u’1 ag0 g2 + x2 + xt

t tt

2

for some x2 ∈ A such that tr(v ’1 x2 ) = 1, by de¬nition of the Dickson invariant