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injective, and we may identify H 1 F, SU(V, h) to a set of isometry classes of

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

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