“

module homomorphism. Its kernel can be identi¬ed to M[K] by mapping m ∈ M[K]

to the map i(m) which carries the trivial coset to m and the nontrivial coset to

’m. Thus, we have an exact sequence of “-modules

i µ

0 ’ M[K] ’ Ind“0 M ’ M ’ 0.

’ ’

“

For ≥ 0, Shapiro™s lemma yields a canonical isomorphism H (F, Ind“0 M ) =

“

H (K, M ). Under this isomorphism, the map induced by i (resp. µ) is the restric-

tion (resp. corestriction) homomorphism (see Brown [?, p. 81]). The cohomology

sequence associated to the sequence above therefore takes the form

δ0

res cor

0 ’ H 0 (F, M[K] ) ’’ H 0 (K, M ) ’’ H 0 (F, M ) ’ . . .

’ ’ ’

...

(30.10)

’1

δ res cor δ

. . . ’ ’ H (F, M[K] ) ’’ H (K, M ) ’’ H (F, M ) ’ . . . .

’’ ’ ’ ’

By substituting M[K] for M in this sequence, we obtain the exact sequence

δ0

res cor

0 ’ H 0 (F, M ) ’’ H 0 (K, M ) ’’ H 0 (F, M[K] ) ’ . . .

’ ’ ’

...

’1

δ res cor δ

. . . ’ ’ H (F, M ) ’’ H (K, M ) ’’ H (F, M[K] ) ’ . . .

’’ ’ ’ ’

since M[K][K] = M by (??) and since M = M[K] as “0 -module.

(30.11) Proposition. Assume K is a ¬eld. Then

H 1 (F, Z[K] ) Z/2Z.

Moreover, for all ≥ 0, the connecting map δ : H (F, M ) ’ H +1 (F, M[K] ) in

(??) is the cup product with the nontrivial element ζK of H 1 (F, Z[K] ), i.e.,

δ (ξ) = ζK ∪ ξ for ξ ∈ H (F, M ).

418 VII. GALOIS COHOMOLOGY

Proof : The ¬rst part follows from the long exact sequence (??) with M = Z. The

second part is veri¬ed by an explicit cochain calculation.

A cocycle representing the nontrivial element ζK ∈ H 1 (F, Z[K] ) is given by the

map

0 if σ ∈ “0 ,

σ’

1 if σ ∈ “ “0 .

Therefore, the map H 1 (F, Z[K] ) ’ H 1 (F, Z/2Z) = H 1 (F, S2 ) induced by reduction

modulo 2 carries ζK to the cocycle associated to K.

(30.12) Corollary. Assume K is a ¬eld. Write [K] for the cocycle in H 1 (F, S2 )

associated to K.

(1) For any “-module M such that 2M = 0, there is a long exact sequence

[K]∪

res cor

0 ’ H 0 (F, M ) ’’ H 0 (K, M ) ’’ H 0 (F, M ) ’ ’ . . .

’ ’ ’’

...

[K]∪ [K]∪

res cor

. . . ’ ’ H (F, M ) ’’ H (K, M ) ’’ H (F, M ) ’ ’ . . . .

’’ ’ ’ ’’

(2) Suppose M is a “-module for which multiplication by 2 is an isomorphism. For

all ≥ 0, there is a split exact sequence

res cor

0 ’ H (F, M[K] ) ’’ H (K, M ) ’’ H (F, M ) ’ 0.

’ ’

1 1

cor: H (K, M ) ’ H (F, M[K] ) and res: H (F, M ) ’

The splitting maps are 2 2

H (K, M ).

Proof : (??) follows from (??) and the description of ζK above.

(??) follows from cor —¦ res = [K : F ] = 2 (see Brown [?, Chapter 3, Proposi-

tion (9.5)]), since multiplication by 2 is an isomorphism.

For the sequel, the case where M = µn (Fsep ) is particularly relevant. The

“-module µn (Fsep )[K] can be viewed as the module of Fsep -points of the group

scheme

NK/F

µn[K] = ker RK/F (µn,K ) ’ ’ ’ µn .

’’

We next give an explicit description of the group H 1 (F, µn[K] ).

(30.13) Proposition. Let F be an arbitrary ¬eld and let n be an integer which is

not divisible by char F . For any ´tale quadratic F -algebra K, there is a canonical

e

isomorphism

{ (x, y) ∈ F — — K — | xn = NK/F (y) }

1

H (F, µn[K] ) .

{ (NK/F (z), z n ) | z ∈ K — }

Proof : Assume ¬rst K = F — F . Then H 1 (F, µn[K] ) = H 1 (F, µn ) F — /F —n . On

the other hand, the map x, (y1 , y2 ) ’ y2 induces an isomorphism from the factor

group on the right side to F — /F —n , and the proof is complete.

Assume next that K is a ¬eld. De¬ne a group scheme T over F as the kernel

of the map Gm —RK/F (Gm,K ) ’ Gm given by (x, y) ’ xn NK/F (y)’1 and de¬ne

θ : RK/F (Gm,K ) ’ T

§30. GALOIS COHOMOLOGY OF ROOTS OF UNITY 419

by θ(z) = NK/F (z), z n . The kernel of θ is µn[K] , and we have an exact sequence

θ

1 ’ µn[K] ’ RK/F (Gm,K ) ’ T ’ 1.

’

By Hilbert™s Theorem 90 and Shapiro™s lemma (??) and (??), we have

H 1 F, RK/F (Gm,K ) = 1,

hence the induced cohomology sequence yields an exact sequence

θ

K — ’ T(F ) ’ H 1 (F, µn[K] ) ’ 1.

’

If K is a ¬eld with absolute Galois group “0 ‚ “, an explicit description of the

isomorphism

{ (x, y) ∈ F — — K — | xn = NK/F (y) } ∼

’ H 1 (F, µn[K] )

’

n | z ∈ K— }

{ NK/F (z), z

is given as follows: for (x, y) ∈ F — — K — such that xn = NK/F (y), choose ξ ∈ Fsep

—

such that ξ n = y. A cocycle representing the image of (x, y) in H 1 (F, µn[K] ) is

given by

σ(ξ)ξ ’1 if σ ∈ “0 ,

σ’ ’1

x σ(ξ)ξ if σ ∈ “ “0 .

Similarly, if K = F — F , the isomorphism

∼

F — /F —n ’ H 1 (F, µn )

’

associates to x ∈ F — the cohomology class of the cocycle σ ’ σ(ξ)ξ ’1 , where

ξ ∈ Fsep is such that ξ n = x.

—

(30.14) Corollary. Suppose K is a quadratic separable ¬eld extension of F . Let

K 1 = { x ∈ K — | NK/F (x) = 1 }. For every odd integer n which is not divisible by

char F , there is a canonical isomorphism

H 1 (F, µn[K] ) K 1 /(K 1 )n .

Proof : For (x, y) ∈ F — — K — such that xn = NK/F (y), let

n’1

y ’1 )n ∈ K 1 .

ψ(x, y) = y · (x 2

A computation shows that ψ induces an isomorphism

{ (x, y) ∈ F — — K — | xn = NK/F (y) } ∼

’ K 1 /(K 1 )n .

’

{ (NK/F (z), z n ) | z ∈ K — }

The corollary follows from (??). (An alternate proof can be derived from (??).)

Finally, we use Corollary (??) to relate H 2 (F, µn[K] ) to central simple F -

algebras with unitary involution with center K.

(30.15) Proposition. Suppose K is a quadratic separable ¬eld extension of F .

Let n be an odd integer which is not divisible by char F . There is a natural bijection

between the group H 2 (F, µn[K] ) and the set of Brauer classes of central simple K-

algebras of exponent dividing n which can be endowed with a unitary involution

whose restriction to F is the identity.

420 VII. GALOIS COHOMOLOGY

Proof : The norm map NK/F : n Br(K) ’ n Br(F ) corresponds to the corestriction

map cor: H 2 (K, µn ) ’ H 2 (F, µn ) under any of the canonical (opposite) identi¬ca-

tions n Br(K) = H 2 (K, µn ) (see Riehm [?]). Therefore, by Theorem (??), Brauer

classes of central simple K-algebras which can be endowed with a unitary involu-

tion whose restriction to F is the identity are in one-to-one correspondence with

the kernel of the corestriction map. Since n is odd, Corollary (??) shows that this

kernel can be identi¬ed with H 2 (F, µn[K] ).

(30.16) De¬nition. Let K be a quadratic separable ¬eld extension of F and let n

be an odd integer which is not divisible by char F . For any central simple F -algebra

with unitary involution (B, „ ) of degree n with center K, we denote by g2 (B, „ ) the

cohomology class in H 2 (F, µn[K] ) corresponding to the Brauer class of B under the

bijection of the proposition above, identifying n Br(K) to H 2 (K, µn ) by the crossed

product construction. This cohomology class is given by

1

cor[B] ∈ H 2 (F, µn[K] ),

g2 (B, „ ) = 2

where cor: H 2 (K, µn ) ’ H 2 (F, µn[K] ) is the corestriction32 map; it is uniquely

determined by the condition

res g2 (B, „ ) = [B] ∈ H 2 (K, µn ),

where res: H 2 (F, µn[K] ) ’ H 2 (K, µn ) is the restriction map.

From the de¬nition, it is clear that for (B, „ ), (B , „ ) central simple F -algebras

with unitary involution of degree n with the same center K, we have

g2 (B, „ ) = g2 (B , „ ) if and only if B B as K-algebras.

Thus, g2 (B, „ ) does not yield any information on the involution „ .

Note that g2 (B, „ ) is the opposite of the Tits class t(B, „ ) de¬ned in a more

general situation in (??), because we are using here the identi¬cation H 2 (K, µn ) =

n Br(K) a¬orded by the crossed product construction instead of the identi¬cation

given by the connecting map of (??).

If the center K of B is not a ¬eld, then (B, „ ) (E — E op , µ) for some central

simple F -algebra E of degree n, where µ is the exchange involution. In this case,

we de¬ne a class g2 (B, „ ) ∈ H 2 (F, µn[K] ) = H 2 (F, µn ) by

g2 (B, „ ) = [E],

the cohomology class corresponding to the Brauer class of E by the crossed product

construction.

30.C. Cohomological invariants of algebras of degree three. As a ¬rst

illustration of Galois cohomology techniques, we combine the preceding results with

those of Chapter ?? to obtain cohomological invariants for cubic ´tale algebras and

e

for central simple algebras with unitary involution of degree 3. Cohomological

invariants will be discussed from a more general viewpoint in §??.

32 Bycontrast, observe that [B] is in the kernel of the corestriction map cor : H 2 (K, n) ’

H 2 (F, n ), by Theorem (??).

§30. GALOIS COHOMOLOGY OF ROOTS OF UNITY 421

´

Etale algebras of degree 3. Cubic ´tale algebras, i.e., ´tale algebras of di-

e e

1

mension 3, are classi¬ed by H (F, S3 ) (see (??)). Let L be a cubic ´tale F -algebra

e

and let φ : “ ’ S3 be a cocycle de¬ning L. Since the “-action on S3 is trivial, the

map φ is a homomorphism which is uniquely determined by L up to conjugation.

We say that L is of type i S3 (for i = 1, 2, 3 or 6) if im φ ‚ S3 is a subgroup of

order i. Thus,

L is of type 1 S3 if and only if L F — F — F,

L is of type 2 S3 if and only if L F —K for some quadratic separable

¬eld extension K of F ,

L is of type 3 S3 if and only if L is a cyclic ¬eld extension of F ,

L is of type 6 S3 if and only if L is a ¬eld extension of F which is not Galois.

Let A3 be the alternating group on 3 elements. The group S3 is the semidirect

product A3 S2 , so the exact sequence

sgn

i

(30.17) 1 ’ A3 ’ S3 ’ ’ S 2 ’ 1

’ ’

is split. In the induced sequence in cohomology

sgn1

i1

1 ’ H 1 (F, A3 ) ’ H 1 (F, S3 ) ’ ’ H 1 (F, S2 ) ’ 1

’ ’’

the map sgn1 associates to an algebra L its discriminant algebra ∆(L). The (unique)

section of sgn1 is given by [K] ’ [F — K], for any quadratic ´tale algebra K. The

e

1

set H (F, A3 ) classi¬es Galois A3 -algebras (see (??)); it follows from the sequence

above (by an argument similar to the one used for H 1 F, AutZ (B) in (??)) that

they can as well be viewed as pairs (L, ψ) where L is cubic ´tale over F and ψ is an

e

∼ 1

isomorphism ∆(L) ’ F —F . The group S2 acts on H (F, A3 ) by (L, φ) ’ (L, ι—¦φ)

’

where ι is the exchange map. Let K be a quadratic ´tale F -algebra. We use the

e

associated cocycle “ ’ S2 to twist the action of “ on the sequence (??). In the

corresponding sequence in cohomology:

sgn1

i1

1 ’ H 1 (F, A3 [K] ) ’ H 1 (F, S3 [K] ) ’ ’ H 1 (F, S2 ) ’ 1

’ ’’

the distinguished element of H 1 (F, S3 [K] ) is the class of F — K and the pointed

∼

set H 1 (F, A3 [K] ) classi¬es pairs (L, ψ) with ψ an isomorphism ∆(L) ’ K. Note

’

that, again, S2 operates on H 1 (F, A3 [K] ). We now de¬ne two cohomological invari-

ants for cubic ´tale F -algebras: f1 (L) ∈ H 1 (F, S2 ) is the class of the discriminant

e

algebra ∆(L) of L, and g1 (L) is the class of L in the orbit space H 1 F, A3 [∆(L)] /S2 .

(30.18) Proposition. Cubic ´tale algebras are classi¬ed by the cohomological in-

e

variants f1 (L) and g1 (L). In particular L is of type 1 S3 if g1 (L) = 0 and f1 (L) = 0,

of type 2 S3 if g1 (L) = 0 and f1 (L) = 0, of type 3 S3 if g1 (L) = 0 and f1 (L) = 0,

and of type 6 S3 if g1 (L) = 0 and f1 (L) = 0.

Proof : The fact that cubic ´tale algebras are classi¬ed by the cohomological invari-

e

ants f1 (L) and g1 (L) follows from the exact sequence above. In particular L is a

¬eld if and only if g1 (L) = 0 and is a cyclic algebra if and only if f1 (L) = 0.

Let F be a ¬eld of characteristic not 3 and let F (ω) = F [X]/(X 2 +X +1), where

√

ω is the image of X in the factor ring. (Thus, F (ω) F ( ’3) if char F = 2.) We

have µ3 = A3 [F (ω)] so that H 1 (F, µ3 ) classi¬es pairs (L, ψ) where L is a cubic ´tale

e

∼

F -algebra and ψ is an isomorphism ∆(L) ’ F (ω). The action of S2 interchanges

’

422 VII. GALOIS COHOMOLOGY

the pairs (L, ψ) and (L, ιF (ω) —¦ ψ). In particular H 1 (F, µ3 ) modulo the action of S2

classi¬es cubic ´tale F -algebras with discriminant algebra F (ω).

e

(30.19) Proposition. Let K be√ quadratic ´tale F -algebra and let x ∈ K 1 be an

a e √

norm 1 in K. Let K( x) = K[t]/(t3 ’ x) and let ξ = 3 x be the image

element of √ 3

of t in K( 3 x). Extend the nontrivial automorphism ιK to an automorphism ι of

√ √

K( 3 x) by setting ι(ξ) = ξ ’1 . Then, the F -algebra L = { u ∈ K( 3 x) | ι(u) = u }

is a cubic ´tale F -algebra with discriminant algebra K — F (ω). Conversely, suppose

e

L is a cubic ´tale F -algebra with discriminant ∆(L), and let K = F (ω) — ∆(L).

e

Then, there exists x ∈ K 1 such that

√

L { u ∈ K( 3 x) | ι(u) = u }.

√

Proof : If x = ±1, then K( 3 x) K — K(ω) and L F — K — F (ω) , hence

√

the ¬rst assertion is clear. Suppose x = ±1. Since every element in K( 3 x) has a

unique expression of the form a + bξ + cξ ’1 with a, b, c ∈ K, it is easily seen that

L = F (·) with · = ξ + ξ ’1 . We have

· 3 ’ 3· = x + x’1

with x + x’1 = ±2, hence Proposition (??) shows that ∆(L) K — F (ω). This

completes the proof of the ¬rst assertion.

To prove the second assertion, we use the fact that cubic ´tale F -algebras

e

with discriminant ∆(L) are in one-to-one correspondence with the orbit space

H 1 (F, A3[∆(L)] )/S2 . For K = F (ω) — ∆(L), we have A3[∆(L)] = µ3[K] , hence Corol-

lary (??) yields a bijection H 1 (F, A3[∆(L)] ) K 1 /(K 1 )3 . If x ∈ K 1 corresponds to

√

the isomorphism class of L, then L { u ∈ K( 3 x) | ι(u) = u }.

Central simple algebras with unitary involution. To every central simple

F -algebra with unitary involution (B, „ ) we may associate the cocycle [K] of its

center K. We let

f1 (B, „ ) = [K] ∈ H 1 (F, S2 ).

Now, assume char F = 2, 3 and let (B, „ ) be a central simple F -algebra with

unitary involution of degree 3. Let K be the center of B. A secondary invariant

g2 (B, „ ) ∈ H 2 (F, µ3[K] ) is de¬ned in (??). The results in §?? show that g2 (B, „ )

has a special form:

(30.20) Proposition. For any central simple F -algebra with unitary involution

(B, „ ) of degree 3 with center K, there exist ± ∈ H 1 (F, Z/3Z[K] ) and β ∈ H 1 (F, µ3 )

such that

g2 (B, „ ) = ± ∪ β.

(E — E op , µ) for some

Proof : Suppose ¬rst that K F — F . Then (B, „ )

central simple F -algebra E of degree 3, where µ is the exchange involution, and

g2 (B, „ ) = [E]. Wedderburn™s theorem on central simple algebras of degree 3 (see

(??)) shows that E is cyclic, hence Proposition (??) yields the required decompo-

sition of g2 (B, „ ).

Suppose next that K is a ¬eld. Albert™s theorem on central simple algebras of

degree 3 with unitary involution (see (??)) shows that B contains a cubic ´tale F -

e

algebra L with discriminant isomorphic to K. By (??), we may ¬nd a corresponding

§31. COHOMOLOGICAL INVARIANTS 423

cocycle ± ∈ H 1 (F, Z/3Z[K] ) = H 1 (F, A3[K] ) (whose orbit under S2 is g1 (L)). The

K-algebra LK = L —F K is cyclic and splits B, hence by (??)

[B] = res(±) ∪ (b) ∈ H 2 (K, µ3 )

for some (b) ∈ H 1 (K, µ3 ), where res: H 1 (F, Z/3Z[K] ) ’ H 1 (K, Z/3Z) is the

restriction map. By taking the image of each side under the corestriction map

cor: H 2 (K, µ3 ) ’ H 2 (F, µ3[K] ), we obtain by the projection (or transfer) formula

(see Brown [?, (3.8), p. 112])

cor[B] = ± ∪ cor(b)

1

hence g2 (B, „ ) = ± ∪ β with β = 2 cor(b). (Here, cor(b) = NK/F (b) is the image

of (b) under the corestriction map cor: H 1 (K, µ3 ) ’ H 1 (F, µ3 ).)

The 3-fold P¬ster form π(„ ) of (??) yields a third cohomological invariant of

(B, „ ) via the map e3 of (??). We set

f3 (B, „ ) = e3 π(„ ) ∈ H 3 (F, µ2 ).

This is a Rost invariant in the sense of §??, see (??).

Since the form π(„ ) classi¬es the unitary involutions on B up to isomorphism

by (??), we have a complete set of cohomological invariants for central simple F -

algebras with unitary involution of degree 3:

(30.21) Theorem. Let F be a ¬eld of characteristic di¬erent from 2, 3. Triples

(B, K, „ ), where K is a quadratic separable ¬eld extension of F and (B, „ ) is a cen-

tral simple F -algebra with unitary involution of degree 3 with center K, are classi¬ed

over F by the three cohomological invariants f1 (B, „ ), g2 (B, „ ) and f3 (B, „ ).

§31. Cohomological Invariants

In this section, we show how cohomology can be used to de¬ne various canonical

maps and to attach invariants to algebraic groups. In §??, we use cohomology

sequences to relate multipliers and spinor norms to connecting homomorphisms.

We also de¬ne the Tits class of a simply connected semisimple group; it is an

invariant of the group in the second cohomology group of its center. In §??, we

take a systematic approach to the de¬nition of cohomological invariants of algebraic

groups and de¬ne invariants of dimension 3.

Unless explicitly mentioned, the base ¬eld F is arbitrary throughout this sec-

tion. However, when using the cohomology of µn , we will need to assume that

char F does not divide n.

31.A. Connecting homomorphisms. Let G be a simply connected semi-

simple group with center C and let G = G/C. The exact sequence

1’C’G’G’1

yields connecting maps in cohomology

δ 0 : H 0 (F, G) ’ H 1 (F, C) δ 1 : H 1 (F, G) ’ H 2 (F, C).

and

We will give an explicit description of δ 0 for each type of classical group and use

δ 1 to de¬ne the Tits class of G.

424 VII. GALOIS COHOMOLOGY

Unitary groups. Let (B, „ ) be a central simple F -algebra with unitary invo-

lution of degree n, and let K be the center of B. As observed in §??, we have an

exact sequence of group schemes

(31.1) 1 ’ µn[K] ’ SU(B, „ ) ’ PGU(B, „ ) ’ 1

since the kernel N of the norm map RK/F (µn,K ) ’ µn,F is µn[K] .

Suppose char F does not divide n, so that µn[K] is smooth. By Proposition (??),

we derive from (??) an exact sequence of Galois modules. The connecting map

δ 0 : PGU(B, „ ) ’ H 1 (F, µn[K] )

can be described as follows: for g ∈ GU(B, „ ),

δ 0 (g · K — ) = µ(g), NrdB (g)

where µ(g) = „ (g)g ∈ F — is the multiplier of g, and [x, y] is the image of (x, y) ∈

F — — K — in the factor group

{ (x, y) ∈ F — — K — | xn = NK/F (y) }

H 1 (F, µn[K] )

{ NK/F (z), z n | z ∈ K — }

(see (??)).

If K F — F , then (B, „ ) (A — Aop , µ) for some central simple F -algebra A

of degree n, where µ is the exchange involution. We have PGU(B, „ ) PGL1 (A)

and SU(B, „ ) SL1 (A), and the exact sequence (??) takes the form

1 ’ µn ’ SL1 (A) ’ PGL1 (A) ’ 1.

The connecting map δ 0 : PGL1 (A) ’ H 1 (F, µn ) = F — /F —n is given by the re-

duced norm map.

Orthogonal groups. Let (V, q) be a quadratic space of odd dimension. There

is an exact sequence of group schemes

1 ’ µ2 ’ Spin(V, q) ’ O+ (V, q) ’ 1

(see §??). If char F = 2, the connecting map δ 0 : O+ (V, q) ’ H 1 (F, µ2 ) = F — /F —2

is the spinor norm.

Let (A, σ, f ) be a central simple F -algebra of even degree 2n with quadratic

pair. The center C of the spin group Spin(A, σ, f ) is determined in §??: if Z is

the center of the Cli¬ord algebra C(A, σ, f ), we have

RZ/F (µ2 ) if n is even,

C=

if n is odd.

µ4[Z]

Therefore, we have exact sequences of group schemes

1 ’ RZ/F (µ2 ) ’ Spin(A, σ, f ) ’ PGO+ (A, σ, f ) ’ 1 if n is even,

1 ’ µ4[Z] ’ Spin(A, σ, f ) ’ PGO+ (A, σ, f ) ’ 1 if n is odd.

Suppose char F = 2. The connecting maps δ 0 in the associated cohomology

sequences are determined in §??. If n is even, the map

δ 0 : PGO+ (A, σ, f ) ’ H 1 F, RZ/F (µ2 ) = H 1 (Z, µ2 ) = Z — /Z —2

coincides with the map S of (??), see Proposition (??). If n is odd, the map

δ 0 : PGO+ (A, σ, f ) ’ H 1 (F, µ4[Z] )

§31. COHOMOLOGICAL INVARIANTS 425

is de¬ned in (??), see Proposition (??). Note that the discussion in §?? does not

use the hypothesis that char F = 2. This hypothesis is needed here because we

apply (??) to derive exact sequences of Galois modules from the exact sequences

of group schemes above.

Symplectic groups. Let (A, σ) be a central simple F -algebra with symplectic

involution. We have an exact sequence of group schemes

1 ’ µ2 ’ Sp(A, σ) ’ PGSp(A, σ) ’ 1

(see §??). If char F = 2, the connecting homomorphism

δ 0 : PGSp(A, σ) ’ H 1 (F, µ2 ) = F — /F —2

is induced by the multiplier map: it takes g · F — ∈ PGSp(A, σ) to µ(g) · F —2 .

The Tits class. Let G be a split simply connected or adjoint semisimple group

over F , let T ‚ G be a split maximal torus, Π a system of simple roots in the root

system of G with respect to T . By (??) and (??), the homomorphism

(31.2) Aut(G) ’ Aut(Dyn(G))

is a split surjection. A splitting i : Aut Dyn(G) ’ Aut(G) can be chosen in such a

way that any automorphism in the image of i leaves the torus T invariant. Assume

that the Galois group “ acts on the Dynkin diagram Dyn(G), or equivalently,

consider a continuous homomorphism

= H 1 F, Aut Dyn(G) .

• ∈ Hom “, Aut Dyn(G)

Denote by γ a cocycle representing the image of • in H 1 F, Aut(Gsep ) under the

map induced by the splitting i. Since γ normalizes T , the twisted group Gγ contains

the maximal torus Tγ . Moreover, the natural action of “ on Tγ leaves Π invariant,

hence Gγ is a quasisplit group. In fact, up to isomorphism Gγ is the unique simply

connected quasisplit group with Dynkin diagram Dyn(G) and with the given action

of “ on Dyn(G) (see (??)). Twisting G in (??) by γ, we obtain:

(31.3) Proposition. Let G be a quasisplit simply connected group. Then the nat-

ural homomorphism Aut(G) ’ Aut(Dyn(G)) is surjective.

Let G be semisimple group over F . By §?? a twisted form G of G corresponds

to an element ξ ∈ H 1 F, Aut(Gsep ) . We say that G is an inner form of G if ξ

belongs to the image of the map

±G : H 1 (F, G) ’’ H 1 F, Aut(Gsep )

induced by the homomorphism Int : G = G/C ’ Aut(G), where C is the center of

G. Since G acts trivially on C, the centers of G and G are isomorphic (as group

schemes of multiplicative type).

(31.4) Proposition. Any semisimple group is an inner twisted form of a unique

quasisplit group up to isomorphism.

Proof : Since the centers of inner twisted forms are isomorphic and all the groups

which are isogenous to a simply connected group correspond to subgroups in its

center, we may assume that the given group G is simply connected. Denote by

Gd the split twisted form of G, so that G corresponds to some element ρ ∈

H 1 F, Aut(Gd ) . Denote by γ ∈ H 1 F, Aut(Gd ) the image of ρ under the

sep sep

426 VII. GALOIS COHOMOLOGY

i

composition induced by Aut(Gd ) ’ Aut Dyn(Gd ) ’ Aut(Gd ), where i is the split-

ting considered above. We prove that the quasisplit group Gd is an inner twisted

γ

form of G = Gd . By (??), there is a bijection

ρ

∼

θρ : H 1 F, Aut(Gsep ) ’ H 1 F, Aut(Gd )

’ sep

taking the trivial cocycle to ρ. Denote by γ0 the element in H 1 F, Aut(Gsep ) such

that θ(γ0 ) = γ. Since ρ and γ have the same image in H 1 F, Aut Dyn(Gd ) , sep

1

the trivial cocycle and γ0 have the same images in H F, Aut Dyn(Gsep ) , hence

Theorem (??) shows that γ0 belongs to the image of H 1 (F, G) ’ H 1 F, Aut(Gsep ) ,

i.e., the group Gγ0 Gd is a quasisplit inner twisted form of G.

γ

Until the end of the subsection we shall assume that G is a simply connected

semisimple group. Denote by ξG the element in H 1 F, Aut(Gsep ) corresponding

to the (unique) quasisplit inner twisted form of G. In general, the map ±G is not

injective. Nevertheless, we have

(31.5) Proposition. There is only one element νG ∈ H 1 (F, G) such that

±G (νG ) = ξG .

Proof : Denote Gq the quasisplit inner twisted form of G. By (??), there is a

bijection between ±’1 (ξG ) and the factor group of Aut Dyn(Gq ) by the image of

G

Aut(Gq ) ’ Aut Dyn(Gq ) . But the latter map is surjective (see (??)).

Let C be the center of G. The exact sequence 1 ’ C ’ G ’ G ’ 1 induces

the connecting map

δ 1 : H 1 (F, G) ’’ H 2 (F, C).

The Tits class of G is the element tG = ’δ 1 (νG ) ∈ H 2 (F, C).

(31.6) Proposition. Let χ ∈ C — be a character. Denote by Fχ the ¬eld of de¬ni-

tion of χ and by Aχ its minimal Tits algebra. The image of the Tits class tG under

the composite map

χ—

res

H 2 (F, C) ’’ H 2 (Fχ , C) ’’ H 2 (Fχ , Gm ) = Br(Fχ )

’

is [Aχ ]. (We use the canonical identi¬cation H 2 (Fχ , Gm ) = Br(Fχ ) given by the

connecting map of (??), which is the opposite of the identi¬cation given by the

crossed product construction.)

Proof : There is a commutative diagram (see §??)

1 ’’ CFχ ’’ G Fχ ’’ G Fχ ’’ 1

¦ ¦ ¦

¦ ¦ ¦

χ

1 ’’ Gm ’’ GL1 (A) ’’ PGL1 (A) ’’ 1

where A = Aχ . Therefore, it su¬ces to prove that the image of res(νG ) under the

composite map

H 1 (Fχ , GFχ ) ’ H 1 Fχ , PGL1 (A) ’ H 2 (Fχ , Gm ) = Br(Fχ )

is ’[A]. The twist of the algebra A by a cocycle representing res(νG ) is the Tits

algebra of the quasisplit group (GνG )Fχ , hence it is trivial. Therefore the image

§31. COHOMOLOGICAL INVARIANTS 427

γ of res(νG ) in H 1 Fχ , PGL1 (A) corresponds to the split form PGLn (Fχ ) of

PGL1 (A), where n = deg(A). By (??), there is a commutative square

H 1 Fχ , PGLn (Fχ ) ’ ’ ’ Br(Fχ )

’’

¦ ¦

¦ ¦g

θγ

H 1 Fχ , PGL1 (A) ’ ’ ’ Br(Fχ )

’’

where g(v) = v + u and u is the image of γ in Br(Fχ ). Pick µ ∈ Z 1 Fχ , PGLn (Fχ )

such that the twisting of PGLn (Fχ ) by µ equals PGL1 (A). In other words, θγ (µ) =

1 and the image of µ in Br(Fχ ) equals [A]. The commutativity of the diagram then

implies that u = ’[A].

(31.7) Example. Let G = SU(B, „ ) where (B, „ ) is a central simple F -algebra

with unitary involution of degree n. Assume that char F does not divide n and let

K be the center of B. Since the center C of SU(B, „ ) is µn[K] , the Tits class tG

belongs to H 2 (F, µn[K] ). Abusing terminology, we call it the Tits class of (B, „ )

and denote it by t(B, „ ), i.e.,

t(B, „ ) = tSU(B,„ ) ∈ H 2 (F, µn[K] ).

Suppose K is a ¬eld. For χ = 1 + nZ ∈ Z/nZ = C — , the ¬eld of de¬nition of χ is

K and the minimal Tits algebra is B, see §??. Therefore, Proposition (??) yields

resK/F t(B, „ ) = [B] ∈ Br(K).

If n is odd, it follows from Corollary (??) that

1

t(B, „ ) = cor[B].

2

On the other hand, if n is even we may consider the character » = n + nZ ∈ C — of

2

raising to the power n . The corresponding minimal Tits algebra is the discriminant

2

algebra D(B, „ ), see §??. By (??) we obtain

»— t(B, „ ) = D(B, „ ) ∈ Br(F ).

If K F — F we have (B, „ ) = (A — Aop , µ) for some central simple F -algebra

A of degree n, where µ is the exchange involution. The commutative diagram with

exact rows

1 ’ ’ ’ µn ’ ’ ’ SLn ’ ’ ’ PGLn ’ ’ ’ 1

’’ ’’ ’’ ’’

¦ ¦

¦ ¦

1 ’ ’ ’ Gm ’ ’ ’ GLn ’ ’ ’ PGLn ’ ’ ’ 1

’’ ’’ ’’ ’’

shows that t(B, „ ) = [A] ∈ H 2 (F, µn ) = n Br(F ).

(31.8) Example. Let G = Spin(V, q), where (V, q) is a quadratic space of odd

dimension. Suppose char F = 2. The center of G is µ2 and the Tits class tG ∈

H 2 (F, µ2 ) = 2 Br(F ) is the Brauer class of the even Cli¬ord algebra C0 (V, q), since

the minimal Tits algebra for the nontrivial character is C0 (V, q), see §??.

(31.9) Example. Let G = Sp(A, σ), where (A, σ) is a central simple F -algebra

with symplectic involution. Suppose char F = 2. The center of G is µ2 and the

Tits class tG ∈ H 2 (F, µ2 ) = 2 Br(F ) is the Brauer class of the algebra A, see §??.

428 VII. GALOIS COHOMOLOGY

(31.10) Example. Let G = Spin(A, σ, f ), where (A, σ, f ) is a central simple F -

algebra of even degree 2n with quadratic pair. Let Z be the center of the Cli¬ord

algebra C(A, σ, f ) and assume that char F = 2.

Suppose ¬rst that n is even. Then the center of G is RZ/F (µ2 ), hence

tG ∈ H 2 F, RZ/F (µ2 ) = H 2 (Z, µ2 ) = 2 Br(Z).

The minimal Tits algebra corresponding to the norm character

» : RZ/F (µ2 ) ’ µ2 ’ Gm

is A, hence

corZ/F (tG ) = [A] ∈ H 2 (F, µ2 ) = 2 Br(F ).

(31.11)

On the other hand, the minimal Tits algebras for the two other nontrivial characters

»± are C(A, σ, f ) (see §??), hence

tG = C(A, σ, f ) ∈ H 2 (Z, µ2 ) = 2 Br(Z).

(31.12)

Now, assume that n is odd. Then the center of G is µ4[Z] , hence

tG ∈ H 2 (F, µ4[Z] ).

By applying Proposition (??), we can compute the image of tG under the squaring

map

»— : H 2 (F, µ4[Z] ) ’ H 2 (F, µ2 ) = 2 Br(F )

and under the restriction map

res : H 2 (F, µ4[Z] ) ’ H 2 (Z, µ4 ) = 4 Br(Z)

(or, equivalently, under the map H 2 (F, µ4[Z] ) ’ H 2 F, RZ/F (Gm,Z ) = Br(Z)

induced by the inclusion µ4[Z] ’ RZ/F (Gm,Z )). We obtain

»— (tG ) = [A] and res(tG ) = C(A, σ, f ) .

Note that the fundamental relations (??) of Cli¬ord algebras readily follow

from the computations above (under the hypothesis that char F = 2). If n is even,

2

(??) shows that C(A, σ, f ) = 1 in Br(Z) and (??) (together with (??)) implies

NZ/F C(A, σ, f ) = [A].

If n is odd we have

2

[AZ ] = res —¦»— (tG ) = res(tG )2 = C(A, σ, f )

and

NZ/F C(A, σ, f ) = cor —¦ res(tG ) = 0,

by (??). (See Exercise ?? for a cohomological proof of the fundamental relations

without restriction on char F .)

§31. COHOMOLOGICAL INVARIANTS 429

31.B. Cohomological invariants of algebraic groups. Let G be an alge-

braic group over a ¬eld F . For any ¬eld extension E of F we consider the pointed

set

H 1 (E, G) = H 1 (E, GE )

of GE -torsors over E. A homomorphism E ’ L of ¬elds over F induces a map of

pointed sets

H 1 (E, G) ’ H 1 (L, G).

Thus, H 1 (?, G) is a functor from the category of ¬eld extensions of F (with mor-

phisms being ¬eld homomorphisms over F ) to the category of pointed sets.

Let M be a torsion discrete Galois module over F , i.e., a discrete module over

the absolute Galois group “ = Gal(Fsep /F ). For a ¬eld extension E of F , M

can be endowed with a structure of a Galois module over E and hence the ordinary

cohomology groups H d (E, M ) are de¬ned. Thus, for any d ≥ 0, we obtain a functor

H d (?, M ) from the category of ¬eld extensions of F to the category of pointed sets

(actually, the category of abelian groups). A cohomological invariant of the group G

of dimension d with coe¬cients in M is a natural transformation of functors

a : H 1 (?, G) ’ H d (?, M ).

In other words, the cohomological invariant a assigns to any ¬eld extension E of F

a map of pointed sets

aE : H 1 (E, G) ’ H d (E, M ),

such that for any ¬eld F -homomorphism E ’ L the following diagram commutes

a

H 1 (E, G) ’ ’ ’ H d (E, M )

’E’

¦ ¦

¦ ¦

a

H 1 (L, G) ’ ’ ’ H d (L, M ).

’L’

The set Invd (G, M ) of all cohomological invariants of the group G of dimension d

with coe¬cients in M forms an abelian group in a natural way.

(31.13) Example. Let G = GL(V ) or SL(V ). By Hilbert™s Theorem 90 (see

(??) and (??)) we have H 1 (E, G) = 1 for any ¬eld extension L of F . Hence

Invd (G, M ) = 0 for any d and any Galois module M .

A group homomorphism ± : G ’ G over F induces a natural homomorphism

±— : Invd (G , M ) ’ Invd (G, M ).

A homomorphism of Galois modules g : M ’ M yields a group homomorphism

g— : Invd (G, M ) ’ Invd (G, M ).

For a ¬eld extension L of F there is a natural restriction homomorphism

res : Invd (G, M ) ’ Invd (GL , M ).

Let L be a ¬nite separable extension of F . If G is an algebraic group over L and M

is a Galois module over F , then the corestriction homomorphism for cohomology

groups and Shapiro™s lemma yield the corestriction homomorphism

cor: Invd (G, M ) ’ Invd RL/F (G), M .

430 VII. GALOIS COHOMOLOGY

Invariants of dimension 1. Let G be an algebraic group over a ¬eld F . As

in §??, write π0 (G) for the factor group of G modulo the connected component

of G. It is an ´tale group scheme over F .

e

Let M be a discrete Galois module over F and let

g : π0 (Gsep ) ’ M

be a “-homomorphism. For any ¬eld extension E of F we then have the following

composition

ag : H 1 (E, G) ’ H 1 E, π0 (G) ’ H 1 (E, M )

E

where the ¬rst map is induced by the canonical surjection G ’ π0 (G) and the

second one by g. We can view ag as an invariant of dimension 1 of the group G

with coe¬cients in M .

(31.14) Proposition. The map

Hom“ π0 (Gsep ), M ’ Inv1 (G, M ) given by g ’ ag

is an isomorphism.

In particular, a connected group has no nonzero invariants of dimension 1.

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

of degree 2n and let G = PGO(A, σ, f ) be the corresponding projective orthog-

onal group. The set H 1 (F, G) classi¬es triples (A , σ , f ) with a central simple

F -algebra A of degree 2n with an quadratic pair (σ , f ) (see §??). We have

π0 (G) Z/2Z and the group Inv1 (G, Z/2Z) is isomorphic to Z/2Z. The nontrivial

invariant

H 1 (F, G) ’ H 1 (F, Z/2Z)

associates to any triple (A , σ , f ) the sum [Z ] + [Z] of corresponding classes of the

discriminant quadratic extensions.

(31.16) Example. Let K be a quadratic ´tale F -algebra, let (B, „ ) be a central

e

simple K-algebra of degree n with a unitary involution and set G = Aut(B, „ ).

The set H 1 (F, G) classi¬es algebras of degree n with a unitary involution (see (??)).

Then π0 (G) Z/2Z and, as in the previous example, the group Inv1 (G, Z/2Z) is

isomorphic to Z/2Z. The nontrivial invariant

H 1 (F, G) ’ H 1 (F, Z/2Z)

associates to any central simple F -algebra with unitary involution (B , „ ) with

center K the class [K ] + [K].

Invariants of dimension 2. For any natural numbers i and n let µ—i (F ) ben

the i-th tensor power of the group µn (F ). If n divides m, there is a natural injection

µ—i (F ) ’ µ—i (F ).

n m

The groups µ—i (F ) form an injective system with respect to the family of injections

n

de¬ned above. We denote the direct limit of this system, for all n prime to the

characteristic of F , by Q/Z(i)(F ). For example, Q/Z(1)(F ) is the group of all

roots of unity in F .

The group Q/Z(i)(Fsep ) is endowed in a natural way with a structure of a

Galois module. We set

H d F, Q/Z(i) = H d F, Q/Z(i)(Fsep ) .

§31. COHOMOLOGICAL INVARIANTS 431

In the case where char F = p > 0, this group can be modi¬ed by adding an

appropriate p-component. In particular, the group H 2 F, Q/Z(1) is canonically

isomorphic to Br(F ), while before the modi¬cation it equals lim H 2 F, µn (Fsep )

’’

which is the subgroup of elements in Br(F ) of exponent prime to p.

Let G be a connected algebraic group over a ¬eld F . Assume that we are given

an exact sequence of algebraic groups

(31.17) 1 ’ Gm,F ’ G ’ G ’ 1.

For any ¬eld extension E of F , this sequence induces a connecting map H 1 (E, G) ’

H 2 (E, Gm,F ) which, when composed with the identi¬cations

H 2 (E, Gm,F ) = Br(E) = H 2 E, Q/Z(1) ,

provides an invariant aE of dimension 2 of the group G. On the other hand, the

sequence (??) de¬nes an element of the Picard group Pic(G). It turns out that the

invariant a depends only on the element of the Picard group and we have a well

de¬ned group homomorphism

β : Pic(G) ’ Inv2 G, Q/Z(1) .

(31.18) Proposition. The map β is an isomorphism.

Since the n-torsion part of Q/Z(1) equals µn , we have

(31.19) Corollary. If n is not divisible by char F , then

Inv2 G, µn (Fsep ) n Pic(G).

(31.20) Example. Let G be a semisimple algebraic group, let π : G ’ G be a

universal covering and set Z = ker(π). There is a natural isomorphism

∼ ∼

Z — ’ Pic(G) ’ Inv2 G, Q/Z(1) .

’ ’

Hence attached to each character χ ∈ Z — is an invariant which we denote by aχ .

The construction is as follows. Consider the group G = (G — Gm,F )/Z where Z is

embedded into the product canonically on the ¬rst factor and by the character χ

on the second. There is an exact sequence

1 ’ Gm,F ’ G ’ G ’ 1.

We de¬ne aχ to be the invariant associated to this exact sequence as above.

The conjugation homomorphism G ’ Aut(G) induces the map

H 1 (F, G) ’ H 1 F, Aut(Gsep ) .

Hence, associated to each γ ∈ H 1 (F, G) is a twisted form Gγ of G (called an inner

form of G). If we choose γ such that Gγ is quasisplit (i.e., Gγ contains a Borel

subgroup de¬ned over F ), then

aχ (γ) = [Aχ ] ∈ Br(F )

F

where Aχ is the Tits algebra associated to the character χ (see §??).

(31.21) Example. Let T be an algebraic torus over F . Then

∼

Pic(T ) ’ H 1 F, T — (Fsep )

’

432 VII. GALOIS COHOMOLOGY

and all the cohomological invariants of dimension 2 of T with coe¬cients in Q/Z(1)

are given by the cup product

H 1 (F, T ) — H 1 F, T — (Fsep ) ’ H 2 (F, Fsep ) = H 2 F, Q/Z(1)

—

associated to the natural pairing

T (Fsep ) — T — (Fsep ) ’ Fsep .

—

Invariants of dimension 3. Let G be an algebraic group over a ¬eld F .

Assume ¬rst that F is separably closed. A loop in G is a group homomorphism

Gm,F ’ G over F . Write G— for the set of all loops in G. In general there is no

group structure on G— , but if f and h are two loops with commuting images, then

the pointwise product f h is also a loop. In particular, for any integer n and any

loop f the nth power f n is de¬ned. For any g ∈ G(F ) and any loop f , write gf for

the loop

Int(g)

f

Gm,F ’ G ’’

’ ’’ G.

Consider the set Q(G) of all functions q : G— ’ Z, such that

(a) q( gf ) = q(f ) for all g ∈ G(F ) and f ∈ G— ,

(b) for any two loops f and h with commuting images, the function

(k, m) ’ q(f k hm )

Z — Z ’ Z,

is a quadratic form.

There is a natural abelian group structure on Q(G).

Assume now that F is an arbitrary ¬eld. There is a natural action of the

absolute Galois group “ on the set of loops in Gsep and hence on Q(Gsep ). We set

Q(G) = Q(Gsep )“ .

(31.22) Example. Let T be an algebraic torus. Then T— (Fsep ) is the group of

cocharacters of T and

“

Q(T ) = S 2 T — (Fsep )

is the group of Galois invariant integral quadratic forms on T— (Fsep ).

(31.23) Example. Let G = GL(V ) and f ∈ G— (Fsep ) be a loop. We can view f as

a representation of Gm,F . By the theory of representations of diagonalizable groups

(see §??), f is uniquely determined by its weights χai , i = 1, 2, . . . , n = dim(V )

where χ is the canonical character of Gm,F . The function qV on G— (Fsep ) de¬ned

by

n

a2

qV (f ) = i

i=1

clearly belongs to Q(G).

A group homomorphism G ’ G over F induces a map of loop sets G— (Fsep ) ’

G— (Fsep ) and hence a group homomorphism

Q(G ) ’ Q(G),

making Q a contravariant functor from the category of algebraic groups over F to

the category of abelian groups.

§31. COHOMOLOGICAL INVARIANTS 433

Let G and G be two algebraic groups over F . The natural embeddings of G

and G in G—G and both projections from the product G—G to its factors induce

a natural isomorphism of Q(G) • Q(G ) with a direct summand of Q(G — G ).

(31.24) Lemma. If G— (Fsep ) = 1, then Q(G) = 0.

Proof : Choose an embedding G ’ GL(V ). Since G— (Fsep ) = 1, the restriction of

the positive function qV (see Example (??)) on this set is nonzero.

Assume now that G is a semisimple algebraic group over F .

(31.25) Lemma. Q(G) is a free abelian group of rank at most the number of

simple factors of Gsep .

Proof : We may assume that F is separably closed. Let T be a maximal torus in G

de¬ned over F . Since any loop in G is conjugate to a loop with values in T , the

restriction homomorphism Q(G) ’ Q(T ) is injective. By Example (??), the group

Q(T ) is free abelian of ¬nite rank, hence so is Q(G).

The Weyl group W acts naturally on Q(T ) and the image of the restriction ho-

momorphism belongs to Q(T )W . Hence any element in Q(G) de¬nes a W -invariant

quadratic form on T— and hence on the Q-vector space T— —Z Q. This space decom-

poses as a direct sum of subspaces according to the decomposition of G into the

product of simple factors and such a quadratic form (with values in Q) is known to

be unique (up to a scalar) on each component. Hence, the rank of Q(G) is at most

the number of simple components.

From Lemma (??) and the proof of Lemma (??) we obtain

(31.26) Corollary. If G is an absolutely simple algebraic group, then Q(G) is an

in¬nite cyclic group with a canonical generator which is a positive function.

(31.27) Corollary. Under the hypotheses of the previous corollary the homomor-

phism Q(G) ’ Q(GL ) is an isomorphism for any ¬eld extension L of F .

Proof : It su¬ces to consider the case L = Fsep . Since the group Q(G) is nontrivial,

the Galois action on the in¬nite cyclic group Q(GL ) must be trivial, and hence

Q(G) = Q(GL )“ = Q(GL ).

(31.28) Example. Let G = SL(V ). As in Example (??), for i = 1, . . . , n =

dim V , one associates integers ai to a loop f . In our case the sum of all the ai is

even (in fact, zero), hence the sum of the squares of the ai is even. Therefore

qV = 1 qV ∈ Q(G).

2

It is easy to show that qV is the canonical generator of Q(G).

(31.29) Corollary. If F is separably closed, then the rank of Q(G) equals the

number of simple factors of G.

Proof : Let G = G1 — · · · — Gm where the Gi are simple groups. By Lemma (??),

rank Q(G) ¤ m. On the other hand, the group Q(G) contains the direct sum of

the Q(Gi ) which is a free group of rank m by Corollary (??).

(31.30) Proposition. Let G and G be semisimple algebraic groups. Then

Q(G — G ) = Q(G) • Q(G ).

434 VII. GALOIS COHOMOLOGY

Proof : Clearly, we may assume that F is separably closed. The group Q(G) •

Q(G ) is a direct summand of the free group Q(G — G ) and has the same rank by

Corollary (??), hence the claim.

Let L be a ¬nite separable ¬eld extension of F and let G be a semisimple group

over L. Then the transfer RL/F (G) is a semisimple group over F .

∼

(31.31) Proposition. There is a natural isomorphism Q(G) ’ Q RL/F (G) .

’

Proof : Choose an embedding ρ : L ’ Fsep and set

“ = Gal Fsep /ρ(L) ‚ “.

The group RL/F (G)sep is isomorphic to the direct product of groups G„ as „ varies

over the set X of all F -embeddings of L into Fsep (see Proposition (??)). Hence,

by Proposition (??),

Q RL/F (G)sep = Q(G„ ).

„ ∈X

The Galois group “ acts naturally on the direct sum, transitively permuting com-

ponents. Hence it is the induced “-module from the “ -module Q(Gρ ). The propo-

sition then follows from the fact that for any “ -module M there is a natural

isomorphism between the group of “ -invariant elements in M and the group of

“-invariant elements in the induced module Map“ (“, M ).

By Theorem (??), a simply connected semisimple group over F is isomorphic

to a product of groups of the form RL/F (G ) where L is a ¬nite separable ¬eld

extension of F and G is an absolutely simple simply connected group over L.

Hence, Corollary (??) and Propositions (??), (??) yield the computation of Q(G)

for any simply connected semisimple group G.

A relation between Q(G) and cohomological invariants of dimension 3 of simply

connected semisimple groups is given by the following

(31.32) Theorem. Let G be a simply connected semisimple algebraic group over

a ¬eld F . Then there is a natural surjective homomorphism

γ(G) : Q(G) ’ Inv3 G, Q/Z(2) .

The naturality of γ in the theorem means, ¬rst of all, that for any group

homomorphism ± : G ’ G the following diagram commutes:

γ(G )

Q(G ) ’ ’ ’ Inv3 G , Q/Z(2)

’’

¦ ¦

¦ ¦—

(31.33) ±

γ(G)

Q(G) ’ ’ ’ Inv3 G, Q/Z(2) .

’’

For any ¬eld extension L of F the following diagram also commutes:

γ(G)

Q(G) ’ ’ ’ Inv3 G, Q/Z(2)

’’

¦ ¦

¦ ¦res

(31.34)

γ(GL )

Q(GL ) ’ ’ ’ Inv3 GL , Q/Z(2) .

’’

§31. COHOMOLOGICAL INVARIANTS 435

In addition, for a ¬nite separable extension L of F and an algebraic group G over L

the following diagram is also commutative:

γ(G)

Inv3 G, Q/Z(2)

Q(G) ’’’

’’

¦ ¦

¦ ¦cor

(31.35)

γ RL/F (G)

Q RL/F (G) ’ ’ ’ ’ ’ Inv3 RL/F (G), Q/Z(2) .

’’’’

Let G be an absolutely simple simply connected group over F . By Corol-

lary (??) and Theorem (??), the group Inv3 G, Q/Z(2) is cyclic, generated by a

canonical element which we denote i(G) and call the Rost invariant of the group G.

The commutativity of diagram (??) and Corollary (??) show that for any ¬eld

extension L of F ,

resL/F i(G) = i(GL ).

Let L be a ¬nite separable ¬eld extension of F and let G be an absolutely

simple simply connected group over L. It follows from the commutativity of (??)

and Proposition (??) that the group Inv3 RL/F (G), Q/Z(2) is cyclic and generated

by corL/F i(G) .

Let G be a simply connected semisimple group over F and let ρ : G ’ SL(V )

be a representation. The triviality of the right-hand group in the top row of the

following commutative diagram (see Example (??)):

Q SL(V ) ’ ’ ’ Inv3 SL(V ), Q/Z(2)

’’

¦ ¦

¦ ¦

Inv3 G, Q/Z(2)

Q(G) ’’’

’’

shows that the image of Q SL(V ) ’ Q(G) belongs to the kernel of

γ : Q(G) ’ Inv3 G, Q/Z(2) .

One can prove that all the elements in the kernel are obtained in this way.

(31.36) Theorem. The kernel of γ : Q(G) ’ Inv 3 G, Q/Z(2) is generated by the

images of Q SL(V ) ’ Q(G) for all representations of G.

(31.37) Corollary. Let G and G be simply connected semisimple groups over F .

Then

Inv3 G — G , Q/Z(2) = Inv3 G, Q/Z(2) • Inv3 G , Q/Z(2) .

(31.38) Corollary. Let L/F be a ¬nite separable ¬eld extension, G be a simply

connected semisimple group over L. Then the corestriction map

cor: Inv3 G, Q/Z(2) ’ Inv3 RL/F (G), Q/Z(2)

is an isomorphism.

These two corollaries reduce the study of the group Inv 3 G, Q/Z(2) to the

case of an absolutely simple simply connected group G.

436 VII. GALOIS COHOMOLOGY

Let ± : G ’ G be a homomorphism of absolutely simple simply connected

groups over F . There is a unique integer n± such that the following diagram

commutes:

=

Z ’ ’ ’ Q(G )

’’

¦ ¦

¦ ¦Q(±)

n±

=

Z ’ ’ ’ Q(G).

’’

If we have another homomorphism β : G ’ G , then clearly nβ± = nβ n± . Assume

that G = GL(V ). It follows from the proof of Lemma (??) that nβ > 0 and

nβ± > 0, hence n± is a natural number for any group homomorphism ±.

Let ρ : G ’ SL(V ) be a representation. As we observed above, nρ · iG =

0. Denote nG the greatest common divisor of nρ for all representations ρ of the

group G. Clearly, nG · i(G) = 0. Theorem (??) then implies

(31.39) Proposition. Let G be an absolutely simple simply connected group. Then

Inv3 G, Q/Z(2) is a ¬nite cyclic group of order nG .

Let n be any natural number prime to char F . The exact sequence

n

1 ’ µ—2 ’ Q/Z(2) ’ Q/Z(2) ’ 1

’

n

yields the following exact sequence of cohomology groups

n

H 2 F, Q/Z(2) ’ H 2 F, Q/Z(2) ’ H 3 (F, µ—2 ) ’

’ n

n

’ H 3 F, Q/Z(2) ’ H 3 F, Q/Z(2) .

’

Since the group H 2 F, Q/Z(2) is n-divisible (see Merkurjev-Suslin [?]), the group

H 3 (F, µ—2 ) is identi¬ed with the subgroup of elements of exponent n in H 3 F, Q/Z(2) .

n

Now let G be an absolutely simple simply connected group over F . By Propo-

sition (??), the values of the invariant i(G) lie in H 3 (F, µ—2 ), so that

nG

Inv3 G, Q/Z(2) = Inv3 (G, µ—2 ).

nG

In the following sections we give the numbers nG for all absolutely simple simply

connected groups. In some cases we construct the Rost invariant directly.

Spin groups of quadratic forms. Let F be a ¬eld of characteristic di¬erent

from 2. Let W F be the Witt ring of F and let IF be the fundamental ideal of

even-dimensional forms. The nth power I n F of this ideal is generated by the classes

of n-fold P¬ster forms.

To any 3-fold P¬ster form a, b, c the Arason invariant associates the class

(a) ∪ (b) ∪ (c) ∈ H 3 (F, Z/2Z) = H 3 (F, µ—2 ),

2

see (??). The Arason invariant extends to a group homomorphism

e3 : I 3 F ’ H 3 (F, Z/2Z)

(see Arason [?]). Note that I 3 F consists precisely of the classes [q] of quadratic

forms q having even dimension, trivial discriminant, and trivial Hasse-Witt invari-

ant (see Merkurjev [?]).

Let q be a non-degenerate quadratic form over F . The group G = Spin(q)

is a simply connected semisimple group if dim q ≥ 3 and is absolutely simple if

dim q = 4. It is a group of type Bn if dim q = 2n + 1 and of type Dn if dim q = 2n.

§31. COHOMOLOGICAL INVARIANTS 437

Conversely, any absolutely simple simply connected group of type Bn is isomorphic

to Spin(q) for some q. (The same property does not hold for Dn .)

The exact sequence

π

1 ’ µ2 ’ Spin(q) ’ O+ (q) ’ 1

(31.40) ’

gives the following exact sequence of pointed sets

δ1

π

H 1 F, Spin(q) ’— H 1 F, O+ (q) ’ H 2 (F, µ2 ) = 2 Br(F ).

’ ’

The set H 1 F, O+ (q) classi¬es quadratic forms of the same dimension and discrim-

inant as q, see (??). The connecting map δ 1 takes such a form q to the Hasse-Witt

invariant e2 [q ] ’ [q] ∈ Br(F ). Thus, the image of π— consists of classes of forms

having the same dimension, discriminant, and Hasse-Witt invariant as q. Therefore,

π— (u) ’ [q] ∈ I 3 F for any u ∈ H 1 F, Spin(q) .

The map

i Spin(q) : H 1 F, Spin(q) ’ H 3 (F, µ—2 )

(31.41) 2

de¬ned by u ’ e3 π— (u) ’ [q] gives rise to an invariant of Spin(q). It turns out

that this is the Rost invariant if dim q ≥ 5. If dim q = 5 or 6 and q is of maximal

Witt index, the anisotropic form representing π— (u) ’ [q] is of dimension less than 8

and hence is trivial by the Arason-P¬ster Hauptsatz. In these cases the invariant

is trivial and nG = 1. Otherwise the invariant is not trivial and nG = 2.

In the case where dim q = 4 the group G is a product of two groups of type A1

if disc q is trivial and otherwise is isomorphic to RL/F SL1 (C0 ) where L/F is the

discriminant ¬eld extension and C0 = C0 (q) (see (??) and §??). In the latter case

the group Inv3 G, Q/Z(2) is cyclic and generated by the invariant described above.

This invariant is trivial if and only if the even Cli¬ord algebra C0 is split.

If dim q = 3, then the described invariant is trivial since it is twice the Rost

invariant. In this case G = SL1 (C0 ) is a group of type A1 and the Rost invariant

is described below.

Type An .

Inner forms. Let G be an absolutely simple simply connected group of inner

type An over F , so that G = SL1 (A) for a central simple F -algebra of degree n + 1.

It turns out that nG = e = exp(A), and the Rost invariant

i(G) : H 1 (F, G) = F — / Nrd(A— ) ’ H 3 (F, µ—2 )

e

is given by the formula

i(G) a · Nrd(A— ) = (a) ∪ [A]

where (a) ∈ H 1 (F, µe ) = F — /F —e is the class of a ∈ F — and [A] ∈ H 2 (F, µe ) =

e Br(F ) is the class of the algebra A.

Outer forms. Let G be an absolutely simple simply connected group of outer

type An over F with n ≥ 2, so that G = SU(B, „ ) where (B, „ ) is a central

simple F -algebra with unitary involution of degree n + 1 and the center K of B

is a quadratic separable ¬eld extension of F . If n is odd, let D = D(B, „ ) be the

discriminant algebra (see §??).

(31.42) Proposition. The number nG equals either exp(B) or 2 exp(B). The ¬rst

case occurs if and only if (n + 1) is a 2-power and either

(1) exp(B) = n + 1 or

438 VII. GALOIS COHOMOLOGY

n+1

(2) exp(B) = and D is split.

2

An element of the set H 1 F, SU(B, „ ) is represented by a pair (s, z) where s ∈

Sym(B, „ )— and z ∈ K — satisfy Nrd(s) = NK/F (z) (see (??)). Since SU(B, „ )K

SL1 (B), it follows from the description of the Rost invariant in the inner case that

= (z) ∪ [B] ∈ H 3 K, Q/Z(2) .

i(G) (s, z)/≈ K

(31.43) Example. Assume that char F = 2 and B is split, i.e., B = EndK (V ) for

some vector space V of dimension n + 1 over K. The involution „ is adjoint to some

hermitian form h on V over K (Theorem (??)). Considering V as a vector space

over F we have a quadratic form q on V given by q(v) = h(v, v) ∈ F for v ∈ V .

Any isometry of h is also an isometry of q, hence we have the embedding

U(B, „ ) ’ O+ (V, q).

Since SU(B, „ ) is simply connected, the restriction of this embedding lifts to a

group homomorphism

± : SU(B, „ ) ’ Spin(V, q)

(see Borel-Tits [?, Proposition 2.24(i), p. 262]). One can show that n± = 1, so that

i(G) is the composition

H 1 F, SU(B, „ ) ’ H 1 F, Spin(V, q) ’ H 3 F, Q/Z(2)

where the latter map is the Rost invariant of Spin(V, q) which was described in (??).

An element of the ¬rst set in the composition is represented by a pair (s, z) ∈

Sym(B, „ )— — K — such that Nrd(s) = NK/F (z). The symmetric element s de¬nes

another hermitian form hs on V by

hs (u, v) = h s’1 (u), v

which in turn de¬nes, as described above, a quadratic form qs on V considered as

a vector space over F . The condition on the reduced norm of s shows that the

discriminants of hs and h are equal, see (??), hence [qs ] ’ [q] ∈ I 3 F . It follows from

the description of the Rost invariant for the group Spin(V, q) (see (??)) that the

invariant of the group G is given by the formula

i(G) (s, z)/≈ = e3 [qs ] ’ [q] .

If dim V is odd (i.e., n is even), the canonical map

H 1 F, SU(B, „ ) ’ H 1 F, GU(B, „ )

is surjective, since every unitary involution „ = Int(u) —¦ „ on B may be written as

„ = Int u NrdB (u) —¦ „,

showing that the conjugacy class of „ is the image of u NrdB (u), NrdB (u)(n/2)+1 .

The invariant i(G) induces an invariant

i GU(B, „ ) : H 1 F, GU(B, „ ) ’ H 3 (F, µ—2 )

2

which can be explicitly described as follows: given a unitary involution „ on B,

represent „ as the adjoint involution with respect to some hermitian form h with

disc h = disc h, and set

i GU(B, „ ) („ ) = e3 [q ] ’ [q]

where q is the quadratic form on V de¬ned by q (v) = h (v, v). Alternately,

consider the quadratic trace form Q„ (x) = TrdB (x2 ) on Sym(B, „ ). If h has a

§31. COHOMOLOGICAL INVARIANTS 439

F [X]/(X 2 ’ ±), Propositions (??) and (??)

diagonalization δ1 , . . . , δn+1 and K

show that

Q„ = (n + 1) 1 ⊥ 2 · ± · ⊥1¤i<j¤n+1 δi δj .

On the other hand,

q = ± · δ1 , . . . , δn+1 .

Since disc h = disc h , we may ¬nd a diagonalization h = δ1 , . . . , δn+1 such that

δ1 . . . δn+1 = δ1 . . . δn+1 . Using the formulas for the Hasse-Witt invariant of a sum

in Lam [?, p. 121], we may show that

e2 ⊥1¤i<j¤n+1 δi δj ’ ⊥1¤i<j¤n+1 δi δj =

e2 δ1 , . . . , δn+1 ’ δ1 , . . . , δn+1 ,

hence

i GU(B, „ ) („ ) = e3 [q ] ’ [q] = e3 [Q„ ] ’ [Q„ ] .

(31.44) Example. Assume that (n + 1) is odd and B has exponent e. Assume

also that char F does not divide 2e. For G = SU(B, „ ) we have nG = 2e. Since e

is odd we have µ—2 = µ—2 — µ—2 , hence the Rost invariant i(G) may be viewed as

e

2e 2

a pair of invariants

i1 (G), i2 (G) : H 1 F, SU(B, „ ) ’ H 3 (F, µ—2 ) — H 3 (F, µ—2 ).

e

2

Since B is split by a scalar extension of odd degree, we may use (??) to determine

i1 (G):

i1 (G) (s, z)/≈ = e3 [QInt(s)—¦„ ] ’ [Q„ ] ∈ H 3 (F, µ—2 ).

2

(By (??), it is easily seen that [QInt(s)—¦„ ] ’ [Q„ ] ∈ I 3 F .)

On the other hand, we have SU(B, „ )K SL1 (B) hence we may use the

invariant of SL1 and Corollary (??) to determine i2 (G):

1

corK/F (s) ∪ [B] ∈ H 3 (F, µ—2 ).

i2 (G) (s, z)/≈ = e

2

Note that the canonical map H 1 F, SU(B, „ ) ’ H 1 F, GU(B, „ ) is sur-

jective, as in the split case (Example (??)), and the invariant i1 (G) induces an

invariant

i GU(B, „ ) : H 1 F, GU(B, „ ) ’ H 3 (F, µ—2 )

2

which maps the conjugacy class of any unitary involution „ to e3 [Q„ ] ’ [Q„ ] .

In the particular case where deg(B, „ ) = 3, we also have a P¬ster form π(„ )

de¬ned in (??) and a cohomological invariant f3 (B, „ ) = e3 π(„ ) , see (??). From

the relation between π(„ ) and Q„ , it follows that

[Q„ ] ’ [Q„ ] = [ 2 ] · π(„ ) ’ π(„ ) ,

hence

i GU(B, „ ) („ ) = e3 π(„ ) ’ π(„ ) = f3 (B, „ ) ’ f3 (B, „ ).

440 VII. GALOIS COHOMOLOGY

Type Cn . Let G be an absolutely simple simply connected group of type Cn

over F , so that G = Sp(A, σ) where A is a central simple algebra of degree 2n

over F with a symplectic involution σ.

Assume ¬rst that the algebra A is split, i.e., G = Sp2n . Since all the nonsingular

alternating forms are pairwise isomorphic, the set H 1 (E, G) is trivial for any ¬eld

extension E of F . Hence nG = 1 and the invariant i(G) is trivial.

Assume now that A is nonsplit, so that exp(A) = 2. Consider the natural

embedding ± : G ’ SL1 (A). One can check that n± = 1, hence the Rost invari-

ant of G is given by the composition of ± and the invariant of SL1 (A), so that

nG = 2. By (??), we have H 1 (F, G) = Symd(A, σ)— /∼, and the following diagram

commutes:

±1

H 1 (F, G) ’ ’ ’ H 1 F, SL1 (A)

’’

Nrp

Symd(A, σ)— /∼ ’ ’ σ F — / Nrd(A— ),

’’ ’

(where Nrpσ is the pfa¬an norm). Hence the invariant

i(G) : H 1 (F, G) ’ H 3 (F, µ—2 )

2

is given by the formula

i(G)(u/∼) = Nrpσ (u) ∪ [A].

The exact sequence

µ

1 ’ Sp(A, σ) ’ GSp(A, σ) ’ Gm ’ 1,

’

where µ is the multiplier map, induces the following exact sequence in cohomology:

H 1 F, Sp(A, σ) ’ H 1 F, GSp(A, σ) ’ 1

since H 1 (F, Gm ) = 1 by Hilbert™s Theorem 90. If deg A is divisible by 4, it turns

out that the invariant i(G) induces an invariant

i GSp(A, σ) : H 1 F, GSp(A, σ) ’ H 3 (F, µ—2 ).

2

Indeed, viewing H 1 F, GSp(A, σ) as the set of conjugacy classes of symplectic

involutions on A (see (??)), the canonical map

Symd(A, σ)— /∼ = H 1 F, Sp(A, σ) ’ H 1 F, GSp(A, σ)

takes u/∼ to the conjugacy class of Int(u) —¦ σ. For z ∈ F — and u ∈ Symd(A, σ)—

we have Nrpσ (zu) = z deg A/2 Nrpσ (u), hence Nrpσ (zu) = Nrpσ (u) in H 1 (F, µ2 )

if deg A is divisible by 4. Therefore, in this case we may set

i GSp(A, σ) Int(u) —¦ σ = i Sp(A, σ) (u/∼) = Nrpσ (u) ∪ [A].

(31.45) Example. Consider the particular case where deg A = 4. Since the quad-

ratic form Nrpσ is an Albert form of A by (??), its Hasse-Witt invariant is [A].

Therefore,

Nrpσ (u) ∪ [A] = e3 Nrpσ (u) · Nrpσ

and it follows by (??) that

i GSp(A, σ) („ ) = e3 jσ („ )

for every symplectic involution „ on A.

§31. COHOMOLOGICAL INVARIANTS 441

(31.46) Example. Let A = EndQ (V ) where V is a vector space of even dimen-

sion over a quaternion division algebra Q, and let σ be a hyperbolic involution

on A. For every nonsingular hermitian form h on V (with respect to the conju-

gation involution on Q), the invariant i GSp(A, σ) (σh ) of the adjoint involution

σh is the cohomological version of the Jacobson discriminant of h, see the notes to

Chapter ??. Indeed, if h has a diagonalization ±1 , . . . , ±n , then we may assume

σh = Int(u) —¦ σ where u is the diagonal matrix

u = diag(±1 , ’±2 , . . . , ±n’1 , ’±n ).

Then Nrpσ (u) = (’1)n/2 ±1 . . . ±n , hence

i GSp(A, σ) (σh ) = (’1)n/2 ±1 . . . ±n ∪ [Q].

Type Dn . Assume that char F = 2. Let G be an absolutely simple simply

connected group of type Dn (n ≥ 5) over F , so that G = Spin(A, σ) where A

is a central simple algebra of degree 2n over F with an orthogonal involution σ.

The case where A is split, i.e., G = Spin(q) for some quadratic form q, has been

considered in (??).

Assume that the algebra A is not split. In this case nG = 4. The exact sequence

similar to (??) yields a map

i1 : F — /F —2 = H 1 (F, µ2 ) ’ H 1 F, Spin(A, σ) .

The image i1 (a · F —2 ) for a ∈ F — corresponds to the torsor Xa given in the Cli¬ord

group “(A, σ) by the equation σ(x)x = a. The Rost invariant i(G) on Xa is given

by the formula

i(G)(Xa ) = (a) ∪ [A]

and therefore it is in general nontrivial. Hence the invariant does not factor through

the image of

H 1 F, Spin(A, σ) ’ H 1 F, O+ (A, σ)

as is the case when A is split.

Exceptional types.

G2 . Let G be an absolutely simple simply connected group of type G2 over F ,

so that G = Aut(C) where C is a Cayley algebra over F . The set H 1 (F, G) classi¬es

Cayley algebras over F . One has nG = 2 and the Rost invariant

i(G) : H 1 (F, G) ’ H 3 (F, µ—2 )

2

is given by the formula

i(G)(C ) = e3 (nC ) + e3 (nC )

where nC is the norm form of the Cayley algebra C (which is a 3-fold P¬ster form)

and e3 is the Arason invariant.

D4 . An absolutely simple simply connected algebraic group of type D4 over F

is isomorphic to Spin(T ) where T = (E, L, σ, ±) is a trialitarian algebra (see §??).

Here E is a central simple algebra with an orthogonal involution σ over a cubic

´tale extension L of F .

e

Assume ¬rst that L splits completely, i.e., L = F — F — F . Then E = A1 —

A2 — A3 where the Ai are central simple algebras of degree 8 over F . In this case

nG = 2 or 4. The ¬rst case occurs if and only if at least one of the algebras Ai is

split.

442 VII. GALOIS COHOMOLOGY

Assume now that L is not a ¬eld but does not split completely, i.e., L = F — K

where K is a quadratic ¬eld extension of F , hence E = A — C where A and C are

central simple algebras of degree 8 over F and K respectively (see §??). In this

case also nG = 2 or 4 and the ¬rst case takes place if and only if A is split.

Finally assume that L is a ¬eld (this is the trialitarian case). In this case

nG = 6 or 12. The ¬rst case occurs if and only if E is split.

F4 . nG = 6. The set H 1 (F, G) classi¬es absolutely simple groups of type F4

and also exceptional Jordan algebras. The cohomological invariant is discussed in

Chapter ??.

E6 . nG = 6 (when G is split).

Isn™t the

statement for E6 E7 . nG = 12 (when G is split).

and E7 true E8 . nG = 60.

whenever the Tits

algebras of G

are split and G

is inner ? (Skip Exercises

G.)

1. Let G be a pro¬nite group and let A be a (continuous) G-group. Show that

there is a natural bijection between the pointed set H 1 (G, A) and the direct

limit of H 1 (G/U, AU ) where U ranges over all open normal subgroups in G.

ˆ ˆ

2. Let Z be the inverse limit of Z/nZ, n ∈ N, and A be a Z-group such that any

element of A has a ¬nite order. Show that there is a natural bijection between

ˆ

the pointed set H 1 (Z, A) and the set of equivalence classes of A where the

equivalence relation is given by a ∼ a if there is b ∈ A such that a = b’1 ·a·σ(b)

ˆ

(σ is the canonical topological generator of Z).

3. Show that Aut(GL2 ) = Aut(SL2 ) — Z/2Z. Describe the twisted forms of GL2 .

4. Let Sn act on (Z/2Z)n through permutations and let G = (Z/2Z)n Sn . Let F

be an arbitrary ¬eld. Show that H 1 (F, G) classi¬es towers F ‚ L ‚ E with

L/F ´tale of dimension n and E/L quadratic ´tale.

e e

5. Let G = GLn /µ2 . Show that there is a natural bijection between H 1 (F, G)

and the set of isomorphism classes of triples (A, V, ρ) where A is a central

simple F -algebra of degree n, V is an F -vector space of dimension n2 and

ρ : A —F A ’ EndF (V ) is an isomorphism of F -algebras.

Hint: For an n-dimensional F -vector space U there is an associated triple

(AU , VU , ρU ) where AU = EndF (U ), VU = U —2 and where

ρ : EndF (U ) —F EndF (U ) ’ EndF (U —2 )

is the natural map. If F is separably closed, then any triple (A, V, ρ) is isomor-

phic to (AU , VU , ρU ). Moreover the homomorphism

GL(U ) ’ { (±, β) ∈ AutF (AU ) — GL(VU ) | ρ —¦ (± — ±) = Ad(β) —¦ µ }

given by γ ’ Ad(γ), γ —2 is surjective with kernel µ2 .

6. Let G be as in Exercise ??. Show that the sequence

2δ 1

»

H 1 (F, G) ’ H 1 (F, PGLn ) ’ ’ H 2 (F, Gm )

’ ’

is exact. Here » is induced from the natural map GLn ’ PGLn and δ 1 is the

connecting homomorphism for (??).

Using this result one may restate Albert™s theorem on the existence of

involutions of the ¬rst kind (Theorem (??)) by saying that the natural inclusion

EXERCISES 443

PGOn ’ G induces a surjection

H 1 (F, PGOn ) ’ H 1 (F, G).

The construction of Exercise ?? in Chapter ?? can be interpreted in terms

of Galois cohomology via the natural homomorphism GL(U ) ’ PGO H(U )

where U is an n-dimensional vector space and H(U ) is the hyperbolic quadratic

space de¬ned in §??.

7. Let K/F be separable quadratic extension of ¬elds. Taking transfers, the exact

sequence (??) induces an exact sequence

1 ’ RK/F (Gm ) ’ RK/F (GLn ) ’ RK/F (PGLn ) ’ 1.

Let N : RK/F (Gm ) ’ Gm be the transfer map and set

G = RK/F (GLn )/ ker N.

Show that there is a natural bijection between H 1 (F, G) and the set of iso-

morphism classes of triples (A, V, ρ) where A is a central simple K-algebra of

degree n, V is an F -vector space of dimension n2 and

ρ : NK/F (A) ’ EndF (V )

is an isomorphism of F -algebras. Moreover show that the sequence

corK/F —¦δ 1

»

H (F, G) ’ H (K, PGLn ) ’ ’ ’ ’ H 2 (F, Gm )

1 1

’ ’ ’ ’’

is exact. Here δ 1 is the connecting homomorphism for the sequence (??) and

» is given by

H 1 (F, G) ’ H 1 F, RK/F (PGLn ) = H 1 (K, PGLn ).

Using this result one may restate the theorem on the existence of involutions of

the second kind (Theorem (??)) by saying that the natural inclusion PGU n =

SUn / ker N ’ G induces a surjection

H 1 (F, PGUn ) ’ H 1 (F, G).

The construction of Exercise ?? in Chapter ?? can be interpreted in terms of

Galois cohomology via the natural homomorphism GL(UK ) ’ PGU H1 (UK )

where U is an n-dimensional F -vector space and H1 (UK ) is the hyperbolic

hermitian space de¬ned in §??.

8. Let (A, σ, f ) be a central simple F -algebra with quadratic pair. Let GL1 (A)

act on the vector space Symd(A, σ) • Sym(A, σ)— by

ρ(a)(x, g) = axσ(a), ag ,

where ag(y) = g σ(a)ya for y ∈ Sym(A, σ). Show that the stabilizer of (1, f )

is O(A, σ, f ) and that the twisted ρ-forms of (1, f ) are the pairs (x, g) such that

x ∈ A— and g y + σ(y) = TrdA (y) for all y ∈ A. Use these results to give

an alternate description of H 1 F, O(A, σ, f ) , and describe the canonical map

induced by the inclusion O(A, σ, f ) ’ GO(A, σ, f ).

9. Let L be a Galois Z/nZ-algebra over a ¬eld F of arbitrary characteristic. Using

the exact sequence 0 ’ Z ’ Q ’ Q/Z ’ 0, associate to L a cohomology class

[L] in H 2 (F, Z) and show that the class (L, a) ∈ H 2 (F, Gm ) corresponding

to the cyclic algebra (L, a) under the crossed product construction is the cup

product [L] ∪ a, for a ∈ F — = H 0 (F, Gm ).

444 VII. GALOIS COHOMOLOGY

10. Let K/F be a separable quadratic extension of ¬elds with nontrivial auto-

morphism ι, and let n be an integer which is not divisible by char F . Use

Proposition (??) to identify H 1 (F, µn[K] ) to the factor group

{ (x, y) ∈ F — — K — | xn = NK/F (y) }

.

{ (NK/F (z), z n ) | z ∈ K — }

For (x, y) ∈ F — — K — such that xn = NK/F (y), let [x, y] ∈ H 1 (F, µ[K] ) be the

corresponding cohomology class.

(a) Suppose n = 2. Since µ2[K] = µ2 , there is a canonical isomorphism

H 1 (F, µ2[K] ) F — /F —2 . Show that this isomorphism takes [x, y] to

NK/F (z) · F —2 , where z ∈ K — is such that x’1 y = zι(z)’1 .

(b) Suppose n = rs for some integers r, s. Consider the exact sequence

j

i

1 ’ µr[K] ’ µn[K] ’ µs[K] ’ 1.

’ ’

Show that the induced maps

j1

i1

H 1 (F, µr[K] ) ’ H 1 (F, µn[K] ) ’ H 1 (F, µs[K] )

’ ’

can be described as follows:

i1 [x, y] = [x, y s ] j 1 [x, y] = [xr , y].

and

(Compare with (??).)

(c) Show that the restriction map

res: H 1 (F, µn[K] ) ’ H 1 (K, µn ) = K — /K —n

takes [x, y] to y · K —n and the corestriction map

cor : H 1 (K, µn ) ’ H 1 (F, µn[K] )

takes z · K —n to [1, zι(z)’1 ].