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