u’1 b = g2 g0 u’1 ag0 g2 + (x0 + x2 ) + (x0 + x2 )t .

tt

Now, tr v ’1 (x0 +x2 ) = 0, hence we may set g = ug2 g0 u’1 and x = u(x0 +x2 ).

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 411

Now, let (A, σ, f ) be a central simple algebra with quadratic pair over a ¬eld F

of characteristic 2. Let G = GL1 (A) act on the vector space W = A/ Alt(A, σ) by

ρ(g) a + Alt(A, σ) = gaσ(g) + Alt(A, σ)

for g ∈ G and a ∈ A. Let ∈ A satisfy f (s) = TrdA ( s) for all s ∈ Sym(A, σ)

(see (??)). We next determine the stabilizer AutG +Alt(A, σ) . For every rational

point g of this stabilizer we have

g σ(g) = + x + σ(x) for some x ∈ A.

Applying σ and using + σ( ) = 1 (see (??)), it follows that gσ(g) = 1. Moreover,

since Alt(A, σ) is orthogonal to Sym(A, σ) for the bilinear form TA (see (??)) we

have

TrdA g σ(g)s = TrdA ( s) for all s ∈ Sym(A, σ)

hence

f σ(g)sg = f (s) for all s ∈ Sym(A, σ).

Therefore, g ∈ O(A, σ, f ). Conversely, if g ∈ O(A, σ, f ) then f σ(g)sg = f (s) for

all s ∈ Sym(A, σ), hence TrdA g σ(g)s = TrdA ( s) for all s ∈ Sym(A, σ), and it

follows that g σ(g) ≡ mod Alt(A, σ). Therefore,

(29.35) AutG + Alt(A, σ) = O(A, σ, f ).

On the other hand, Lemma (??) shows that the twisted ρ-forms of + Alt(A, σ)

are the elements a + Alt(A, σ) such that a + σ(a) ∈ A— . Let

Q(A, σ) = { a + Alt(A, σ) | a + σ(a) ∈ A— } ‚ A/ Alt(A, σ)

and de¬ne an equivalence relation ∼ on Q(A, σ) by a + Alt(A, σ) ∼ a + Alt(A, σ)

if and only if a ≡ gaσ(g) mod Alt(A, σ) for some g ∈ A— . Proposition (??) and

Hilbert™s Theorem 90 yield a bijection

H 1 F, O(A, σ, f )

(29.36) Q(A, σ)/∼ ←’

which maps the base point of H 1 F, O(A, σ, f ) to the equivalence class of +

Alt(A, σ). (Compare with (??) and (??).) Note that if A = EndF (V ) the set

Q(A, σ) is in one-to-one correspondence with the set of nonsingular quadratic forms

on V , see §??.

In order to give a similar description of the set H 1 F, O+ (A, σ, f ) (still as-

suming char F = 2), we consider the set

’1

Q0 (A, σ, ) = { a ∈ A | a + σ(a) ∈ A— and SrdA a + σ(a) a = SrdA ( ) }

and the set of equivalence classes

Q+ (A, σ, ) = { [a] | a ∈ Q0 (A, σ, ) }

where [a] = [a ] if and only if a = a + x + σ(x) for some x ∈ A such that

’1

TrdA a + σ(a) x = 0.

We thus have a natural map Q+ (A, σ, ) ’ Q(A, σ) which maps [a] to a+Alt(A, σ).

For simplicity of notation, we set

Q+ (A, σ, )sep = Q+ (Asep , σsep , ) and O(A, σ, f )sep = O(Asep , σsep , fsep ).

412 VII. GALOIS COHOMOLOGY

Since Q+ (A, σ, ) is not contained in a vector space, we cannot apply the general

principle (??). Nevertheless, we may let A— act on Q+ (A, σ, )sep by

sep

for g ∈ A— and a ∈ Q0 (A, σ, )sep .

g[a] = [gaσ(g)] sep

As observed in (??), for g ∈ A— we have g σ(g) = + x + σ(x) for some

sep

x ∈ Asep if and only if g ∈ O(A, σ, f )sep . Moreover the de¬nition of the Dickson

invariant in (??) yields ∆(g) = Trd(x), hence we have g[ ] = [ ] if and only if

g ∈ O+ (A, σ, f )sep . On the other hand, Lemma (??) shows that the A— -orbit of

sep

+

[ ] is Q (A, σ, )sep , hence the action on yields a bijection

A— / O+ (A, σ, f )sep Q+ (A, σ, )sep .

(29.37) ←’

sep

Therefore, by (??) and Hilbert™s Theorem 90 we obtain a bijection between the

pointed set H 1 F, O+ (A, σ, f ) and the orbit set of A— in Q+ (A, σ, )“ , with the

sep

orbit of [ ] as base point.

Claim. Q+ (A, σ, )“ = Q+ (A, σ, ).

sep

Let a ∈ A satisfy γ[a] = [a] for all γ ∈ “. This means that for all γ ∈ “ there

exists xγ ∈ Asep such that

’1

γ(a) = a + xγ + σ(xγ ) and Trd a + σ(a) xγ = 0.

The map γ ’ γ(a) ’ a is a 1-cocycle of “ in Alt(A, σ)sep . For any ¬nite Ga-

lois extension L/F , the normal basis theorem (see Bourbaki [?, §10]) shows that

Alt(A, σ) —F L is an induced “-module, hence H 1 “, Alt(A, σ)sep = 0. Therefore,

there exists y ∈ Alt(A, σ)sep such that

γ(a) ’ a = xγ + σ(xγ ) = y ’ γ(y) for all γ ∈ “.

Choose z0 ∈ Asep such that y = z0 + σ(z0 ). Then a + z0 + σ(z0 ) is invariant under

“, hence a + z0 + σ(z0 ) ∈ A. Moreover, xγ + z0 + γ(z0 ) ∈ Sym(A, σ)sep , hence the

’1

condition Trd a + σ(a) xγ = 0 implies

’1 ’1

γ Trd a + σ(a) z0 = Trd a + σ(a) z0 ,

’1

i.e., Trd a + σ(a) z0 ∈ F . Let z1 ∈ A satisfy

’1 ’1

Trd a + σ(a) z0 = Trd a + σ(a) z1 .

Then a + z0 + σ(z0 ) + z1 + σ(z1 ) ∈ A and

[a] = a + (z0 + z1 ) + σ(z0 + z1 ) ∈ Q+ (A, σ, ),

proving the claim.

In conclusion, we obtain from (??) via (??) and Hilbert™s Theorem 90 a canon-

ical bijection

H 1 F, O+ (A, σ, f )

Q+ (A, σ, )/∼

(29.38) ←’

where the equivalence relation ∼ is de¬ned by the action of A— , i.e.,

[a] ∼ [a ] if and only if [a ] = gaσ(g) for some g ∈ A— .

§30. GALOIS COHOMOLOGY OF ROOTS OF UNITY 413

§30. Galois Cohomology of Roots of Unity

Let F be an arbitrary ¬eld. As in the preceding section, let “ = Gal(Fsep /F )

be the absolute Galois group of F . Let n be an integer which is not divisible by

char F . The Kummer sequence is the exact sequence of group schemes

( )n

(30.1) 1 ’ µn ’ Gm ’ ’ Gm ’ 1.

’’

Since H 1 (F, Gm ) = 1 by Hilbert™s Theorem 90, the induced long exact sequence in

cohomology yields isomorphisms

2

H 1 (F, µn ) F — /F —n and H 2 (F, µn ) nH (F, Gm ),

where, for any abelian group H, n H denotes the n-torsion subgroup of H. Since

H 2 (F, Gm ) Br(F ) (see (??)), we also have

H 2 (F, µn ) n Br(F ).

This isomorphism suggests deep relations between central simple algebras and the

cohomology of µn , which are formalized through the cyclic algebra construction in

§??.

If char F = 2, we may identify µ2 —µ2 with µ2 through the map (’1)a —(’1)b ’

(’1)ab and de¬ne a cup product

∪ : H i (F, µ2 ) — H j (F, µ2 ) ’ H i+j (F, µ2 ).

For ± ∈ F — , we set (±) ∈ H 1 (F, µ2 ) for the image of ±·F —2 under the isomorphism

H 1 (F, µ2 ) F — /F —2 .

The following theorem shows that the Galois cohomology of µ2 also has a

far-reaching relationship with quadratic forms:

(30.2) Theorem. Let F be a ¬eld of characteristic di¬erent from 2. For ± 1 , . . . ,

±n ∈ F — , the cup product (±1 ) ∪ · · · ∪ (±n ) ∈ H n (F, µ2 ) depends only on the

isometry class of the P¬ster form ±1 , . . . , ±n . We may therefore de¬ne a map

en on the set of isometry classes of n-fold P¬ster forms by setting

en ± 1 , . . . , ±n = (±1 ) ∪ · · · ∪ (±n ).

Moreover, the map en is injective: n-fold P¬ster forms π, π are isometric if and

only if en (π) = en (π ).

Reference: The ¬rst assertion appears in Elman-Lam [?, (3.2)], the second in

Arason-Elman-Jacob [?, Theorem 1] for n ¤ 4 (see also Lam-Leep-Tignol [?, The-

orem A5] for n = 3). The second assertion was proved by Rost (unpublished) for

n ¤ 6, and a proof for all n was announced by Voevodsky in 1996. (In this book,

the statement above is not used for n > 3.)

By combining the interpretations of Galois cohomology in terms of algebras

and in terms of quadratic forms, we translate the results of §?? to obtain in §?? a

complete set of cohomological invariants for central simple F -algebras with unitary

involution of degree 3. We also give a cohomological classi¬cation of cubic ´tale

e

F -algebras. The cohomological invariants discussed in §?? use cohomology groups

with twisted coe¬cients which are introduced in §??. Cohomological invariants

will be discussed in greater generality in §??.

414 VII. GALOIS COHOMOLOGY

Before carrying out this programme, we observe that there is an analogue of the

Kummer sequence in characteristic p. If char F = p, the Artin-Schreier sequence is

the exact sequence of group schemes

„˜

0 ’ Z/pZ ’ Ga ’ Ga ’ 0

’

where „˜(x) = xp ’ x. The normal basis theorem (Bourbaki [?, §10]) shows that the

additive group of any ¬nite Galois extension L/F is an induced Gal(L/F )-module,

hence H Gal(L/F ), L = 0 for all > 0. Therefore,

H (F, Ga ) = 0 for all >0

and the cohomology sequence induced by the Artin-Schreier exact sequence yields

H 1 (F, Z/pZ) F/„˜(F ) and H (F, Z/pZ) = 0 for

(30.3) ≥ 2.

30.A. Cyclic algebras. The construction of cyclic algebras, already intro-

duced in §?? in the particular case of degree 3, has a close relation with Galois

cohomology which is described next.

Let n be an arbitrary integer and let L be a Galois (Z/nZ)-algebra over F . We

set ρ = 1 + nZ ∈ Z/nZ. For a ∈ F — , the cyclic algebra (L, a) is

(L, a) = L • Lz • · · · • Lz n’1

where z n = a and z = ρ( )z for ∈ L. Every cyclic algebra (L, a) is central

simple of degree n over F . Moreover, it is easy to check, using the Skolem-Noether

theorem, that every central simple F -algebra of degree n which contains L has the

form (L, a) for some a ∈ F — (see Albert [?, Chapter 7, §1]).

We now give a cohomological interpretation of this construction. Let [L] ∈

1

H (F, Z/nZ) be the cohomology class corresponding to L by (??). Since the “-

action on Z/nZ is trivial, we have

H 1 (F, Z/nZ) = Z 1 (F, Z/nZ) = Hom(“, Z/nZ),

so [L] is a continuous homomorphism “ ’ Z/nZ. If L is a ¬eld (viewed as a sub¬eld

of Fsep ), this homomorphism is surjective and its kernel is the absolute Galois group

of L. For σ ∈ “, de¬ne f (σ) ∈ {0, 1, . . . , n ’ 1} by the condition

[L](σ) = f (σ) + nZ ∈ Z/nZ.

Since [L] is a homomorphism, we have f (σ„ ) ≡ f (σ) + f („ ) mod n.

Now, assume that n is not divisible by char F . We may then use the Kummer

sequence to identify F — /F —n = H 1 (F, µn ) and n Br(F ) = H 2 (F, µn ). For this last

identi¬cation, we actually have two canonical (and opposite) choices (see §??); we

choose the identi¬cation a¬orded by the crossed product construction. Thus, the

image in H 2 (F, Gm ) of the class (L, a) ∈ H 2 (F, µn ) corresponding to (L, a) is

—

represented by the cocycle h : “ — “ ’ Fsep de¬ned as follows:

f (σ)+f („ )’f (σ„ ) /n

h(σ, „ ) = z f (σ) · z f („ ) · z ’f (σ„ ) = a .

(See Pierce [?, p. 277] for the case where L is a ¬eld.)

The bilinear pairing (Z/nZ) — µn (Fsep ) ’ µn (Fsep ) which maps (i + nZ, ζ) to

i

ζ induces a cup product

∪ : H 1 (F, Z/nZ) — H 1 (F, µn ) ’ H 2 (F, µn ).

§30. GALOIS COHOMOLOGY OF ROOTS OF UNITY 415

(30.4) Proposition. The homomorphism [L] ∈ H 1 (F, Z/nZ), the class (a) ∈

H 1 (F, µn ) corresponding to a · F —n under the identi¬cation H 1 (F, µn ) = F — /F —n

and the class (L, a) ∈ H 2 (F, µn ) corresponding to (L, a) by the crossed product

construction are related by

(L, a) = [L] ∪ (a).

Proof : Since the canonical map H 2 (F, µn ) ’ H 2 (F, Gm ) is injective, it su¬ces to

compare the images of [L] ∪ (a) and of (L, a) in H 2 (F, Gm ). Let ξ ∈ Fsep satisfy

ξ n = a. The class (a) is then represented by the cocycle σ ’ σ(ξ)ξ ’1 , and the cup

—

product [L] ∪ (a) by the cocycle g : “ — “ ’ Fsep de¬ned by

[L](σ) f (σ)

g(σ, „ ) = σ „ (ξ)ξ ’1 = σ „ (ξ)ξ ’1 for σ, „ ∈ “.

Consider the function c : “ ’ Fsep given by cσ = σ(ξ)f (σ) . We have

—

f (σ)+f („ )’f (σ„ ) /n

g(σ, „ )cσ σ(c„ )c(σ„ )’1 = σ„ (ξ)f (σ)+f („ )’f (σ„ ) = a .

Therefore, the cocyles g and h are cohomologous in H 2 (F, Gm ).

(30.5) Corollary. For a, b ∈ F — ,

in H 2 (F, µn ).

(L, a) — (L, b) = (L, ab)

In order to determine when two cyclic algebras (L, a), (L, b) are isomorphic, we

¬rst give a criterion for a cyclic algebra to be split:

(30.6) Proposition. Let L be a Galois (Z/nZ)-algebra over F and a ∈ F — . The

cyclic algebra (L, a) is split if and only if a ∈ NL/F (L— ).

Proof : A direct proof (without using cohomology) can be found in Albert [?, Theo-

rem 7.6] or (when L is a ¬eld) in Pierce [?, p. 278]. We next sketch a cohomological

proof. Let A = EndF L. We embed L into A by identifying ∈ L with the map

x ’ x. Let ρ ∈ A be given by the action of ρ = 1 + nZ ∈ Z/nZ on L. From

Dedekind™s lemma on the independence of automorphisms, we have

A = L • Lρ • · · · • Lρn’1

(so that A is a cyclic algebra A = (L, 1)). Let L1 = { u ∈ L | NL/F (u) = 1 }. For

u ∈ L1 , de¬ne ψ(u) ∈ Aut(A) by

i ii

ψ(u)( i iρ )= i iu ρ for 0, ... ∈ L.

n’1

Clearly, ψ(u) is the identity on L, hence ψ(u) ∈ Aut(A, L). In fact, every auto-

morphism of A which preserves L has the form ψ(u) for some u ∈ L1 , and the

restriction map Aut(A, L) ’ Aut(L) is surjective by the Skolem-Noether theorem,

hence there is an exact sequence

ψ

1 ’ L1 ’ Aut(A, L) ’ Aut(L) ’ 1.

’

More generally, there is an exact sequence of group schemes

ψ

1 ’ G1 ’ Aut(A, L) ’ Aut(L) ’ 1,

m,L ’

where G1 m,L is the kernel of the norm map NL/F : RL/F (Gm,L ) ’ Gm,L . Since

the restriction map Aut(A, L) ’ Aut(L) is surjective, the induced cohomology

sequence shows that the map ψ 1 : H 1 (F, G1 ) ’ H 1 F, Aut(A, L) has trivial

m,L

416 VII. GALOIS COHOMOLOGY

kernel. On the other hand, by Shapiro™s lemma and Hilbert™s Theorem 90, the

cohomology sequence induced by the exact sequence

NL/F

1 ’ G1 ’ RL/F (Gm,L ) ’’ ’ Gm ’ 1

’

m,L

yields an isomorphism H 1 (F, G1 ) F — /NL/F (L— ).

m,L

1

Recall from (??) that H F, Aut(A, L) is in one-to-one correspondence with

the set of isomorphism classes of pairs (A , L ) where A is a central simple F -

algebra of degree n and L is an ´tale F -subalgebra of A of dimension n. The

e

map ψ associates to a · NL/F (L ) ∈ F — /NL/F (L— ) the isomorphism class of the

1 —

pair (L, a), L . The algebra (L, a) is split if and only if this isomorphism class is

the base point in H 1 F, Aut(A, L) . Since ψ 1 has trivial kernel, the proposition

follows.

(30.7) Corollary. Two cyclic algebras (L, a) and (L, b) are isomorphic if and only

if a/b ∈ NL/F (L— ).

Proof : This readily follows from (??) and (??).

30.B. Twisted coe¬cients. The automorphism group of Z is the group S2

of two elements, generated by the automorphism x ’ ’x. We may use cocycles in

Z 1 (F, S2 ) = H 1 (F, S2 ) to twist the (trivial) action of “ = Gal(Fsep /F ) on Z, hence

also on every “-module. Since H 1 (F, S2 ) is in one-to-one correspondence with the

isomorphism classes of quadratic ´tale F -algebras by (??), we write Z[K] for the

e

module Z with the action of “ twisted by the cocycle corresponding to a quadratic

´tale F -algebra K. Thus, Z[F —F ] = Z and, if K is a ¬eld with absolute Galois

e

group “0 ‚ “, the “-action on Z[K] is given by

x if σ ∈ “0

σx =

’x if σ ∈ “ “0 .

We may similarly twist the “-action of any “-module M . We write M[K] for the

twisted module; thus

M[K] = Z[K] —Z M.

Clearly, M[K] = M if 2M = 0.

Since S2 is commutative, there is a group structure on the set H 1 (F, S2 ), which

can be transported to the set of isomorphism classes of quadratic ´tale F -algebras.

e

For K, K quadratic ´tale F -algebras, the sum [K]+[K ] is the class of the quadratic

e

´tale F -algebra

e

(30.8) K — K = { x ∈ K — K | (ιK — ιK )(x) = x }

where ιK , ιK are the nontrivial automorphisms of K and K respectively. We say

that K — K is the product algebra of K and K . If char F = 2, we have

H 1 (F, S2 ) = H 1 (F, µ2 ) F — /F —2 .

To any ± ∈ F — , the corresponding quadratic ´tale algebra is

e

√

F ( ±) = F [X]/(X 2 ’ ±).

In this case

√ √

√

F ( ±) — F ( ± ) = F ( ±± ).

§30. GALOIS COHOMOLOGY OF ROOTS OF UNITY 417

If char F = 2, we have

H 1 (F, S2 ) = H 1 (F, Z/2Z) F/„˜(F ).

To any ± ∈ F , the corresponding quadratic ´tale algebra is

e

F „˜’1 (±) = F [X]/(X 2 + X + ±).

In this case

F „˜’1 (±) — F „˜’1 (± ) = F „˜’1 (± + ± ) .

A direct computation shows:

(30.9) Proposition. Let K, K be quadratic ´tale F -algebras. For any “-module

e

M,

M[K][K ] = M[K—K ] .

In particular, M[K][K] = M .

Now, let K be a quadratic ´tale F -algebra which is a ¬eld and let “0 ‚ “ be the

e

absolute Galois group of K. Let M be a “-module. Recall from §?? the induced

“-module Ind“0 M , which in this case can be de¬ned as

“

Ind“0 M = Map(“/“0 , M )

“

(see (??)). The map µ : Ind“0 M ’ M which takes f to x∈“/“0 f (x) is a “-

“

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 ,