Ind“0 : A0 “Tors “0 ’ (Ind“0 A0 )“Tors “ .

“ “

On the other hand, the “0 -homomorphism π : Ind“0 A0 ’ A0 yields a functor

“

π— : (Ind“0 A0 )“Tors “ ’ A0 “Tors “0 ,

“

as explained above.

(28.17) Proposition. Let “0 be a closed subgroup of the pro¬nite group “, let A0

be a “0 -group and A = Ind“0 A0 . The functors Ind“0 and π— de¬ne an equivalence

“ “

of categories

A0 “Tors “0 ≡ A“Tors “ .

Proof : Let P0 ∈ A0 “Tors “0 and let P = Ind“0 P0 be the induced A-torsor. Consider

“

the map

given by g(p, a0 ) = p(1)a0 .

g : P — A 0 ’ P0

For any a ∈ A one has

g (p, a0 )a = g pa , π(a’1 )a0 = g pa , a(1)’1 a0 = p(1)a0 = g(p, a0 ),

i.e., g is compatible with the right A-action on P — A0 and hence factors through

a map on the orbit space

g : π— (P ) = (P — A0 )/A ’ P0 .

It is straightforward to check that g is a homomorphism of A0 -torsors and hence is

necessarily an isomorphism. Thus, π— —¦ Ind“0 is naturally equivalent to the identity

“

on A0 “Tors “0 .

On the other hand, let P ∈ A“Tors “ . We denote the orbit in π— (P ) = (P —

A0 )/A of a pair (p, a0 ) by (p, a0 )A . Consider the map

h : P ’ Ind“0 π— (P )

“

which carries p ∈ P to the map hp de¬ned by hp (σ) = (σp, 1)A . For any a ∈ A one

has

A A

hpa (σ) = σ(pa ), 1 = σ(p)σa , 1 for σ ∈ “.

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 391

σa

Since σ(p)σa , 1 = σ(p), σa(1) , the A-orbits of σ(p)σa , 1 and σ(p), σa(1)

coincide. We have σ(p), σa(1) = σ(p), 1) σa(1), hence

hpa (σ) = hp (σ) σa(1) = ha (σ).

p

Thus h is a homomorphism (hence an isomorphism) of A-torsors, showing that

Ind“0 —¦π— is naturally equivalent to the identity on A“Tors “ .

“

By combining the preceding proposition with (??), we obtain:

(28.18) Corollary. With the same notation as in (??), there is a natural bijection

of pointed sets between H 1 (“0 , A0 ) and H 1 (“, A).

(28.19) Remark. If A0 is a “-group (not just a “0 -group), there is a simpler

description of the “-group Ind“0 A0 : let “/“0 denote the set of left cosets of “0

“

in “. On the group Map(“/“0 , A0 ) of continuous maps “/“0 ’ A0 , consider the

“-action given by σf (x) = σf (σ ’1 x). The “-group Map(“/“0 , A0 ) is naturally

isomorphic to Ind“0 A0 . For, there are mutually inverse isomorphisms

“

± : Ind“0 A0 ’ Map(“/“0 , A0 ) given by ±(a)(σ · “0 ) = σa(σ ’1 )

“

and

β : Map(“/“0 , A0 ) ’ Ind“0 A0 given by β(f )(σ) = σf (σ ’1 · “0 ).

“

(28.20) Example. Let A be a “-group and n be an integer, n ≥ 1. We let the

symmetric group Sn act by permutations on the product An of n copies of A, and

we let “ act trivially on Sn . Any continuous homomorphism ρ : “ ’ Sn is a 1-

cocycle in Z 1 (“, Sn ). It yields a 1-cocycle ± : “ ’ Aut An via the action of Sn on

An , and we may consider the twisted group (An )± .

Assume that “ acts transitively via ρ on the set X = {1, 2, . . . , n}. Let “0 ‚

“ be the stabilizer of 1 ∈ X. The set X is then identi¬ed with “/“0 . It is

straightforward to check that (An )± is identi¬ed with Map(“/“0 , A) = Ind“0 A.

“

Consider the semidirect product An Sn and the exact sequence

1 ’ A n ’ An Sn ’ Sn ’ 1.

By (??) and (??), there is a canonical bijection between the ¬ber of the map

H 1 (“, An Sn ) ’ H 1 (“, Sn ) over [ρ] and the orbit set in H 1 (“0 , A) of the group

(Sn )“ , which is the centralizer of the image of ρ in Sn .

ρ

§29. Galois Cohomology of Algebraic Groups

In this section, the pro¬nite group “ is the absolute Galois group of a ¬eld F ,

i.e., “ = Gal(Fsep /F ) where Fsep is a separable closure of F . If A is a discrete

“-group, we write H i (F, A) for H i (“, A).

Let G be a group scheme over F . The Galois group “ acts continuously on

the discrete group G(Fsep ). Hence H i F, G(Fsep ) is de¬ned for i = 0, 1, and it is

de¬ned for all i ≥ 2 if G is a commutative group scheme. We use the notation

H i (F, G) = H i F, G(Fsep ) .

In particular, H 0 (F, G) = G(F ).

Every group scheme homomorphism f : G ’ H induces a “-homomorphism

G(Fsep ) ’ H(Fsep ) and hence a homomorphism of groups (resp. of pointed sets)

f i : H i (F, G) ’ H i (F, H)

392 VII. GALOIS COHOMOLOGY

for i = 0 (resp. i = 1). If 1 ’ N ’ G ’ S ’ 1 is an exact sequence of algebraic

group schemes such that the induced sequence of “-homomorphisms

1 ’ N (Fsep ) ’ G(Fsep ) ’ S(Fsep ) ’ 1

is exact (this is always the case if N is smooth, see (??)), we have a connecting

map δ 0 : S(F ) ’ H 1 (F, N ), and also, if N lies in the center of G, a connecting

map δ 1 : H 1 (F, S) ’ H 2 (F, N ). We may thus apply the techniques developed in

the preceding section.

Our main goal is to give a description of the pointed set H 1 (F, G) for various

algebraic groups G. We ¬rst explain the main technical tool.

Let G be a group scheme over F and let ρ : G ’ GL(W ) be a representation

with W a ¬nite dimensional F -space. Fix an element w ∈ W , and identify W with

an F -subspace of Wsep = W —F Fsep . An element w ∈ Wsep is called a twisted ρ-

form of w if w = ρsep (g)(w) for some g ∈ G(Fsep ). As in §??, consider the category

A(ρ, w) whose objects are the twisted ρ-forms of w and whose maps w ’ w are

the elements g ∈ G(Fsep ) such that ρsep (g)(w ) = w . This category is a connected

groupoid. On the other hand, let A(ρ, w) denote the groupoid whose objects are the

twisted ρ-forms of w which lie in W , and whose maps w ’ w are the elements

g ∈ G(F ) such that ρ(g)(w ) = w . Thus, if X denotes the “-set of objects of

A(ρ, w), the set X “ = H 0 (“, X) is the set of objects of A(ρ, w). Moreover, the set

of orbits of G(F ) in X “ is the set of isomorphism classes Isom A(ρ, w) . It is a

pointed set with the isomorphism class of w as base point.

Let AutG (w) denote the stabilizer of w; it is a subgroup of the group scheme

G. Since G(Fsep ) acts transitively on X, the “-set X is identi¬ed with the set of left

cosets of G(Fsep ) modulo AutG (w)(Fsep ). Corollary (??) yields a natural bijection

of pointed sets between the kernel of H 1 F, AutG (w) ’ H 1 (F, G) and the orbit

set X “ /G(F ). We thus obtain:

(29.1) Proposition. If H 1 (F, G) = 1, there is a natural bijection of pointed sets

∼

Isom A(ρ, w) ’ H 1 F, AutG (w)

’

which maps the isomorphism class of w to the base point of H 1 F, AutG (w) .

The bijection is given by the following rule: for w ∈ A(ρ, w), choose g ∈

G(Fsep ) such that ρsep (g)(w) = w , and let ±σ = g ’1 · σ(g). The map ± : “ ’

AutG (w)(Fsep ) is a 1-cocycle corresponding to w . On the other hand, since

H 1 (F, G) = 1, any 1-cocycle ± ∈ Z 1 F, AutG (w) is cohomologous to the base

point in Z 1 (F, G), hence ±σ = g ’1 · σ(g) for some g ∈ G(Fsep ). The corresponding

object in A(ρ, w) is ρsep (g)(w).

In order to apply the proposition above, we need examples of group schemes

G for which H 1 (F, G) = 1. Hilbert™s Theorem 90, which is discussed in the next

subsection, provides such examples. We then apply (??) to give descriptions of the

¬rst cohomology set for various algebraic groups.

29.A. Hilbert™s Theorem 90 and Shapiro™s lemma.

(29.2) Theorem (Hilbert™s Theorem 90). For any separable associative F -algebra

A,

H 1 F, GL1 (A) = 1.

In particular H 1 (F, Gm ) = 1.

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 393

Proof : Let ± ∈ Z 1 (“, A— ). We de¬ne a new action of “ on Asep by putting

sep

γ — a = ±γ · γ(a) for γ ∈ “ and a ∈ Asep .

This action is continuous and semilinear, i.e. γ — (ax) = (γ — a)γ(x) for γ ∈ “,

a ∈ Asep and x ∈ Fsep . Therefore, we may apply the Galois descent Lemma (??):

if

U = { a ∈ Asep | γ — a = a for all γ ∈ “ },

the map

f : U —F Fsep ’ Asep given by f (u — x) = ux

is an isomorphism of Fsep -vector spaces. For γ ∈ “, a ∈ Asep and a0 ∈ A we have

γ — (aa0 ) = (γ — a)a0

since γ(a0 ) = a0 . Therefore, U is a right A-submodule of Asep , hence U — Fsep is a

right Asep -module, and f is an isomorphism of right Asep -modules.

Since A is separable, we have A = A1 — · · · — Am for some ¬nite dimensional

simple F -algebras A1 , . . . , Am , and the A-module U decomposes as U = U1 — · · · —

Um where each Ui is a right Ai -module. Since modules over simple algebras are

classi¬ed by their reduced dimension (see (??)), and since U — Fsep Asep , we have

Ui Ai for i = 1, . . . , m, hence the right A-modules U and A are isomorphic.

Choose an A-module isomorphism g : A ’ U . The composition f —¦ (g — IdFsep ) is

an Asep -module automorphism of Asep and is therefore left multiplication by the

invertible element a = g(1) ∈ A— . Since a ∈ U we have

sep

a = γ — a = ±γ · γ(a) for all γ ∈ “,

hence ±γ = a · γ(a)’1 , showing that ± is a trivial cocycle.

(29.3) Remark. It follows from (??) that H 1 Gal(L/F ), GLn (L) = 1 for any

¬nite Galois ¬eld extension L/F , a result due to Speiser [?] (and applied by Speiser

to irreducible representations of ¬nite groups). Suppose further that L is cyclic

Galois over F , with θ a generator of the Galois group G = Gal(L/F ). Let c be

a cocycle with values in Gm (L) = L— . Since cθi = cθ · . . . · θi’1 (cθ ), the cocycle

is determined by its value on θ, and NL/F (cθ ) = 1. Conversely any ∈ L— with

NL/F ( ) = 1 de¬nes a cocycle such that cθ = . Thus, by (??), any ∈ L— such

that NL/F ( ) = 1 is of the form = aθ(a)’1 . This is the classical Theorem 90 of

Hilbert (see [?, §54]).

(29.4) Corollary. Suppose A is a central simple F -algebra. The connecting map

in the cohomology sequence associated to the exact sequence

Nrd

1 ’ SL1 (A) ’ GL1 (A) ’ ’ Gm ’ 1

’

induces a canonical bijection of pointed sets

H 1 F, SL1 (A) F — / Nrd(A— ).

Let V be a ¬nite dimensional F -vector space. It follows from (??) that H 1 F, GL(V ) =

1 since GL1 (A) = GL(V ) for A = EndF (V ). A similar result holds for ¬‚ags:

(29.5) Corollary. Let F : V = V0 ⊃ V1 ⊃ · · · ⊃ Vk be a ¬‚ag of ¬nite dimensional

F -vector spaces and let G be its group scheme of automorphisms over F . Then

H 1 (F, G) = 1.

394 VII. GALOIS COHOMOLOGY

Proof : Let ± ∈ Z 1 “, G(Fsep ) . We de¬ne a new action of “ on Vsep by

γ — v = ±γ (γv) for γ ∈ “ and v ∈ Vsep .

This action is continuous and semilinear, hence we are in the situation of Galois

descent. Moreover, the action preserves (Vi )sep for i = 0, . . . , k. Let

Vi = { v ∈ (Vi )sep | γ — v = v for all γ ∈ “ }.

Each Vi is an F -vector space and we may identify (Vi )sep = (Vi )sep by (??). Clearly,

∼

F : V = V0 ⊃ V1 ⊃ · · · ⊃ Vk is a ¬‚ag (see (??)). Let f : F ’ F be an ’

∼

isomorphism of ¬‚ags, i.e., an isomorphism of F -vector spaces V ’ V such that

’

f (Vi ) = Vi for all i. Extend f by linearity to an isomorphism of Fsep -vector

∼

spaces Vsep ’ Vsep = Vsep , and write also f for this extension. Then f is an

’

automorphism of Fsep , hence f ∈ G(Fsep ). Moreover, for v ∈ V we have f (v) ∈ V ,

hence σ f (v) = ±’1 f (v) . Therefore,

σ

σ

f (v) = σ f (σ ’1 v) = ±’1 f (v) for all σ ∈ “.

σ

It follows that ±σ = f —¦ σf ’1 for all σ ∈ “, hence ± is cohomologous to the trivial

cocycle.

Corollary (??) also follows from the fact that, if H is a parabolic subgroup of a

connected reductive group G, then the map H 1 (F, H) ’ H 1 (F, G) is injective (see

Serre [?, III, 2.1, Exercice 1]).

The next result is classical and independently due to Eckmann, Faddeev, and

Shapiro. It determines the cohomology sets with coe¬cients in a corestriction

RL/F (G).

Let L/F be a ¬nite separable extension of ¬elds and let G be a group scheme

de¬ned over L. By ¬xing an embedding L ’ Fsep , we consider L as a sub¬eld of

Fsep . Let “0 = Gal(Fsep /L) ‚ “ and let A = L[G], so that

RL/F (G)(Fsep ) = G(L —F Fsep ) = HomAlg L (A, L — Fsep ).

For h ∈ HomAlg L (A, L — Fsep ), de¬ne •h : “ ’ HomAlg L (A, Fsep ) = G(Fsep ) by

•h (γ) = γ —¦ h, where γ( — x) = γ(x) for γ ∈ “, ∈ L and x ∈ Fsep . The map

•h is continuous and satis¬es •h (γ0 —¦ γ) = γ0 —¦ •h (γ) for γ0 ∈ “0 and γ ∈ “, hence

•h ∈ Ind“0 G(Fsep ). Since L —F Fsep Map(“/“0 , Fsep ) by (??), the map h ’ •h

“

de¬nes an isomorphism of “-groups

∼

RL/F (G)(Fsep ) ’ Ind“0 G(Fsep ).

’ “

The following result readily follows by (??):

(29.6) Lemma (Eckmann, Faddeev, Shapiro). Let L/F be a ¬nite separable ex-

tension of ¬elds and let G be a group scheme de¬ned over L. There is a natural

bijection of pointed sets

∼

H 1 F, RL/F (G) ’ H 1 (L, G).

’

The same result clearly holds for H 0 -groups, since

H 0 F, RL/F (G) = RL/F (G)(F ) = G(L) = H 0 (L, G).

If G is a commutative group scheme, there is a group isomorphism

∼

H i F, RL/F (G) ’ H i (L, G)

’

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 395

for all i ≥ 0. (See Brown [?, Chapter 3, Proposition (6.2)].) It is the composition

fi

res

H F, RL/F (G) ’’ H L, RL/F (G)L ’ H i (L, G)

i i

’ ’

where f : RL/F (G)L ’ G is the group scheme homomorphism corresponding to the

identity on RL/F (G) under the bijection

∼

HomF RL/F (G), RL/F (G) ’ HomL RL/F (G)L , G

’

of (??).

(29.7) Remark. If L/F is an ´tale algebra (not necessarily a ¬eld), one de¬nes

e

the pointed set H (L, G) as the product of the H 1 (Li , G) where the Li are the ¬eld

1

extensions of F such that L = Li . Lemma (??) remains valid in this setting (see

Remark (??) for the de¬nition of RL/F ).

29.B. Classi¬cation of algebras. We now apply Proposition (??) and Hilbert™s

Theorem 90 to show how ´tale and central simple algebras are classi¬ed by H 1 -

e

cohomology sets.

Let A be a ¬nite dimensional algebra over F . Multiplication in A yields a linear

map w : A —F A ’ A. Let W = HomF (A — A, A) and G = GL(A), the linear group

of A where A is viewed as an F -vector space. Consider the representation

ρ : G ’ GL(W )

given by the formula

ρ(g)(•)(x — y) = g —¦ • g ’1 (x) — g ’1 (y)

for g ∈ G, • ∈ W and x, y ∈ A. A linear map g ∈ G is an algebra automorphism

of A if and only if ρ(g)(w) = w, hence the group scheme AutG (w) coincides with

the group scheme Autalg (A) of all algebra automorphisms of A. A twisted ρ-form

of w is an algebra structure A on the F -vector space A such that the Fsep -algebras

Asep and Asep are isomorphic. Thus, by Proposition (??) there is a bijection

F -isomorphism classes of F -algebras A

H 1 F, Autalg (A) .

such that the Fsep -algebras

(29.8) ←’

Asep and Asep are isomorphic

∼

The bijection is given explicitly as follows: if β : Asep ’ Asep is an Fsep -isomor-

’

phism, the corresponding cocycle is ±γ = β ’1 —¦ (Id — γ) —¦ β —¦ (Id — γ ’1 ). Conversely,

given a cocycle ± ∈ Z 1 “, Autalg (A) , we set

A = { x ∈ Asep | ±γ —¦ (Id — γ)(x) = x for all γ ∈ “ }.

We next apply this general principle to ´tale algebras and to central simple

e

algebras.

´

Etale algebras. The F -algebra A = F — · · · — F (n copies) is ´tale of dimen-

e

sion n. If { ei | i = 1, . . . , n } is the set of primitive idempotents of A, any F -algebra

automorphism of A is determined by the images of the ei . Thus Autalg (A) is the

constant symmetric group Sn . Proposition (??) shows that the ´tale F -algebras of

e

dimension n are exactly the twisted forms of A. Therefore, the preceding discussion

with A = F — · · · — F yields a natural bijection

F -isomorphism classes of

H 1 (F, Sn ).

(29.9) ←’

´tale F -algebras of degree n

e

396 VII. GALOIS COHOMOLOGY

Since the “-action on Sn is trivial, the pointed set H 1 (F, Sn ) coincides with the

set of conjugacy classes of continuous maps “ ’ Sn and hence also classi¬es iso-

morphism classes of “-sets X consisting of n elements (see (??)). The cocycle

γ : “ ’ Sn corresponds to the ´tale algebra L = Map(X, Fsep )“ where “ acts on

e

the set X via γ.

The sign map sgn : Sn ’ {±1} = S2 induces a map in cohomology

sgn1 : H 1 (F, Sn ) ’ H 1 (F, S2 ).

In view of (??) this map sends (the isomorphism class of) an ´tale algebra L to

e

(the isomorphism class of) its discriminant ∆(L).

Another interpretation of H 1 (F, Sn ) is given in Example (??):

H 1 (“, Sn ) Isom(Sn “Gal F ).

In fact, we may associate to any ´tale F -algebra L of dimension n its Galois S n -

e

closure Σ(L) (see (??)). This construction induces a canonical bijection between

the isomorphism classes of ´tale algebras of dimension n and isomorphism classes

e

of Galois Sn -algebras. Note however that Σ is not a functor: an F -algebra ho-

momorphism L1 ’ L2 which is not injective does not induce any homomorphism

Σ(L1 ) ’ Σ(L2 ).

Central simple algebras. Let A = Mn (F ), the matrix algebra of degree n.

Since every central simple F -algebra is split by Fsep , and since every F -algebra A

such that Asep Mn (Fsep ) is central simple (see (??)), the twisted forms of A are

exactly the central simple F -algebras of degree n. The Skolem-Noether theorem

(??) shows that every automorphism of A is inner, hence Autalg (A) = PGLn .

Therefore, as in (??), there is a natural bijection

F -isomorphism classes of

H 1 (F, PGLn ).

←’

central simple F -algebras of degree n

Consider the exact sequence:

(29.10) 1 ’ Gm ’ GLn ’ PGLn ’ 1.

By twisting all the groups by a cocycle in H 1 (F, PGLn ) corresponding to a central

simple F -algebra B of degree n, we get the exact sequence

1 ’ Gm ’ GL1 (B) ’ PGL1 (B) ’ 1.

Since H 1 F, GL1 (B) = 1 by Hilbert™s Theorem 90, it follows from Corollary (??)

that the connecting map

δ 1 : H 1 (F, PGLn ) ’ H 2 (F, Gm )

with respect to (??) is injective. The map δ 1 is de¬ned here as follows: if ±γ ∈

AutFsep Mn (Fsep ) is a 1-cocycle, choose cγ ∈ GLn (Fsep ) such that ±γ = Int(cγ )

(by Skolem-Noether). Then

cγ,γ = cγ · γcγ · c’1 ∈ Z 2 (F, Gm )

γγ

is the corresponding 2-cocycle. The δ 1 for di¬erent n™s ¬t together to induce an

injective homomorphism Br(F ) ’ H 2 (F, Gm ). To prove that this homomorphism

is surjective, we may reduce to the case of ¬nite Galois extensions, since for every

2-cocycle cγ,γ with values in Gm there is a ¬nite Galois extension L/F such that

c : “ — “ ’ Fsep factors through a 2-cocycle in Z 2 Gal(L/F ), L— . Thus, the

—

following proposition completes the proof that Br(F ) H 2 (F, Gm ):

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 397

(29.11) Proposition. Let L/F be a ¬nite Galois extension of ¬elds of degree n,

and let G = Gal(L/F ). The map

δ 1 : H 1 G, PGLn (L) ’ H 2 (G, L— )

is bijective.

Proof : Injectivity follows by the same argument as for the connecting map with

respect to (??). To prove surjectivity, choose c ∈ Z 2 (G, L— ) and let V be the

n-dimensional L-vector space

V= eσ L.

σ∈G

Numbering the elements of G, we may identify V = Ln , hence EndL (V ) = Mn (L)

and Aut EndL (V ) = PGLn (L). For σ ∈ G, let aσ ∈ EndL (V ) be de¬ned by

aσ (e„ ) = eσ„ cσ,„ .

We have

aσ —¦ σ(a„ )(eν ) = eσ„ ν cσ,„ ν σ(c„,ν ) = eσ„ ν cσ„,ν cσ,„ = aσ„ (eν )cσ,„

for all σ, „ , ν ∈ G, hence Int(aσ ) ∈ Aut EndL (V ) is a 1-cocycle whose image

under δ 1 is represented by the cocycle cσ,„ .

With the same notation as in the proof above, a central simple F -algebra Ac

corresponding to the 2-cocycle c ∈ Z 2 (G, L— ) is given by

Ac = { f ∈ EndL (V ) | aσ —¦ σ(f ) = f —¦ aσ for all σ ∈ G }.

This construction is closely related to the crossed product construction, which we

brie¬‚y recall: on the L-vector space

C= Lzσ

σ∈G

with basis (zσ )σ∈G , de¬ne multiplication by

zσ = σ( )zσ and zσ z„ = cσ,„ zσ„

for σ, „ ∈ G and ∈ L. The cocycle condition ensures that C is an associative

algebra, and it can be checked that C is central simple of degree n over F (see, e.g.,

Pierce [?, 14.1]).

For σ ∈ G and for ∈ L, de¬ne yσ , u ∈ EndL (V ) by

yσ (e„ ) = e„ σ c„,σ and u (e„ ) = e„ „ ( ) for „ ∈ G.

Computations show that yσ , u ∈ Ac , and

u —¦ yσ = yσ —¦ uσ( ) , yσ —¦ y„ = y„ σ —¦ uc„,σ .

uop —¦ yσ is an F -algebra

Therefore, the map C ’ Aop which sends op

σ zσ to

c σ

homomorphism, hence an isomorphism since C and Aop are central simple of de-

c

gree n. Thus, C Ac , showing that the isomorphism Br(F ) H 2 (F, Gm ) de¬ned

op

by the crossed product construction is the opposite of the isomorphism induced by

δ1 .

398 VII. GALOIS COHOMOLOGY

29.C. Algebras with a distinguished subalgebra. The same idea as in

§?? applies to pairs (A, L) consisting of an F -algebra A and a subalgebra L ‚ A.

∼ ∼

An isomorphism of pairs (A , L ) ’ (A, L) is an F -isomorphism A ’ A which

’ ’

∼

restricts to an isomorphism L ’ L. Let G ‚ GL(B) be the group scheme of

’

automorphisms of the ¬‚ag of vector spaces A ⊃ L. The group G acts on the space

HomF (A—F A, A) as in §?? and the group scheme AutG (m) where m : A—F A ’ A

is the multiplication map coincides with the group scheme Autalg (A, L) of all F -

algebra automorphisms of the pair (A, L). Since H 1 (F, G) = 1 by (??), there is by

Proposition (??) a bijection

F -isomorphism classes of pairs

H 1 F, Autalg (A, L) .

of F -algebras (A , L )

(29.12) ←’

such that (A , L )sep (A, L)sep

The map H 1 F, Autalg (A, L) ’ H 1 F, Autalg (A) induced by the inclusion of

Aut(A, L) in Aut(A) maps the isomorphism class of a pair (A , L ) to the isomor-

phism class of A . On the other hand, the map

H 1 F, Aut(A, L) ’ H 1 (F, Autalg (L)

induced by the restriction map Aut(A, L) ’ Aut(L) takes the isomorphism class

of (A , L ) to the isomorphism class of L .

Let AutL (A) be the kernel of the restriction map Aut(A, L) ’ Autalg (L).

In order to describe the set H 1 F, AutL (A) as a set of isomorphism classes as

in (??), let

W = HomF (A —F A, A) • HomF (L, A).

The group G = GL(A) acts on W as follows:

ρ(g)(ψ, •)(x — y, z) = g —¦ ψ g ’1 (x) — g ’1 (y) , g —¦ •(z)

for g ∈ G, ψ ∈ HomF (A —F A, A), • ∈ HomF (L, A), x, y ∈ A and z ∈ L. The

multiplication map m : A —F A ’ A and the inclusion i : L ’ A de¬ne an element

w = (m, i) ∈ W , and the group AutG (w) coincides with AutL (A). A twisted

form of w is a pair (A , •) where A is an F -algebra isomorphic to A over Fsep and

• : L ’ A is an F -algebra embedding of L in A . By Proposition (??) there is a

natural bijection

F -isomorphism classes of

H 1 F, AutL (A) .

pairs (A , •) isomorphic to

(29.13) ←’

the pair (A, i)sep over Fsep

The canonical map H 1 F, AutL (A) ’ H 1 F, Aut(A, L) takes the isomorphism

class of a pair (A , •) to the isomorphism class of the pair A , •(L) .

The preceding discussion applies in particular to separable F -algebras. If B is

a separable F -algebra with center Z, the restriction homomorphism

Autalg (B) = Autalg (B, Z) ’ Autalg (Z)

gives rise to the map of pointed sets

H 1 F, Autalg (B) ’ H 1 F, Autalg (Z)

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 399

which takes the class of a twisted form B of B to the class of its center Z .

On the other hand, the natural isomorphism AutZ (B) RZ/F PGL1 (B) and

Lemma (??) give a bijection of pointed sets

H 1 F, AutZ (B) H 1 F, RZ/F PGL1 (B) H 1 Z, PGL1 (B)

which takes the class of a pair (B , •) to the class of the Z-algebra B —Z Z (where

the tensor product is taken with respect to •).

29.D. Algebras with involution. Let (A, σ) be a central simple algebra

with involution (of any kind) over a ¬eld F . In this section, we give interpretations

for the cohomology sets

H 1 F, Aut(A, σ) , H 1 F, Sim(A, σ) , H 1 F, Iso(A, σ) .

We shall discuss separately the unitary, the symplectic and the orthogonal case,

but we ¬rst outline the general principles.

Let W = HomF (A — A, A) • EndF (A) and G = GL(A), the linear group of A

where A is viewed as an F -vector space. Consider the representation

ρ : G ’ GL(W )

de¬ned by

ρ(g)(•, ψ)(x — y, z) = g —¦ • g ’1 (x) — g ’1 (y) , g —¦ ψ —¦ g ’1 (z)

for g ∈ G, • ∈ HomF (A — A, A), ψ ∈ EndF (A) and x, y, z ∈ A. Let w =

(m, σ) ∈ W , where m is the multiplication map of A. The subgroup AutG (w)

of G coincides with the group scheme Aut(A, σ) of F -algebra automorphisms of

A commuting with σ. A twisted form of w is the structure of an algebra with

involution isomorphic over Fsep to (Asep , σsep ). Hence, by Proposition (??) there is

a natural bijection

F -isomorphism classes of

H 1 F, Aut(A, σ) .

F -algebras with involution (A , σ )

(29.14) ←’

isomorphic to (Asep , σsep ) over Fsep

Next, let W = EndF (A) and G = GL1 (A), the linear group of A (i.e., the

group of invertible elements in A). Consider the representation

ρ : G ’ GL(W )

de¬ned by

ρ (a)(ψ) = Int(a) —¦ ψ —¦ Int(a)’1 ,

for a ∈ G and ψ ∈ EndF (A). The subgroup AutG (σ) of G coincides with the

group scheme Sim(A, σ) of similitudes of (A, σ). A twisted ρ -form of σ is an invo-

lution of A which, over Fsep , is conjugate to σsep = σ — IdFsep . By Proposition (??)

and Hilbert™s Theorem 90 (see (??)), we get a bijection

conjugacy classes of involutions

H 1 F, Sim(A, σ) .

on A which over Fsep are

(29.15) ←’

conjugate to σsep

The canonical homomorphism Int : Sim(A, σ) ’ Aut(A, σ) induces a map

Int1 : H 1 F, Sim(A, σ) ’ H 1 F, Aut(A, σ)

400 VII. GALOIS COHOMOLOGY

which maps the conjugacy class of an involution σ to the isomorphism class of

(A, σ ).

Finally, recall from §?? that the group scheme Iso(A, σ) is de¬ned as the

stabilizer of 1 ∈ A under the action of GL1 (A) on A given by

ρ (a)(x) = a · x · σ(a)

for a ∈ GL1 (A) and x ∈ A. Twisted ρ -forms of 1 are elements s ∈ A for which

there exists a ∈ A— such that s = a · σ(a). We write Sym(A, σ) for the set of

sep

these elements,

Sym(A, σ) = { s ∈ A | s = a · σ(a) for some a ∈ A— } ‚ Sym(A, σ) © A— ,

sep

and de¬ne an equivalence relation on Sym(A, σ) by

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

s∼s

The equivalence classes are exactly the ρ -isomorphism classes of twisted forms of

1, hence Proposition (??) yields a canonical bijection

H 1 F, Iso(A, σ) .

(29.16) Sym(A, σ) /∼ ←’

The inclusion i : Iso(A, σ) ’ Sim(A, σ) induces a map

i1 : H 1 F, Iso(A, σ) ’ H 1 F, Sim(A, σ)

which maps the equivalence class of s ∈ Sym(A, σ) to the conjugacy class of the

involution Int(s) —¦ σ.

We now examine the various types of involutions separately.

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

involution. Let K be the center of B, which is a quadratic ´tale F -algebra, and let

e

n = deg(B, „ ). (The algebra B is thus central simple of degree n if K is a ¬eld, and

it is a direct product of two central simple F -algebras of degree n if K F — F .)

From (??), we readily derive a canonical bijection

F -isomorphism classes of

H 1 F, Aut(B, „ ) .

central simple F -algebras ←’

with unitary involution of degree n

We have an exact sequence of group schemes

f

1 ’ PGU(B, „ ) ’ Aut(B, „ ) ’ S2 ’ 1

’

where f is the restriction homomorphism to Autalg (K) = S2 . We may view the

group PGU(B, „ ) as the automorphism group of the pair (B, „ ) over K. As in

Proposition (??) (see also (??)) we obtain a natural bijection

F -isomorphism classes of triples (B , „ , •)

consisting of a central simple F -algebra

H 1 F, PGU(B, „ ) .

←’

with unitary involution (B , „ ) of degree n

∼

and an F -algebra isomorphism • : Z(B ) ’ K ’

By Proposition (??) the group Autalg (K) acts transitively on each ¬ber of the

map

H 1 F, PGU(B, „ ) ’ H 1 F, Aut(B, „ ) .

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 401

The ¬ber over a pair (B , „ ) consists of the triples (B , „ , IdK ) and (B , „ , ι), where

ι is the nontrivial automorphism of K/F . These triples are isomorphic if and only

if PGU(B , „ ) AutF (B , „ ).

After scalar extension to Fsep , we have Bsep Mn (Fsep ) — Mn (Fsep )op , and all

the unitary involutions on Bsep are conjugate to the exchange involution µ by (??).

Therefore, (??) specializes to a bijection

conjugacy classes of

H 1 F, GU(B, „ ) .

unitary involutions on B ←’

which are the identity on F

The exact sequence

1 ’ RK/F (Gm,K ) ’ GU(B, „ ) ’ PGU(B, „ ) ’ 1

induces a connecting map in cohomology

δ 1 : H 1 F, PGU(B, „ ) ’ H 2 F, RK/F (Gm,K ) = H 2 (K, Gm,K ) = Br(K)

where the identi¬cation H 2 F, RK/F (Gm,K ) = H 2 (K, Gm,K ) is given by Shapiro™s

lemma and the identi¬cation H 2 (K, Gm,K ) = Br(K) by the connecting map in the

cohomology sequence associated to

1 ’ Gm,K ’ GLn,K ’ PGLn,K ’ 1,

see §??. Under δ 1 , the class of a triple (B , „ , •) is mapped to the Brauer class

[B —K K] · [B]’1 , where the tensor product is taken with respect to •.

Our next goal is to give a description of H 1 F, U(B, „ ) . Every symmetric

element s ∈ Sym Mn (Fsep ) — Mn (Fsep )op , µ has the form

s = (m, mop ) = (m, 1op ) · µ(m, 1op )

for some m ∈ Mn (Fsep ). Therefore, the set Sym(B, „ ) of (??) is the set of sym-

metric units,

Sym(B, „ ) = Sym(B, „ )— (= Sym(B, „ ) © B — ),

and (??) yields a canonical bijection

Sym(B, „ )— /∼ H 1 F, U(B, „ ) .

(29.17) ←’

By associating with every symmetric unit u ∈ Sym(B, „ )— the hermitian form

u’1 : B — B ’ B

de¬ned by u’1 (x, y) = „ (x)u’1 y, it follows that H 1 F, U(B, „ ) classi¬es her-

mitian forms on B-modules of rank 1 up to isometry.

In order to describe the set H 1 F, SU(B, „ ) , consider the representation

ρ : GL1 (B) ’ GL(B • K)

given by

ρ(b)(x, y) = b · x · „ (b), Nrd(b)y

for b ∈ GL1 (B), x ∈ B and y ∈ K. Let w = (1, 1) ∈ B • K. The group AutG (w)

coincides with SU(B, „ ). Clearly, twisted forms of w are contained in the set31

SSym(B, „ )— = { (s, z) ∈ Sym(B, „ )— — K — | NrdB (s) = NK/F (z) }.

31 This set plays an essential rˆle in the Tits construction of exceptional simple Jordan alge-

o

bras (see § ??).

402 VII. GALOIS COHOMOLOGY

Over Fsep , we have Bsep Mn (Fsep ) — Mn (Fsep )op and we may identify „sep to the

exchange involution µ. Thus, for every (s, z) ∈ SSym(Bsep , „sep )— , there are m ∈

Mn (Fsep ) and z1 , z2 ∈ Fsep such that s = (m, mop ), z = (z1 , z2 ) and det m = z1 z2 .

—

Let m1 ∈ GLn (Fsep ) be any matrix such that det m1 = z1 , and let m2 = m’1 m. 1

Then

s = (m1 , mop ) · µ(m1 , mop ) and z = Nrd(m1 , mop ),

2 2 2

hence (s, z) = ρsep (m1 , mop )(w). Therefore, SSym(B, „ )— is the set of twisted

2

ρ-forms of w.

De¬ne an equivalence relation ≈ on SSym(B, „ )— by

if and only if s = b · s · „ (b) and z = NrdB (b)z for some b ∈ B —

(s, z) ≈ (s , z )

so that the equivalence classes under ≈ are exactly the ρ-isomorphism classes of

twisted forms. Proposition (??) yields a canonical bijection

SSym(B, „ )— /≈ H 1 F, SU(B, „ ) .

(29.18) ←’

The natural map of pointed sets

H 1 F, SU(B, „ ) ’ H 1 F, U(B, „ )

takes the class of (s, z) ∈ SSym(B, „ )— to the class of s ∈ Sym(B, „ )— .

There is an exact sequence

Nrd

1 ’ SU(B, „ ) ’ U(B, „ ) ’ ’ G1

’ m,K ’ 1

where

NK/F

G1

m,K = ker RK/F (Gm,K ) ’ ’ ’ Gm,F

’’

(hence G1 (F ) = K 1 is the group of norm 1 elements in K). The connecting map

m,K

G1 (F ) ’ H 1 F, SU(B, „ )

m,K

takes x ∈ G1 (F ) ‚ K — to the class of the pair (1, x).

m,K

(29.19) Example. Suppose K is a ¬eld and let (V, h) be a hermitian space over K

(with respect to the nontrivial automorphism ι of K/F ). We write simply U(V, h)

for U EndK (V ), σh and SU(V, h) for SU EndK (V ), σh . As in (??) and (??), we

have canonical bijections

—

H 1 F, U(V, h) ,

Sym EndK (V ), σh /∼ ←’

—

H 1 F, SU(V, h) .

SSym EndK (V ), σh /≈ ←’

—

The set Sym EndK (V ), σh /∼ is also in one-to-one correspondence correspon-

dence with the set of isometry classes of nonsingular hermitian forms on V , by

—

mapping s ∈ Sym EndK (V ), σh to the hermitian form hs : V — V ’ K de¬ned

by

hs (x, y) = h s’1 (x), y = h x, s’1 (y)

for x, y ∈ V . Therefore, we have a canonical bijection of pointed sets

isometry classes of

H 1 F, U(V, h)

nonsingular hermitian ←’

forms on V

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 403

where the base point of H 1 F, U(V, h) corresponds to the isometry class of h.

To give a similar interpretation of H 1 F, SU(V, h) , observe that for every

unitary involution „ on EndK (V ) and every y ∈ K — such that NK/F (y) = 1 there

exists u ∈ U EndK (V ), „ such that det(u) = y. Indeed, „ is the adjoint involution

with respect to some hermitian form h . If (e1 , . . . , en ) is an orthogonal basis of V

for h , we may take for u the endomorphism which leaves ei invariant for i = 1, . . . ,

n ’ 1 and maps en to en y. From this observation, it follows that the canonical map

— —

SSym EndK (V ), σh /≈ ’ Sym EndK (V ), σh /∼

—

given by (s, z) ’ s is injective. For, suppose (s, z), (s , z ) ∈ SSym EndK (V ), σh

are such that s ∼ s , and let b ∈ EndK (V )— satisfy

s = b · s · σh (b).

Since det(s) = NK/F (z) and det(s ) = NK/F (z ), it follows that

NK/F (z ) = NK/F z det(b) .

Choose u ∈ U EndK (V ), Int(s ) —¦ σh such that det(u) = z z ’1 det(b)’1 . Then

s = u · s · σh (u) = ub · s · σh (ub) and z = z det(ub),

hence (s , z ) ≈ (s, z).

As a consequence, the canonical map H 1 F, SU(V, h) ’ H 1 F, U(V, h) is

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

—

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

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

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

Therefore, we have a canonical bijection of pointed sets

isometry classes of nonsingular

H 1 F, SU(V, h) .

hermitian forms h on V ←’

with disc h = disc h

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

lution (a, bop ) ’ (b, aop ). We have

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

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

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

H 1 F, U(B, µ) = H 1 F, SU(B, µ) = 1.

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

we may identify

Aut Mn (F ) — Mn (F )op , µ = PGLn S2

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

The exact sequence

1 ’ PGLn ’ PGLn S2 ’ S2 ’ 1

induces the following exact sequence in cohomology:

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

404 VII. GALOIS COHOMOLOGY

This cohomology sequence corresponds to

central simple central simple F -algebras quadratic

F -algebras of with unitary involution ´tale

e

’ ’

degree n of degree n F -algebras

A ’ (A — Aop , µ) B ’ Z(B)

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

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

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

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

e

1

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

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

∼

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

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

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

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

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

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

(29.22)

F -isomorphism classes of

H 1 F, PGSp(A, σ)

central simple F -algebras of degree 2n ←’

with symplectic involution

conjugacy classes of

H 1 F, GSp(A, σ) .

symplectic involutions

(29.23) ←’

on A

The exact sequence

1 ’ Gm ’ GSp(A, σ) ’ PGSp(A, σ) ’ 1

yields a connecting map in cohomology

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

The commutative diagram

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

¦ ¦

¦ ¦

1 ’’ Gm ’’ GL1 (A) ’’ PGL1 (A) ’’ 1

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

[A ] · [A]’1 .

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

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

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

we have

x + σ(x) = xu ’ (xu)t u’1 .

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 405

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

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

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

x + σ(x) = a · σ(a),

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

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

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

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

Sym(A, σ) = Symd(A, σ)— ,

and (??) yields a bijection of pointed sets

Symd(A, σ)— /∼ H 1 F, Sp(A, σ) .

(29.24) ←’

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

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

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

forms on V are isometric to a, we have

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

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

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

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

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

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

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

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

F -isomorphism classes of

H 1 F, PGO(A, σ)

central simple F -algebras of degree n ←’

with orthogonal involution

conjugacy classes of

H 1 F, GO(A, σ) .

orthogonal involutions ←’

on A

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

connecting map

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

in the cohomology sequence arising from the exact sequence

1 ’ Gm ’ GO(A, σ) ’ PGO(A, σ) ’ 1

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

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

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

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

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

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

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

/

406 VII. GALOIS COHOMOLOGY

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

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

Sym(A, σ) ‚ Sym(A, σ)— Alt(A, σ).

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

GLn (Fsep ), then

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

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

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

Sym(A, σ)— H 1 F, O(A, σ)

Alt(A, σ) /∼ ←’

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

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

form

Sym(A, σ)— /∼ H 1 F, O(A, σ) .

(29.26) ←’

Assuming char F = 2, let

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

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

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

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

sets

H 1 F, O+ (A, σ) .

SSym(A, σ)— /≈

(29.27) ←’

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

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

of the dual space of V . Consider the representation

ρ : G = GL(V ) ’ GL(W )

de¬ned by

ρ(±)(f )(x) = f ±’1 (x)

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

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

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

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

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

Proposition (??) yields a canonical bijection

isometry classes of

H 1 F, O(V, q) .

n-dimensional nonsingular

(29.28) ←’

quadratic spaces over F

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

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

ρ : G = GL(V ) — GL C(V, q) ’ GL(W )

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 407

where

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

and

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

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

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

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

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

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

W + = W • EndF (Z)

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

Z

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

∼

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

’

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

bijection

isometry classes of

n-dimensional nonsingular

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

(29.29) ←’

quadratic spaces (V , q ) over F

such that disc q = disc q

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

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

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

H 1 F, O+ (V, q) ’ H 1 F, O(V, q)

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

∆

1 ’ O+ (V, q) ’ O(V, q) ’ Z/2Z ’ 0

(29.30) ’

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

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

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

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

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

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

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

substituting for (??) the exact sequence

det

1 ’ O+ (V, q) ’ O(V, q) ’’ µ2 ’ 1.

’

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

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

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

408 VII. GALOIS COHOMOLOGY

Proposition (??) yields a bijection

isometry classes of

n-dimensional nonsingular

H 1 F, O(V, q) .

←’

quadratic spaces (V , q ) over F

such that disc q = disc q

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

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

√ √

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

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

disc q = disc q. Therefore, we have

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

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

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

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

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

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

of (??):

ρ : G = GL(A) — GL Sym(A, σ) ’ GL(W )

where

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

with ρ given by

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

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

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

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

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

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

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

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

f »(s) = p(s) for s ∈ Sym(A, σ).

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

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

∼ ∼

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

’ ’

de¬ne ψ, •, p by

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

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

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

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

form. If

∼

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

’

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 409

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

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

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

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

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

is a canonical bijection

F -isomorphism classes of

H 1 F, PGO(A, σ, f ) .

central simple F -algebras with ←’

quadratic pair of degree 2n

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

e

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

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

d

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

’

where d is the natural homomorphism

PGO(A, σ, f ) ’ Autalg (Z) S2 .

Thus the map

H 1 F, PGO(A, σ, f ) ’ H 1 (F, S2 )

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

obtain a natural bijection

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

with a central simple F -algebra A

←’ H 1 F, PGO+ (A, σ, f ) .

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

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

the centers of the Cli¬ord algebras

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

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

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

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

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

bijections

conjugacy classes of

H 1 F, GO(A, σ, f ) ,

←’

quadratic pairs on A

conjugacy classes of triples (σ , f , •)

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

H 1 F, GO+ (A, σ, f ) .

←’

and • : Z ’ Z is an isomorphism of

the centers of the Cli¬ord algebras

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

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

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

410 VII. GALOIS COHOMOLOGY

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

observation on involutions on algebras over separably closed ¬elds.

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

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

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

b = gaσ(g) + x + σ(x).

’1 ’1

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

’1

Trd b + σ(b) x = 0.

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

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

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

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

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

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

u’1 b ≡ g0 u’1 ag0

t

mod Alt(A, σ)

hence

b ≡ g1 aσ(g1 ) mod Alt(A, σ)

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

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

b = g1 aσ(g1 ) + x1 + σ(x1 ).

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

u’1 b = g0 u’1 ag0 + x0 + xt .

t

(29.33) 0

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

v = g0 u’1 a + (u’1 a)t g0 .

t

From (??), we derive

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

t

(29.34)

’1

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

t

The left side is Srd b + σ(b)

to

’1

u’1 a + (u’1 a)t u’1 a = a + σ(a) a.

’1 ’1

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

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

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

’1

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

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

’1 t

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

t tt

2

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

in (??), and by substituting in (??),

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 )

“