(denoted there by G) is ¬nite and carries the trivial action of “.) We let A“Tors “

denote the category of A-torsors, where the maps are the A- and “-equivariant

functions. This category is a groupoid, since the maps are isomorphisms.

§28. COHOMOLOGY OF PROFINITE GROUPS 389

To construct examples of A-torsors, we may proceed as follows: for ± ∈

1

Z (“, A), let P± be the set A with the “- and A-actions

and xa = xa

σ x = ±σ σx for σ ∈ “ and x, a ∈ A.

It turns out that every A-torsor is isomorphic to some P± :

(28.14) Proposition. The map ± ’ P± induces a bijection

∼

H 1 (“, A) ’ Isom(A“Tors “ ).

’

Proof : If ±, ± ∈ Z 1 (“, A) are cohomologous, let a ∈ A satisfy ±σ = a·±σ ·σa’1 for

∼

all σ ∈ “. Multiplication on the left by a is an isomorphism of torsors P± ’ P± .

’

We thus have a well-de¬ned map H 1 (“, A) ’ Isom(A“Tors “ ). The inverse map is

given as follows: Let P ∈ A“Tors “ . For a ¬xed x ∈ P , the map ± : “ ’ A de¬ned

by

σ(x) = x±σ for σ ∈ “

is a 1-cocycle. Replacing x with xa changes ±σ into the cohomologous cocycle

a’1 ±σ σa.

(28.15) Example. Let “ be the absolute Galois group of a ¬eld F , and let G be

a ¬nite group which we endow with the trivial action of “. By combining (??) with

(??), we obtain a canonical bijection

∼

H 1 (“, G) ’ Isom(G“Gal F ),

’

hence H 1 (“, G) classi¬es the Galois G-algebras over F up to isomorphism.

Functoriality. Let f : A ’ B be a “-homomorphism of “-groups. We de¬ne

a functor

f— : A“Tors “ ’ B“Tors “

as follows: for P ∈ A“Tors “ , consider the product P — B with the diagonal action

of “. The groups A and B act on P — B by

(p, b)a = pa , f (a’1 )b and (p, b) b = (p, bb )

for p ∈ P , a ∈ A and b, b ∈ B, and these two actions commute. Hence there is an

induced right action of B on the set of A-orbits f— (P ) = (P — B)/A, making f— (P )

a B-torsor.

(28.16) Proposition. The following diagram commutes:

f1

H 1 (“, A) H 1 (“, B)

’’’

’’

¦ ¦

¦ ¦

f—

Isom(A“Tors “ ) ’ ’ ’ Isom(B“Tors “ ),

’’

where the vertical maps are the natural bijections of (??).

Proof : Let ± ∈ Z 1 (“, A). Every A-orbit in (P± — B)/A can be represented by a

unique element of the form (1, b) with b ∈ B. The map which takes the orbit (1, b)A

∼

to b ∈ Pf 1 (±) is an isomorphism of B-torsors (P± — B)/A ’ Pf 1 (±) .

’

390 VII. GALOIS COHOMOLOGY

Induced torsors. Let “0 be a closed subgroup of “ and let A0 be a “0 -

group. The induced “-group Ind“0 A0 is de¬ned as the group of all continuous

“

maps f : “ ’ A0 such that f (γ0 γ) = γ0 f (γ) for all γ0 ∈ “0 , γ ∈ “:

Ind“0 A0 = { f ∈ Map(“, A0 ) | f (γ0 γ) = γ0 f (γ) for γ0 ∈ “0 , γ ∈ “ }.

“

The “-action on Ind“0 A0 is given by σf (γ) = f (γσ) for σ, γ ∈ “. We let

“

π : Ind“0 A0 ’ A0 be the map which takes f ∈ Ind“0 A0 to f (1). This map satis¬es

“

“

“

σ σ

π( f ) = π(f ) for all σ ∈ “0 , f ∈ Ind“0 A0 . It is therefore a “0 -homomorphism.

(Compare with (??).)

By applying this construction to A0 -torsors, we obtain (Ind“0 A0 )-torsors: for

“

P0 ∈ A0 “Tors “0 , the “-group Ind“0 P0 carries a right action of Ind“0 A0 de¬ned by

“ “

for p ∈ Ind“0 P0 , f ∈ Ind“0 A0 and γ ∈ “.

pf (γ) = p(γ)f (γ) “ “

This action makes Ind“0 P0 an (Ind“0 A0 )-torsor, called the induced torsor. We

“ “

thus have a functor

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