6. Let L/F be a ¬nite Galois ¬eld extension with the Galois group G. Show that

RL/F (µn,L ) is the Cartier dual to (Z/nZ)[G]et , where the “-action is induced

by the natural homomorphism “ ’ G.

7. Let p = char F and ±p the kernel of the pth power homomorphism Ga ’ Ga .

Show that (±p )D ±p .

8. Let f : G ’ H be an algebraic group scheme homomorphism with G connected.

Prove that if falg is surjective then H is also connected.

9. If N and G/N are connected then G is also connected.

1

10. Show that F [PGLn (F )] is isomorphic to the subalgebra of F [Xij , det X ] con-

sisting of all homogeneous rational functions of degree 0.

11. Let B be a quaternion algebra with a unitary involution „ over an ´tale quad-

e

ratic extension of F . Prove that SU(B, „ ) SL1 (A) for some quaternion

algebra A over F .

12. Show that Spin+ (A, σ, f ) and Spin’ (A, σ, f ) are isomorphic if and only if

GO’ (A, σ, f ) = ….

13. Show that the automorphism x ’ x’t of SL2 is inner.

14. Nrd(X) ’ 1 is irreducible.

15. Let F be a ¬eld of characteristic 2 and ±1 , ±2 ∈ F — . Let G be the algebraic

group scheme of isometries of the bilinear form ±1 x1 y1 + ±2 x2 y2 , so that F [G]

is the factor algebra of the polynomial ring F [x11 , x12 , x21 , x22 ] by the ideal

generated by the entries of

t

x11 x12 ±1 0 x11 x12 ±1 0

· · ’ .

x21 x22 0 ±2 x21 x22 0 ±2

(a) Show that x11 x22 + x12 x21 + 1 and x11 + x22 are nilpotent in F [G].

(b) Assuming ±1 ±’1 ∈ F —2 , show that

/

2

F [G]red = F [x11 , x21 ]/(x2 + ±2 ±’1 x2 + 1),

11 21

1

and that F [G]red F is not reduced. Therefore, there is no smooth alge-

alg

braic group associated to G.

(c) Assuming ±1 ±’1 ∈ F —2 , show that the additive group Ga is the smooth

2

algebraic group associated to G.

NOTES 381

16. Let F be a perfect ¬eld of characteristic 2 and let b be a nonsingular symmetric

nonalternating bilinear form on a vector space V of dimension n.

(a) Show that there is a unique vector e ∈ V such that b(v, v) = b(v, e)2 for all

v ∈ V . Let V = e⊥ be the hyperplane of all vectors which are orthogonal

to e. Show that e ∈ V if and only if n is even, and that the restriction b

of b to V is an alternating form.

(b) Show that the smooth algebraic group O(V, b)red associated to the orthog-

onal group of the bilinear space (V, b) stabilizes e.

(c) Suppose n is odd. Show that the alternating form b is nonsingular and

that the restriction map O(V, b)red ’ Sp(V , b ) is an isomorphism.

(d) Suppose n is even. Show that the radical of b is eF . Let V = V /eF

and let b be the nonsingular alternating form on V induced by b . Show

that the restriction map ρ : O(V, b)red ’ Sp(V , b ) is surjective. Show

that every u ∈ ker ρ induces on V a linear transformation of the form

v ’ v + e• (v ) for some linear form • ∈ (V )— such that • (e) = 0. The

form • therefore induces a linear form • ∈ (V )— ; show that the map

j : ker ρ ’ (V )— which maps u to • is a homomorphism. Show that

there is an exact sequence

j

i

0 ’ Ga ’ ker ρ ’ (V )— ’ 0

’ ’

where i maps » ∈ F to the endomorphism v ’ v+e»b(v, e) of V . Conclude

that ker ρ is the maximal solvable connected normal subgroup of O(V, b)red .

Notes

§§??“??. Historical comments on the theory of algebraic groups are given by

Springer in his survey article [?] and we restrict to comments closely related to

material given in this chapter. The functorial approach to algebraic groups was

developed in the S´minaire du Bois Marie 62/64, directed by M. Demazure and

e

A. Grothendieck [?]. The ¬rst systematic presentation of this approach is given

in the treatise of Demazure-Gabriel [?]. As mentioned in the introduction to this

chapter, the classical theory (mostly over an algebraically closed ¬eld) can be found

for example in Borel [?] and Humphreys [?]. See also the new edition of the book

of Springer [?]. Relations between algebraic structures and exceptional algebraic

groups (at least from the point of view of Lie algebras) are described in the books

of Jacobson [?], Seligman [?] and the survey of Faulkner and Ferrar [?].

§??. In his commentary (Collected Papers, Vol. II, [?, pp. 548“549]) to [?],

Weil makes interesting historical remarks on the relations between classical groups

and algebras with involution. In particular he attributes the idea to view classical

algebraic groups as groups of automorphisms of algebras with involution to Siegel.

§??. Most of this comes from Weil [?] (see also the Tata notes of Kneser [?]

and the book of Platonov-Rapinchuk [?]). One di¬erence is that we use ideas from

Tits [?], to give a characteristic free presentation, and that we also consider types

G2 and F4 (see the paper [?] of Hijikata). For type D4 (also excluded by Weil), we

need the theory developed in Chapter ??. The use of groupoids (categories with

isomorphisms as morphisms) permits one to avoid the explicit use of non-abelian

Galois cohomology, which will not be introduced until the following chapter.

382 VI. ALGEBRAIC GROUPS

§??. A discussion of the maximal possible indexes of Tits algebras can be found

in Merkurjev-Panin-Wadsworth [?], [?] and Merkurjev [?].

CHAPTER VII

Galois Cohomology

In the preceding chapters, we have met groupoids M = M(F ) of “algebraic

objects” over a base ¬eld F , for example ¬nite dimensional F -algebras or algebraic

groups of a certain type. If over a separable closure Fsep of F the groupoid M(Fsep )

is connected, i.e., all objects over Fsep are isomorphic, then in many cases the objects

of M are classi¬ed up to isomorphism by a cohomology set H 1 Gal(Fsep /F ), A ,

where A is the automorphism group of a ¬xed object of M(Fsep ). The aim of this

chapter is to develop the general theory of such cohomology sets, to reinterpret

some earlier results in this setting and to give techniques (like twisting) which will

be used in later parts of this book.

There are four sections. The basic techniques are explained in §??, and §??

gives an explicit description of the cohomology sets of various algebraic groups in

terms of algebras with involution. In §?? we focus on the cohomology groups of µ n ,

which are used in §?? to reinterpret various invariants of algebras with involution

or of algebraic groups, and to de¬ne higher cohomological invariants.

§28. Cohomology of Pro¬nite Groups

In this chapter, we let “ denote a pro¬nite group, i.e., a group which is the

inverse limit of a system of ¬nite groups. For instance, “ may be the absolute

Galois group of a ¬eld (this is the main case of interest for the applications in

§§??“??), or a ¬nite group (with the discrete topology). An action of “ on the left

on a discrete topological space is called continuous if the stabilizer of each point is

an open subgroup of “; discrete topological spaces with a continuous left action of

“ are called “-sets. (Compare with §??, where only ¬nite “-sets are considered.) A

group A which is also a “-set is called a “-group if “ acts by group homomorphisms,

i.e.,

σ(a1 · a2 ) = σa1 · σa2 for σ ∈ “, a1 , a2 ∈ A.

A “-group which is commutative is called a “-module.

In this section, we review some general constructions of nonabelian cohomology:

in the ¬rst subsection, we de¬ne cohomology sets H i (“, A) for i = 0 if A is a “-set,

for i = 0, 1 if A is a “-group and for i = 0, 1, 2, . . . if A is a “-module, and

we relate these cohomology sets by exact sequences in the second subsection. The

third subsection discusses the process of twisting, and the fourth subsection gives

an interpretation of H 1 (“, A) in terms of torsors.

28.A. Cohomology sets. For any “-set A, we set

H 0 (“, A) = A“ = { a ∈ A | σa = a for σ ∈ “ }.

If A is a “-group, the subset H 0 (“, A) is a subgroup of A.

383

384 VII. GALOIS COHOMOLOGY

Let A be a “-group. A 1-cocycle of “ with values in A is a continuous map

± : “ ’ A such that, denoting by ±σ the image of σ ∈ “ in A,

±σ„ = ±σ · σ±„ for σ, „ ∈ “.

We denote by Z 1 (“, A) the set of all 1-cocycles of “ with values in A. The constant

map ±σ = 1 is a distinguished element in Z 1 (“, A), which is called the trivial 1-

cocycle. Two 1-cocycles ±, ± ∈ Z 1 (“, A) are said to be cohomologous or equivalent

if there exists a ∈ A satisfying

±σ = a · ±σ · σa’1 for all σ ∈ “.

Let H 1 (“, A) be the set of equivalence classes of 1-cocycles. It is a pointed set

whose distinguished element (or base point) is the cohomology class of the trivial

1-cocycle.

For instance, if the action of “ on A is trivial, then Z 1 (“, A) is the set of all

continuous group homomorphisms from “ to A; two homomorphisms ±, ± are

cohomologous if and only if ± = Int(a) —¦ ± for some a ∈ A.

If A is a “-module the set Z 1 (“, A) is an abelian group for the operation

(±β)σ = ±σ · βσ . This operation is compatible with the equivalence relation on

1-cocycles, hence it induces an abelian group structure on H 1 (“, A).

Now, let A be a “-module. A 2-cocycle of “ with values in A is a continuous

map ± : “ — “ ’ A satisfying

σ±„,ρ · ±σ,„ ρ = ±σ„,ρ ±σ,„ for σ, „ , ρ ∈ “.

The set of 2-cocycles of “ with values in A is denoted by Z 2 (“, A). This set is an

abelian group for the operation (±β)σ,„ = ±σ,„ · βσ,„ . Two 2-cocycles ±, ± are said

to be cohomologous or equivalent if there exists a continuous map • : “ ’ A such

that

±σ,„ = σ•„ · •’1 · •σ · ±σ,„ for all σ, „ ∈ “.

σ„

The equivalence classes of 2-cocycles form an abelian group denoted H 2 (“, A).

Higher cohomology groups H i (“, A) (for i ≥ 3) will be used less frequently in the

sequel; we refer to Brown [?] for their de¬nition.

Functorial properties. Let f : A ’ B be a homomorphism of “-sets, i.e., a

map such that f (σa) = σf (a) for σ ∈ “ and a ∈ A. If a ∈ A is ¬xed by “, then so

is f (a) ∈ B. Therefore, f restricts to a map

f 0 : H 0 (“, A) ’ H 0 (“, B).

If A, B are “-groups and f is a group homomorphism, then f 0 is a group

homomorphism. Moreover, there is an induced map

f 1 : H 1 (“, A) ’ H 1 (“, B)

which carries the cohomology class of any 1-cocycle ± to the cohomology class of the

1-cocycle f 1 (±) de¬ned by f 1 (±)σ = f (±σ ). In particular, f 1 is a homomorphism

of pointed sets, in the sense that f 1 maps the distinguished element of H 1 (“, A) to

the distinguished element of H 1 (“, B).

If A, B are “-modules, then f 1 is a group homomorphism. Moreover, f induces

homomorphisms

f i : H i (“, A) ’ H i (“, B)

for all i ≥ 0.

§28. COHOMOLOGY OF PROFINITE GROUPS 385

Besides those functorial properties in A, the sets H i (“, A) also have functorial

properties in “. We just consider the case of subgroups: let “0 ‚ “ be a closed

subgroup and let A be a “-group. The action of “ restricts to a continuous action

of “0 . The obvious inclusion A“ ‚ A“0 is called restriction:

res: H 0 (“, A) ’ H 0 (“0 , A).

If A is a “-group, the restriction of a 1-cocycle ± ∈ Z 1 (“, A) to “0 is a 1-cocycle

of “0 with values in A. Thus, there is a restriction map of pointed sets

res: H 1 (“, A) ’ H 1 (“0 , A).

Similarly, if A is a “-module, there is for all i ≥ 2 a restriction map

res : H i (“, A) ’ H i (“0 , A).

28.B. Cohomology sequences. By de¬nition, the kernel ker(µ) of a map of

pointed sets µ : N ’ P is the subset of all n ∈ N such that µ(n) is the base point

of P . A sequence of maps of pointed sets

ρ µ

M’ N’ P

’ ’

ρ

is said to be exact if im(ρ) = ker(µ). Thus, the sequence M ’ N ’ 1 is exact if

’

µ

and only if ρ is surjective. The sequence 1 ’ N ’ P is exact if and only if the

’

base point of N is the only element mapped by µ to the base point of P . Note that

this condition does not imply that µ is injective.

The exact sequence associated to a subgroup. Let B be a “-group and

let A ‚ B be a “-subgroup (i.e., σa ∈ A for all σ ∈ “, a ∈ A). Let B/A be the

“-set of left cosets of A in B, i.e.,

B/A = { b · A | b ∈ B }.

The natural projection of B onto B/A induces a map of pointed sets B “ ’ (B/A)“ .

Let b · A ∈ (B/A)“ , i.e., σb · A = b · A for all σ ∈ “. The map ± : “ ’ A de¬ned

by ±σ = b’1 · σb ∈ A is a 1-cocycle with values in A, whose class [±] in H 1 (“, A) is

independent of the choice of b in b · A. Hence we have a map of pointed sets

δ 0 : (B/A)“ ’ H 1 (“, A), b · A ’ [±] where ±σ = b’1 · σb.

(28.1) Proposition. The sequence

δ0

1 ’ A“ ’ B “ ’ (B/A)“ ’ H 1 (“, A) ’ H 1 (“, B)

’

is exact.

Proof : For exactness at (B/A)“ , suppose that the 1-cocycle ±σ = b’1 · σb ∈ A is

trivial in H 1 (“, A) i.e., ±σ = a’1 · σa for some a ∈ A. Then ba’1 ∈ B “ and the

coset b · A = ba’1 · A in B/A is equal to the image of ba’1 ∈ B “ .

If ± ∈ Z 1 (“, A) satis¬es ±σ = b’1 · σb for some b ∈ B, then b · A ∈ (B/A)“ and

[±] = δ 0 (b · A).

The group B “ acts naturally (by left multiplication) on the pointed set (B/A)“ .

(28.2) Corollary. There is a natural bijection between ker H 1 (“, A) ’ H 1 (“, B)

and the orbit set of the group B “ in (B/A)“ .

386 VII. GALOIS COHOMOLOGY

Proof : A coset b · A ∈ (B/A)“ determines the element

δ 0 (b · A) = [b’1 · σb] ∈ ker H 1 (“, A) ’ H 1 (“, B) .

One checks easily that δ 0 (b · A) = δ 0 (b · A) if and only if the cosets b · A and b · A

lie in the same B “ -orbit in (B/A)“ .

The exact sequence associated to a normal subgroup. Assume for the

rest of this subsection that the “-subgroup A of B is normal in B, and set C = B/A.

It is a “-group.

(28.3) Proposition. The sequence

δ0

1 ’ A“ ’ B “ ’ C “ ’ H 1 (“, A) ’ H 1 (“, B) ’ H 1 (“, C)

’

is exact.

Proof : Let β ∈ Z 1 (“, B) where [β] lies in the kernel of the last map. Then βσ · A =

b’1 · σb · A = b’1 · A · σb for some b ∈ B. Hence βσ = b’1 · ±σ · σb for ± ∈ Z 1 (“, A)

and [β] is the image of [±] in H 1 (“, B).

The group C “ acts on H 1 (“, A) as follows: for c = b·A ∈ C “ and ± ∈ Z 1 (“, A),

set c[±] = [β] where βσ = b · ±σ · σb’1 .

(28.4) Corollary. There is a natural bijection between ker H 1 (“, B) ’ H 1 (“, C)

and the orbit set of the group C “ in H 1 (“, A).

The exact sequence associated to a central subgroup. Now, assume

that A lies in the center of B. Then A is an abelian group and one can de¬ne a

map of pointed sets

δ 1 : H 1 (“, C) ’ H 2 (“, A)

as follows. Given any γ ∈ Z 1 (“, C), choose a map β : “ ’ B such that βσ maps

to γσ for all σ ∈ “. Consider the function ± : “ — “ ’ A given by

’1

±σ,„ = βσ · σβ„ · βσ„ .

One can check that ± ∈ Z 2 (“, A) and that its class in H 2 (“, A) does not depend

on the choices of γ ∈ [γ] and β. We de¬ne δ 1 [γ] = [±].

(28.5) Proposition. The sequence

δ0 δ1

1 ’ A“ ’ B “ ’ C “ ’ H 1 (“, A) ’ H 1 (“, B) ’ H 1 (“, C) ’ H 2 (“, A)

’ ’

is exact.

Proof : Assume that for γ ∈ Z 1 (“, C) and β, ± as above we have

±σ,„ = βσ · σβ„ · βσ„ = aσ · σa„ · a’1

’1

σ„

for some aσ ∈ A. Then βσ = βσ · a’1 is a 1-cocycle in Z 1 (“, B) and γ is the image

σ

of β .

The group H 1 (“, A) acts naturally on H 1 (“, B) by (± · β)σ = ±σ · βσ .

(28.6) Corollary. There is a natural bijection between the kernel of the connecting

map δ 1 : H 1 (“, C) ’ H 2 (“, A) and the orbit set of the group H 1 (“, A) in H 1 (“, B).

Proof : Two elements of H 1 (“, B) have the same image in H 1 (“, C) if and only if

they are in the same orbit under the action of H 1 (“, A).

§28. COHOMOLOGY OF PROFINITE GROUPS 387

(28.7) Remark. If the exact sequence of “-homomorphisms

1’A’B’C ’1

is split by a “-map C ’ B, then the connecting maps δ 0 and δ 1 are trivial.

28.C. Twisting. Let A be a “-group. We let “ act on the group Aut A of

automorphisms of A by

σ

f (a) = σ f (σ ’1 a) for σ ∈ “, a ∈ A and f ∈ Aut A.

(Compare with §??.) The subgroup (Aut A)“ of Aut A consists of all “-automor-

phisms of A.

For a ¬xed 1-cocycle ± ∈ Z 1 (“, Aut A) we de¬ne a new action of “ on A by

σ — a = ±σ (σa), for σ ∈ “ and a ∈ A.

The group A with this new “-action is denoted by A± . We say that A± is obtained

by twisting A by the 1-cocycle ±.

If 1-cocycles ±, ± ∈ Z 1 (“, Aut A) are related by ±σ = f —¦ ±σ —¦ σf ’1 for some

∼

f ∈ Aut A, then f de¬nes an isomorphism of “-groups A± ’ A± . Therefore,

’

cohomologous cocycles de¬ne isomorphic twisted “-groups. However, the isomor-

∼

phism A± ’ A± is not canonical, hence we cannot de¬ne a twisted group A[±] for

’

[±] ∈ H 1 (“, A).

Now, let ± ∈ Z 1 (“, A) and let ± be the image of ± in Z 1 (“, Aut A) under the

map Int : A ’ Aut A. We also write A± for the twist A± of A. By de¬nition we

then have

σ — a = ±σ · σa · ±’1 , for a ∈ A± and σ ∈ “.

σ

(28.8) Proposition. Let A be a “-group and ± ∈ Z 1 (“, A). Then the map

θ± : H 1 (“, A± ) ’ H 1 (“, A) given by (γσ ) ’ (γσ · ±σ )

is a well-de¬ned bijection which takes the trivial cocycle of H 1 (“, A± ) to [±].

Proof : Let γ be a cocycle with values in A± . We have γσ„ = γσ ±σ σ(γ„ )±’1 , hence

σ

γσ„ · ±σ„ = γσ · ±σ · σ(γ„ ±„ )

and γ± ∈ Z 1 (“, A). If γ ∈ Z 1 (“, A± ) is cohomologous to γ, let a ∈ A satisfy

γσ = a · γσ · (σ — a’1 ). Then γσ ±σ = a · γσ ±σ · σa’1 , hence γ ± is cohomologous

to γ±. This shows that θ± is a well-de¬ned map. To prove that θ± is a bijection,

observe that the map σ ’ ±’1 is a 1-cocycle in Z 1 (“, A± ). The induced map

σ

θ±’1 : H 1 (“, A) ’ H 1 (“, A± ) is the inverse of θ± .

(28.9) Remark. If A is abelian, we have A = A± for ± ∈ Z 1 (“, A), and θ± is

translation by [±].

Functoriality. Let f : A ’ B be a “-homomorphism and let β = f 1 (±) ∈

Z 1 (“, B) for ± ∈ Z 1 (“, A). Then the map f , considered as a map f± : A± ’ Bβ , is

a “-homomorphism, and the following diagram commutes:

θ

H 1 (“, A± ) ’ ’ ’ H 1 (“, A)

’±’

¦ ¦

1¦ ¦1

f± f

θβ

H 1 (“, Bβ ) ’ ’ ’ H 1 (“, B).

’’

In particular, θ± induces a bijection between ker f± and the ¬ber (f 1 )’1 ([β]).

1

388 VII. GALOIS COHOMOLOGY

Let A be a “-subgroup of a “-group B, let ± ∈ Z 1 (“, A), and let β ∈ Z 1 (“, B)

be the image of ±. Corollary (??) implies:

(28.10) Proposition. There is a natural bijection between the ¬ber of H 1 (“, A) ’

H 1 (“, B) over [β] and the orbit set of the group (Bβ )“ in (Bβ /A± )“ .

Now, assume that A is a normal “-subgroup of B and let C = B/A. Let

β ∈ Z 1 (“, B) and let γ ∈ Z 1 (“, C) be the image of β. The conjugation map

B ’ Aut A associates to β a 1-cocycle ± ∈ Z 1 (“, Aut A). Corollary (??) implies:

(28.11) Proposition. There is a natural bijection between the ¬ber of H 1 (“, B) ’

H 1 (“, C) over [γ] and the orbit set of the group (Cγ )“ in H 1 (“, A± ).

Assume further that A lies in the center of B and let γ ∈ Z 1 (“, C), where

C = B/A. The conjugation map C ’ Aut B induces a 1-cocycle β ∈ Z 1 (“, Aut B).

Let µ be the image of [γ] under the map δ 1 : H 1 (“, C) ’ H 2 (“, A).

(28.12) Proposition. The following diagram

θγ

H 1 (“, Cγ ) ’ ’ ’ H 1 (“, C)

’’

¦ ¦

1¦ ¦1

δ δ

γ

g

H 2 (“, A) ’ ’ ’ H 2 (“, A)

’’

1

commutes, where δγ is the connecting map with respect to the exact sequence

1 ’ A ’ Bβ ’ Cγ ’ 1

and g is multiplication by µ.

Proof : Let ± ∈ Z 1 (“, Cγ ). Choose xσ ∈ ±σ and yσ ∈ γσ . Then

’1

µσ,„ = yσ · σy„ · yσ„

and

δ 1 θγ (±) = xσ yσ · σ(x„ y„ ) · yσ„ x’1 = xσ yσ · σx„ · yσ · µσ,„ · x’1

’1 ’1

σ„ σ„

σ,„

= xσ · (σ —¦ x„ ) · x’1 · µσ,„ = δγ (x)σ,„ · µσ,„ ,

1

σ„

hence δ 1 θγ (±) = δγ (±) · µ = g δγ (±) .

1 1

As in Corollary (??), one obtains:

(28.13) Corollary. There is a natural bijection between the ¬ber over µ of the map

δ 1 : H 1 (“, C) ’ H 2 (“, A) and the orbit set of the group H 1 (“, A) in H 1 (“, Bβ ).

28.D. Torsors. Let A be a “-group and let P be a nonempty “-set on which

A acts on the right. Suppose that

σ(xa ) = σ(x)σa for σ ∈ “, x ∈ P and a ∈ A.

We say that P is an A-torsor (or a principal homogeneous set under A) if the action

of A on P is simply transitive, i.e., for any pair x, y of elements of P there exists