(31.2) Aut(G) ’ Aut(Dyn(G))

is a split surjection. A splitting i : Aut Dyn(G) ’ Aut(G) can be chosen in such a

way that any automorphism in the image of i leaves the torus T invariant. Assume

that the Galois group “ acts on the Dynkin diagram Dyn(G), or equivalently,

consider a continuous homomorphism

= H 1 F, Aut Dyn(G) .

• ∈ Hom “, Aut Dyn(G)

Denote by γ a cocycle representing the image of • in H 1 F, Aut(Gsep ) under the

map induced by the splitting i. Since γ normalizes T , the twisted group Gγ contains

the maximal torus Tγ . Moreover, the natural action of “ on Tγ leaves Π invariant,

hence Gγ is a quasisplit group. In fact, up to isomorphism Gγ is the unique simply

connected quasisplit group with Dynkin diagram Dyn(G) and with the given action

of “ on Dyn(G) (see (??)). Twisting G in (??) by γ, we obtain:

(31.3) Proposition. Let G be a quasisplit simply connected group. Then the nat-

ural homomorphism Aut(G) ’ Aut(Dyn(G)) is surjective.

Let G be semisimple group over F . By §?? a twisted form G of G corresponds

to an element ξ ∈ H 1 F, Aut(Gsep ) . We say that G is an inner form of G if ξ

belongs to the image of the map

±G : H 1 (F, G) ’’ H 1 F, Aut(Gsep )

induced by the homomorphism Int : G = G/C ’ Aut(G), where C is the center of

G. Since G acts trivially on C, the centers of G and G are isomorphic (as group

schemes of multiplicative type).

(31.4) Proposition. Any semisimple group is an inner twisted form of a unique

quasisplit group up to isomorphism.

Proof : Since the centers of inner twisted forms are isomorphic and all the groups

which are isogenous to a simply connected group correspond to subgroups in its

center, we may assume that the given group G is simply connected. Denote by

Gd the split twisted form of G, so that G corresponds to some element ρ ∈

H 1 F, Aut(Gd ) . Denote by γ ∈ H 1 F, Aut(Gd ) the image of ρ under the

sep sep

426 VII. GALOIS COHOMOLOGY

i

composition induced by Aut(Gd ) ’ Aut Dyn(Gd ) ’ Aut(Gd ), where i is the split-

ting considered above. We prove that the quasisplit group Gd is an inner twisted

γ

form of G = Gd . By (??), there is a bijection

ρ

∼

θρ : H 1 F, Aut(Gsep ) ’ H 1 F, Aut(Gd )

’ sep

taking the trivial cocycle to ρ. Denote by γ0 the element in H 1 F, Aut(Gsep ) such

that θ(γ0 ) = γ. Since ρ and γ have the same image in H 1 F, Aut Dyn(Gd ) , sep

1

the trivial cocycle and γ0 have the same images in H F, Aut Dyn(Gsep ) , hence

Theorem (??) shows that γ0 belongs to the image of H 1 (F, G) ’ H 1 F, Aut(Gsep ) ,

i.e., the group Gγ0 Gd is a quasisplit inner twisted form of G.

γ

Until the end of the subsection we shall assume that G is a simply connected

semisimple group. Denote by ξG the element in H 1 F, Aut(Gsep ) corresponding

to the (unique) quasisplit inner twisted form of G. In general, the map ±G is not

injective. Nevertheless, we have

(31.5) Proposition. There is only one element νG ∈ H 1 (F, G) such that

±G (νG ) = ξG .

Proof : Denote Gq the quasisplit inner twisted form of G. By (??), there is a

bijection between ±’1 (ξG ) and the factor group of Aut Dyn(Gq ) by the image of

G

Aut(Gq ) ’ Aut Dyn(Gq ) . But the latter map is surjective (see (??)).

Let C be the center of G. The exact sequence 1 ’ C ’ G ’ G ’ 1 induces

the connecting map

δ 1 : H 1 (F, G) ’’ H 2 (F, C).

The Tits class of G is the element tG = ’δ 1 (νG ) ∈ H 2 (F, C).

(31.6) Proposition. Let χ ∈ C — be a character. Denote by Fχ the ¬eld of de¬ni-

tion of χ and by Aχ its minimal Tits algebra. The image of the Tits class tG under

the composite map

χ—

res

H 2 (F, C) ’’ H 2 (Fχ , C) ’’ H 2 (Fχ , Gm ) = Br(Fχ )

’

is [Aχ ]. (We use the canonical identi¬cation H 2 (Fχ , Gm ) = Br(Fχ ) given by the

connecting map of (??), which is the opposite of the identi¬cation given by the

crossed product construction.)

Proof : There is a commutative diagram (see §??)

1 ’’ CFχ ’’ G Fχ ’’ G Fχ ’’ 1

¦ ¦ ¦

¦ ¦ ¦

χ

1 ’’ Gm ’’ GL1 (A) ’’ PGL1 (A) ’’ 1

where A = Aχ . Therefore, it su¬ces to prove that the image of res(νG ) under the

composite map

H 1 (Fχ , GFχ ) ’ H 1 Fχ , PGL1 (A) ’ H 2 (Fχ , Gm ) = Br(Fχ )

is ’[A]. The twist of the algebra A by a cocycle representing res(νG ) is the Tits

algebra of the quasisplit group (GνG )Fχ , hence it is trivial. Therefore the image

§31. COHOMOLOGICAL INVARIANTS 427

γ of res(νG ) in H 1 Fχ , PGL1 (A) corresponds to the split form PGLn (Fχ ) of

PGL1 (A), where n = deg(A). By (??), there is a commutative square

H 1 Fχ , PGLn (Fχ ) ’ ’ ’ Br(Fχ )

’’

¦ ¦

¦ ¦g

θγ

H 1 Fχ , PGL1 (A) ’ ’ ’ Br(Fχ )

’’

where g(v) = v + u and u is the image of γ in Br(Fχ ). Pick µ ∈ Z 1 Fχ , PGLn (Fχ )

such that the twisting of PGLn (Fχ ) by µ equals PGL1 (A). In other words, θγ (µ) =

1 and the image of µ in Br(Fχ ) equals [A]. The commutativity of the diagram then

implies that u = ’[A].

(31.7) Example. Let G = SU(B, „ ) where (B, „ ) is a central simple F -algebra

with unitary involution of degree n. Assume that char F does not divide n and let

K be the center of B. Since the center C of SU(B, „ ) is µn[K] , the Tits class tG

belongs to H 2 (F, µn[K] ). Abusing terminology, we call it the Tits class of (B, „ )

and denote it by t(B, „ ), i.e.,

t(B, „ ) = tSU(B,„ ) ∈ H 2 (F, µn[K] ).

Suppose K is a ¬eld. For χ = 1 + nZ ∈ Z/nZ = C — , the ¬eld of de¬nition of χ is

K and the minimal Tits algebra is B, see §??. Therefore, Proposition (??) yields

resK/F t(B, „ ) = [B] ∈ Br(K).

If n is odd, it follows from Corollary (??) that

1

t(B, „ ) = cor[B].

2

On the other hand, if n is even we may consider the character » = n + nZ ∈ C — of

2

raising to the power n . The corresponding minimal Tits algebra is the discriminant

2

algebra D(B, „ ), see §??. By (??) we obtain

»— t(B, „ ) = D(B, „ ) ∈ Br(F ).

If K F — F we have (B, „ ) = (A — Aop , µ) for some central simple F -algebra

A of degree n, where µ is the exchange involution. The commutative diagram with

exact rows

1 ’ ’ ’ µn ’ ’ ’ SLn ’ ’ ’ PGLn ’ ’ ’ 1

’’ ’’ ’’ ’’

¦ ¦

¦ ¦

1 ’ ’ ’ Gm ’ ’ ’ GLn ’ ’ ’ PGLn ’ ’ ’ 1

’’ ’’ ’’ ’’

shows that t(B, „ ) = [A] ∈ H 2 (F, µn ) = n Br(F ).

(31.8) Example. Let G = Spin(V, q), where (V, q) is a quadratic space of odd

dimension. Suppose char F = 2. The center of G is µ2 and the Tits class tG ∈

H 2 (F, µ2 ) = 2 Br(F ) is the Brauer class of the even Cli¬ord algebra C0 (V, q), since

the minimal Tits algebra for the nontrivial character is C0 (V, q), see §??.

(31.9) Example. Let G = Sp(A, σ), where (A, σ) is a central simple F -algebra

with symplectic involution. Suppose char F = 2. The center of G is µ2 and the

Tits class tG ∈ H 2 (F, µ2 ) = 2 Br(F ) is the Brauer class of the algebra A, see §??.

428 VII. GALOIS COHOMOLOGY

(31.10) Example. Let G = Spin(A, σ, f ), where (A, σ, f ) is a central simple F -

algebra of even degree 2n with quadratic pair. Let Z be the center of the Cli¬ord

algebra C(A, σ, f ) and assume that char F = 2.

Suppose ¬rst that n is even. Then the center of G is RZ/F (µ2 ), hence

tG ∈ H 2 F, RZ/F (µ2 ) = H 2 (Z, µ2 ) = 2 Br(Z).

The minimal Tits algebra corresponding to the norm character

» : RZ/F (µ2 ) ’ µ2 ’ Gm

is A, hence

corZ/F (tG ) = [A] ∈ H 2 (F, µ2 ) = 2 Br(F ).

(31.11)

On the other hand, the minimal Tits algebras for the two other nontrivial characters

»± are C(A, σ, f ) (see §??), hence

tG = C(A, σ, f ) ∈ H 2 (Z, µ2 ) = 2 Br(Z).

(31.12)

Now, assume that n is odd. Then the center of G is µ4[Z] , hence

tG ∈ H 2 (F, µ4[Z] ).

By applying Proposition (??), we can compute the image of tG under the squaring

map

»— : H 2 (F, µ4[Z] ) ’ H 2 (F, µ2 ) = 2 Br(F )

and under the restriction map

res : H 2 (F, µ4[Z] ) ’ H 2 (Z, µ4 ) = 4 Br(Z)

(or, equivalently, under the map H 2 (F, µ4[Z] ) ’ H 2 F, RZ/F (Gm,Z ) = Br(Z)

induced by the inclusion µ4[Z] ’ RZ/F (Gm,Z )). We obtain

»— (tG ) = [A] and res(tG ) = C(A, σ, f ) .

Note that the fundamental relations (??) of Cli¬ord algebras readily follow

from the computations above (under the hypothesis that char F = 2). If n is even,

2

(??) shows that C(A, σ, f ) = 1 in Br(Z) and (??) (together with (??)) implies

NZ/F C(A, σ, f ) = [A].

If n is odd we have

2

[AZ ] = res —¦»— (tG ) = res(tG )2 = C(A, σ, f )

and

NZ/F C(A, σ, f ) = cor —¦ res(tG ) = 0,

by (??). (See Exercise ?? for a cohomological proof of the fundamental relations

without restriction on char F .)

§31. COHOMOLOGICAL INVARIANTS 429

31.B. Cohomological invariants of algebraic groups. Let G be an alge-

braic group over a ¬eld F . For any ¬eld extension E of F we consider the pointed

set

H 1 (E, G) = H 1 (E, GE )

of GE -torsors over E. A homomorphism E ’ L of ¬elds over F induces a map of

pointed sets

H 1 (E, G) ’ H 1 (L, G).

Thus, H 1 (?, G) is a functor from the category of ¬eld extensions of F (with mor-

phisms being ¬eld homomorphisms over F ) to the category of pointed sets.

Let M be a torsion discrete Galois module over F , i.e., a discrete module over

the absolute Galois group “ = Gal(Fsep /F ). For a ¬eld extension E of F , M

can be endowed with a structure of a Galois module over E and hence the ordinary

cohomology groups H d (E, M ) are de¬ned. Thus, for any d ≥ 0, we obtain a functor

H d (?, M ) from the category of ¬eld extensions of F to the category of pointed sets

(actually, the category of abelian groups). A cohomological invariant of the group G

of dimension d with coe¬cients in M is a natural transformation of functors

a : H 1 (?, G) ’ H d (?, M ).

In other words, the cohomological invariant a assigns to any ¬eld extension E of F

a map of pointed sets

aE : H 1 (E, G) ’ H d (E, M ),

such that for any ¬eld F -homomorphism E ’ L the following diagram commutes

a

H 1 (E, G) ’ ’ ’ H d (E, M )

’E’

¦ ¦

¦ ¦

a

H 1 (L, G) ’ ’ ’ H d (L, M ).

’L’

The set Invd (G, M ) of all cohomological invariants of the group G of dimension d

with coe¬cients in M forms an abelian group in a natural way.

(31.13) Example. Let G = GL(V ) or SL(V ). By Hilbert™s Theorem 90 (see

(??) and (??)) we have H 1 (E, G) = 1 for any ¬eld extension L of F . Hence

Invd (G, M ) = 0 for any d and any Galois module M .

A group homomorphism ± : G ’ G over F induces a natural homomorphism

±— : Invd (G , M ) ’ Invd (G, M ).

A homomorphism of Galois modules g : M ’ M yields a group homomorphism

g— : Invd (G, M ) ’ Invd (G, M ).

For a ¬eld extension L of F there is a natural restriction homomorphism

res : Invd (G, M ) ’ Invd (GL , M ).

Let L be a ¬nite separable extension of F . If G is an algebraic group over L and M

is a Galois module over F , then the corestriction homomorphism for cohomology

groups and Shapiro™s lemma yield the corestriction homomorphism

cor: Invd (G, M ) ’ Invd RL/F (G), M .

430 VII. GALOIS COHOMOLOGY

Invariants of dimension 1. Let G be an algebraic group over a ¬eld F . As

in §??, write π0 (G) for the factor group of G modulo the connected component

of G. It is an ´tale group scheme over F .

e

Let M be a discrete Galois module over F and let

g : π0 (Gsep ) ’ M

be a “-homomorphism. For any ¬eld extension E of F we then have the following

composition

ag : H 1 (E, G) ’ H 1 E, π0 (G) ’ H 1 (E, M )

E

where the ¬rst map is induced by the canonical surjection G ’ π0 (G) and the

second one by g. We can view ag as an invariant of dimension 1 of the group G

with coe¬cients in M .

(31.14) Proposition. The map

Hom“ π0 (Gsep ), M ’ Inv1 (G, M ) given by g ’ ag

is an isomorphism.

In particular, a connected group has no nonzero invariants of dimension 1.

(31.15) Example. Let (A, σ, f ) be a central simple F -algebra with quadratic pair

of degree 2n and let G = PGO(A, σ, f ) be the corresponding projective orthog-

onal group. The set H 1 (F, G) classi¬es triples (A , σ , f ) with a central simple

F -algebra A of degree 2n with an quadratic pair (σ , f ) (see §??). We have

π0 (G) Z/2Z and the group Inv1 (G, Z/2Z) is isomorphic to Z/2Z. The nontrivial

invariant

H 1 (F, G) ’ H 1 (F, Z/2Z)

associates to any triple (A , σ , f ) the sum [Z ] + [Z] of corresponding classes of the

discriminant quadratic extensions.

(31.16) Example. Let K be a quadratic ´tale F -algebra, let (B, „ ) be a central

e

simple K-algebra of degree n with a unitary involution and set G = Aut(B, „ ).

The set H 1 (F, G) classi¬es algebras of degree n with a unitary involution (see (??)).

Then π0 (G) Z/2Z and, as in the previous example, the group Inv1 (G, Z/2Z) is

isomorphic to Z/2Z. The nontrivial invariant

H 1 (F, G) ’ H 1 (F, Z/2Z)

associates to any central simple F -algebra with unitary involution (B , „ ) with

center K the class [K ] + [K].

Invariants of dimension 2. For any natural numbers i and n let µ—i (F ) ben

the i-th tensor power of the group µn (F ). If n divides m, there is a natural injection

µ—i (F ) ’ µ—i (F ).

n m

The groups µ—i (F ) form an injective system with respect to the family of injections

n

de¬ned above. We denote the direct limit of this system, for all n prime to the

characteristic of F , by Q/Z(i)(F ). For example, Q/Z(1)(F ) is the group of all

roots of unity in F .

The group Q/Z(i)(Fsep ) is endowed in a natural way with a structure of a

Galois module. We set

H d F, Q/Z(i) = H d F, Q/Z(i)(Fsep ) .

§31. COHOMOLOGICAL INVARIANTS 431

In the case where char F = p > 0, this group can be modi¬ed by adding an

appropriate p-component. In particular, the group H 2 F, Q/Z(1) is canonically

isomorphic to Br(F ), while before the modi¬cation it equals lim H 2 F, µn (Fsep )

’’

which is the subgroup of elements in Br(F ) of exponent prime to p.

Let G be a connected algebraic group over a ¬eld F . Assume that we are given

an exact sequence of algebraic groups

(31.17) 1 ’ Gm,F ’ G ’ G ’ 1.

For any ¬eld extension E of F , this sequence induces a connecting map H 1 (E, G) ’

H 2 (E, Gm,F ) which, when composed with the identi¬cations

H 2 (E, Gm,F ) = Br(E) = H 2 E, Q/Z(1) ,

provides an invariant aE of dimension 2 of the group G. On the other hand, the

sequence (??) de¬nes an element of the Picard group Pic(G). It turns out that the

invariant a depends only on the element of the Picard group and we have a well

de¬ned group homomorphism

β : Pic(G) ’ Inv2 G, Q/Z(1) .

(31.18) Proposition. The map β is an isomorphism.

Since the n-torsion part of Q/Z(1) equals µn , we have

(31.19) Corollary. If n is not divisible by char F , then

Inv2 G, µn (Fsep ) n Pic(G).

(31.20) Example. Let G be a semisimple algebraic group, let π : G ’ G be a

universal covering and set Z = ker(π). There is a natural isomorphism

∼ ∼

Z — ’ Pic(G) ’ Inv2 G, Q/Z(1) .

’ ’

Hence attached to each character χ ∈ Z — is an invariant which we denote by aχ .

The construction is as follows. Consider the group G = (G — Gm,F )/Z where Z is

embedded into the product canonically on the ¬rst factor and by the character χ

on the second. There is an exact sequence

1 ’ Gm,F ’ G ’ G ’ 1.

We de¬ne aχ to be the invariant associated to this exact sequence as above.

The conjugation homomorphism G ’ Aut(G) induces the map

H 1 (F, G) ’ H 1 F, Aut(Gsep ) .

Hence, associated to each γ ∈ H 1 (F, G) is a twisted form Gγ of G (called an inner

form of G). If we choose γ such that Gγ is quasisplit (i.e., Gγ contains a Borel

subgroup de¬ned over F ), then

aχ (γ) = [Aχ ] ∈ Br(F )

F

where Aχ is the Tits algebra associated to the character χ (see §??).

(31.21) Example. Let T be an algebraic torus over F . Then

∼

Pic(T ) ’ H 1 F, T — (Fsep )

’

432 VII. GALOIS COHOMOLOGY

and all the cohomological invariants of dimension 2 of T with coe¬cients in Q/Z(1)

are given by the cup product

H 1 (F, T ) — H 1 F, T — (Fsep ) ’ H 2 (F, Fsep ) = H 2 F, Q/Z(1)

—

associated to the natural pairing

T (Fsep ) — T — (Fsep ) ’ Fsep .

—

Invariants of dimension 3. Let G be an algebraic group over a ¬eld F .

Assume ¬rst that F is separably closed. A loop in G is a group homomorphism

Gm,F ’ G over F . Write G— for the set of all loops in G. In general there is no

group structure on G— , but if f and h are two loops with commuting images, then

the pointwise product f h is also a loop. In particular, for any integer n and any

loop f the nth power f n is de¬ned. For any g ∈ G(F ) and any loop f , write gf for

the loop

Int(g)

f

Gm,F ’ G ’’

’ ’’ G.

Consider the set Q(G) of all functions q : G— ’ Z, such that

(a) q( gf ) = q(f ) for all g ∈ G(F ) and f ∈ G— ,

(b) for any two loops f and h with commuting images, the function

(k, m) ’ q(f k hm )

Z — Z ’ Z,

is a quadratic form.

There is a natural abelian group structure on Q(G).

Assume now that F is an arbitrary ¬eld. There is a natural action of the

absolute Galois group “ on the set of loops in Gsep and hence on Q(Gsep ). We set

Q(G) = Q(Gsep )“ .

(31.22) Example. Let T be an algebraic torus. Then T— (Fsep ) is the group of

cocharacters of T and

“

Q(T ) = S 2 T — (Fsep )

is the group of Galois invariant integral quadratic forms on T— (Fsep ).

(31.23) Example. Let G = GL(V ) and f ∈ G— (Fsep ) be a loop. We can view f as

a representation of Gm,F . By the theory of representations of diagonalizable groups

(see §??), f is uniquely determined by its weights χai , i = 1, 2, . . . , n = dim(V )

where χ is the canonical character of Gm,F . The function qV on G— (Fsep ) de¬ned

by

n

a2

qV (f ) = i

i=1

clearly belongs to Q(G).

A group homomorphism G ’ G over F induces a map of loop sets G— (Fsep ) ’

G— (Fsep ) and hence a group homomorphism

Q(G ) ’ Q(G),

making Q a contravariant functor from the category of algebraic groups over F to