§31. COHOMOLOGICAL INVARIANTS 433

Let G and G be two algebraic groups over F . The natural embeddings of G

and G in G—G and both projections from the product G—G to its factors induce

a natural isomorphism of Q(G) • Q(G ) with a direct summand of Q(G — G ).

(31.24) Lemma. If G— (Fsep ) = 1, then Q(G) = 0.

Proof : Choose an embedding G ’ GL(V ). Since G— (Fsep ) = 1, the restriction of

the positive function qV (see Example (??)) on this set is nonzero.

Assume now that G is a semisimple algebraic group over F .

(31.25) Lemma. Q(G) is a free abelian group of rank at most the number of

simple factors of Gsep .

Proof : We may assume that F is separably closed. Let T be a maximal torus in G

de¬ned over F . Since any loop in G is conjugate to a loop with values in T , the

restriction homomorphism Q(G) ’ Q(T ) is injective. By Example (??), the group

Q(T ) is free abelian of ¬nite rank, hence so is Q(G).

The Weyl group W acts naturally on Q(T ) and the image of the restriction ho-

momorphism belongs to Q(T )W . Hence any element in Q(G) de¬nes a W -invariant

quadratic form on T— and hence on the Q-vector space T— —Z Q. This space decom-

poses as a direct sum of subspaces according to the decomposition of G into the

product of simple factors and such a quadratic form (with values in Q) is known to

be unique (up to a scalar) on each component. Hence, the rank of Q(G) is at most

the number of simple components.

From Lemma (??) and the proof of Lemma (??) we obtain

(31.26) Corollary. If G is an absolutely simple algebraic group, then Q(G) is an

in¬nite cyclic group with a canonical generator which is a positive function.

(31.27) Corollary. Under the hypotheses of the previous corollary the homomor-

phism Q(G) ’ Q(GL ) is an isomorphism for any ¬eld extension L of F .

Proof : It su¬ces to consider the case L = Fsep . Since the group Q(G) is nontrivial,

the Galois action on the in¬nite cyclic group Q(GL ) must be trivial, and hence

Q(G) = Q(GL )“ = Q(GL ).

(31.28) Example. Let G = SL(V ). As in Example (??), for i = 1, . . . , n =

dim V , one associates integers ai to a loop f . In our case the sum of all the ai is

even (in fact, zero), hence the sum of the squares of the ai is even. Therefore

qV = 1 qV ∈ Q(G).

2

It is easy to show that qV is the canonical generator of Q(G).

(31.29) Corollary. If F is separably closed, then the rank of Q(G) equals the

number of simple factors of G.

Proof : Let G = G1 — · · · — Gm where the Gi are simple groups. By Lemma (??),

rank Q(G) ¤ m. On the other hand, the group Q(G) contains the direct sum of

the Q(Gi ) which is a free group of rank m by Corollary (??).

(31.30) Proposition. Let G and G be semisimple algebraic groups. Then

Q(G — G ) = Q(G) • Q(G ).

434 VII. GALOIS COHOMOLOGY

Proof : Clearly, we may assume that F is separably closed. The group Q(G) •

Q(G ) is a direct summand of the free group Q(G — G ) and has the same rank by

Corollary (??), hence the claim.

Let L be a ¬nite separable ¬eld extension of F and let G be a semisimple group

over L. Then the transfer RL/F (G) is a semisimple group over F .

∼

(31.31) Proposition. There is a natural isomorphism Q(G) ’ Q RL/F (G) .

’

Proof : Choose an embedding ρ : L ’ Fsep and set

“ = Gal Fsep /ρ(L) ‚ “.

The group RL/F (G)sep is isomorphic to the direct product of groups G„ as „ varies

over the set X of all F -embeddings of L into Fsep (see Proposition (??)). Hence,

by Proposition (??),

Q RL/F (G)sep = Q(G„ ).

„ ∈X

The Galois group “ acts naturally on the direct sum, transitively permuting com-

ponents. Hence it is the induced “-module from the “ -module Q(Gρ ). The propo-

sition then follows from the fact that for any “ -module M there is a natural

isomorphism between the group of “ -invariant elements in M and the group of

“-invariant elements in the induced module Map“ (“, M ).

By Theorem (??), a simply connected semisimple group over F is isomorphic

to a product of groups of the form RL/F (G ) where L is a ¬nite separable ¬eld

extension of F and G is an absolutely simple simply connected group over L.

Hence, Corollary (??) and Propositions (??), (??) yield the computation of Q(G)

for any simply connected semisimple group G.

A relation between Q(G) and cohomological invariants of dimension 3 of simply

connected semisimple groups is given by the following

(31.32) Theorem. Let G be a simply connected semisimple algebraic group over

a ¬eld F . Then there is a natural surjective homomorphism

γ(G) : Q(G) ’ Inv3 G, Q/Z(2) .

The naturality of γ in the theorem means, ¬rst of all, that for any group

homomorphism ± : G ’ G the following diagram commutes:

γ(G )

Q(G ) ’ ’ ’ Inv3 G , Q/Z(2)

’’

¦ ¦

¦ ¦—

(31.33) ±

γ(G)

Q(G) ’ ’ ’ Inv3 G, Q/Z(2) .

’’

For any ¬eld extension L of F the following diagram also commutes:

γ(G)

Q(G) ’ ’ ’ Inv3 G, Q/Z(2)

’’

¦ ¦

¦ ¦res

(31.34)

γ(GL )

Q(GL ) ’ ’ ’ Inv3 GL , Q/Z(2) .

’’

§31. COHOMOLOGICAL INVARIANTS 435

In addition, for a ¬nite separable extension L of F and an algebraic group G over L

the following diagram is also commutative:

γ(G)

Inv3 G, Q/Z(2)

Q(G) ’’’

’’

¦ ¦

¦ ¦cor

(31.35)

γ RL/F (G)

Q RL/F (G) ’ ’ ’ ’ ’ Inv3 RL/F (G), Q/Z(2) .

’’’’

Let G be an absolutely simple simply connected group over F . By Corol-

lary (??) and Theorem (??), the group Inv3 G, Q/Z(2) is cyclic, generated by a

canonical element which we denote i(G) and call the Rost invariant of the group G.

The commutativity of diagram (??) and Corollary (??) show that for any ¬eld

extension L of F ,

resL/F i(G) = i(GL ).

Let L be a ¬nite separable ¬eld extension of F and let G be an absolutely

simple simply connected group over L. It follows from the commutativity of (??)

and Proposition (??) that the group Inv3 RL/F (G), Q/Z(2) is cyclic and generated

by corL/F i(G) .

Let G be a simply connected semisimple group over F and let ρ : G ’ SL(V )

be a representation. The triviality of the right-hand group in the top row of the

following commutative diagram (see Example (??)):

Q SL(V ) ’ ’ ’ Inv3 SL(V ), Q/Z(2)

’’

¦ ¦

¦ ¦

Inv3 G, Q/Z(2)

Q(G) ’’’

’’

shows that the image of Q SL(V ) ’ Q(G) belongs to the kernel of

γ : Q(G) ’ Inv3 G, Q/Z(2) .

One can prove that all the elements in the kernel are obtained in this way.

(31.36) Theorem. The kernel of γ : Q(G) ’ Inv 3 G, Q/Z(2) is generated by the

images of Q SL(V ) ’ Q(G) for all representations of G.

(31.37) Corollary. Let G and G be simply connected semisimple groups over F .

Then

Inv3 G — G , Q/Z(2) = Inv3 G, Q/Z(2) • Inv3 G , Q/Z(2) .

(31.38) Corollary. Let L/F be a ¬nite separable ¬eld extension, G be a simply

connected semisimple group over L. Then the corestriction map

cor: Inv3 G, Q/Z(2) ’ Inv3 RL/F (G), Q/Z(2)

is an isomorphism.

These two corollaries reduce the study of the group Inv 3 G, Q/Z(2) to the

case of an absolutely simple simply connected group G.

436 VII. GALOIS COHOMOLOGY

Let ± : G ’ G be a homomorphism of absolutely simple simply connected

groups over F . There is a unique integer n± such that the following diagram

commutes:

=

Z ’ ’ ’ Q(G )

’’

¦ ¦

¦ ¦Q(±)

n±

=

Z ’ ’ ’ Q(G).

’’

If we have another homomorphism β : G ’ G , then clearly nβ± = nβ n± . Assume

that G = GL(V ). It follows from the proof of Lemma (??) that nβ > 0 and

nβ± > 0, hence n± is a natural number for any group homomorphism ±.

Let ρ : G ’ SL(V ) be a representation. As we observed above, nρ · iG =

0. Denote nG the greatest common divisor of nρ for all representations ρ of the

group G. Clearly, nG · i(G) = 0. Theorem (??) then implies

(31.39) Proposition. Let G be an absolutely simple simply connected group. Then

Inv3 G, Q/Z(2) is a ¬nite cyclic group of order nG .

Let n be any natural number prime to char F . The exact sequence

n

1 ’ µ—2 ’ Q/Z(2) ’ Q/Z(2) ’ 1

’

n

yields the following exact sequence of cohomology groups

n

H 2 F, Q/Z(2) ’ H 2 F, Q/Z(2) ’ H 3 (F, µ—2 ) ’

’ n

n

’ H 3 F, Q/Z(2) ’ H 3 F, Q/Z(2) .

’

Since the group H 2 F, Q/Z(2) is n-divisible (see Merkurjev-Suslin [?]), the group

H 3 (F, µ—2 ) is identi¬ed with the subgroup of elements of exponent n in H 3 F, Q/Z(2) .

n

Now let G be an absolutely simple simply connected group over F . By Propo-

sition (??), the values of the invariant i(G) lie in H 3 (F, µ—2 ), so that

nG

Inv3 G, Q/Z(2) = Inv3 (G, µ—2 ).

nG

In the following sections we give the numbers nG for all absolutely simple simply

connected groups. In some cases we construct the Rost invariant directly.

Spin groups of quadratic forms. Let F be a ¬eld of characteristic di¬erent

from 2. Let W F be the Witt ring of F and let IF be the fundamental ideal of

even-dimensional forms. The nth power I n F of this ideal is generated by the classes

of n-fold P¬ster forms.

To any 3-fold P¬ster form a, b, c the Arason invariant associates the class

(a) ∪ (b) ∪ (c) ∈ H 3 (F, Z/2Z) = H 3 (F, µ—2 ),

2

see (??). The Arason invariant extends to a group homomorphism

e3 : I 3 F ’ H 3 (F, Z/2Z)

(see Arason [?]). Note that I 3 F consists precisely of the classes [q] of quadratic

forms q having even dimension, trivial discriminant, and trivial Hasse-Witt invari-

ant (see Merkurjev [?]).

Let q be a non-degenerate quadratic form over F . The group G = Spin(q)

is a simply connected semisimple group if dim q ≥ 3 and is absolutely simple if

dim q = 4. It is a group of type Bn if dim q = 2n + 1 and of type Dn if dim q = 2n.

§31. COHOMOLOGICAL INVARIANTS 437

Conversely, any absolutely simple simply connected group of type Bn is isomorphic

to Spin(q) for some q. (The same property does not hold for Dn .)

The exact sequence

π

1 ’ µ2 ’ Spin(q) ’ O+ (q) ’ 1

(31.40) ’

gives the following exact sequence of pointed sets

δ1

π

H 1 F, Spin(q) ’— H 1 F, O+ (q) ’ H 2 (F, µ2 ) = 2 Br(F ).

’ ’

The set H 1 F, O+ (q) classi¬es quadratic forms of the same dimension and discrim-

inant as q, see (??). The connecting map δ 1 takes such a form q to the Hasse-Witt

invariant e2 [q ] ’ [q] ∈ Br(F ). Thus, the image of π— consists of classes of forms

having the same dimension, discriminant, and Hasse-Witt invariant as q. Therefore,

π— (u) ’ [q] ∈ I 3 F for any u ∈ H 1 F, Spin(q) .

The map

i Spin(q) : H 1 F, Spin(q) ’ H 3 (F, µ—2 )

(31.41) 2

de¬ned by u ’ e3 π— (u) ’ [q] gives rise to an invariant of Spin(q). It turns out

that this is the Rost invariant if dim q ≥ 5. If dim q = 5 or 6 and q is of maximal

Witt index, the anisotropic form representing π— (u) ’ [q] is of dimension less than 8

and hence is trivial by the Arason-P¬ster Hauptsatz. In these cases the invariant

is trivial and nG = 1. Otherwise the invariant is not trivial and nG = 2.

In the case where dim q = 4 the group G is a product of two groups of type A1

if disc q is trivial and otherwise is isomorphic to RL/F SL1 (C0 ) where L/F is the

discriminant ¬eld extension and C0 = C0 (q) (see (??) and §??). In the latter case

the group Inv3 G, Q/Z(2) is cyclic and generated by the invariant described above.

This invariant is trivial if and only if the even Cli¬ord algebra C0 is split.

If dim q = 3, then the described invariant is trivial since it is twice the Rost

invariant. In this case G = SL1 (C0 ) is a group of type A1 and the Rost invariant

is described below.

Type An .

Inner forms. Let G be an absolutely simple simply connected group of inner

type An over F , so that G = SL1 (A) for a central simple F -algebra of degree n + 1.

It turns out that nG = e = exp(A), and the Rost invariant

i(G) : H 1 (F, G) = F — / Nrd(A— ) ’ H 3 (F, µ—2 )

e

is given by the formula

i(G) a · Nrd(A— ) = (a) ∪ [A]

where (a) ∈ H 1 (F, µe ) = F — /F —e is the class of a ∈ F — and [A] ∈ H 2 (F, µe ) =

e Br(F ) is the class of the algebra A.

Outer forms. Let G be an absolutely simple simply connected group of outer

type An over F with n ≥ 2, so that G = SU(B, „ ) where (B, „ ) is a central

simple F -algebra with unitary involution of degree n + 1 and the center K of B

is a quadratic separable ¬eld extension of F . If n is odd, let D = D(B, „ ) be the

discriminant algebra (see §??).

(31.42) Proposition. The number nG equals either exp(B) or 2 exp(B). The ¬rst

case occurs if and only if (n + 1) is a 2-power and either

(1) exp(B) = n + 1 or

438 VII. GALOIS COHOMOLOGY

n+1

(2) exp(B) = and D is split.

2

An element of the set H 1 F, SU(B, „ ) is represented by a pair (s, z) where s ∈

Sym(B, „ )— and z ∈ K — satisfy Nrd(s) = NK/F (z) (see (??)). Since SU(B, „ )K

SL1 (B), it follows from the description of the Rost invariant in the inner case that

= (z) ∪ [B] ∈ H 3 K, Q/Z(2) .

i(G) (s, z)/≈ K

(31.43) Example. Assume that char F = 2 and B is split, i.e., B = EndK (V ) for

some vector space V of dimension n + 1 over K. The involution „ is adjoint to some

hermitian form h on V over K (Theorem (??)). Considering V as a vector space

over F we have a quadratic form q on V given by q(v) = h(v, v) ∈ F for v ∈ V .

Any isometry of h is also an isometry of q, hence we have the embedding

U(B, „ ) ’ O+ (V, q).

Since SU(B, „ ) is simply connected, the restriction of this embedding lifts to a

group homomorphism

± : SU(B, „ ) ’ Spin(V, q)

(see Borel-Tits [?, Proposition 2.24(i), p. 262]). One can show that n± = 1, so that

i(G) is the composition

H 1 F, SU(B, „ ) ’ H 1 F, Spin(V, q) ’ H 3 F, Q/Z(2)

where the latter map is the Rost invariant of Spin(V, q) which was described in (??).

An element of the ¬rst set in the composition is represented by a pair (s, z) ∈

Sym(B, „ )— — K — such that Nrd(s) = NK/F (z). The symmetric element s de¬nes

another hermitian form hs on V by

hs (u, v) = h s’1 (u), v

which in turn de¬nes, as described above, a quadratic form qs on V considered as

a vector space over F . The condition on the reduced norm of s shows that the

discriminants of hs and h are equal, see (??), hence [qs ] ’ [q] ∈ I 3 F . It follows from

the description of the Rost invariant for the group Spin(V, q) (see (??)) that the

invariant of the group G is given by the formula

i(G) (s, z)/≈ = e3 [qs ] ’ [q] .

If dim V is odd (i.e., n is even), the canonical map

H 1 F, SU(B, „ ) ’ H 1 F, GU(B, „ )

is surjective, since every unitary involution „ = Int(u) —¦ „ on B may be written as

„ = Int u NrdB (u) —¦ „,

showing that the conjugacy class of „ is the image of u NrdB (u), NrdB (u)(n/2)+1 .

The invariant i(G) induces an invariant

i GU(B, „ ) : H 1 F, GU(B, „ ) ’ H 3 (F, µ—2 )

2

which can be explicitly described as follows: given a unitary involution „ on B,

represent „ as the adjoint involution with respect to some hermitian form h with

disc h = disc h, and set

i GU(B, „ ) („ ) = e3 [q ] ’ [q]

where q is the quadratic form on V de¬ned by q (v) = h (v, v). Alternately,

consider the quadratic trace form Q„ (x) = TrdB (x2 ) on Sym(B, „ ). If h has a

§31. COHOMOLOGICAL INVARIANTS 439

F [X]/(X 2 ’ ±), Propositions (??) and (??)

diagonalization δ1 , . . . , δn+1 and K

show that

Q„ = (n + 1) 1 ⊥ 2 · ± · ⊥1¤i<j¤n+1 δi δj .

On the other hand,

q = ± · δ1 , . . . , δn+1 .

Since disc h = disc h , we may ¬nd a diagonalization h = δ1 , . . . , δn+1 such that

δ1 . . . δn+1 = δ1 . . . δn+1 . Using the formulas for the Hasse-Witt invariant of a sum

in Lam [?, p. 121], we may show that

e2 ⊥1¤i<j¤n+1 δi δj ’ ⊥1¤i<j¤n+1 δi δj =

e2 δ1 , . . . , δn+1 ’ δ1 , . . . , δn+1 ,

hence

i GU(B, „ ) („ ) = e3 [q ] ’ [q] = e3 [Q„ ] ’ [Q„ ] .

(31.44) Example. Assume that (n + 1) is odd and B has exponent e. Assume

also that char F does not divide 2e. For G = SU(B, „ ) we have nG = 2e. Since e

is odd we have µ—2 = µ—2 — µ—2 , hence the Rost invariant i(G) may be viewed as

e

2e 2

a pair of invariants

i1 (G), i2 (G) : H 1 F, SU(B, „ ) ’ H 3 (F, µ—2 ) — H 3 (F, µ—2 ).

e

2

Since B is split by a scalar extension of odd degree, we may use (??) to determine

i1 (G):

i1 (G) (s, z)/≈ = e3 [QInt(s)—¦„ ] ’ [Q„ ] ∈ H 3 (F, µ—2 ).

2

(By (??), it is easily seen that [QInt(s)—¦„ ] ’ [Q„ ] ∈ I 3 F .)

On the other hand, we have SU(B, „ )K SL1 (B) hence we may use the

invariant of SL1 and Corollary (??) to determine i2 (G):

1

corK/F (s) ∪ [B] ∈ H 3 (F, µ—2 ).

i2 (G) (s, z)/≈ = e

2

Note that the canonical map H 1 F, SU(B, „ ) ’ H 1 F, GU(B, „ ) is sur-

jective, as in the split case (Example (??)), and the invariant i1 (G) induces an

invariant

i GU(B, „ ) : H 1 F, GU(B, „ ) ’ H 3 (F, µ—2 )

2

which maps the conjugacy class of any unitary involution „ to e3 [Q„ ] ’ [Q„ ] .

In the particular case where deg(B, „ ) = 3, we also have a P¬ster form π(„ )

de¬ned in (??) and a cohomological invariant f3 (B, „ ) = e3 π(„ ) , see (??). From

the relation between π(„ ) and Q„ , it follows that

[Q„ ] ’ [Q„ ] = [ 2 ] · π(„ ) ’ π(„ ) ,

hence

i GU(B, „ ) („ ) = e3 π(„ ) ’ π(„ ) = f3 (B, „ ) ’ f3 (B, „ ).

440 VII. GALOIS COHOMOLOGY

Type Cn . Let G be an absolutely simple simply connected group of type Cn

over F , so that G = Sp(A, σ) where A is a central simple algebra of degree 2n

over F with a symplectic involution σ.

Assume ¬rst that the algebra A is split, i.e., G = Sp2n . Since all the nonsingular

alternating forms are pairwise isomorphic, the set H 1 (E, G) is trivial for any ¬eld

extension E of F . Hence nG = 1 and the invariant i(G) is trivial.

Assume now that A is nonsplit, so that exp(A) = 2. Consider the natural

embedding ± : G ’ SL1 (A). One can check that n± = 1, hence the Rost invari-

ant of G is given by the composition of ± and the invariant of SL1 (A), so that

nG = 2. By (??), we have H 1 (F, G) = Symd(A, σ)— /∼, and the following diagram

commutes:

±1

H 1 (F, G) ’ ’ ’ H 1 F, SL1 (A)

’’

Nrp