’’ ’

(where Nrpσ is the pfa¬an norm). Hence the invariant

i(G) : H 1 (F, G) ’ H 3 (F, µ—2 )

2

is given by the formula

i(G)(u/∼) = Nrpσ (u) ∪ [A].

The exact sequence

µ

1 ’ Sp(A, σ) ’ GSp(A, σ) ’ Gm ’ 1,

’

where µ is the multiplier map, induces the following exact sequence in cohomology:

H 1 F, Sp(A, σ) ’ H 1 F, GSp(A, σ) ’ 1

since H 1 (F, Gm ) = 1 by Hilbert™s Theorem 90. If deg A is divisible by 4, it turns

out that the invariant i(G) induces an invariant

i GSp(A, σ) : H 1 F, GSp(A, σ) ’ H 3 (F, µ—2 ).

2

Indeed, viewing H 1 F, GSp(A, σ) as the set of conjugacy classes of symplectic

involutions on A (see (??)), the canonical map

Symd(A, σ)— /∼ = H 1 F, Sp(A, σ) ’ H 1 F, GSp(A, σ)

takes u/∼ to the conjugacy class of Int(u) —¦ σ. For z ∈ F — and u ∈ Symd(A, σ)—

we have Nrpσ (zu) = z deg A/2 Nrpσ (u), hence Nrpσ (zu) = Nrpσ (u) in H 1 (F, µ2 )

if deg A is divisible by 4. Therefore, in this case we may set

i GSp(A, σ) Int(u) —¦ σ = i Sp(A, σ) (u/∼) = Nrpσ (u) ∪ [A].

(31.45) Example. Consider the particular case where deg A = 4. Since the quad-

ratic form Nrpσ is an Albert form of A by (??), its Hasse-Witt invariant is [A].

Therefore,

Nrpσ (u) ∪ [A] = e3 Nrpσ (u) · Nrpσ

and it follows by (??) that

i GSp(A, σ) („ ) = e3 jσ („ )

for every symplectic involution „ on A.

§31. COHOMOLOGICAL INVARIANTS 441

(31.46) Example. Let A = EndQ (V ) where V is a vector space of even dimen-

sion over a quaternion division algebra Q, and let σ be a hyperbolic involution

on A. For every nonsingular hermitian form h on V (with respect to the conju-

gation involution on Q), the invariant i GSp(A, σ) (σh ) of the adjoint involution

σh is the cohomological version of the Jacobson discriminant of h, see the notes to

Chapter ??. Indeed, if h has a diagonalization ±1 , . . . , ±n , then we may assume

σh = Int(u) —¦ σ where u is the diagonal matrix

u = diag(±1 , ’±2 , . . . , ±n’1 , ’±n ).

Then Nrpσ (u) = (’1)n/2 ±1 . . . ±n , hence

i GSp(A, σ) (σh ) = (’1)n/2 ±1 . . . ±n ∪ [Q].

Type Dn . Assume that char F = 2. Let G be an absolutely simple simply

connected group of type Dn (n ≥ 5) over F , so that G = Spin(A, σ) where A

is a central simple algebra of degree 2n over F with an orthogonal involution σ.

The case where A is split, i.e., G = Spin(q) for some quadratic form q, has been

considered in (??).

Assume that the algebra A is not split. In this case nG = 4. The exact sequence

similar to (??) yields a map

i1 : F — /F —2 = H 1 (F, µ2 ) ’ H 1 F, Spin(A, σ) .

The image i1 (a · F —2 ) for a ∈ F — corresponds to the torsor Xa given in the Cli¬ord

group “(A, σ) by the equation σ(x)x = a. The Rost invariant i(G) on Xa is given

by the formula

i(G)(Xa ) = (a) ∪ [A]

and therefore it is in general nontrivial. Hence the invariant does not factor through

the image of

H 1 F, Spin(A, σ) ’ H 1 F, O+ (A, σ)

as is the case when A is split.

Exceptional types.

G2 . Let G be an absolutely simple simply connected group of type G2 over F ,

so that G = Aut(C) where C is a Cayley algebra over F . The set H 1 (F, G) classi¬es

Cayley algebras over F . One has nG = 2 and the Rost invariant

i(G) : H 1 (F, G) ’ H 3 (F, µ—2 )

2

is given by the formula

i(G)(C ) = e3 (nC ) + e3 (nC )

where nC is the norm form of the Cayley algebra C (which is a 3-fold P¬ster form)

and e3 is the Arason invariant.

D4 . An absolutely simple simply connected algebraic group of type D4 over F

is isomorphic to Spin(T ) where T = (E, L, σ, ±) is a trialitarian algebra (see §??).

Here E is a central simple algebra with an orthogonal involution σ over a cubic

´tale extension L of F .

e

Assume ¬rst that L splits completely, i.e., L = F — F — F . Then E = A1 —

A2 — A3 where the Ai are central simple algebras of degree 8 over F . In this case

nG = 2 or 4. The ¬rst case occurs if and only if at least one of the algebras Ai is

split.

442 VII. GALOIS COHOMOLOGY

Assume now that L is not a ¬eld but does not split completely, i.e., L = F — K

where K is a quadratic ¬eld extension of F , hence E = A — C where A and C are

central simple algebras of degree 8 over F and K respectively (see §??). In this

case also nG = 2 or 4 and the ¬rst case takes place if and only if A is split.

Finally assume that L is a ¬eld (this is the trialitarian case). In this case

nG = 6 or 12. The ¬rst case occurs if and only if E is split.

F4 . nG = 6. The set H 1 (F, G) classi¬es absolutely simple groups of type F4

and also exceptional Jordan algebras. The cohomological invariant is discussed in

Chapter ??.

E6 . nG = 6 (when G is split).

Isn™t the

statement for E6 E7 . nG = 12 (when G is split).

and E7 true E8 . nG = 60.

whenever the Tits

algebras of G

are split and G

is inner ? (Skip Exercises

G.)

1. Let G be a pro¬nite group and let A be a (continuous) G-group. Show that

there is a natural bijection between the pointed set H 1 (G, A) and the direct

limit of H 1 (G/U, AU ) where U ranges over all open normal subgroups in G.

ˆ ˆ

2. Let Z be the inverse limit of Z/nZ, n ∈ N, and A be a Z-group such that any

element of A has a ¬nite order. Show that there is a natural bijection between

ˆ

the pointed set H 1 (Z, A) and the set of equivalence classes of A where the

equivalence relation is given by a ∼ a if there is b ∈ A such that a = b’1 ·a·σ(b)

ˆ

(σ is the canonical topological generator of Z).

3. Show that Aut(GL2 ) = Aut(SL2 ) — Z/2Z. Describe the twisted forms of GL2 .

4. Let Sn act on (Z/2Z)n through permutations and let G = (Z/2Z)n Sn . Let F

be an arbitrary ¬eld. Show that H 1 (F, G) classi¬es towers F ‚ L ‚ E with

L/F ´tale of dimension n and E/L quadratic ´tale.

e e

5. Let G = GLn /µ2 . Show that there is a natural bijection between H 1 (F, G)

and the set of isomorphism classes of triples (A, V, ρ) where A is a central

simple F -algebra of degree n, V is an F -vector space of dimension n2 and

ρ : A —F A ’ EndF (V ) is an isomorphism of F -algebras.

Hint: For an n-dimensional F -vector space U there is an associated triple

(AU , VU , ρU ) where AU = EndF (U ), VU = U —2 and where

ρ : EndF (U ) —F EndF (U ) ’ EndF (U —2 )

is the natural map. If F is separably closed, then any triple (A, V, ρ) is isomor-

phic to (AU , VU , ρU ). Moreover the homomorphism

GL(U ) ’ { (±, β) ∈ AutF (AU ) — GL(VU ) | ρ —¦ (± — ±) = Ad(β) —¦ µ }

given by γ ’ Ad(γ), γ —2 is surjective with kernel µ2 .

6. Let G be as in Exercise ??. Show that the sequence

2δ 1

»

H 1 (F, G) ’ H 1 (F, PGLn ) ’ ’ H 2 (F, Gm )

’ ’

is exact. Here » is induced from the natural map GLn ’ PGLn and δ 1 is the

connecting homomorphism for (??).

Using this result one may restate Albert™s theorem on the existence of

involutions of the ¬rst kind (Theorem (??)) by saying that the natural inclusion

EXERCISES 443

PGOn ’ G induces a surjection

H 1 (F, PGOn ) ’ H 1 (F, G).

The construction of Exercise ?? in Chapter ?? can be interpreted in terms

of Galois cohomology via the natural homomorphism GL(U ) ’ PGO H(U )

where U is an n-dimensional vector space and H(U ) is the hyperbolic quadratic

space de¬ned in §??.

7. Let K/F be separable quadratic extension of ¬elds. Taking transfers, the exact

sequence (??) induces an exact sequence

1 ’ RK/F (Gm ) ’ RK/F (GLn ) ’ RK/F (PGLn ) ’ 1.

Let N : RK/F (Gm ) ’ Gm be the transfer map and set

G = RK/F (GLn )/ ker N.

Show that there is a natural bijection between H 1 (F, G) and the set of iso-

morphism classes of triples (A, V, ρ) where A is a central simple K-algebra of

degree n, V is an F -vector space of dimension n2 and

ρ : NK/F (A) ’ EndF (V )

is an isomorphism of F -algebras. Moreover show that the sequence

corK/F —¦δ 1

»

H (F, G) ’ H (K, PGLn ) ’ ’ ’ ’ H 2 (F, Gm )

1 1

’ ’ ’ ’’

is exact. Here δ 1 is the connecting homomorphism for the sequence (??) and

» is given by

H 1 (F, G) ’ H 1 F, RK/F (PGLn ) = H 1 (K, PGLn ).

Using this result one may restate the theorem on the existence of involutions of

the second kind (Theorem (??)) by saying that the natural inclusion PGU n =

SUn / ker N ’ G induces a surjection

H 1 (F, PGUn ) ’ H 1 (F, G).

The construction of Exercise ?? in Chapter ?? can be interpreted in terms of

Galois cohomology via the natural homomorphism GL(UK ) ’ PGU H1 (UK )

where U is an n-dimensional F -vector space and H1 (UK ) is the hyperbolic

hermitian space de¬ned in §??.

8. Let (A, σ, f ) be a central simple F -algebra with quadratic pair. Let GL1 (A)

act on the vector space Symd(A, σ) • Sym(A, σ)— by

ρ(a)(x, g) = axσ(a), ag ,

where ag(y) = g σ(a)ya for y ∈ Sym(A, σ). Show that the stabilizer of (1, f )

is O(A, σ, f ) and that the twisted ρ-forms of (1, f ) are the pairs (x, g) such that

x ∈ A— and g y + σ(y) = TrdA (y) for all y ∈ A. Use these results to give

an alternate description of H 1 F, O(A, σ, f ) , and describe the canonical map

induced by the inclusion O(A, σ, f ) ’ GO(A, σ, f ).

9. Let L be a Galois Z/nZ-algebra over a ¬eld F of arbitrary characteristic. Using

the exact sequence 0 ’ Z ’ Q ’ Q/Z ’ 0, associate to L a cohomology class

[L] in H 2 (F, Z) and show that the class (L, a) ∈ H 2 (F, Gm ) corresponding

to the cyclic algebra (L, a) under the crossed product construction is the cup

product [L] ∪ a, for a ∈ F — = H 0 (F, Gm ).

444 VII. GALOIS COHOMOLOGY

10. Let K/F be a separable quadratic extension of ¬elds with nontrivial auto-

morphism ι, and let n be an integer which is not divisible by char F . Use

Proposition (??) to identify H 1 (F, µn[K] ) to the factor group

{ (x, y) ∈ F — — K — | xn = NK/F (y) }

.

{ (NK/F (z), z n ) | z ∈ K — }

For (x, y) ∈ F — — K — such that xn = NK/F (y), let [x, y] ∈ H 1 (F, µ[K] ) be the

corresponding cohomology class.

(a) Suppose n = 2. Since µ2[K] = µ2 , there is a canonical isomorphism

H 1 (F, µ2[K] ) F — /F —2 . Show that this isomorphism takes [x, y] to

NK/F (z) · F —2 , where z ∈ K — is such that x’1 y = zι(z)’1 .

(b) Suppose n = rs for some integers r, s. Consider the exact sequence

j

i

1 ’ µr[K] ’ µn[K] ’ µs[K] ’ 1.

’ ’

Show that the induced maps

j1

i1

H 1 (F, µr[K] ) ’ H 1 (F, µn[K] ) ’ H 1 (F, µs[K] )

’ ’

can be described as follows:

i1 [x, y] = [x, y s ] j 1 [x, y] = [xr , y].

and

(Compare with (??).)

(c) Show that the restriction map

res: H 1 (F, µn[K] ) ’ H 1 (K, µn ) = K — /K —n

takes [x, y] to y · K —n and the corestriction map

cor : H 1 (K, µn ) ’ H 1 (F, µn[K] )

takes z · K —n to [1, zι(z)’1 ].

11. Show that for n dividing 24, µn — µn and Z/nZ are isomorphic as Galois

modules.

12. Let (A, σ) be a central simple algebra over F with a symplectic involution σ.

Show that the map

Symd(A, σ)— /∼ = H 1 F, Sp(A, σ) ’ H 1 F, SL1 (A) = F — / Nrd(A— )

induced by the inclusion Sp(A, σ) ’ SL1 (A) takes a ∈ Sym(A, σ)— to its

pfa¬an norm NrpA (a) modulo Nrd(A— ).

13. Let A be a central simple algebra over F . For any c ∈ F — write Xc for the set

of all x ∈ A— such that Nrd(x) = c. Prove that

sep

(a) Xc is a SL1 (Asep )-torsor.

(b) Any SL1 (Asep )-torsor is isomorphic to Xc for some c.

(c) Xc Xd if and only if cd’1 ∈ Nrd(A— ).

14. Describe H 1 F, Spin(V, q) in terms of twisted forms of tensors.

15. Let (A, σ, f ) be a central simple F -algebra with quadratic pair of even degree 2n

over an arbitrary ¬eld F . Let Z be the center of the Cli¬ord algebra C(A, σ, f )

and let „¦(A, σ, f ) be the extended Cli¬ord group.

(a) Show that the connecting map

δ 1 : H 1 F, PGO+ (A, σ, f ) ’ H 2 F, RZ/F (Gm,Z ) = Br(Z)

EXERCISES 445

in the cohomology sequence associated to

χ

1 ’ RZ/F (Gm,Z ) ’ „¦(A, σ, f ) ’ PGO+ (A, σ, f ) ’ 1

’

’1

maps the 4-tuple (A , σ , f , •) to C(A , σ , f )—Z Z C(A, σ, f ) , where

the tensor product is taken with respect to •.

(b) Show that the multiplication homomorphism

Spin(A, σ, f ) — RZ/F (Gm,Z ) ’ „¦(A, σ, f )

induces an isomorphism

„¦(A, σ, f ) Spin(A, σ, f ) — RZ/F (Gm,Z ) /C

where C is isomorphic to the center of Spin(A, σ, f ). Similarly, show that

GO+ (A, σ, f ) O+ (A, σ, f ) — Gm /µ2

where µ2 is embedded diagonally in the product.

(c) Assume that n is even. Let ± : „¦(A, σ, f ) ’ GO+ (A, σ, f ) be the ho-

momorphism which, under the isomorphism in (??), is the vector rep-

resentation χ on Spin(A, σ, f ) and the norm map on RZ/F (Gm,Z ). By

relating via ± the exact sequence in (??) to a similar exact sequence for

GO+ (A, σ, f ), show that for all 4-tuple (A , σ , f , •) representing an ele-

ment of H 1 F, PGO+ (A, σ, f ) ,

’1

= [A ][A]’1

NZ/F C(A , σ , f ) —Z Z C(A, σ, f ) in Br(F ).

In particular, NZ/F C(A, σ, f ) = [A].

Similarly, using the homomorphism „¦(A, σ, f ) ’ RZ/F (Gm,Z ) which is

trivial on Spin(A, σ, f ) and the squaring map on RZ/F (Gm,Z ), show that

’1 2

C(A , σ , f ) —Z Z C(A, σ, f ) = 1.

2

In particular, C(A, σ, f ) = 1. (Compare with (??).)

(d) Assume that n is odd. Let G = O+ (A, σ, f ) — RZ/F (Gm,Z ) /µ2 . Using

the homomorphism ± : „¦(A, σ, f ) ’ G which is the vector representation

χ on Spin(A, σ, f ) and the squaring map on RZ/F (Gm,Z ), show that for all

4-tuple (A , σ , f , •) representing an element of H 1 F, PGO+ (A, σ, f ) ,

’1 2

= [AZ ][AZ ]’1

C(A , σ , f ) —Z Z C(A, σ, f ) in Br(Z).

2

In particular, C(A, σ, f ) = [AZ ].

Using the character of „¦(A, σ, f ) which is trivial on Spin(A, σ, f ) and is

the norm on RZ/F (Gm,Z ), show that

’1

NZ/F C(A , σ , f ) —Z Z C(A, σ, f ) = 1.

In particular, NZ/F C(A, σ, f ) = 1. (Compare with (??).)

16. (Qu´guiner [?]) Let (B, „ ) be a central simple F -algebra with unitary involution

e

of degree n. Let K be the center of B and let „ = Int(u) —¦ „ for some unit u ∈

Sym(B, „ ). Assume that char F does not divide n. Show that the Tits classes

t(B, „ ) and t(B, „ ) in H 2 (F, µn[K] ) are related by t(B, „ ) = t(B, „ ) + ζK ∪

NrdB (u) where ζK is the nontrivial element of H 1 (F, Z[K] ) and NrdB (u) =

NrdB (u) · F —n ∈ F — /F —n = H 1 (F, µn ). (Compare with (??).)

446 VII. GALOIS COHOMOLOGY

Notes

§??. The concept of a nonabelian cohomology set H 1 (“, A) has its origin in the

theory of principal homogeneous spaces (or torsors) due to Grothendieck [?], see

also Frenkel [?] and Serre [?]. The ¬rst steps in the theory of principal homogeneous

spaces attached to an algebraic group (in fact a commutative group variety) are

found in Weil [?].

Galois descent was implicitly used by Chˆtelet [?], in the case where A is an

a

elliptic curve (see also [?]). An explicit formulation (and proof) of Galois descent

in algebraic geometry was ¬rst given by Weil [?]. The idea of twisting the action

of the Galois group using automorphisms appears also in this paper, see Weil™s

commentaries in [?, pp. 543“544].

No Galois cohomology appears in the paper [?] on principal homogeneous spaces

mentioned above. The fact that Weil™s group of classes of principal homogeneous

spaces for a commutative group variety A over a ¬eld F stands in bijection with

the Galois 1-cohomology set H 1 (F, A) was noticed by Serre; details are given in

Lang and Tate [?], see also Tate™s Bourbaki talk [?].

The ¬rst systematic treatment of Galois descent, including nonabelian cases

(linear groups, in particular PGLn with application to the Brauer group), appeared

in Serre™s book “Corps locaux” [?], which was based on a course at the Coll`ge de

e

France in 1958/59. Twisted forms of algebraic structures viewed as tensors are

mentioned as examples. Applications to quadratic forms are given in Springer [?].

Another early application is the realization by Weil [?], following an observation of

“un amateur de cocycles tr`s connu”33 , of Siegel™s idea that classical groups can be

e

described as automorphism groups of algebras with involution (Weil [?, pp. 548“

549]).

Since then this simple but very useful formalism found many applications. See

the latest revised and completed edition of the Lecture Notes of Serre [?] and his

Bourbaki talk [?] for more information and numerous references. A far-reaching

generalization of nonabelian Galois cohomology, which goes beyond Galois exten-

sions and applies in the setting of schemes, was given by Grothendieck [?].

Our presentation in this section owes much to Serre™s Lecture Notes [?] and to

the paper [?] of Borel and Serre. The technique of changing base points by twisting

coe¬cients in cohomology, which we use systematically, was ¬rst developed there.

Note that the term “co-induced module” is used by Serre [?] and by Brown [?] for

the modules which we call “induced”, following Serre [?].

§??. Lemma (??), the so-called “Shapiro lemma”, was independently proved

by Eckmann [?, Theorem 4], D. K. Faddeev [?], and Arnold Shapiro. Shapiro™s

proof appears in Hochschild-Nakayama [?, Lemma 1.1].

Besides algebras and quadratic forms, Severi-Brauer varieties also have a nice

interpretation in terms of Galois cohomology: the group scheme PGL n occurs not

only as the automorphism group of a split central simple algebra of degree n, but

also as the automorphism group of the projective space Pn’1 . The Severi-Brauer

variety SB(A) attached to a central simple algebra A is a twisted form of the

projective space, given by the cocycle of A (see Artin [?]).

For any quadratic space (V, q) of even dimension 2n, the Cli¬ord functor de¬nes

a homomorphism C : PGO(V, q) ’ Autalg C0 (V, q) (see (??)). The induced map

in cohomology C 1 : H 1 F, PGO(V, q) ’ H 1 F, Autalg C0 (V, q) associates to

33 also referred to as “Mr. P. (the famous winner of many cocycle races)”

NOTES 447

every central simple F -algebra with quadratic pair of degree 2n a separable F -

algebra of dimension 22n’1 ; this is the de¬nition of the Cli¬ord algebra of a central

simple algebra with quadratic pair by Galois descent.

§??. Although the cyclic algebra construction is classical, the case considered

here, where L is an arbitrary Galois Z/nZ-algebra, is not so common in the lit-

erature. It can be found however in Albert [?, Chapter VII]. Note that if L is a

¬eld, its Galois Z/nZ-algebra structure designates a generator of the Galois group

Gal(L/F ).

The exact sequence (??) was observed by Arason-Elman [?, Appendix] and by

Serre [?, Chapter I, §2, Exercise 2]. (This exercise is not in the 1973 edition.) The

special case where M = µ2 (Fsep ) (Corollary (??)) plays a crucial rˆle in Arason [?].

o

The cohomological invariants f1 , g2 , f3 for central simple F -algebras with

unitary involution of degree 3 are discussed in Haile-Knus-Rost-Tignol [?, Corol-

lary 32]. It is also shown in [?] that these invariants are not independent and that

the invariant g2 (B, „ ) gives information on the ´tale F -subalgebras of B. To state

e

precise results, recall from (??) that cubic ´tale F -algebras with discriminant ∆

e

1

are classi¬ed by the orbit set H (F, A3[∆] )/S2 . Suppose char F = 2, 3 and let

F (ω) = F [X]/(X 2 + X + 1), so that µ3 = A3[F (ω)] . Let (B, „ ) be a central simple

F -algebra with unitary involution of degree 3 and let L be a cubic ´tale F -algebra

e

1

with discriminant ∆. Let K be the center of B and let cL ∈ H (F, A3[∆] ) be a co-

homology class representing L. The algebra B contains a subalgebra isomorphic to

L if and only if g2 (B, „ ) = cL ∪d for some d ∈ H 1 (F, A3[K—F (ω)—∆] ). (Compare with

Proposition (??).) If this condition holds, then B also contains an ´tale subalgebra

e

L with associated cohomology class d (hence with discriminant K — F (ω) — ∆).

Moreover, there exists an involution „ such that Sym(B, „ ) contains L and L .

See [?, Proposition 31].

§??. Let (A, σ) be a central simple algebra with orthogonal involution of even

degree over a ¬eld F of characteristic di¬erent from 2. The connecting homomor-

phism

δ 1 : H 1 F, O+ (A, σ) ’ H 2 (F, µ2 ) = 2 Br(F )

in the cohomology sequence associated to the exact sequence

1 ’ µ2 ’ Spin(A, σ) ’ O+ (A, σ) ’ 1

is described in Garibaldi-Tignol-Wadsworth [?]. Recall from (??) the bijection

H 1 F, O+ (A, σ) SSym(A, σ)— /≈.

For (s, z) ∈ SSym(A, σ)— , consider the algebra A = M2 (A) EndA (A2 ) with the

involution σ adjoint to the hermitian form 1, ’s’1 , i.e.,

’σ(c)s’1

a b σ(a)

σ = for a, b, c, d ∈ A.

sσ(d)s’1

c d ’sσ(b)

01

∈ A . We have s ∈ Skew(A , σ ) and NrdA (s ) = NrdA (s) = z 2 .

Let s =

s0