H 1 (F, Sn ).

(29.9) ←’

´tale F -algebras of degree n

e

396 VII. GALOIS COHOMOLOGY

Since the “-action on Sn is trivial, the pointed set H 1 (F, Sn ) coincides with the

set of conjugacy classes of continuous maps “ ’ Sn and hence also classi¬es iso-

morphism classes of “-sets X consisting of n elements (see (??)). The cocycle

γ : “ ’ Sn corresponds to the ´tale algebra L = Map(X, Fsep )“ where “ acts on

e

the set X via γ.

The sign map sgn : Sn ’ {±1} = S2 induces a map in cohomology

sgn1 : H 1 (F, Sn ) ’ H 1 (F, S2 ).

In view of (??) this map sends (the isomorphism class of) an ´tale algebra L to

e

(the isomorphism class of) its discriminant ∆(L).

Another interpretation of H 1 (F, Sn ) is given in Example (??):

H 1 (“, Sn ) Isom(Sn “Gal F ).

In fact, we may associate to any ´tale F -algebra L of dimension n its Galois S n -

e

closure Σ(L) (see (??)). This construction induces a canonical bijection between

the isomorphism classes of ´tale algebras of dimension n and isomorphism classes

e

of Galois Sn -algebras. Note however that Σ is not a functor: an F -algebra ho-

momorphism L1 ’ L2 which is not injective does not induce any homomorphism

Σ(L1 ) ’ Σ(L2 ).

Central simple algebras. Let A = Mn (F ), the matrix algebra of degree n.

Since every central simple F -algebra is split by Fsep , and since every F -algebra A

such that Asep Mn (Fsep ) is central simple (see (??)), the twisted forms of A are

exactly the central simple F -algebras of degree n. The Skolem-Noether theorem

(??) shows that every automorphism of A is inner, hence Autalg (A) = PGLn .

Therefore, as in (??), there is a natural bijection

F -isomorphism classes of

H 1 (F, PGLn ).

←’

central simple F -algebras of degree n

Consider the exact sequence:

(29.10) 1 ’ Gm ’ GLn ’ PGLn ’ 1.

By twisting all the groups by a cocycle in H 1 (F, PGLn ) corresponding to a central

simple F -algebra B of degree n, we get the exact sequence

1 ’ Gm ’ GL1 (B) ’ PGL1 (B) ’ 1.

Since H 1 F, GL1 (B) = 1 by Hilbert™s Theorem 90, it follows from Corollary (??)

that the connecting map

δ 1 : H 1 (F, PGLn ) ’ H 2 (F, Gm )

with respect to (??) is injective. The map δ 1 is de¬ned here as follows: if ±γ ∈

AutFsep Mn (Fsep ) is a 1-cocycle, choose cγ ∈ GLn (Fsep ) such that ±γ = Int(cγ )

(by Skolem-Noether). Then

cγ,γ = cγ · γcγ · c’1 ∈ Z 2 (F, Gm )

γγ

is the corresponding 2-cocycle. The δ 1 for di¬erent n™s ¬t together to induce an

injective homomorphism Br(F ) ’ H 2 (F, Gm ). To prove that this homomorphism

is surjective, we may reduce to the case of ¬nite Galois extensions, since for every

2-cocycle cγ,γ with values in Gm there is a ¬nite Galois extension L/F such that

c : “ — “ ’ Fsep factors through a 2-cocycle in Z 2 Gal(L/F ), L— . Thus, the

—

following proposition completes the proof that Br(F ) H 2 (F, Gm ):

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 397

(29.11) Proposition. Let L/F be a ¬nite Galois extension of ¬elds of degree n,

and let G = Gal(L/F ). The map

δ 1 : H 1 G, PGLn (L) ’ H 2 (G, L— )

is bijective.

Proof : Injectivity follows by the same argument as for the connecting map with

respect to (??). To prove surjectivity, choose c ∈ Z 2 (G, L— ) and let V be the

n-dimensional L-vector space

V= eσ L.

σ∈G

Numbering the elements of G, we may identify V = Ln , hence EndL (V ) = Mn (L)

and Aut EndL (V ) = PGLn (L). For σ ∈ G, let aσ ∈ EndL (V ) be de¬ned by

aσ (e„ ) = eσ„ cσ,„ .

We have

aσ —¦ σ(a„ )(eν ) = eσ„ ν cσ,„ ν σ(c„,ν ) = eσ„ ν cσ„,ν cσ,„ = aσ„ (eν )cσ,„

for all σ, „ , ν ∈ G, hence Int(aσ ) ∈ Aut EndL (V ) is a 1-cocycle whose image

under δ 1 is represented by the cocycle cσ,„ .

With the same notation as in the proof above, a central simple F -algebra Ac

corresponding to the 2-cocycle c ∈ Z 2 (G, L— ) is given by

Ac = { f ∈ EndL (V ) | aσ —¦ σ(f ) = f —¦ aσ for all σ ∈ G }.

This construction is closely related to the crossed product construction, which we

brie¬‚y recall: on the L-vector space

C= Lzσ

σ∈G

with basis (zσ )σ∈G , de¬ne multiplication by

zσ = σ( )zσ and zσ z„ = cσ,„ zσ„

for σ, „ ∈ G and ∈ L. The cocycle condition ensures that C is an associative

algebra, and it can be checked that C is central simple of degree n over F (see, e.g.,

Pierce [?, 14.1]).

For σ ∈ G and for ∈ L, de¬ne yσ , u ∈ EndL (V ) by

yσ (e„ ) = e„ σ c„,σ and u (e„ ) = e„ „ ( ) for „ ∈ G.

Computations show that yσ , u ∈ Ac , and

u —¦ yσ = yσ —¦ uσ( ) , yσ —¦ y„ = y„ σ —¦ uc„,σ .

uop —¦ yσ is an F -algebra

Therefore, the map C ’ Aop which sends op

σ zσ to

c σ

homomorphism, hence an isomorphism since C and Aop are central simple of de-

c

gree n. Thus, C Ac , showing that the isomorphism Br(F ) H 2 (F, Gm ) de¬ned

op

by the crossed product construction is the opposite of the isomorphism induced by

δ1 .

398 VII. GALOIS COHOMOLOGY

29.C. Algebras with a distinguished subalgebra. The same idea as in

§?? applies to pairs (A, L) consisting of an F -algebra A and a subalgebra L ‚ A.

∼ ∼

An isomorphism of pairs (A , L ) ’ (A, L) is an F -isomorphism A ’ A which

’ ’

∼

restricts to an isomorphism L ’ L. Let G ‚ GL(B) be the group scheme of

’

automorphisms of the ¬‚ag of vector spaces A ⊃ L. The group G acts on the space

HomF (A—F A, A) as in §?? and the group scheme AutG (m) where m : A—F A ’ A

is the multiplication map coincides with the group scheme Autalg (A, L) of all F -

algebra automorphisms of the pair (A, L). Since H 1 (F, G) = 1 by (??), there is by

Proposition (??) a bijection

F -isomorphism classes of pairs

H 1 F, Autalg (A, L) .

of F -algebras (A , L )

(29.12) ←’

such that (A , L )sep (A, L)sep

The map H 1 F, Autalg (A, L) ’ H 1 F, Autalg (A) induced by the inclusion of

Aut(A, L) in Aut(A) maps the isomorphism class of a pair (A , L ) to the isomor-

phism class of A . On the other hand, the map

H 1 F, Aut(A, L) ’ H 1 (F, Autalg (L)

induced by the restriction map Aut(A, L) ’ Aut(L) takes the isomorphism class

of (A , L ) to the isomorphism class of L .

Let AutL (A) be the kernel of the restriction map Aut(A, L) ’ Autalg (L).

In order to describe the set H 1 F, AutL (A) as a set of isomorphism classes as

in (??), let

W = HomF (A —F A, A) • HomF (L, A).

The group G = GL(A) acts on W as follows:

ρ(g)(ψ, •)(x — y, z) = g —¦ ψ g ’1 (x) — g ’1 (y) , g —¦ •(z)

for g ∈ G, ψ ∈ HomF (A —F A, A), • ∈ HomF (L, A), x, y ∈ A and z ∈ L. The

multiplication map m : A —F A ’ A and the inclusion i : L ’ A de¬ne an element

w = (m, i) ∈ W , and the group AutG (w) coincides with AutL (A). A twisted

form of w is a pair (A , •) where A is an F -algebra isomorphic to A over Fsep and

• : L ’ A is an F -algebra embedding of L in A . By Proposition (??) there is a

natural bijection

F -isomorphism classes of

H 1 F, AutL (A) .

pairs (A , •) isomorphic to

(29.13) ←’

the pair (A, i)sep over Fsep

The canonical map H 1 F, AutL (A) ’ H 1 F, Aut(A, L) takes the isomorphism

class of a pair (A , •) to the isomorphism class of the pair A , •(L) .

The preceding discussion applies in particular to separable F -algebras. If B is

a separable F -algebra with center Z, the restriction homomorphism

Autalg (B) = Autalg (B, Z) ’ Autalg (Z)

gives rise to the map of pointed sets

H 1 F, Autalg (B) ’ H 1 F, Autalg (Z)

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 399

which takes the class of a twisted form B of B to the class of its center Z .

On the other hand, the natural isomorphism AutZ (B) RZ/F PGL1 (B) and

Lemma (??) give a bijection of pointed sets

H 1 F, AutZ (B) H 1 F, RZ/F PGL1 (B) H 1 Z, PGL1 (B)

which takes the class of a pair (B , •) to the class of the Z-algebra B —Z Z (where

the tensor product is taken with respect to •).

29.D. Algebras with involution. Let (A, σ) be a central simple algebra

with involution (of any kind) over a ¬eld F . In this section, we give interpretations

for the cohomology sets

H 1 F, Aut(A, σ) , H 1 F, Sim(A, σ) , H 1 F, Iso(A, σ) .

We shall discuss separately the unitary, the symplectic and the orthogonal case,

but we ¬rst outline the general principles.

Let W = HomF (A — A, A) • EndF (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 )

de¬ned by

ρ(g)(•, ψ)(x — y, z) = g —¦ • g ’1 (x) — g ’1 (y) , g —¦ ψ —¦ g ’1 (z)

for g ∈ G, • ∈ HomF (A — A, A), ψ ∈ EndF (A) and x, y, z ∈ A. Let w =

(m, σ) ∈ W , where m is the multiplication map of A. The subgroup AutG (w)

of G coincides with the group scheme Aut(A, σ) of F -algebra automorphisms of

A commuting with σ. A twisted form of w is the structure of an algebra with

involution isomorphic over Fsep to (Asep , σsep ). Hence, by Proposition (??) there is

a natural bijection

F -isomorphism classes of

H 1 F, Aut(A, σ) .

F -algebras with involution (A , σ )

(29.14) ←’

isomorphic to (Asep , σsep ) over Fsep

Next, let W = EndF (A) and G = GL1 (A), the linear group of A (i.e., the

group of invertible elements in A). Consider the representation

ρ : G ’ GL(W )

de¬ned by

ρ (a)(ψ) = Int(a) —¦ ψ —¦ Int(a)’1 ,

for a ∈ G and ψ ∈ EndF (A). The subgroup AutG (σ) of G coincides with the

group scheme Sim(A, σ) of similitudes of (A, σ). A twisted ρ -form of σ is an invo-

lution of A which, over Fsep , is conjugate to σsep = σ — IdFsep . By Proposition (??)

and Hilbert™s Theorem 90 (see (??)), we get a bijection

conjugacy classes of involutions

H 1 F, Sim(A, σ) .

on A which over Fsep are

(29.15) ←’

conjugate to σsep

The canonical homomorphism Int : Sim(A, σ) ’ Aut(A, σ) induces a map

Int1 : H 1 F, Sim(A, σ) ’ H 1 F, Aut(A, σ)

400 VII. GALOIS COHOMOLOGY

which maps the conjugacy class of an involution σ to the isomorphism class of

(A, σ ).

Finally, recall from §?? that the group scheme Iso(A, σ) is de¬ned as the

stabilizer of 1 ∈ A under the action of GL1 (A) on A given by

ρ (a)(x) = a · x · σ(a)

for a ∈ GL1 (A) and x ∈ A. Twisted ρ -forms of 1 are elements s ∈ A for which

there exists a ∈ A— such that s = a · σ(a). We write Sym(A, σ) for the set of

sep

these elements,

Sym(A, σ) = { s ∈ A | s = a · σ(a) for some a ∈ A— } ‚ Sym(A, σ) © A— ,

sep

and de¬ne an equivalence relation on Sym(A, σ) by

if and only if s = a · s · σ(a) for some a ∈ A— .

s∼s

The equivalence classes are exactly the ρ -isomorphism classes of twisted forms of

1, hence Proposition (??) yields a canonical bijection

H 1 F, Iso(A, σ) .

(29.16) Sym(A, σ) /∼ ←’

The inclusion i : Iso(A, σ) ’ Sim(A, σ) induces a map

i1 : H 1 F, Iso(A, σ) ’ H 1 F, Sim(A, σ)

which maps the equivalence class of s ∈ Sym(A, σ) to the conjugacy class of the

involution Int(s) —¦ σ.

We now examine the various types of involutions separately.

Unitary involutions. Let (B, „ ) be a central simple F -algebra with unitary

involution. Let K be the center of B, which is a quadratic ´tale F -algebra, and let

e

n = deg(B, „ ). (The algebra B is thus central simple of degree n if K is a ¬eld, and

it is a direct product of two central simple F -algebras of degree n if K F — F .)

From (??), we readily derive a canonical bijection

F -isomorphism classes of

H 1 F, Aut(B, „ ) .

central simple F -algebras ←’

with unitary involution of degree n

We have an exact sequence of group schemes

f

1 ’ PGU(B, „ ) ’ Aut(B, „ ) ’ S2 ’ 1

’

where f is the restriction homomorphism to Autalg (K) = S2 . We may view the

group PGU(B, „ ) as the automorphism group of the pair (B, „ ) over K. As in

Proposition (??) (see also (??)) we obtain a natural bijection

F -isomorphism classes of triples (B , „ , •)

consisting of a central simple F -algebra

H 1 F, PGU(B, „ ) .

←’

with unitary involution (B , „ ) of degree n

∼

and an F -algebra isomorphism • : Z(B ) ’ K ’

By Proposition (??) the group Autalg (K) acts transitively on each ¬ber of the

map

H 1 F, PGU(B, „ ) ’ H 1 F, Aut(B, „ ) .

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 401

The ¬ber over a pair (B , „ ) consists of the triples (B , „ , IdK ) and (B , „ , ι), where

ι is the nontrivial automorphism of K/F . These triples are isomorphic if and only

if PGU(B , „ ) AutF (B , „ ).

After scalar extension to Fsep , we have Bsep Mn (Fsep ) — Mn (Fsep )op , and all

the unitary involutions on Bsep are conjugate to the exchange involution µ by (??).

Therefore, (??) specializes to a bijection

conjugacy classes of

H 1 F, GU(B, „ ) .

unitary involutions on B ←’

which are the identity on F

The exact sequence

1 ’ RK/F (Gm,K ) ’ GU(B, „ ) ’ PGU(B, „ ) ’ 1

induces a connecting map in cohomology

δ 1 : H 1 F, PGU(B, „ ) ’ H 2 F, RK/F (Gm,K ) = H 2 (K, Gm,K ) = Br(K)

where the identi¬cation H 2 F, RK/F (Gm,K ) = H 2 (K, Gm,K ) is given by Shapiro™s

lemma and the identi¬cation H 2 (K, Gm,K ) = Br(K) by the connecting map in the

cohomology sequence associated to

1 ’ Gm,K ’ GLn,K ’ PGLn,K ’ 1,

see §??. Under δ 1 , the class of a triple (B , „ , •) is mapped to the Brauer class

[B —K K] · [B]’1 , where the tensor product is taken with respect to •.

Our next goal is to give a description of H 1 F, U(B, „ ) . Every symmetric

element s ∈ Sym Mn (Fsep ) — Mn (Fsep )op , µ has the form

s = (m, mop ) = (m, 1op ) · µ(m, 1op )

for some m ∈ Mn (Fsep ). Therefore, the set Sym(B, „ ) of (??) is the set of sym-

metric units,

Sym(B, „ ) = Sym(B, „ )— (= Sym(B, „ ) © B — ),

and (??) yields a canonical bijection

Sym(B, „ )— /∼ H 1 F, U(B, „ ) .

(29.17) ←’

By associating with every symmetric unit u ∈ Sym(B, „ )— the hermitian form

u’1 : B — B ’ B

de¬ned by u’1 (x, y) = „ (x)u’1 y, it follows that H 1 F, U(B, „ ) classi¬es her-

mitian forms on B-modules of rank 1 up to isometry.

In order to describe the set H 1 F, SU(B, „ ) , consider the representation

ρ : GL1 (B) ’ GL(B • K)

given by

ρ(b)(x, y) = b · x · „ (b), Nrd(b)y

for b ∈ GL1 (B), x ∈ B and y ∈ K. Let w = (1, 1) ∈ B • K. The group AutG (w)

coincides with SU(B, „ ). Clearly, twisted forms of w are contained in the set31

SSym(B, „ )— = { (s, z) ∈ Sym(B, „ )— — K — | NrdB (s) = NK/F (z) }.

31 This set plays an essential rˆle in the Tits construction of exceptional simple Jordan alge-

o

bras (see § ??).

402 VII. GALOIS COHOMOLOGY

Over Fsep , we have Bsep Mn (Fsep ) — Mn (Fsep )op and we may identify „sep to the

exchange involution µ. Thus, for every (s, z) ∈ SSym(Bsep , „sep )— , there are m ∈

Mn (Fsep ) and z1 , z2 ∈ Fsep such that s = (m, mop ), z = (z1 , z2 ) and det m = z1 z2 .

—

Let m1 ∈ GLn (Fsep ) be any matrix such that det m1 = z1 , and let m2 = m’1 m. 1

Then

s = (m1 , mop ) · µ(m1 , mop ) and z = Nrd(m1 , mop ),

2 2 2

hence (s, z) = ρsep (m1 , mop )(w). Therefore, SSym(B, „ )— is the set of twisted

2

ρ-forms of w.

De¬ne an equivalence relation ≈ on SSym(B, „ )— by

if and only if s = b · s · „ (b) and z = NrdB (b)z for some b ∈ B —

(s, z) ≈ (s , z )

so that the equivalence classes under ≈ are exactly the ρ-isomorphism classes of

twisted forms. Proposition (??) yields a canonical bijection

SSym(B, „ )— /≈ H 1 F, SU(B, „ ) .

(29.18) ←’

The natural map of pointed sets

H 1 F, SU(B, „ ) ’ H 1 F, U(B, „ )

takes the class of (s, z) ∈ SSym(B, „ )— to the class of s ∈ Sym(B, „ )— .

There is an exact sequence

Nrd

1 ’ SU(B, „ ) ’ U(B, „ ) ’ ’ G1

’ m,K ’ 1

where

NK/F

G1

m,K = ker RK/F (Gm,K ) ’ ’ ’ Gm,F

’’

(hence G1 (F ) = K 1 is the group of norm 1 elements in K). The connecting map

m,K

G1 (F ) ’ H 1 F, SU(B, „ )

m,K

takes x ∈ G1 (F ) ‚ K — to the class of the pair (1, x).

m,K

(29.19) Example. Suppose K is a ¬eld and let (V, h) be a hermitian space over K

(with respect to the nontrivial automorphism ι of K/F ). We write simply U(V, h)

for U EndK (V ), σh and SU(V, h) for SU EndK (V ), σh . As in (??) and (??), we

have canonical bijections

—

H 1 F, U(V, h) ,

Sym EndK (V ), σh /∼ ←’

—

H 1 F, SU(V, h) .

SSym EndK (V ), σh /≈ ←’

—

The set Sym EndK (V ), σh /∼ is also in one-to-one correspondence correspon-

dence with the set of isometry classes of nonsingular hermitian forms on V , by

—

mapping s ∈ Sym EndK (V ), σh to the hermitian form hs : V — V ’ K de¬ned

by

hs (x, y) = h s’1 (x), y = h x, s’1 (y)

for x, y ∈ V . Therefore, we have a canonical bijection of pointed sets

isometry classes of

H 1 F, U(V, h)

nonsingular hermitian ←’

forms on V

§29. GALOIS COHOMOLOGY OF ALGEBRAIC GROUPS 403

where the base point of H 1 F, U(V, h) corresponds to the isometry class of h.

To give a similar interpretation of H 1 F, SU(V, h) , observe that for every

unitary involution „ on EndK (V ) and every y ∈ K — such that NK/F (y) = 1 there

exists u ∈ U EndK (V ), „ such that det(u) = y. Indeed, „ is the adjoint involution

with respect to some hermitian form h . If (e1 , . . . , en ) is an orthogonal basis of V

for h , we may take for u the endomorphism which leaves ei invariant for i = 1, . . . ,

n ’ 1 and maps en to en y. From this observation, it follows that the canonical map

— —

SSym EndK (V ), σh /≈ ’ Sym EndK (V ), σh /∼

—

given by (s, z) ’ s is injective. For, suppose (s, z), (s , z ) ∈ SSym EndK (V ), σh

are such that s ∼ s , and let b ∈ EndK (V )— satisfy

s = b · s · σh (b).

Since det(s) = NK/F (z) and det(s ) = NK/F (z ), it follows that

NK/F (z ) = NK/F z det(b) .

Choose u ∈ U EndK (V ), Int(s ) —¦ σh such that det(u) = z z ’1 det(b)’1 . Then

s = u · s · σh (u) = ub · s · σh (ub) and z = z det(ub),

hence (s , z ) ≈ (s, z).

As a consequence, the canonical map H 1 F, SU(V, h) ’ H 1 F, U(V, h) is