and is isomorphic to Autalg (J) where J is a split simple exceptional Jordan algebra

of dimension 27.

Reference: See Chevalley-Schafer [?], Freudenthal [?, Satz 4.11], Springer [?] or [?,

14.27, 14.28]. The proof given in [?] is over R, the proofs in [?] and [?] assume that

F is a ¬eld of characteristic di¬erent from 2 and 3. Springer™s proof [?] holds for

any ¬eld.

360 VI. ALGEBRAIC GROUPS

For a simple split group of type G2 we have

(25.14) Theorem. A split simple group of type G2 is simply connected and adjoint

and is isomorphic to Autalg (C) where C is a split Cayley algebra.

Reference: See Jacobson [?], Freudenthal [?] or Springer [?]. The proof in [?] is

over R, the one in [?] assumes that F is a ¬eld of characteristic zero and [?] gives

a proof for arbitrary ¬elds.

More on (??), resp. (??) can be found in the notes at the end of Chapter IX,

resp. VIII.

25.B. Automorphisms of split semisimple groups. Let G be a split semi-

simple group over F , let T ‚ G be a split maximal torus, and Π a system of simple

roots in ¦(G). For any • ∈ Aut(G) there is g ∈ G(F ) such that for ψ = Int(g) —¦ •,

one has ψ(T ) = T and ψ(Π) = Π, hence ψ induces an automorphism of Dyn(¦).

Thus, we obtain a homomorphism Aut(G) ’ Aut Dyn(¦) .

On the other hand, we have a homomorphism Int : G(F ) ’ Aut(G) taking

g ∈ G(F ) to the inner automorphism Int(gR ) of G(R) for any R ∈ Alg F where gR

is the image of g under G(F ) ’ G(R).

(25.15) Proposition. If G is a split semisimple adjoint group, the sequence

Int

1 ’ G(F ) ’’ Aut(G) ’ Aut Dyn(¦) ’ 1

’

is split exact.

Let G be a split semisimple group (not necessarily adjoint) and let C = C(G)

be the kernel of adG . Then G = G/C is an adjoint group with ¦(G) = ¦(G) and

we have a natural homomorphism Aut(G) ’ Aut(G). It turns out to be injective

and its image contains Int G(F ) .

(25.16) Theorem. Let G be a split semisimple group. Then there is an exact

sequence

1 ’ G(F ) ’ Aut(G) ’ Aut Dyn(¦)

where the last map is surjective and the sequence splits provided G is simply con-

nected or adjoint.

(25.17) Corollary. Let G be a split simply connected semisimple group. Then the

natural map Aut(G) ’ Aut(G) is an isomorphism.

§26. Semisimple Groups over an Arbitrary Field

In this section we give the classi¬cation of semisimple groups over an arbitrary

¬eld which do not contain simple components of types D4 , E6 , E7 or E8 . We recall

that a category A is a groupoid if all morphisms in A are isomorphisms. A groupoid

A is connected if all its objects are isomorphic.

Let “ be a pro¬nite group and let A be a groupoid. A “-embedding of A is a

functor i : A ’ A where A is a connected groupoid, such that for every X, Y in A,

there is a continuous “-action on the set HomA (i X, i Y ) with the discrete topology,

compatible with the composition law in A, and such that the functor i induces a

bijection

∼

HomA (X, Y ) ’ HomA (i X, i Y )“ .

’

It follows from the de¬nition that a “-embedding is a faithful functor.

§26. SEMISIMPLE GROUPS OVER AN ARBITRARY FIELD 361

(26.1) Examples. (1) Let 1 An = 1 An (F ) be the category of all central simple

algebras over F of degree n + 1 with morphisms being isomorphisms of F -algebras.

Then for the group “ = Gal(Fsep /F ) the natural functor

j : 1 An (F ) ’ 1 An (Fsep ), A ’ Asep = A —F Fsep

is a “-embedding.

(2) Let G be an algebraic group over a ¬eld F and let A = A(F ) be the groupoid

of all twisted forms of G (objects are algebraic groups G over F such that Gsep

Gsep and morphisms are algebraic group isomorphisms over F ). Then for “ =

Gal(Fsep /F ) the natural functor

j : A(F ) ’ A(Fsep ), G ’ Gsep

is a “-embedding.

Let i : A ’ A and let j : B ’ B be two “-embeddings and let S : A ’ B be a

functor. A “-extension of S (with respect to i and j) is a functor S : A ’ B such

that j —¦ S = S —¦ i and for all γ ∈ “, X, Y ∈ A, and f ∈ HomA (i X, i Y ) one has

γ S(f ) = S(γf ).

We call a continuous map γ ∈ “ ’ fγ ∈ AutA (i X) a 1-cocycle if

fγ —¦ γfρ = fγρ

for all γ, ρ ∈ “ and, we say that a “-embedding i: A ’ A satis¬es the descent

condition if for any object X ∈ A and for any 1-cocycle fγ ∈ AutA (i X) there exist

an object Y ∈ A and a morphism h : i Y ’ i X in A such that

fγ = h —¦ γh’1

for all γ ∈ “.

(26.2) Proposition. Let i : A ’ A and j : B ’ B be two “-embeddings and let

S : A ’ B be a functor having a “-extension S : A ’ B. Assume that the “-

embedding i satis¬es the descent condition. If S is an equivalence of categories,

then so is S.

Proof : Since i, j, and S are faithful functors, the functor S is also faithful. Let

g ∈ HomB (S X, S Y ) be any morphism for some X and Y in A. Since S is an

equivalence of categories, we can ¬nd f ∈ HomA (i X, i Y ) such that S(f ) = j(g).

For any γ ∈ “ one has

S(γf ) = γ S(f ) = γ j(g) = j(g) = S(f ),

hence γf = f . By the de¬nition of a “-embedding, there exists h ∈ HomA (X, Y )

such that i h = f . The equality j S(h) = S i(h) = S(f ) = j g shows that S(h) = g,

i.e., S is full as a functor. In view of Maclane [?, p. 91] it remains to check that any

object Z ∈ B is isomorphic to S(Y ) for some Y ∈ A. Take any object X ∈ A. Since

B is a connected groupoid, the objects j S(X) and j Z are isomorphic. Choose any

isomorphism g : j S(X) ’ j Z in B and set

gγ = g ’1 —¦ γg ∈ AutB j S(X)

for γ ∈ “. Clearly, gγ is a 1-cocycle. Since S is bijective on morphisms, there exists

a 1-cocycle

fγ ∈ AutA (i X)

362 VI. ALGEBRAIC GROUPS

such that S(fγ ) = gγ for any γ ∈ “. By the descent condition for the “-embedding i

one can ¬nd Y ∈ A and a morphism h : i Y ’ i X in A such that fγ = h —¦ γh’1 for

any γ ∈ “. Consider the composition

l = g —¦ S(h) : j S(Y ) = S i(Y ) ’ S i(X) = j S(X) ’ j Z

in B. For any γ ∈ “ one has

’1

γl = γg —¦ γ S(h) = g —¦ gγ —¦ S(γh) = g —¦ gγ —¦ S(fγ —¦ h) = g —¦ S(h) = l.

By the de¬nition of a “-embedding, l = j(m) for some isomorphism m : S(Y ) ’ Z

in B, i.e., Z is isomorphic to S(Y ).

(26.3) Remark. Since A and B are connected groupoids, in order to check that a

functor S : A ’ B is an equivalence of categories, it su¬ces to show that for some

object X ∈ A the map

AutA (X) ’ AutB S(X)

is an isomorphism, see Proposition (??).

We now introduce a class of “-embeddings satisfying the descent condition. All

“-embeddings occurring in the sequel will be in this class.

Let F be a ¬eld and “ = Gal(Fsep /F ). Consider a collection of F -vector spaces

V , V (2) , . . . , V (n) , and W (not necessarily of ¬nite dimension). The group “

(1)

(i)

acts on GL(Vsep ) and GL(Wsep ) in a natural way. Let

(1) (n)

ρ : H = GL(Vsep ) — · · · — GL(Vsep ) ’ GL(Wsep )

be a “-equivariant group homomorphism.

Fix an element w ∈ W ‚ Wsep and consider the category A = A(ρ, w) whose

objects are elements w ∈ Wsep such that there exists h ∈ H with ρ(h)(w) = w

(for example w is always an object of A). The set HomA (w , w ) consists of all

those h ∈ H such that ρ(h)(w ) = w . The composition law in A is induced by the

multiplication in H. Clearly, A is a connected groupoid.

Let A = A(ρ, w) be the subcategory of A consisting of all w ∈ W which are

objects in A. Clearly, for any w , w ∈ A the set HomA (w , w ) is “-invariant with

respect to the natural action of “ on H, and we set

HomA (w , w ) = HomA (w , w )“ ‚ H “ .

Clearly, the embedding functor i : A(ρ, w) ’ A(ρ, w) is a “-embedding.

(26.4) Proposition. The “-embedding i : A(ρ, w) ’ A(ρ, w) satis¬es the descent

condition.

Proof : Let w ∈ W be an object in A and let fγ ∈ AutA (w ) ‚ H be a 1-cocycle.

(i) (i)

Let fγ be the i-th component of fγ in GL(Vsep ). We introduce a new “-action

(i) (i) (i)

on each Vsep by the formula γ — v = fγ (γv) where γ ∈ “, v ∈ Vsep . Clearly,

(i)

γ — (xv) = γx · (γ — v) for any x ∈ Fsep . Let U (i) be the F -subspace in Vsep of

“-invariant elements with respect to the new action. In view of Lemma (??) the

natural maps

θ(i) : Fsep —F U (i) ’ Vsep ,

(i)

x — u ’ xu

§26. SEMISIMPLE GROUPS OVER AN ARBITRARY FIELD 363

are Fsep -isomorphisms of vector spaces. For any x ∈ Fsep and u ∈ U (i) one has

(i)’1

γu = fγ (u), hence (with respect to the usual “-action)

(fγ —¦ γ —¦ θ(i) )(x — u) = (fγ —¦ γ)(xu)

(i) (i)

’1

(i) (i) (i)

= fγ (γx · γu) = γx · fγ fγ (u) = γx · u

= θ(i) (γx — u) = θ(i) —¦ γ (x — u).

In other words, fγ = θ —¦ γθ’1 where γθ = γ —¦ θ —¦ γ ’1 .

The F -vector spaces U (i) and V (i) have the same dimension and are there-

fore F -isomorphic. Choose any F -isomorphism ±(i) : V (i) ’ U (i) and consider the

(i) (i)

composition β (i) = θ(i) —¦ ±sep ∈ GL(Vsep ). Clearly,

fγ = β —¦ γβ ’1 .

(26.5)

Consider the element w = ρ(β ’1 )(w ) ∈ Wsep . By de¬nition, w is an object

of A and β represents a morphism w ’ w in A. We show that w ∈ W , i.e.,

w ∈ A. Indeed, for any γ ∈ “ one has

γ(w ) = ρ(γβ ’1 )(w ) = ρ(β ’1 —¦ fγ )(w ) = ρ(β ’1 )(w ) = w

since ρ(fγ )(w ) = w . Finally, the equation (??) shows that the functor i satis¬es

the descent condition.

(26.6) Corollary. The functors j in Examples (??), (??) and (??), (??) satisfy

the descent condition.

Proof : For (??) we consider the F -vector space W = HomF (A —F A, A) where

A = Mn+1 (F ) is the split algebra and w = m ∈ W is the multiplication map of A.

For (??), let A = F [G]. Consider the F -vector space

W = HomF (A —F A, A) • Hom(A, A —F A),

the element w = (m, c) ∈ W where m is the multiplication and c is the comultipli-

cation on A. In each case we have a natural representation

ρsep : GL(Asep ) ’ GL(Wsep )

(see Example (??), (??)).

We now restrict our attention to Example (??),(??), since the argument for

Example (??), (??) is similar (and even simpler). By Proposition (??) there is a

“-embedding

i : A(ρsep , w) ’ A(ρsep , w)

satisfying the descent condition. We have a functor

T : A(ρsep , w) ’ A(F )

taking w = (m , c ) ∈ A(ρsep , w) to the F -vector space A with the Hopf algebra

structure given by m and comultiplication c . Clearly A has a Hopf algebra struc-

ture (with some co-inverse map i and co-unit u ) since over Fsep it is isomorphic

to the Hopf algebra Asep . A morphism between w and w , being an element of

GL(A), de¬nes an isomorphism of the corresponding Hopf algebra structures on A.

The functor T has an evident “-extension

T : A(ρsep , w) ’ A(Fsep ),

364 VI. ALGEBRAIC GROUPS

which is clearly an equivalence of groupoids. Since the functor i satis¬es the descent

condition, so does the functor j.

26.A. Basic classi¬cation results. Let G be a semisimple algebraic group

over an arbitrary ¬eld F . Choose any maximal torus T ‚ G. Then Tsep is a split

maximal torus in Gsep , hence we have a root system ¦(Gsep ), which we call the

root system of G and denote ¦(G). The absolute Galois group “ = Gal(Fsep /F )

acts naturally on ¦(G) and hence on the Dynkin diagram Dyn ¦(G) .

The group G is said to be simply connected (resp. adjoint) if the split group

Gsep is so.

(26.7) Theorem. For any semisimple group G there exists (up to an isomor-

phism) a unique simply connected group G and a unique adjoint group G such that

there are central isogenies G ’ G ’ G.

Proof : Let C ‚ G be the kernel of the adjoint representation adG . Then G =

G/C im(adG ) is an adjoint group with the same root system as G. Denote

by Gd a split twisted form of G and by Gd its simply connected covering. Consider

the groupoid A(F ) (resp. B(F )) of all twisted forms of Gd (resp. Gd ). The group

G is an object of A(F ). Clearly, the natural functors

i : A(F ) ’ A(Fsep ), j : B(F ) ’ B(Fsep )

are “-embeddings where “ = Gal(Fsep /F ). The natural functor

S(F ) : B(F ) ’ A(F ), G ’ G = G /C(G )

has the “-extension S(Fsep ). By Corollary (??) the functor S(Fsep ) is an equivalence

of groupoids. By Proposition (??) and Corollary (??) S(F ) is also an equivalence

of groupoids. Hence there exists a unique (up to isomorphism) simply connected

group G such that G/C(G) G.

Let π : G ’ G and π : G ’ G be central isogenies. Since Gsep is a split group

there exists a central isogeny ρ : Gsep ’ Gsep (see (??)). Remark (??) shows that

after modifying ρ by an automorphism of Gsep one can assume that πsep —¦ ρ = πsep .

Take any γ ∈ “. Since γρ : Gsep ’ Gsep is a central isogeny, by (??) there exists

± ∈ Aut(Gsep ) such that γρ = ρ —¦ ±. Then

πsep = γπsep = γ(πsep —¦ ρ) = πsep —¦ γρ = πsep —¦ ρ —¦ ± = πsep —¦ ±,

hence ± belongs to the kernel of Aut(Gsep ) ’ Aut(Gsep ), which is trivial by Corol-

lary (??), i.e., ± = Id and γρ = ρ. Then ρ = δsep for a central isogeny δ : G ’ G.

The group G in Theorem (??) is isomorphic to G/N where N is a subgroup of

—

C = C(G). Note that the Galois group “ acts on Tsep , leaving invariant the subset

—

¦ = ¦(G) ‚ Tsep , and hence acts on the lattices Λ, Λr , and on the group Λ/Λr .

Note that the “-action on Λ/Λr factors through the natural action of Aut Dyn(¦) .

The group C is ¬nite of multiplicative type, Cartier dual to (Λ/Λr )et (see p. ??).

Therefore, the classi¬cation problem of semisimple groups reduces to the classi-

¬cation of simply connected groups and “-submodules in Λ/Λr . Note that the

classi¬cations of simply connected and adjoint groups are equivalent.

A semisimple group G is called absolutely simple if Gsep is simple. For example,

a split simple group is absolutely simple.

§26. SEMISIMPLE GROUPS OVER AN ARBITRARY FIELD 365

(26.8) Theorem. A simply connected (resp. adjoint) semisimple group over F is

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

extension and G is an absolutely simple simply connected (resp. adjoint) group

over L.

Proof : Let ∆ be the set of connected components of the Dynkin diagram of G.

The absolute Galois group “ acts in a natural way on ∆ making it a ¬nite “-set.

Since G is a simply connected or an adjoint group and Gsep is split, it follows

from Proposition (??) that Gsep is the product of its simple components over Fsep

indexed by the elements of ∆:

Gsep = Gδ .

δ∈∆

Set Aδ = Fsep [Gδ ], then F [G]sep is the tensor product over Fsep of all Aδ , δ ∈ ∆.

Since “ permutes the connected components of the Dynkin diagram of G, there

exist F -algebra isomorphisms

γ : Aδ ’ Aγδ

such that γ(xa) = γ(x)γ(a) for all x ∈ Fsep and a ∈ Aδ , and the “-action on

F [G]sep is given by the formula

γ(—aδ ) = —aδ where aγδ = γ(aδ ).

Consider the ´tale F -algebra L = Map(∆, Fsep )“ corresponding to the ¬nite

e

“-set ∆ (see Theorem (??)). Then ∆ can be identi¬ed with the set of all F -algebra

homomorphisms L ’ Fsep . In particular,

Lsep = L —F Fsep = eδ Lsep

δ∈∆

where the eδ are idempotents, and each eδ Lsep Fsep .

We will de¬ne a group scheme G over L such that G RL/F (G ). Let S be

an L-algebra. The structure map ± : L ’ S gives a decomposition of the identity,

1 = δ∈∆ fδ where the fδ are the orthogonal idempotents in Ssep = S —F Fsep ,

which are the images of the eδ under ±sep : Lsep ’ Ssep ; they satisfy γfδ = fγδ for

all γ ∈ “. For any δ ∈ ∆ consider the group isomorphism

γ : Gδ (fδ Ssep ) ’ Gγδ (fγδ Ssep )

taking a homomorphism u ∈ HomAlg Fsep (Aδ , fδ Ssep ) to

γ —¦ u —¦ γ ’1 ∈ HomAlg Fsep (Aγδ , fγδ Ssep ) = Gγδ (fγδ Ssep ).

The collection of γ de¬nes a “-action on the product

Gδ (fδ Ssep ).

δ∈∆

We de¬ne G (S) to be the group of “-invariant elements in this product. Clearly,

G is a contravariant functor Alg L ’ Groups.

Let S = R —F L where R is an F -algebra. Then

Ssep (R —F eδ Lsep ) = fδ Ssep

δ∈∆ δ∈∆

366 VI. ALGEBRAIC GROUPS

where each fδ = 1 — eδ ∈ S —L Lsep and fδ Ssep Rsep . Hence

“

= G(Rsep )“ = G(R),

G (R —F L) = Gδ (Rsep )

δ∈∆

therefore G = RL/F (G ).

By writing L as a product of ¬elds, L = Li , we obtain

G RLi /F (Gi )

where the Gi are components of G . By comparing the two sides of this isomorphism

over Fsep , we see that Gi is a semisimple group over Li . A count of the number of

connected components of Dynkin diagrams shows that the Gi are absolutely simple

groups.

The collection of ¬eld extensions Li /F and absolutely simple groups Gi in The-

orem (??) is uniquely determined by G. Thus the theorem reduces the classi¬cation

problem to the classi¬cation of absolutely simple simply connected groups. In what

follows we classify such groups of types An , Bn , Cn , Dn (n = 4), F4 , and G2 .

Classi¬cation of simple groups of type An . As in Chapter ??, consider

the groupoid An = An (F ), n > 1, of central simple algebras of degree n + 1 over

some ´tale quadratic extension of F with a unitary involution which is the identity

e

over F , where the morphisms are the F -algebra isomorphisms which preserve the

involution, consider also the groupoid A1 = A1 (F ) of quaternion algebras over F

where morphisms are F -algebra isomorphisms.

Let An = An (F ) (resp. An = An (F )) be the groupoid of simply connected (resp.

adjoint) absolutely simple groups of type An (n ≥ 1) over F , where morphisms are

group isomorphisms. By §?? and Theorem (??) we have functors

Sn : An (F ) ’ An (F ) and Sn : An (F ) ’ An (F )

de¬ned by Sn (B, „ ) = SU(B, „ ), Sn (B, „ ) = PGU(B, „ ) if n ≥ 2, and Sn (Q) =

SL1 (Q), Sn (Q) = PGL1 (Q) if n = 1. Observe that if B = A — Aop and „ is the

exchange involution, then SU(B, „ ) = SL1 (A) and

PGU(B, „ ) = PGL1 (A).

(26.9) Theorem. The functors Sn : An (F ) ’ An (F ) and Sn : An (F ) ’ An (F )

are equivalences of categories.

Proof : Since the natural functor An (F ) ’ An (F ) is an equivalence (see the proof of

Theorem (??)), it su¬ces to prove that Sn is an equivalence. Let “ = Gal(Fsep /F ).

The ¬eld extension functor j : An (F ) ’ An (Fsep ) is clearly a “-embedding. We

show that j satis¬es the descent condition. Assume ¬rst that n ≥ 2. Let (B, „ ) be

some object in An (F ) (a split object, for example). Consider the F -vector space

W = HomF (B —F B, B) • HomF (B, B),

and the element w = (m, „ ) ∈ W where m is the multiplication on B. The natural

representation

ρ : GL(B) ’ GL(W ).

induces a “-equivariant homomorphism

ρsep : GL(Bsep ) ’ GL(Wsep ).

§26. SEMISIMPLE GROUPS OVER AN ARBITRARY FIELD 367

By Proposition (??) the “-embedding

i : A(ρsep , w) ’ A(ρsep , w)

satis¬es the descent condition. We have a functor

T = T(F ) : A(ρsep , w) ’ An (F )

taking w ∈ A(ρsep , w) to the F -vector space B with the algebra structure and

involution de¬ned by w . A morphism from w to w is an element of GL(B)

and it de¬nes an isomorphism of the corresponding algebra structures on B. The

functor T has an evident “-extension

T = T(Fsep ) : A(ρsep , w) ’ An (Fsep )

which is clearly an equivalence of groupoids. Since the functor i satis¬es the descent

condition, so does the functor j.

Assume now that n = 1. Let Q be a quaternion algebra over F . Consider the

F -vector space

W = HomF (Q —F Q, Q),

the multiplication map w ∈ W , and the natural representation

ρ : GL(Q) ’ GL(W ).

By Proposition (??) there is a “-embedding i satisfying the descent condition and a

functor T as above taking w ∈ A(ρsep , w) to the F -vector space Q with the algebra

structure de¬ned by w . The functor T has an evident “-extension which is an

equivalence of groupoids. As above, we conclude that the functor j satis¬es the

descent condition.

For the rest of the proof we again treat the cases n ≥ 2 and n = 1 separately.

Assume that n ≥ 2. By Remark (??) it su¬ces to show that for any (B, „ ) ∈ An (F )

the functor Sn , for F be separably closed, induces a group isomorphism

(26.10) AutF (B, „ ) ’ Aut PGU(B, „ ) .

The restriction of this homomorphism to the subgroup PGU(B, „ ) of index 2, is

the conjugation homomorphism. It induces an isomorphism of this group with the

group of inner automorphisms Int PGU(B, „ ) , a subgroup of Aut PGU(B, „ ) ,

which is also of index 2 (Theorem (??)). We may take the split algebra B =

op

Mn+1 (F ) — Mn+1 (F )op and „ the exchange involution. Then (x, y op ) ’ (y t , xt )

is an outer automorphism of (B, „ ). Its image in Aut PGU(B, „ ) = PGLn+1 is

the class of x ’ x’t , which is known to be an outer automorphism if (and only if)

n ≥ 2. Hence (??) is an isomorphism.

Finally, consider the case n = 1. As above, it su¬ces to show that, for a

quaternion algebra Q over a separably closed ¬eld F , the natural map

PGL1 (Q) = AutF (Q) ’ Aut PGL1 (Q)

is an isomorphism. But this follows from the fact that any automorphism of an

adjoint simple group of type A1 is inner (Theorem (??)).

(26.11) Remark. Let A be a central simple algebra of degree n + 1 over F . Then

Sn (A — Aop , µ) = SL1 (A), where µ is the exchange involution. In particular, two

groups SL1 (A1 ) and SL1 (A2 ) are isomorphic if and only if

(A1 — Aop , µ1 ) (A2 — Aop , µ2 ),

1 2

368 VI. ALGEBRAIC GROUPS

Aop .

i.e. A1 A2 or A1 2

Let B be a central simple algebra of degree n + 1 over an ´tale quadratic

e

extension L/F . The kernel C of the universal covering

SU(B, „ ) ’ PGU(B, „ )

is clearly equal to

NL/F

ker RL/F (µn+1,L ) ’’’’ µn+1,F .

It is a ¬nite group of multiplicative type, Cartier dual to Z/(n + 1)Z et . An abso-

lutely simple group of type An is isomorphic to SU(B, „ )/Nk where k divides n + 1

and Nk is the unique subgroup of order k in C.

Classi¬cation of simple groups of type Bn . For n ≥ 1, let Bn = Bn (F )

be the groupoid of oriented quadratic spaces of dimension 2n + 1, i.e., the groupoid

of triples (V, q, ζ), where (V, q) is a regular quadratic space of trivial discriminant

and ζ ∈ C(V, q) is an orientation (so ζ = 1 if char F = 2). Let B n = B n (F ) (resp.

B n = B n (F )) be the groupoid of simply connected (resp. adjoint) absolutely simple

groups of type Bn (n ≥ 1) over F . By §?? and Theorem (??) we have functors

Sn : Bn (F ) ’ B n (F ) and Sn : Bn (F ) ’ B n (F )

de¬ned by Sn (V, q, ζ) = Spin(V, q), Sn (V, q, ζ) = O+ (V, q).

(26.12) Theorem. The functors Sn : Bn (F ) ’ B n (F ) and Sn : Bn (F ) ’ B n (F )

are equivalences of categories.

Proof : Since the natural functor B n (F ) ’ B n (F ) is an equivalence, it su¬ces to

prove that Sn is an equivalence. Let “ = Gal(Fsep /F ). The ¬eld extension functor

j : Bn (F ) ’ Bn (Fsep ) is clearly a “-embedding. We show ¬rst that the functor j

satis¬es the descent condition. Let (V, q) be some regular quadratic space over F

of trivial discriminant and dimension n + 1. Consider the F -vector space

W = S 2 (V — ) • F,

the element w = (q, 1) ∈ W , and the natural representation

ρ : GL(V ) ’ GL(W ), ρ(g)(x, ±) = g(x), det x · ±

where g(x) is given by the natural action of GL(V ) on S 2 (V — ). By Proposition (??)

the “-embedding

i : A(ρsep , w) ’ A(ρsep , w)

satis¬es the descent condition. Thus, to prove that j satis¬es the descent condition,

it su¬ces to show that the functors i and j are equivalent. First recall that:

(a) If (q , ») ∈ A(ρsep , w), then q has trivial discriminant.

(b) (q, ») (q , » ) in A(ρsep , w) if and only if q q .

(c) AutA(ρsep ,w) (q, ») = O+ (V, q) = AutBn (F ) (V, q, ζ) (see (??)).

We construct a functor

T = T(F ) : A(ρsep , w) ’ Bn (F )

as follows. If char F = 2 we put T(q , ») = (V, q , 1). Now, assume that the

characteristic of F is not 2. Choose an orthogonal basis (v1 , v2 , . . . , v2n+1 ) of V for

the form q, such that the central element ζ = v1 · v2 · . . . · v2n+1 ∈ C(V, q) satis¬es

ζ 2 = 1, i.e., ζ is an orientation. Take any (q , ») ∈ A(ρsep , w) and f ∈ GL(Vsep )

§26. SEMISIMPLE GROUPS OVER AN ARBITRARY FIELD 369

such that qsep f (v) = qsep (v) for any v ∈ Vsep and det f = ». Then the central

element

ζ = f (v1 ) · f (v2 ) · . . . · f (v2n+1 ) ∈ C(Vsep , qsep )

satis¬es ζ 2 = 1. In particular, ζ ∈ C(V, q ). It is easy to see that ζ does not

depend on the choice of f . Set T(q , ») = (V, q , ζ ). It is immediate that T(F ) is

a well-de¬ned equivalence of categories. Thus, the functor j satis¬es the descent

condition.

To complete the proof of the theorem, it su¬ces by Proposition (??) (and Re-

mark (??)) to show that, for any (V, q, ζ) ∈ Bn (F ), the functor Sn over a separably

closed ¬eld F induces a group isomorphism

O+ (V, q) ’ Aut O+ (V, q) .

This holds since automorphisms of O+ (V, q) are inner (Theorem (??)).

(26.13) Remark. If char F = 2, the theorem can be reformulated in terms of

algebras with involution. Namely, the groupoid Bn is naturally equivalent to to the

groupoid Bn of central simple algebras over F of degree 2n + 1 with involution of

the ¬rst kind, where morphisms are isomorphisms of algebras which are compatible

with the involutions (see (??)).

Classi¬cation of simple groups of type Cn . Consider the groupoid Cn =

Cn (F ), n ≥ 1, of central simple F -algebras of degree 2n with symplectic involu-

tion, where morphisms are F -algebra isomorphisms which are compatible with the

involutions.

Let C n = C n (F ) (resp. C n = C n (F )) be the groupoid of simply connected

(resp. adjoint) simple groups of type Cn (n ≥ 1) over F , where morphisms are

group isomorphisms. By (??) and Theorem (??) we have functors

Sn : Cn (F ) ’ C n (F ) and Sn : Cn (F ) ’ C n (F )

de¬ned by Sn (A, σ) = Sp(A, σ), Sn (A, σ) = PGSp(A, σ).

(26.14) Theorem. The functors Sn : Cn (F ) ’ C n (F ) and Sn : Cn (F ) ’ C n (F )

are equivalences of categories.

Proof : Since the natural functor C n (F ) ’ C n (F ) is an equivalence, it su¬ces to

prove that Sn is an equivalence. Let “ = Gal(Fsep /F ). The ¬eld extension functor

j : Cn (F ) ’ Cn (Fsep ) is clearly a “-embedding. We ¬rst show that the functor j

satis¬es the descent condition. Let (A, σ) be some object in Cn (F ) (a split one, for

example). Consider the F -vector space

W = HomF (A —F A, A) • HomF (A, A),

the element w = (m, σ) ∈ W where m is the multiplication on A, and the natural

representation

ρ : GL(A) ’ GL(W ).

By Proposition (??) the “-embedding

i : A(ρsep , w) ’ A(ρsep , w)

satis¬es the descent condition. We have the functor

T = T(F ) : A(ρsep , w) ’ Cn (F )

370 VI. ALGEBRAIC GROUPS

taking w ∈ A(ρsep , w) to the F -vector space A with the algebra structure and

involution de¬ned by w . A morphism from w to w is an element of GL(A)

and it de¬nes an isomorphism of the corresponding algebra structures on A. The

functor T has an evident “-extension

T = T(Fsep ) : A(ρsep , w) ’ Cn (Fsep ),

which is clearly an equivalence of groupoids. Since the functor i satis¬es the descent

condition, so does the functor j.

To complete the proof of the theorem, it su¬ces by Remark (??) to show that

for any (A, σ) ∈ Cn (F ) the functor Sn over a separably closed ¬eld F induces a

group isomorphism

PGSp(A, σ) = AutF (A, σ) ’ Aut PGSp(A, σ) .

This follows from the fact that automorphisms of PGSp are inner (Theorem (??)).

Classi¬cation of semisimple groups of type Dn , n = 4. Consider the

groupoid Dn = Dn (F ), n ≥ 2, of central simple F -algebras of degree 2n with

quadratic pair, where morphisms are F -algebra isomorphisms compatible with the

quadratic pairs.

Denote by D n = D n (F ) (resp. D n = D n (F )) the groupoid of simply connected

(resp. adjoint) semisimple (simple if n > 2) groups of type Dn (n ≥ 2) over F ,

where morphisms are group isomorphisms. By §?? and Theorem (??) we have

functors

Sn : Dn (F ) ’ D n (F ) and S n : Dn (F ) ’ D n (F )

de¬ned by Sn (A, σ, f ) = Spin(A, σ, f ), S n (A, σ, f ) = PGO+ (A, σ, f ).

(26.15) Theorem. If n = 4, the functors Sn : Dn (F ) ’ D n (F ) and S n : Dn (F ) ’

D n (F ) are equivalences of categories.

Proof : Since the natural functor D n (F ) ’ D n (F ) is an equivalence, it su¬ces to

prove that S n is an equivalence. Let “ = Gal(Fsep /F ). The ¬eld extension functor

j : Dn (F ) ’ Dn (Fsep ) is clearly a “-embedding. We show ¬rst that the functor j

satis¬es the descent condition. Let (A, σ, f ) be some object in Dn (F ) (a split one,

for example). Let A+ be the space Sym(A, σ). Consider the F -vector space

W = HomF (A+ , A) • HomF (A —F A, A) • HomF (A, A) • (A+ )— ,

which contains the element w = (i, m, σ, f ) where i : A+ ’ A is the inclusion and

m is the multiplication on A; we have a natural representation

ρ : GL(A) — GL(A+ ) ’ GL(W ).

ρ(g, h)(», x, y, p) = g —¦ » —¦ h’1 , g(x), g(y , p —¦ h’1 )

where g(x) and g(y) are obtained by applying the natural action of GL(A) on the

second and third summands of W . By Proposition (??) the “-embedding

i : A(ρsep , w) ’ A(ρsep , w)

satis¬es the descent condition. We have the functor

T = T(F ) : A(ρsep , w) ’ Dn (F )

§26. SEMISIMPLE GROUPS OVER AN ARBITRARY FIELD 371

which takes w ∈ A(ρsep , w) to the F -vector space A with the algebra structure and

quadratic pair de¬ned by w . A morphism from w to w is an element of GL(A) —

GL(A+ ) and it de¬nes an isomorphism between the corresponding structures on A.

The functor T has an evident “-extension

T = T(Fsep ) : A(ρsep , w) ’ Dn (Fsep ),

which is clearly an equivalence of groupoids. Since the functor i satis¬es the de-

scent condition, so does the functor j. For the proof of the theorem it su¬ces by

Proposition (??) (and Remark (??)) to show that for any (A, σ, f ) ∈ Dn (F ) the

functor Sn for a separably closed ¬eld F induces a group isomorphism

PGO(A, σ, f ) = AutF (A, σ, f ) ’ Aut PGO+ (A, σ, f ) .

(26.16)

The restriction of this homomorphism to the subgroup PGO+ (A, σ, f ), which is of

index 2, induces an isomorphism of this subgroup with the group of inner auto-

morphisms Int PGO+ (A, σ, f ) , which is a subgroup in Aut PGO+ (A, σ) also of

index 2 (since n = 4, see Theorem (??)). A straightforward computation shows that

any element in PGO’ (A, σ, f ) induces an outer automorphism of PGO+ (A, σ, f ).

Hence (??) is an isomorphism.

(26.17) Remark. The case of D4 is exceptional, in the sense that the group of

automorphisms of the Dynkin diagram of D4 is S3 . Triality is needed and we refer

to Theorem (??) below for an analogue of Theorem (??) for D4 .

Let C be the kernel of the adjoint representation of Spin(A, σ, f ). If n is even,

then C is the Cartier dual to (Z/2Z • Z/2Z)et , where the absolute Galois group “

acts by the permutation of summands. This action factors through Aut Dyn(Dn ) .

On the other hand, the “-action on the center Z of the Cli¬ord algebra C(A, σ, f )

given by the composition

Z/2Z

“ ’ AutFsep C(Asep , σsep , fsep ) ’ AutFsep (Zsep )

also factors through Aut Dyn(Dn ) . Hence the Cartier dual to C is isomorphic

to (Z/2Z)[G]et , where G = Gal(Z/F ) and “ acts by the natural homomorphism

“ ’ G. By Exercise ??,

C = RZ/F (µ2,Z ).

If n is odd, then C is the Cartier dual to (Z/4Z)et and “ acts on M = Z/4Z

through G identi¬ed with the automorphism group of Z/4Z. We have an exact

sequence

0 ’ Z/4Z ’ (Z/4Z)[G] ’ M ’ 0,

where Z/4Z is considered with the trivial “-action. By Cartier duality,

NZ/F

C = ker RZ/F (µ4,Z ) ’’ ’ µ4,F .

’

If n is odd, then C has only one subgroup of order 2 which corresponds to

+ —

Z/2Z • Z/2Z. If σ has nontrivial dis-

O (A, σ, f ). If n is even, then Csep

—

criminant (i.e., Z is not split), then “ acts non-trivially on Csep , hence there is

still only one proper subgroup of C corresponding to GO+ (A, σ, f ). In the case

—

where the discriminant is trivial (so Z is split), “ acts trivially on Csep , and there

are three proper subgroups of C, one of which corresponds again to O+ (A, σ, f ).

372 VI. ALGEBRAIC GROUPS

The two other groups correspond to Spin± (A, σ, f ), which are the images of the

compositions

Spin(A, σ, f ) ’ GL1 C(A, σ, f ) ’ GL1 C ± (A, σ, f )

where C(A, σ, f ) = C + (A, σ, f ) — C ’ (A, σ, f ).

Classi¬cation of simple groups of type F4 . Consider the groupoid F4 =

F4 (F ) of exceptional Jordan algebras of dimension 27 over F (see §?? below if

char F = 2 and §?? if char F = 2), where morphisms are F -algebra isomorphisms.

Denote by F 4 = F 4 (F ) the groupoid of simple groups of type F4 over F , where

morphisms are group isomorphisms. By Theorem (??) we have a functor

S : F4 (F ) ’ F 4 (F ), S(J) = Autalg (J).

(26.18) Theorem. The functor S : F4 (F ) ’ F 4 (F ) is an equivalence of cate-

gories.

Proof : Let “ = Gal(Fsep /F ). The ¬eld extension functor j : F4 (F ) ’ F4 (Fsep )

is clearly a “-embedding. We ¬rst show that the functor j satis¬es the descent

condition. Let J be some object in F4 (F ) (a split one, for example). If char F = 2,

consider the F -vector space

W = HomF (J —F J, J),

the multiplication element w ∈ W and the natural representation

ρ : GL(J) ’ GL(W ).

By Proposition (??) the “-embedding

i : A(ρsep , w) ’ A(ρsep , w)

satis¬es the descent condition. We have the functor

T = T(F ) : A(ρsep , w) ’ F4 (F )

taking w ∈ A(ρsep , w) to the F -vector space J with the Jordan algebra structure

de¬ned by w . A morphism from w to w is an element of GL(J) and it de¬nes an

isomorphism of the corresponding Jordan algebra structures on J. The functor T

has an evident “-extension

T = T(Fsep ) : A(ρsep , w) ’ F4 (Fsep ),

which is clearly an equivalence of groupoids. Since the functor i satis¬es the descent

condition, so does the functor j.

For the proof of the theorem it su¬ces by Proposition (??) to show that for any

J ∈ F4 (F ) the functor S for a separably closed ¬eld F induces a group isomorphism

Autalg (J) ’ Aut Autalg (J) .

This follows from the fact that automorphisms of simple groups of type F4 are inner

(Theorem (??)).

If char F = 2, an exceptional Jordan algebra of dimension 27 is (see §?? below)

a datum (J, N, #, T, 1) consisting of a space J of dimension 27, a cubic form N : J ’

F , the adjoint # : J ’ J of N , which is a quadratic map, a bilinear trace form T ,

and a distinguished element 1, satisfying certain properties (given in §??). In this

case we consider the F -vector space

W = S 3 (J — ) • S 2 (J — ) — J • S 2 (J — ) • F

and complete the argument as in the preceding cases.

§26. SEMISIMPLE GROUPS OVER AN ARBITRARY FIELD 373

Classi¬cation of simple groups of type G2 . Consider the groupoid G2 =

G2 (F ) of Cayley algebras over F , where morphisms are F -algebra isomorphisms.

Denote by G 2 = G 2 (F ) the groupoid of simple groups of type G2 over F , where

morphisms are group isomorphisms. By Theorem (??) there is a functor

S : G2 (F ) ’ G 2 (F ), S(C) = Autalg (C).

(26.19) Theorem. The functor S : G2 (F ) ’ G 2 (F ) is an equivalence of cate-

gories.

Proof : Let “ = Gal(Fsep /F ). The ¬eld extension functor j : G2 (F ) ’ G2 (Fsep )

is clearly a “-embedding. We ¬rst show that the functor j satis¬es the descent

condition. Let C be some object in G2 (F ) (a split one, for example). Consider

the F -vector space W = HomF (C —F C, C), the multiplication element w ∈ W ,

and the natural representation ρ : GL(C) ’ GL(W ). By Proposition (??) the

“-embedding i : A(ρsep , w) ’ A(ρsep , w) satis¬es the descent condition. We have a

functor

T = T(F ) : A(ρsep , w) ’ G2 (F )

which takes w ∈ A(ρsep , w) to the F -vector space C with the Cayley algebra

structure de¬ned by w . A morphism from w to w is an element of GL(C) and it

de¬nes an isomorphism between the corresponding Cayley algebra structures on C.

The functor T has an evident “-extension

T = T(Fsep ) : A(ρsep , w) ’ G2 (Fsep ),

which is clearly an equivalence of groupoids. Since the functor i satis¬es the descent

condition, so does the functor j.

For the proof of the theorem it su¬ces by Proposition (??) to show that for any

J ∈ G2 (F ) the functor S for a separably closed ¬eld F induces a group isomorphism

Autalg (C) ’ Aut Autalg (C) .

This follows from the fact that automorphisms of simple groups of type G2 are

inner (Theorem (??)).

26.B. Algebraic groups of small dimension. Some Dynkin diagrams of

small ranks coincide:

(26.20) A1 = B 1 = C 1

(26.21) D2 = A 1 + A 1

(26.22) B2 = C 2

(26.23) A3 = D 3

We describe explicitly the corresponding isomorphisms for adjoint groups (ana-

logues for algebras are in §??):

A1 = B1 = C1 . Let (V, q) be a regular quadratic form of dimension 3 over a

¬eld F . Then C0 (V, q) is a quaternion algebra over F . The canonical homomor-

phism

O+ (V, q) ’ PGL1 C0 (V, q) = PGSp C0 (V, q), σ q

is injective (see §??) and hence is an isomorphism of adjoint simple groups of

types B1 , A1 and C1 since by dimension count its image has the same dimension

374 VI. ALGEBRAIC GROUPS

as the target group, and since these groups are connected they must coincide, by

Propositions (??) and (??). (We will use this argument several times below.)

Let Q be a quaternion algebra over F and let Q0 = { x ∈ Q | TrdQ (x) = 0 }.

For x ∈ Q0 , we have x2 ∈ F , and the squaring map s : Q0 ’ F is a quadratic form

of discriminant 1 on Q0 (see §??). Consider the conjugation homomorphism

f : GL1 (Q) ’ O+ (Q0 , s).

Since Q0 generates Q, ker(f ) = Gm and the injection

PGSp(Q, σ) = PGL1 (Q) ’ O+ (Q0 , s)

is an isomorphism of adjoint simple groups of types C1 , A1 , and B1 .

D2 = A1 + A1 . Let A be a central simple algebra over F of degree 4 with a

quadratic pair (σ, f ). Then C(A, σ, f ) is a quaternion algebra over a quadratic ´tale

e

extension Z of F . We have the canonical injection

PGO+ (A, σ, f ) ’ AutZ C(A, σ, f ) = RZ/F PGL1 C(A, σ, f )

which is an isomorphism between adjoint groups of type D2 and those of type

A1 + A 1 .

Conversely, let Q be a quaternion algebra over an ´tale quadratic extension

e

Z/F . The norm A = NZ/F (Q) is a central simple algebra of degree 4 over F with

a canonical quadratic pair (σ, f ) (see §??). We have the natural homomorphism

g : RZ/F GL1 (Q) ’ GO+ (A, σ, f ), x ∈ Q — ’ x — x ∈ A— .

R R

—

One checks that x — x ∈ R— if and only if x ∈ ZR , hence g ’1 (Gm ) = RZ/F (Gm,Z ).

By factoring out these subgroups we obtain an injective homomorphism

RZ/F PGL1 (Q) ’ PGO+ (A, σ, f )

which is actually an isomorphism from an adjoint group of type A1 + A1 to one of

type D2 .

B2 = C2 . Let (V, q) be a regular quadratic form of dimension 5. Then C0 (V, q)

is a central simple algebra of degree 4 with (canonical) symplectic involution „ .

There is a canonical injective homomorphism (see §??)

O+ (V, q) ’ PGSp C0 (V, q), „

which is in fact an isomorphism from an adjoint simple groups of type B2 to one

of type C2 .

Conversely, for a central simple algebra A of degree 4 over F with a symplectic

involution σ, the F -vector space

Symd(A, σ)0 = { x ∈ Symd(A, σ) | TrpA (x) = 0 }

admits the quadratic form sσ (x) = x2 ∈ F (see §??). Consider the conjugation

homomorphism

f : GSp(A, σ) ’ O+ Symd(A, σ)0 , sσ , a ’ Int(a).

Since Symd(A, σ) generates A, one has ker(f ) = Gm . Hence, the injection

PGSp(A, σ) ’ O+ V, q

is an isomorphism from an adjoint simple group of type C2 to one of type B2 .

§27. TITS ALGEBRAS OF SEMISIMPLE GROUPS 375

A3 = D3 . Let A be a central simple algebra of degree 6 over F with an orthog-

onal pair (σ, f ). Then C(A, σ, f ) is a central simple algebra of degree 4 over an ´tale

e

quadratic extension Z/F with a unitary involution σ. The natural homomorphism

PGO+ (A, σ, f ) ’ PGU C(A, σ, f ), σ

is injective (see §??) and hence is an isomorphism from an adjoint simple group of

type D3 to one of type A3 .

Conversely, let B be a central simple algebra of degree 4 over an ´tale quadratic

e

extension Z/F with a unitary involution „ . Then the discriminant algebra D(B, „ )

is a central simple algebra of degree 6 over F with canonical quadratic pair („ , f ).

Consider the natural homomorphism

GU(B, „ ) ’ GO+ D(B, „ ), „ , f .

One checks (in the split case) that g ’1 (Gm ) = GL1 (Z). By factoring out these

subgroups we obtain an injection

PGU(B, „ ) ’ PGO+ D(B, „ ), „ , f

which is an isomorphism from an adjoint simple group of type A3 to one of type D3 .

§27. Tits Algebras of Semisimple Groups

The Cli¬ord algebra, the discriminant algebra, the »-powers of a central simple

algebra all arise as to be so-called Tits algebras of the appropriate semisimple

groups. In this section we de¬ne Tits algebras and classify them for simple groups

of the classical series.

For this we need some results on the classi¬cation of representations of split

semisimple groups. Let G be a split semisimple group over F . Choose a split

maximal torus T ‚ G. Fix a system of simple roots in ¦(G), so we have the

corresponding cone Λ+ ‚ Λ of dominant weights.

Let ρ : G ’ GL(V ) be a representation. By the representation theory of diag-

onalizable groups (??) one can associate to the representation ρ|T a ¬nite number

of weights, elements of T — . If ρ is irreducible, among the weights there is a largest

(with respect to the ordering on Λ). It lies in Λ+ and is called the highest weight

of ρ (Humphreys [?]).

(27.1) Theorem. The map

Isomorphism classes of

T — © Λ+

←’

irreducible representations of G

taking the class of a representation ρ to its highest weight, is a bijection.

Reference: Tits [?, Th.2.5]

(27.2) Remark. If G is a simply connected group (i.e., T — = Λ), then T — © Λ+ =

Λ+ .

(27.3) Remark. The classi¬cation of irreducible representations of a split semi-

simple groups does not depend on the base ¬eld in the sense that an irreducible

representation remains irreducible over an arbitrary ¬eld extension and any irre-

ducible representation over an extension comes from the base ¬eld.

376 VI. ALGEBRAIC GROUPS

27.A. De¬nition of the Tits algebras. Let G be a semisimple (not neces-

sarily split) group over F and let T ‚ G be a maximal torus. Choose a system of

—

simple roots Π ‚ ¦ = ¦(G). The group “ acts on Tsep and is the identity on ¦, Λ,

Λr (but not Π).

—

There is another action of “ on Tsep , called the —-action, which is de¬ned as

follows. Take any γ ∈ “. Since the Weyl group W acts simply transitively on the

set of systems of simple roots and γΠ is clearly a system of simple roots, there is a

unique w ∈ W such that w(γΠ) = Π. We set γ — ± = w(γ±) ∈ Π for any ± ∈ Π.

This action, de¬ned on Π, extends to an action on Λ which is the identity on Π, ¦,

Λr , Λ+ . Note that since W acts trivially on Λ/Λr , the —-action on Λ/Λr coincides

with the usual one.

Choose a ¬nite Galois extension F ‚ L ‚ Fsep splitting T and hence G. The —-

action of “ then factors through Gal(L/F ). Let ρ : GL ’ GL(V ) be an irreducible

representation over L (so V is an L-vector space) with highest weight » ∈ Λ+ © T —

(see Theorem (??)). For any γ ∈ “ we can de¬ne the L-space γ V as V as an abelian

group and with the L-action x —¦ v = γ ’1 (x) · v, for all x ∈ L, v ∈ V . Then v ’ v

viewed as a map V ’ γ V is γ-semilinear. Denote it iγ .

Let A = F [G] and let ρ : V ’ V —L AL be the comodule structure for ρ (see

p. ??). The composite

i’1 iγ —(γ—Id)

ρ

γ

γV ’ ’ V ’ V —L (L —F A) ’ ’ ’ ’ γ V —L (L —F A)

’ ’ ’ ’ ’’

gives the comodule structure for some irreducible representation

γρ: GL ’ GL(γ V ).

(Observe that the third map is well-de¬ned because both iγ and γ — Id are γ-

semilinear.) Clearly, the weights of γ ρ are obtained from the weights of ρ by ap-

plying γ. Hence, the highest weight of γ ρ is γ — ».

Assume now that » ∈ Λ+ © T — is invariant under the —-action. Consider the

conjugation representation

g ’ ± ’ ρ(g) —¦ ± —¦ ρ(g)’1 .

Int(ρ) : G ’ GL EndF (V ) ,

Let EndG (V ) be the subalgebra of G-invariant elements in EndF (V ) under Int(ρ).

Then

(27.4) EndG (V ) —F L EndGL (V —F L) EndGL γ∈Gal(L/F ) γ V

since V —F L is L-isomorphic to γ V via v — x ’ (γ ’1 x · v)γ . Since the represen-

tation γ ρ is of highest weight γ — » = », it follows from Theorem (??) that γ ρ ρ,

i.e., all the G-modules γ V are isomorphic to V . Hence, the algebras in (??) are

isomorphic to

EndGL (V n ) = Mn EndGL (V )

where n = [L : F ].

(27.5) Lemma. EndGL (V ) L.

Proof : Since ρ is irreducible, EndGL (V ) is a division algebra over L by Schur™s

lemma. But ρalg remains irreducible by Remark (??), hence EndGL (V ) —L Falg is

also a division algebra and therefore EndGL (V ) = L.

§27. TITS ALGEBRAS OF SEMISIMPLE GROUPS 377

It follows from the lemma that EndG (V ) —F L Mn (L), hence EndG (V ) is a

central simple algebra over F of degree n. Denote its centralizer in EndF (V ) by

A» . This is a central simple algebra over F of degree dimL V . It is clear that A»

is independent of the choice of L. The algebra A» is called the Tits algebra of the

group G corresponding to the dominant weight ».

Since the image of ρ commutes with EndG (V ), it actually lies in A» . Thus we

obtain a representation

ρ : G ’ GL1 (A» ).

By the double centralizer theorem (see (??)), the centralizer of EndG (V ) —F L

in EndF (V ) —F L is A» —F L. On the other hand it contains EndL (V ) (where the

image of ρ lies). By dimension count we have

A» —F L = EndL (V )

and hence the representation (ρ )L is isomorphic to ρ. Thus, ρ can be considered

as a descent of ρ from L to F . The restriction of ρ to the center C = C(G) ‚ G

is given by the restriction of » on C, i.e., is the character of the center C given by

the class of » in C — = T — /Λr ‚ Λ/Λr .

The following lemma shows the uniqueness of the descent ρ .

(27.6) Lemma. Let µi : G ’ GL1 (Ai ), i = 1, 2 be two homomorphisms where

the Ai are central simple algebras over F . Assume that the representations (µ i )sep

are isomorphic and irreducible. Then there is an F -algebra isomorphism ± : A 1 ’

A2 such that GL1 (±) —¦ µ1 = µ2 .

Proof : Choose a ¬nite Galois ¬eld extension L/F splitting G and the Ai , Ai —F L

∼

EndL (Vi ). An L-isomorphism V1 ’ V2 of GL -representations gives rise to an

’

algebra isomorphism

∼

± : EndF (V1 ) ’ EndF (V2 )

’

taking EndG (V1 ) to EndG (V2 ). Clearly, Ai lies in the centralizer of EndG (Vi ) in

EndF (Vi ). By dimension count Ai coincides with the centralizer, hence ±(A1 ) =

A2 .

Let π : G ’ G be a central isogeny with G simply connected. Then the Tits

algebra built out of a representation ρ of GL is the Tits algebra of the group GL

corresponding to the representation ρ —¦ πL . Hence, in order to classify Tits algebras

one can restrict to simply connected groups.

Assume that G is a simply connected semisimple group. For any » ∈ Λ/Λr

consider the corresponding (unique) minimal weight χ(») ∈ Λ+ . The uniqueness

shows that χ(γ») = γ — χ(») for any γ ∈ “. Hence, if » ∈ (Λ/Λr )“ , then clearly

χ(») ∈ Λ“ (with respect to the —-action); the Tits algebra Aχ(») is called a minimal

+

Tits algebra and is denoted simply by A» . For example, if » = 0, then A» = F .

(27.7) Theorem. The map

β : (Λ/Λr )“ ’ Br F, » ’ [A» ]

is a homomorphism.

Reference: Tits [?, Cor. 3.5].

378 VI. ALGEBRAIC GROUPS

If » ∈ Λ/Λr is not necessarily “-invariant, let

“0 = { γ ∈ “ | γ(») = » } ‚ “

and F» = (Fsep )“0 . Then » ∈ (Λ/Λr )“0 and one gets a Tits algebra A» , which is a

central simple algebra over F» , for the group GF» . The ¬eld F» is called the ¬eld

of de¬nition of ».

27.B. Simply connected classical groups. We give here the classi¬cation

of the minimal Tits algebras of the absolutely simple simply connected groups of

classical type.

Type An , n ≥ 1. Let ¬rst G = SL1 (A) where A is a central simple algebra of

degree n + 1 over F . Then C = µn+1 , C — = Z/(n + 1)Z with the trivial “-action.

For any i = 0, 1, . . . , n, consider the natural representation

ρi : G ’ GL1 (»i A).

In the split case ρi is the i-th exterior power representation with the highest weight

e1 + e2 + · · · + ei in the notation of §??, which is a minimal weight. Hence, the

»-powers »i A, for i = 0, 1, . . . , n, (see §??) are the minimal Tits algebras of G.

Now let G = SU(B, „ ) where B is a central simple algebra of degree n + 1 with

a unitary involution over a quadratic separable ¬eld extension K/F . The group

“ acts on C — = Z/(n + 1)Z by x ’ ’x through Gal(K/F ). The only nontrivial

element in (C — )“ is » = n+1 + (n + 1)Z (n should be odd). There is a natural

2

homomorphism

ρ : G ’ GL1 D(B, „ )

which in the split case is the external n+1 -power. Hence, the discriminant algebra

2

(see §??) D(B, „ ) is the minimal Tits algebra corresponding to » for the group G.

The ¬elds of de¬nition Fµ of the other nontrivial characters µ = i + (n + 1)Z ∈

C — , (i = (n+1) ), coincide with K. Hence, by extending the base ¬eld to K one sees

2

that Aµ »i B.

Type Bn , n ≥ 1. Let G = Spin(V, q), here (V, q) is a regular quadratic form

of dimension 2n + 1. Then C = µ2 , C — = Z/2Z = {0, »}. The embedding

G ’ GL1 C0 (V, q)

in the split case is the spinor representation with highest weight 1 (e1 +e2 +· · ·+en )

2

in the notation of §??, which is a minimal weight. Hence, the even Cli¬ord algebra

C0 (V, q) is the minimal Tits algebra A» .

Type Cn , n ≥ 1. Let G = Sp(A, σ) where A is a central simple algebra of

degree 2n with a symplectic involution σ. Then C = µ2 , C — = Z/2Z = {0, »}. The

embedding

G ’ GL1 (A)

in the split case is the representation with highest weight e1 in the notation of §??,

which is a minimal weight. Hence, A is the minimal Tits algebra A» .

§27. TITS ALGEBRAS OF SEMISIMPLE GROUPS 379

Type Dn , n ≥ 2, n = 4. Let G = Spin(A, σ, f ) where A is a central simple

algebra of degree 2n with a quadratic pair (σ, f ), C — = {0, », »+ , »’ } where »

factors through O+ (A, σ, f ). The composition

Spin(A, σ, f ) ’ GO+ (A, σ, f ) ’ GL1 (A)

in the split case is the representation with highest weight e1 in the notation of §??,

which is a minimal weight. Hence, A is the minimal Tits algebra A» .

Assume further that the discriminant of σ is trivial (i.e., the center Z of the

Cli¬ord algebra is split). The group “ then acts trivially on C — . The natural

compositions

Spin(A, σ, f ) ’ GL1 C(A, σ, f ) ’ GL1 C ± (A, σ, f )

in the split case are the representations with highest weights 1 (e1 + · · · + en’1 ± en )

2

which are minimal weights. Hence, C ± (A, σ, f ) are the minimal Tits algebras A»± .

If disc(σ) is not trivial then “ interchanges »+ and »’ , hence the ¬eld of

de¬nition of »± is Z. By extending the base ¬eld to Z one sees that A»± =

C(A, σ, f ). Again, the case of D4 is exceptional, because of triality, and we give on

p. ?? a description of the minimal Tits algebra in this case.

27.C. Quasisplit groups. A semisimple group G is called quasisplit if there

is a maximal torus T ‚ G and a system Π of simple roots in the root system ¦

of G with respect to T which is “-invariant with respect to the natural action, or

—

equivalently, if the —-action on Tsep coincides with the natural one. For example,

split groups are quasisplit.

Let G be a quasisplit semisimple group. The natural action of “ on the

system Π of simple roots, which is invariant under “, de¬nes an action of “ on

Dyn(G) = Dyn(¦) by automorphisms of the Dynkin diagram. Simply connected

and adjoint split groups are classi¬ed by their Dynkin diagrams. The following

statement generalizes this result for quasisplit groups.

(27.8) Proposition. Two quasisplit simply connected (resp. adjoint) semisimple

groups G and G are isomorphic if and only if there is a “-bijection between Dyn(G)

and Dyn(G ). For any Dynkin diagram D and any (continuous) “-action on D

there is a quasisplit simply connected (resp. adjoint) semisimple group G and a

“-bijection between Dyn(G) and D.

—

The “-action on Dyn(G) is trivial if and only if “ acts trivially on Tsep , hence T

and G are split. Therefore, if Aut(Dyn(G)) = 1 (i.e., Dyn(G) has only irreducible

components Bn , Cn , E7 , E8 , F4 , G2 ) and G is quasisplit, then G is actually split.

(27.9) Example. The case An , n > 1. A non-trivial action of the Galois group

“ on the cyclic group Aut(An ) of order two factors through the Galois group of

a quadratic ¬eld extension L/F . The corresponding quasisplit simply connected

simple group of type An is isomorphic to SU(V, h), where (V, h) is a non-degenerate

hermitian form over L/F of dimension n + 1 and maximal Witt index.

(27.10) Example. The case Dn , n > 1, n = 4. As in the previous example, to give

a nontrivial “-action on Dn is to give a quadratic Galois ¬eld extension L/F . The

corresponding quasisplit simply connected simple group of type Dn is isomorphic

to Spin(V, q), where (V, q) is a non-degenerate quadratic form of dimension 2n and

Witt index n ’ 1 with the discriminant quadratic extension L/F .

380 VI. ALGEBRAIC GROUPS

Exercises

1. If L is an ´tale F -algebra, then Autalg (L) is an ´tale group scheme correspond-

e e

ing to the ¬nite group AutFsep (Lsep ) with the natural Gal(Fsep /F )-action.

2. Let G be an algebraic group scheme. Prove that the following statements are

equivalent:

(a) G is ´tale,

e

0

(b) G = 1,

(c) G is smooth and ¬nite,

(d) Lie(G) = 0.

3. Prove that Hdiag is algebraic if and only if H is a ¬nitely generated abelian

group.

4. Let H be a ¬nitely generated abelian group, and let H ‚ H be the subgroup

of elements of order prime to char F . Prove that (Hdiag )0 (H/H )diag and

π0 (Hdiag ) Hdiag .

5. Prove that an algebraic group scheme G is ¬nite if and only if dim G = 0.

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

exactly one a ∈ A such that y = xa . (Compare with (??), where the “-group A

(denoted there by G) is ¬nite and carries the trivial action of “.) We let A“Tors “

denote the category of A-torsors, where the maps are the A- and “-equivariant

functions. This category is a groupoid, since the maps are isomorphisms.

§28. COHOMOLOGY OF PROFINITE GROUPS 389

To construct examples of A-torsors, we may proceed as follows: for ± ∈

1

Z (“, A), let P± be the set A with the “- and A-actions

and xa = xa

σ x = ±σ σx for σ ∈ “ and x, a ∈ A.

It turns out that every A-torsor is isomorphic to some P± :

(28.14) Proposition. The map ± ’ P± induces a bijection

∼

H 1 (“, A) ’ Isom(A“Tors “ ).

’

Proof : If ±, ± ∈ Z 1 (“, A) are cohomologous, let a ∈ A satisfy ±σ = a·±σ ·σa’1 for

∼

all σ ∈ “. Multiplication on the left by a is an isomorphism of torsors P± ’ P± .

’

We thus have a well-de¬ned map H 1 (“, A) ’ Isom(A“Tors “ ). The inverse map is

given as follows: Let P ∈ A“Tors “ . For a ¬xed x ∈ P , the map ± : “ ’ A de¬ned

by

σ(x) = x±σ for σ ∈ “

is a 1-cocycle. Replacing x with xa changes ±σ into the cohomologous cocycle

a’1 ±σ σa.

(28.15) Example. Let “ be the absolute Galois group of a ¬eld F , and let G be

a ¬nite group which we endow with the trivial action of “. By combining (??) with

(??), we obtain a canonical bijection

∼

H 1 (“, G) ’ Isom(G“Gal F ),

’

hence H 1 (“, G) classi¬es the Galois G-algebras over F up to isomorphism.

Functoriality. Let f : A ’ B be a “-homomorphism of “-groups. We de¬ne

a functor

f— : A“Tors “ ’ B“Tors “

as follows: for P ∈ A“Tors “ , consider the product P — B with the diagonal action

of “. The groups A and B act on P — B by

(p, b)a = pa , f (a’1 )b and (p, b) b = (p, bb )

for p ∈ P , a ∈ A and b, b ∈ B, and these two actions commute. Hence there is an

induced right action of B on the set of A-orbits f— (P ) = (P — B)/A, making f— (P )

a B-torsor.

(28.16) Proposition. The following diagram commutes:

f1

H 1 (“, A) H 1 (“, B)

’’’

’’

¦ ¦

¦ ¦

f—

Isom(A“Tors “ ) ’ ’ ’ Isom(B“Tors “ ),

’’

where the vertical maps are the natural bijections of (??).

Proof : Let ± ∈ Z 1 (“, A). Every A-orbit in (P± — B)/A can be represented by a

unique element of the form (1, b) with b ∈ B. The map which takes the orbit (1, b)A

∼

to b ∈ Pf 1 (±) is an isomorphism of B-torsors (P± — B)/A ’ Pf 1 (±) .

’

390 VII. GALOIS COHOMOLOGY

Induced torsors. Let “0 be a closed subgroup of “ and let A0 be a “0 -

group. The induced “-group Ind“0 A0 is de¬ned as the group of all continuous

“

maps f : “ ’ A0 such that f (γ0 γ) = γ0 f (γ) for all γ0 ∈ “0 , γ ∈ “:

Ind“0 A0 = { f ∈ Map(“, A0 ) | f (γ0 γ) = γ0 f (γ) for γ0 ∈ “0 , γ ∈ “ }.

“

The “-action on Ind“0 A0 is given by σf (γ) = f (γσ) for σ, γ ∈ “. We let

“

π : Ind“0 A0 ’ A0 be the map which takes f ∈ Ind“0 A0 to f (1). This map satis¬es

“

“

“

σ σ

π( f ) = π(f ) for all σ ∈ “0 , f ∈ Ind“0 A0 . It is therefore a “0 -homomorphism.

(Compare with (??).)

By applying this construction to A0 -torsors, we obtain (Ind“0 A0 )-torsors: for

“

P0 ∈ A0 “Tors “0 , the “-group Ind“0 P0 carries a right action of Ind“0 A0 de¬ned by

“ “

for p ∈ Ind“0 P0 , f ∈ Ind“0 A0 and γ ∈ “.

pf (γ) = p(γ)f (γ) “ “

This action makes Ind“0 P0 an (Ind“0 A0 )-torsor, called the induced torsor. We

“ “