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