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(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

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