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system of G with respect to T . By (??) and (??), the homomorphism
(31.2) Aut(G) ’ Aut(Dyn(G))
is a split surjection. A splitting i : Aut Dyn(G) ’ Aut(G) can be chosen in such a
way that any automorphism in the image of i leaves the torus T invariant. Assume
that the Galois group “ acts on the Dynkin diagram Dyn(G), or equivalently,
consider a continuous homomorphism
= H 1 F, Aut Dyn(G) .
• ∈ Hom “, Aut Dyn(G)
Denote by γ a cocycle representing the image of • in H 1 F, Aut(Gsep ) under the
map induced by the splitting i. Since γ normalizes T , the twisted group Gγ contains
the maximal torus Tγ . Moreover, the natural action of “ on Tγ leaves Π invariant,
hence Gγ is a quasisplit group. In fact, up to isomorphism Gγ is the unique simply
connected quasisplit group with Dynkin diagram Dyn(G) and with the given action
of “ on Dyn(G) (see (??)). Twisting G in (??) by γ, we obtain:
(31.3) Proposition. Let G be a quasisplit simply connected group. Then the nat-
ural homomorphism Aut(G) ’ Aut(Dyn(G)) is surjective.
Let G be semisimple group over F . By §?? a twisted form G of G corresponds
to an element ξ ∈ H 1 F, Aut(Gsep ) . We say that G is an inner form of G if ξ
belongs to the image of the map
±G : H 1 (F, G) ’’ H 1 F, Aut(Gsep )
induced by the homomorphism Int : G = G/C ’ Aut(G), where C is the center of
G. Since G acts trivially on C, the centers of G and G are isomorphic (as group
schemes of multiplicative type).
(31.4) Proposition. Any semisimple group is an inner twisted form of a unique
quasisplit group up to isomorphism.
Proof : Since the centers of inner twisted forms are isomorphic and all the groups
which are isogenous to a simply connected group correspond to subgroups in its
center, we may assume that the given group G is simply connected. Denote by
Gd the split twisted form of G, so that G corresponds to some element ρ ∈
H 1 F, Aut(Gd ) . Denote by γ ∈ H 1 F, Aut(Gd ) the image of ρ under the
sep sep
426 VII. GALOIS COHOMOLOGY

i
composition induced by Aut(Gd ) ’ Aut Dyn(Gd ) ’ Aut(Gd ), where i is the split-
ting considered above. We prove that the quasisplit group Gd is an inner twisted
γ
form of G = Gd . By (??), there is a bijection
ρ

θρ : H 1 F, Aut(Gsep ) ’ H 1 F, Aut(Gd )
’ sep

taking the trivial cocycle to ρ. Denote by γ0 the element in H 1 F, Aut(Gsep ) such
that θ(γ0 ) = γ. Since ρ and γ have the same image in H 1 F, Aut Dyn(Gd ) , sep
1
the trivial cocycle and γ0 have the same images in H F, Aut Dyn(Gsep ) , hence
Theorem (??) shows that γ0 belongs to the image of H 1 (F, G) ’ H 1 F, Aut(Gsep ) ,
i.e., the group Gγ0 Gd is a quasisplit inner twisted form of G.
γ

Until the end of the subsection we shall assume that G is a simply connected
semisimple group. Denote by ξG the element in H 1 F, Aut(Gsep ) corresponding
to the (unique) quasisplit inner twisted form of G. In general, the map ±G is not
injective. Nevertheless, we have
(31.5) Proposition. There is only one element νG ∈ H 1 (F, G) such that
±G (νG ) = ξG .
Proof : Denote Gq the quasisplit inner twisted form of G. By (??), there is a
bijection between ±’1 (ξG ) and the factor group of Aut Dyn(Gq ) by the image of
G
Aut(Gq ) ’ Aut Dyn(Gq ) . But the latter map is surjective (see (??)).

Let C be the center of G. The exact sequence 1 ’ C ’ G ’ G ’ 1 induces
the connecting map
δ 1 : H 1 (F, G) ’’ H 2 (F, C).
The Tits class of G is the element tG = ’δ 1 (νG ) ∈ H 2 (F, C).
(31.6) Proposition. Let χ ∈ C — be a character. Denote by Fχ the ¬eld of de¬ni-
tion of χ and by Aχ its minimal Tits algebra. The image of the Tits class tG under
the composite map
χ—
res
H 2 (F, C) ’’ H 2 (Fχ , C) ’’ H 2 (Fχ , Gm ) = Br(Fχ )

is [Aχ ]. (We use the canonical identi¬cation H 2 (Fχ , Gm ) = Br(Fχ ) given by the
connecting map of (??), which is the opposite of the identi¬cation given by the
crossed product construction.)
Proof : There is a commutative diagram (see §??)
1 ’’ CFχ ’’ G Fχ ’’ G Fχ ’’ 1
¦ ¦ ¦
¦ ¦ ¦
χ


1 ’’ Gm ’’ GL1 (A) ’’ PGL1 (A) ’’ 1
where A = Aχ . Therefore, it su¬ces to prove that the image of res(νG ) under the
composite map
H 1 (Fχ , GFχ ) ’ H 1 Fχ , PGL1 (A) ’ H 2 (Fχ , Gm ) = Br(Fχ )
is ’[A]. The twist of the algebra A by a cocycle representing res(νG ) is the Tits
algebra of the quasisplit group (GνG )Fχ , hence it is trivial. Therefore the image
§31. COHOMOLOGICAL INVARIANTS 427


γ of res(νG ) in H 1 Fχ , PGL1 (A) corresponds to the split form PGLn (Fχ ) of
PGL1 (A), where n = deg(A). By (??), there is a commutative square
H 1 Fχ , PGLn (Fχ ) ’ ’ ’ Br(Fχ )
’’
¦ ¦
¦ ¦g
θγ



H 1 Fχ , PGL1 (A) ’ ’ ’ Br(Fχ )
’’
where g(v) = v + u and u is the image of γ in Br(Fχ ). Pick µ ∈ Z 1 Fχ , PGLn (Fχ )
such that the twisting of PGLn (Fχ ) by µ equals PGL1 (A). In other words, θγ (µ) =
1 and the image of µ in Br(Fχ ) equals [A]. The commutativity of the diagram then
implies that u = ’[A].

(31.7) Example. Let G = SU(B, „ ) where (B, „ ) is a central simple F -algebra
with unitary involution of degree n. Assume that char F does not divide n and let
K be the center of B. Since the center C of SU(B, „ ) is µn[K] , the Tits class tG
belongs to H 2 (F, µn[K] ). Abusing terminology, we call it the Tits class of (B, „ )
and denote it by t(B, „ ), i.e.,
t(B, „ ) = tSU(B,„ ) ∈ H 2 (F, µn[K] ).
Suppose K is a ¬eld. For χ = 1 + nZ ∈ Z/nZ = C — , the ¬eld of de¬nition of χ is
K and the minimal Tits algebra is B, see §??. Therefore, Proposition (??) yields
resK/F t(B, „ ) = [B] ∈ Br(K).
If n is odd, it follows from Corollary (??) that
1
t(B, „ ) = cor[B].
2

On the other hand, if n is even we may consider the character » = n + nZ ∈ C — of
2
raising to the power n . The corresponding minimal Tits algebra is the discriminant
2
algebra D(B, „ ), see §??. By (??) we obtain
»— t(B, „ ) = D(B, „ ) ∈ Br(F ).
If K F — F we have (B, „ ) = (A — Aop , µ) for some central simple F -algebra
A of degree n, where µ is the exchange involution. The commutative diagram with
exact rows
1 ’ ’ ’ µn ’ ’ ’ SLn ’ ’ ’ PGLn ’ ’ ’ 1
’’ ’’ ’’ ’’
¦ ¦
¦ ¦

1 ’ ’ ’ Gm ’ ’ ’ GLn ’ ’ ’ PGLn ’ ’ ’ 1
’’ ’’ ’’ ’’
shows that t(B, „ ) = [A] ∈ H 2 (F, µn ) = n Br(F ).
(31.8) Example. Let G = Spin(V, q), where (V, q) is a quadratic space of odd
dimension. Suppose char F = 2. The center of G is µ2 and the Tits class tG ∈
H 2 (F, µ2 ) = 2 Br(F ) is the Brauer class of the even Cli¬ord algebra C0 (V, q), since
the minimal Tits algebra for the nontrivial character is C0 (V, q), see §??.
(31.9) Example. Let G = Sp(A, σ), where (A, σ) is a central simple F -algebra
with symplectic involution. Suppose char F = 2. The center of G is µ2 and the
Tits class tG ∈ H 2 (F, µ2 ) = 2 Br(F ) is the Brauer class of the algebra A, see §??.
428 VII. GALOIS COHOMOLOGY


(31.10) Example. Let G = Spin(A, σ, f ), where (A, σ, f ) is a central simple F -
algebra of even degree 2n with quadratic pair. Let Z be the center of the Cli¬ord
algebra C(A, σ, f ) and assume that char F = 2.
Suppose ¬rst that n is even. Then the center of G is RZ/F (µ2 ), hence

tG ∈ H 2 F, RZ/F (µ2 ) = H 2 (Z, µ2 ) = 2 Br(Z).

The minimal Tits algebra corresponding to the norm character

» : RZ/F (µ2 ) ’ µ2 ’ Gm

is A, hence

corZ/F (tG ) = [A] ∈ H 2 (F, µ2 ) = 2 Br(F ).
(31.11)

On the other hand, the minimal Tits algebras for the two other nontrivial characters
»± are C(A, σ, f ) (see §??), hence

tG = C(A, σ, f ) ∈ H 2 (Z, µ2 ) = 2 Br(Z).
(31.12)

Now, assume that n is odd. Then the center of G is µ4[Z] , hence

tG ∈ H 2 (F, µ4[Z] ).

By applying Proposition (??), we can compute the image of tG under the squaring
map

»— : H 2 (F, µ4[Z] ) ’ H 2 (F, µ2 ) = 2 Br(F )

and under the restriction map

res : H 2 (F, µ4[Z] ) ’ H 2 (Z, µ4 ) = 4 Br(Z)

(or, equivalently, under the map H 2 (F, µ4[Z] ) ’ H 2 F, RZ/F (Gm,Z ) = Br(Z)
induced by the inclusion µ4[Z] ’ RZ/F (Gm,Z )). We obtain

»— (tG ) = [A] and res(tG ) = C(A, σ, f ) .

Note that the fundamental relations (??) of Cli¬ord algebras readily follow
from the computations above (under the hypothesis that char F = 2). If n is even,
2
(??) shows that C(A, σ, f ) = 1 in Br(Z) and (??) (together with (??)) implies

NZ/F C(A, σ, f ) = [A].

If n is odd we have
2
[AZ ] = res —¦»— (tG ) = res(tG )2 = C(A, σ, f )

and

NZ/F C(A, σ, f ) = cor —¦ res(tG ) = 0,

by (??). (See Exercise ?? for a cohomological proof of the fundamental relations
without restriction on char F .)
§31. COHOMOLOGICAL INVARIANTS 429


31.B. Cohomological invariants of algebraic groups. Let G be an alge-
braic group over a ¬eld F . For any ¬eld extension E of F we consider the pointed
set
H 1 (E, G) = H 1 (E, GE )
of GE -torsors over E. A homomorphism E ’ L of ¬elds over F induces a map of
pointed sets
H 1 (E, G) ’ H 1 (L, G).
Thus, H 1 (?, G) is a functor from the category of ¬eld extensions of F (with mor-
phisms being ¬eld homomorphisms over F ) to the category of pointed sets.
Let M be a torsion discrete Galois module over F , i.e., a discrete module over
the absolute Galois group “ = Gal(Fsep /F ). For a ¬eld extension E of F , M
can be endowed with a structure of a Galois module over E and hence the ordinary
cohomology groups H d (E, M ) are de¬ned. Thus, for any d ≥ 0, we obtain a functor
H d (?, M ) from the category of ¬eld extensions of F to the category of pointed sets
(actually, the category of abelian groups). A cohomological invariant of the group G
of dimension d with coe¬cients in M is a natural transformation of functors
a : H 1 (?, G) ’ H d (?, M ).
In other words, the cohomological invariant a assigns to any ¬eld extension E of F
a map of pointed sets
aE : H 1 (E, G) ’ H d (E, M ),
such that for any ¬eld F -homomorphism E ’ L the following diagram commutes
a
H 1 (E, G) ’ ’ ’ H d (E, M )
’E’
¦ ¦
¦ ¦
a
H 1 (L, G) ’ ’ ’ H d (L, M ).
’L’
The set Invd (G, M ) of all cohomological invariants of the group G of dimension d
with coe¬cients in M forms an abelian group in a natural way.
(31.13) Example. Let G = GL(V ) or SL(V ). By Hilbert™s Theorem 90 (see
(??) and (??)) we have H 1 (E, G) = 1 for any ¬eld extension L of F . Hence
Invd (G, M ) = 0 for any d and any Galois module M .
A group homomorphism ± : G ’ G over F induces a natural homomorphism
±— : Invd (G , M ) ’ Invd (G, M ).
A homomorphism of Galois modules g : M ’ M yields a group homomorphism
g— : Invd (G, M ) ’ Invd (G, M ).
For a ¬eld extension L of F there is a natural restriction homomorphism
res : Invd (G, M ) ’ Invd (GL , M ).
Let L be a ¬nite separable extension of F . If G is an algebraic group over L and M
is a Galois module over F , then the corestriction homomorphism for cohomology
groups and Shapiro™s lemma yield the corestriction homomorphism
cor: Invd (G, M ) ’ Invd RL/F (G), M .
430 VII. GALOIS COHOMOLOGY


Invariants of dimension 1. Let G be an algebraic group over a ¬eld F . As
in §??, write π0 (G) for the factor group of G modulo the connected component
of G. It is an ´tale group scheme over F .
e
Let M be a discrete Galois module over F and let
g : π0 (Gsep ) ’ M
be a “-homomorphism. For any ¬eld extension E of F we then have the following
composition
ag : H 1 (E, G) ’ H 1 E, π0 (G) ’ H 1 (E, M )
E
where the ¬rst map is induced by the canonical surjection G ’ π0 (G) and the
second one by g. We can view ag as an invariant of dimension 1 of the group G
with coe¬cients in M .
(31.14) Proposition. The map
Hom“ π0 (Gsep ), M ’ Inv1 (G, M ) given by g ’ ag
is an isomorphism.
In particular, a connected group has no nonzero invariants of dimension 1.
(31.15) Example. Let (A, σ, f ) be a central simple F -algebra with quadratic pair
of degree 2n and let G = PGO(A, σ, f ) be the corresponding projective orthog-
onal group. The set H 1 (F, G) classi¬es triples (A , σ , f ) with a central simple
F -algebra A of degree 2n with an quadratic pair (σ , f ) (see §??). We have
π0 (G) Z/2Z and the group Inv1 (G, Z/2Z) is isomorphic to Z/2Z. The nontrivial
invariant
H 1 (F, G) ’ H 1 (F, Z/2Z)
associates to any triple (A , σ , f ) the sum [Z ] + [Z] of corresponding classes of the
discriminant quadratic extensions.
(31.16) Example. Let K be a quadratic ´tale F -algebra, let (B, „ ) be a central
e
simple K-algebra of degree n with a unitary involution and set G = Aut(B, „ ).
The set H 1 (F, G) classi¬es algebras of degree n with a unitary involution (see (??)).
Then π0 (G) Z/2Z and, as in the previous example, the group Inv1 (G, Z/2Z) is
isomorphic to Z/2Z. The nontrivial invariant
H 1 (F, G) ’ H 1 (F, Z/2Z)
associates to any central simple F -algebra with unitary involution (B , „ ) with
center K the class [K ] + [K].
Invariants of dimension 2. For any natural numbers i and n let µ—i (F ) ben
the i-th tensor power of the group µn (F ). If n divides m, there is a natural injection
µ—i (F ) ’ µ—i (F ).
n m
The groups µ—i (F ) form an injective system with respect to the family of injections
n
de¬ned above. We denote the direct limit of this system, for all n prime to the
characteristic of F , by Q/Z(i)(F ). For example, Q/Z(1)(F ) is the group of all
roots of unity in F .
The group Q/Z(i)(Fsep ) is endowed in a natural way with a structure of a
Galois module. We set
H d F, Q/Z(i) = H d F, Q/Z(i)(Fsep ) .
§31. COHOMOLOGICAL INVARIANTS 431


In the case where char F = p > 0, this group can be modi¬ed by adding an
appropriate p-component. In particular, the group H 2 F, Q/Z(1) is canonically
isomorphic to Br(F ), while before the modi¬cation it equals lim H 2 F, µn (Fsep )
’’
which is the subgroup of elements in Br(F ) of exponent prime to p.
Let G be a connected algebraic group over a ¬eld F . Assume that we are given
an exact sequence of algebraic groups
(31.17) 1 ’ Gm,F ’ G ’ G ’ 1.
For any ¬eld extension E of F , this sequence induces a connecting map H 1 (E, G) ’
H 2 (E, Gm,F ) which, when composed with the identi¬cations
H 2 (E, Gm,F ) = Br(E) = H 2 E, Q/Z(1) ,
provides an invariant aE of dimension 2 of the group G. On the other hand, the
sequence (??) de¬nes an element of the Picard group Pic(G). It turns out that the
invariant a depends only on the element of the Picard group and we have a well
de¬ned group homomorphism
β : Pic(G) ’ Inv2 G, Q/Z(1) .
(31.18) Proposition. The map β is an isomorphism.
Since the n-torsion part of Q/Z(1) equals µn , we have
(31.19) Corollary. If n is not divisible by char F , then
Inv2 G, µn (Fsep ) n Pic(G).




(31.20) Example. Let G be a semisimple algebraic group, let π : G ’ G be a
universal covering and set Z = ker(π). There is a natural isomorphism
∼ ∼
Z — ’ Pic(G) ’ Inv2 G, Q/Z(1) .
’ ’
Hence attached to each character χ ∈ Z — is an invariant which we denote by aχ .
The construction is as follows. Consider the group G = (G — Gm,F )/Z where Z is
embedded into the product canonically on the ¬rst factor and by the character χ
on the second. There is an exact sequence
1 ’ Gm,F ’ G ’ G ’ 1.
We de¬ne aχ to be the invariant associated to this exact sequence as above.
The conjugation homomorphism G ’ Aut(G) induces the map
H 1 (F, G) ’ H 1 F, Aut(Gsep ) .
Hence, associated to each γ ∈ H 1 (F, G) is a twisted form Gγ of G (called an inner
form of G). If we choose γ such that Gγ is quasisplit (i.e., Gγ contains a Borel
subgroup de¬ned over F ), then
aχ (γ) = [Aχ ] ∈ Br(F )
F

where Aχ is the Tits algebra associated to the character χ (see §??).
(31.21) Example. Let T be an algebraic torus over F . Then

Pic(T ) ’ H 1 F, T — (Fsep )

432 VII. GALOIS COHOMOLOGY


and all the cohomological invariants of dimension 2 of T with coe¬cients in Q/Z(1)
are given by the cup product
H 1 (F, T ) — H 1 F, T — (Fsep ) ’ H 2 (F, Fsep ) = H 2 F, Q/Z(1)



associated to the natural pairing
T (Fsep ) — T — (Fsep ) ’ Fsep .



Invariants of dimension 3. Let G be an algebraic group over a ¬eld F .
Assume ¬rst that F is separably closed. A loop in G is a group homomorphism
Gm,F ’ G over F . Write G— for the set of all loops in G. In general there is no
group structure on G— , but if f and h are two loops with commuting images, then
the pointwise product f h is also a loop. In particular, for any integer n and any
loop f the nth power f n is de¬ned. For any g ∈ G(F ) and any loop f , write gf for
the loop
Int(g)
f
Gm,F ’ G ’’
’ ’’ G.
Consider the set Q(G) of all functions q : G— ’ Z, such that
(a) q( gf ) = q(f ) for all g ∈ G(F ) and f ∈ G— ,
(b) for any two loops f and h with commuting images, the function
(k, m) ’ q(f k hm )
Z — Z ’ Z,
is a quadratic form.
There is a natural abelian group structure on Q(G).
Assume now that F is an arbitrary ¬eld. There is a natural action of the
absolute Galois group “ on the set of loops in Gsep and hence on Q(Gsep ). We set
Q(G) = Q(Gsep )“ .
(31.22) Example. Let T be an algebraic torus. Then T— (Fsep ) is the group of
cocharacters of T and

Q(T ) = S 2 T — (Fsep )
is the group of Galois invariant integral quadratic forms on T— (Fsep ).
(31.23) Example. Let G = GL(V ) and f ∈ G— (Fsep ) be a loop. We can view f as
a representation of Gm,F . By the theory of representations of diagonalizable groups
(see §??), f is uniquely determined by its weights χai , i = 1, 2, . . . , n = dim(V )
where χ is the canonical character of Gm,F . The function qV on G— (Fsep ) de¬ned
by
n
a2
qV (f ) = i
i=1

clearly belongs to Q(G).
A group homomorphism G ’ G over F induces a map of loop sets G— (Fsep ) ’
G— (Fsep ) and hence a group homomorphism
Q(G ) ’ Q(G),
making Q a contravariant functor from the category of algebraic groups over F to

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