10 I . TH E NOTIO N O F "INVARIANT "
The correspondenc e betwee n classe s o f n-etal e algebra s ove r k an d element s o f
Homconj(Tfc, S n) i s just Galoi s theory , an d fo r suc h a n algebr a E w e writ e tpE for
the correspondin g homomorphis m T^ Sn, se e [BS94 , 1.3.1] ; not e tha t i t i s only
defined u p t o conjugation . Th e orbit s o f im(/?# ) i n {1,... ,n } correspon d t o th e
factors o f E whic h ar e fields. Fo r instance , E i s a field i f an d onl y i f im(/?# ) i s a
transitive subgrou p o f S
n
, an d E i s split i f and onl y i f /?# = 1.
Here i s a list o f groups G corresponding t o othe r functor s A fro m Sectio n 2 :
2.2: Th e spectru m o f a G-Galoi s algebr a i s a G-torso r an d conversely , se e
[KMRT98, p . 290] . Henc e a G-Galoi s algebr a ove r k correspond s t o a n
element oiH l(k,G).
Note that w e have two interpretations o f H1^, S
n
): a s classes of rank
n etal e algebras and a s classes of S n-Galois algebras . On e passes from th e
first interpretatio n t o th e secon d a s follows : i f L i s an 5 n-Galois algebra ,
the fixed point s o f ^ - i o n L mak e u p a n etal e algebr a E o f ran k n.
Conversely, i f E i s etale of rank n , on e may construct th e corresponding L
as a suitable quotien t o f E ® 0 E (n-fol d tenso r product) ; i n geometri c
language, on e remove s th e "fa t diagonal " fro m th e produc t o f n copie s of
Spec£.
2.3: Bas e poin t th e matri x algebr a M n; the n G = Aut(M n) = PGL
n
.
2.4: G = O
n
= O(qo), se e above .
2.5: G = SO(g 0), wher e q
0
= ( 1 , . . . , 1, 6).
The exac t sequenc e 1 SO (go)— O(qo) {±1} - 1 give s a n
embedding of
Hl(k,
SO(go) )
m
H
l{k^
O(go) ) which can be used to identif y
i-T1
(/c,SO(g0)) wit h Quad n5, se e [KMRT98 , p . 407].
2.6: G = U
n
, wher e U
n
i s the unitar y grou p of rank n relativ e to k±/ko (an d
relative t o th e hermitia n for m ( 1 , . . . , 1)^, say) . Th e fac t tha t Herm
n
(&;)
has onl y on e elemen t USE /c* 2 corresponds t o th e fac t tha t U
n
/k i s the n
isomorphic t o GL
n
, henc e ha s a trivial H 1.
2.8: Bas e point th e split octonion s [SV00 , p. 19]; G = split grou p of type G 2.
2.9: Bas e poin t th e spli t Alber t algebr a [KMRT98 , p . 517]; G = spli t grou p
of type F4.
REMARK
3.3 . Fo r n = 1, 2, 3, the n-Pfiste r form s hav e the alternativ e descrip -
tions:
Pfisteri 9* Quadi 9* Et2 = #*(* {±1}),
Pfister2 9 * Quad35l ^ Cent . Simpl
2
2* ff 1 (*,PGL2) = i^ 1(*^0
3
) ,
Pfister3 ^ O c t ^ i f
1(
* , G
2
) .
However, fo r n 4, on e ca n sho w tha t ther e i s no algebrai c grou p G ove r ko such
that th e functo r Pfister
n
i s isomorphic t o th e functo r k 1— H1(k, G).
NOTATION
3.4 . Whe n th e functo r A i s th e functo r k
H- »
H 1{k,G), w e writ e
Inv(G,#) fo r Inv{A,H).
4. Example s o f functor s H
4.1.
ABELIA N GALOI S COHOMOLOGY .
Le t C b e a discret e r fco-module. Fo r
any extension
/C//CQ ,
we have a natural ma p T^ Tk0 (define d u p to inne r conjuga -
tion), s o that th e cohomolog y group s H
l(Yk,
C) , i = 0,1,.. . mak e sense , cf . [Se65 ,
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