3. TORSO R INTERPRETATIO N O F TH E ABOV E EXAMPLE S 9
EXAMPLE 2. 7 (Pfiste r forms) . Fo r n 0,
Pristern(/c) = {isomorphis m classe s o f n-Pfister form s ove r k} .
Recall that a 1-Pfister for m i s a binary quadrati c for m o f type (1, a). A n n-Pfiste r
form i s a tensor produc t o f n 1-Pfister forms . It s ran k i s 2 n.
EXAMPLE
2. 8 (Octonio n algebras) .
Oct(k) = {isomorphis m classe s o f octonion A:-algebras } .
Recall tha t a n octonion algebra O ove r k i s a n 8-dimensiona l nonassociativ e k-
algebra whic h i s a composition algebr a [SVOO , 1.6].
Note that, i f char(/co) ^ 2 , the functor Oc t i s isomorphic t o the functo r Pfister 3
by
octonion algebr a i— norm form ,
see [KMRT98 , 33.18, 33.19] or [SVOO , 1.7]. Thi s extend s t o characteristi c 2 with a
suitable definitio n o f Pfister forms , se e e.g. [Se95 , §10.2].
EXAMPLE
2. 9 (Alber t algebras) .
Alb(A:) = {isomorphis m classe s of Alber t fc-algebras} .
Recall tha t a n Albert k-algebra i s a 27-dimensiona l exceptiona l Jorda n algebra ,
cf. [SVOO , Ch. 5] , [PR94], or [KMRT98 , Ch . IX].
3. Torso r interpretatio n o f th e abov e example s
3.1. TORSORS . Al l of the examples in the previous section (excep t for Exampl e
2.7 fo r n 3) hav e a n interpretatio n i n term s o f th e functo r Torsors a o f (isomor -
phism classe s of ) G-torsors, wher e G i s a smoot h linea r algebrai c grou p ove r /co- b
Note tha t Torsors a (k) i s the sam e a s the firs t nonabelian Galois cohomology set of
G, usuall y denote d b y H^k.G), se e e.g . [Se65 , Ch . Ill ] o r [KMRT98 , Ch . VII] .
For instance , i n Example 2. 4 (quadrati c form s o f rank n) on e selects a base point qo
in Quad n(/co), e.g., th e for m (1,1,..., 1). On e defines G to be the orthogonal group
of qo , i.e., G = Aut(^o ) = On ; thi s i s a smoot h linea r algebrai c grou p ove r fco- If
q i s a n arbitrar y elemen t o f Quad
n
(/c), th e variet y P
q
= Isom(g
0
,g) I s a G-torso r
over k. Th e ma p q i— Pq i s a bijectio n o f Quad
n
(fc) wit h Torsorsa (k), i.e. , wit h
ff^G), cf . [Se65 , Ch. III].
In al l thes e cases , whe n on e ha s selecte d a bas e poin t qo for th e functo r A (s o
that A ca n b e identifie d wit h th e functo r k i— i7
1(/c,G),
wit h G Aut(go)), thi s
base poin t correspond s t o th e elemen t "1" of i7
1(/co,G).
EXAMPLE
3.2 . Fo r A Et
n
a s i n 2.1, we may tak e th e spli t algebr a
Esplit
= fc
0
x x Jfc0
as a base point. The n G Aut(i? spllt) = S
n
(permutation s o f {1, 2,..., n}), viewed
as a n algebrai c grou p o f dimension zero .
Let k b e a n algebrai c closur e o f k, an d le t k
s
b e th e separabl e closur e o f k i n
k; put c T/ e = Gal(k s/k) = Aut(k/k). Th e se t i^ 1 (/c, Sn) ma y b e identifie d wit h
the set Hom conj(r/C, S n) o f continuous homomorphism s F^ S
n
u p to conjugation .
If i t i s usefu l t o mentio n ko explicitly , w e us e th e heavie r notatio n Torsors ^ /^ .
c
The symbol s /c , ks, an d T^ wil l b e use d withou t furthe r explanatio n throughou t thi s par t o f
the book .
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