22. COHOMOLOGICA L INVARIANT S O F ALBER T ALGEBRA S 49
where q G Quadn(/c) i s such tha t qn = h.
REMARK
21.5. Not e that, i n the case n = 2 , an element of Hermn(/c) i s uniquely
determined b y it s tw o invariant s Wi^s) £ H 2(k) an d 1^2,(5) £ H 3(k); thi s follow s
from 20.3 .
THEOREM
21.6. Every element o/Invfc 0(Hermn, Z/2Z ) can be written uniquely
as
n
hh^ x0 + y~]w iiX.(h)
with xo G H(ko) and X{ G (S) -H(ko) for i 1.
PROOF.
An y invarian t o f Herm
n
give s on e o f Quad
n
b y 21.1 , whic h ca n b e
written uniquel y a s
n
q i— XQ + 2_, xi ' wi(°)- i f # •—• ft, with X{ G H(ko).
Moreover, thi s invarian t shoul d giv e a constant invarian t ove r hi. Thi s show s tha t
the
Xi
(i 1) belon g to (6) -H{k0) = ker[H(k
0
) - #(fci)]. D
22. Cohomologica l invariant s o f Alber t algebra s
We will now determin e Inv(Alb , Z/2Z), cf . 2.9 .
EXAMPLE
22. 1 (reduce d Alber t algebras) . Le t q
2
an d qs b e 2 - an d 3-Pfiste r
forms ove r k. Le t O be a n octonio n algebr a with nor m q%. Choos e a\ an d a
2
i n k*
such tha t # 2 is isomorphic t o (1, —ai) 0 (1, —cx2) an d se t
0 = ( ^ - « 2 )
in Ms(O). Ther e is a /c-linear map * defined fo r x G Ms(0) b y x* = g-x^
-g~x,
wher e
t i s the transpose, an d

is the canonical conjugation o n O. Th e subspace of M$(0)
consisting o f those x wit h x x* endowe d wit h th e produc t (x,y) 1— (xy + yx)/2
is an Alber t algebr a whic h we denote b y J . I t i s determined u p to isomorphis m b y
q2 an d q
3
.
This constructio n give s a ma p Oc t x Pfister 2 —• Alb correspondin g t o th e nat -
ural inclusio n
G2 x PGL
2
- F 4.
The Alber t algebra s obtaine d fro m thi s constructio n ar e called reduced, see [SV00 ,
p. 125], [KMRT98 , p . 517], or [P R 94].
LEMMA
22.2 . Suppose that U{ is an odd-dimensional quadratic form and fi, f[
are i-Pfister forms for some finite set of indices i. IfJ2
uifi =
^2
uifi
^
n
^
e
Witt
ring of k, then fi = f[ for all i.
PROOF.
Writ e / fo r th e augmentatio n idea l o f th e Wit t rin g W(k), cf . 27.1.
Let n b e th e smalles t o f th e indices . Modul o 7 n + 1, Yl uifi * s congruent t o u
n
fni
which i s in tur n congruen t t o /
n
, an d similarl y fo r Yl uifi- Sinc e f
n
an d f'
n
ar e n-
Pfister form s which are congruent modul o /n + 1 , the y are isomorphic by a theorem of
Arason-Pfister, cf . [La m 73, X.3.4]. Th e lemm a follows b y induction o n the numbe r
of indices.
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