50
VI. COHOMOLOGICA L INVARIANT S O F O,, , SO,, ,
Every Alber t algebr a A ha s a trac e Tr : A •» k an d a quadrati c for m qA define d
by qA(%) ~ Tr(x 2 )/2. Fo r th e reduce d Alber t algebr a J define d i n 22.1, we hav e
(22.3) qj ^ (2 , 2, 2) © (q
3
® (-a
1212
,-a ,a a )),
cf. [SV00 , p . 118] . Thi s quadrati c for m determine s th e isomorphis m clas s o f J b y
[SV00, 5.8.1].
T H E O R E M 22.4 . [Se95 , Th . 10] Let A be an Albert algebra over k. There exist
3- and 5-Pfister forms q
3
and q$ over k such that
qA®q* = (2 , 2 , 2 ) 0 ^ .
Moreover, this property characterizes q
3
and q$ up to isomorphism and q$ is divisible
by qs (i.e. , q$ is isomorphi c t o a tenso r produc t o f q
3
wit h a 2-Pfiste r form) .
PROOF. Firs t suppos e tha t A i s reduce d an d q
3
, q$ ar e a s i n Exampl e 22.1.
By (22.3) , th e form s q
3
an d # 5 = q
2
q3 satisf y th e condition s o f th e theorem . I f tw o
other form s q f
3
and q'
b
also satisf y thes e conditions , the n b y (22.3 ) w e hav e
( 2 , 2 , 2 ) e f e - g 3 ) = ( 2 , 2 , 2 ) © ( ^ - ^ )
in th e Wit t ring . Lemm a 22. 2 implie s tha t q
3
= q
3
an d q$ = q'
5l
whic h finishes th e
case wher e A i s reduced .
If th e algebr a A i s no t reduced , ther e i s a cubi c extensio n k' o f k suc h tha t
A 0k k f i s reduce d [SV00 , §6.1]. A descen t propert y fo r Pfiste r form s du e t o Ros t
[Rost 99, §3 ] transfer s th e resul t fro m A 0 ^ k' dow n t o A.
For a n alternativ e approach , se e [P R 95].
Define invariant s /
3
, f$ G Inv(Alb , Z/2Z) b y settin g fa e^qi) fo r i 3,5 ,
where qi is the Pfiste r for m associate d t o a n Alber t algebr a b y the precedin g theore m
and a e Inv(Pfister
i5
Z/2Z ) i s a s i n §18.
T H E O R E M 22.5 . Inv
fco
(Alb, Z/2Z ) is a free H(k
0
)-module with basis {1, /3 , /
5
} .
P R O O F . W e hav e a morphis m o f functor s Pfister
3
x Pfister
2
» Alb give n b y
sending (33 , q2) t o the reduce d algebr a constructe d i n Example 22.1. I n this manner ,
a define s a n invarian t ap f o f Pfister 3 x Pfister
2
. Recal l tha t H 1(/c, Z/2Z) = /c*//c* 2,
and s o ther e i s a morphis m o f functor s iJ 1(*,Z/2Z ) —• Pfisteri give n b y (a) 1—
(1, —a). Simila r statement s hol d fo r direc t sum s o f n copie s o f Z/2 Z an d Pfister
n
.
Consequently, fo r G = Z/2 Z x x Z/2Z ( 5 copies), w e hav e morphism s o f functor s
Hl(*,G)— Pfister3 x Pfister
2
Alb. An y invarian t a £ Inv(Alb , Z/2Z) restrict s
to a n invarian t o n #*(* , G) , henc e i s o f th e for m describe d i n 16.4. Th e argument s
in th e proo f o f 18.1sho w tha t
(22.6) a
P f
(q
3
, q
2
) = A
0
+ A
2
-e2(q2) + A
3
-e3(q3) + A
23
-e3{q3) -e
2
(q2)
where th e A' s ar e i n H(ko).
The invarian t ap f i n tur n define s a n invarian t a
2
i n Invfc
0
(Pfister2, Z/2Z ) b y
setting
a2(?2) = «Pf(hyp
3
,g2)
for hyp
3
th e hyperboli c 3-Pfiste r form , whic h i s a tenso r produc t o f thre e copie s o f
( 1 , - 1 ) . Sinc e e3(hyp
3
) i s 0 , thi s invarian t i s give n b y
a2(?2) = A
0
+ A
2
-e2(?2).
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