22. COHOMOLOGICA L INVARIANT S O F ALBER T ALGEBRA S 5 1
The Alber t algebr a constructe d fro m hyp
3
an d an y 2-Pfiste r for m qi i s well-know n
to b e split . Thu s a
2
i s constant , an d sinc e e2(hyp
2
) = 0 w e hav e
0 = a
2
(g
2
) - a
2
(hyp
2
) = A
2
-e2(g2).
By 18.1, this implie s A
2
= 0 . Thu s (22.6 ) ma y b e rewritte n a s
(22.7) a
P
f(q3,q2) = A
0
+ \
3
-e3(q3) + A
23
-e3(q3) -e
2
(g2).
The invarian t a is determined b y it s value s o n th e reduce d Alber t algebras . For ,
let a! b e a n invarian t i n Invfc
0
(Alb, Z/2Z ) whic h agree s wit h a o n reduce d algebras .
Let A b e a nonreduce d Alber t algebr a ove r som e extensio n k o f ko . Ther e i s a cubi c
extension k' o f k ove r whic h A i s reduced [SV00 , §6.1]. S o Res £ (a{A)) = a{A®
k
k')
is equa l t o a'(A 0k k f) Resjf a'{A). Sinc e \k' : k] i s odd , th e ma p H(k) H{k')
is injective , henc e a(A) i s equa l t o a'{A).
Since a agree s wit h A o + A 3 -f3 - f A2 3 -/^ fo r reduce d Alber t algebras , th e tw o
invariants ar e th e same . Thu s th e se t {1, ^3, f$} span s Inv(Alb , Z/2Z) .
Now suppos e tha t a Ao + A 3 -f
3
+ A
2
3 -/s i s 0 . The n ap f = 0 . I n particular ,
Ao is 0 . W e thu s ge t a n invarian t o f Prister 3 b y
q3 ^ a
Pf
(?3,hyp2) = A
3
-e3(73).
Since thi s invarian t i s 0 , w e hav e A 3 = 0 b y 18.1.
The tenso r produc t define s a morphis m
Pfister3 x Pfister
2
Pfister
5
.
For ever y extensio n k o f &o , this i s surjective . Henc e i t induce s a n injectiv e ma p
Invfc0(Pfister5, Z/2Z ) -+ Inv
fco
(Pfister3 x Pfister
2
, Z/2Z) .
The invarian t ap f i s give n b y
apf (93,42) = A
23
-e2(g'2)-e3(g3),
that is , i t i s th e imag e o f A
2
3-es. Sinc e thi s i s 0 b y hypothesis , w e hav e A
2
3 = 0
by 18.1 . Thu s al l o f th e coefficient s specifyin g a ar e 0 . Thi s show s tha t th e se t
{1, f
3
, fa} i s linearl y independent .
For invariant s mo d 3 o f Alb , se e [Rost91] an d [KMRT98 , Chap . IX] .
EXERCISE 22. 8 (cu p product s o f invariant s o f F4) . Sho w tha t th e rin g structur e o f
Inv(i4, Z/2Z) (wit h produc t th e cu p product ) i s give n b y th e formula s
/3-/3 = ( - l ) - ( - l ) - ( - l ) 7 3
h-h = (-!)•(-! ) -(-1)75
/
5
. /
5 =
( _ l ) . ( - l ) . ( _ l ) . ( _ l ) . ( - l ) . /
6
.
[Hint: Us e th e fac t that , i f A i s a Jorda n algebr a ove r /c , there ar e fiv e element s xi, ... ,
x5 oik* suc h tha t /3(A ) = (xi ) -(x
2
) -(x
3
) an d /5(A ) = (xi ) -(x
2
) '(x3) -(x
4
) '(x
5
).]
EXERCISE 22. 9 (invariant s o f EQ). Le t EQ be a spli t grou p o f typ e E&, either simpl y
connected o r adjoint . Ther e ar e standar d embedding s G2 F4 EQ. Sho w tha t th e
composite ma p
e: TorsorsG
2
* Torsors£
6
is "almost " a bijection , i n th e followin g sense :
(1) e is injective ;
(2) fo r ever y /c-i?6-torso r x , ther e i s a n odd-degre e extensio n k! jk ove r whic h x
belongs t o th e imag e o f e .
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