30. APPLICATION S 71
Theorem 29. 4 gives:
] T \ a(qE)-\b(-ujn) = 0 forzm .
a+b=i
This give s a recursiv e proces s fo r computin g th e highe r exterio r power s o f q
E
i n
terms o f the lowe r ones , se e e.g. (30.3 ) an d (30.4 ) i n th e case s n = 4 and i = 3 , 4.
REMARK 29.5 . Defin e SE £ WGV(£; ) a s g # cj
n
; on e ha s rank(S# ) = m.
It i s no t alway s tru e (cf . 31.8) tha t SE i s a n "effective " elemen t o f WGr(k), i.e. ,
that i t ca n b e represente d b y a genuin e quadrati c for m o f ran k m . Still , Th . 29. 4
shows that th e A-power s of SE behav e as if SE wer e effective; thi s has been recentl y
completed b y Ros t [Ros t 02], wh o ha s show n tha t ther e exist s a n ra-Pfister for m
qm(E) suc h tha t
qm{E) = ©£
0
A
z
(te - v n) i n WGr(k).
(Note tha t th e existenc e o f qm{E) woul d b e obviou s i f SE wer e effective. ) Thi s
gives a Pfister for m invarian t o f etale algebras .
30. Application s
The fact tha t multiquadrati c algebra s detect th e equality of Witt invariant s ca n
be use d t o perfor m explici t computations . Fo r instance , le t /? : Sn— SN b e a ho-
momorphism o f symmetric groups . Thi s give s a morphism o f functors Et
n
Etjy,
and w e write E^ fo r th e imag e o f E, whic h i s well-defined u p t o isomorphism. 13 I n
classical language , E^ correspond s t o a "resolvent " o f E. Fo r instance , th e obviou s
map S 4 S 3 allows one to attac h t o a quartic algebr a E it s "cubi c resolvent " E3.
One ma y as k th e followin g question : Is there a formula giving the trace form
QE^ in terms of qE ? The answer is yes: Indeed , the map E 1— » qE i s a Witt invarian t
of E ove r ko the prim e field. Henc e b y Theore m 29. 2 on e ca n writ e qE as a
linear combination of the X lqE wit h coefficient s i n W(ko). An d thes e coefficient s
can b e compute d b y lookin g a t wha t happen s whe n E i s multiquadratic. Her e ar e
three examples .
30.1.
TH E CUBI C RESOLVEN T O F A QUARTI C EQUATION .
Le t E b e a n etal e
algebra o f rank 4 and le t E' b e th e algebr a o f rank 3 deduced fro m i t vi a th e ma p
SA S3 mentione d above . On e want s t o comput e th e trac e for m qE' i n term s
of th e trac e for m q qE- An y trac e for m o f ran k 3 i s equa l t o (1,2,2c/) wher e
d i s th e discriminant , se e Theore m 31.10 . Sinc e 6 4 S3 i s compatibl e wit h th e
sign homomorphism s S
n
Z/2Z , d i s als o th e discriminan t o f E. Henc e w e ge t
QE' = (1, 2, 2d) wit h d = d(q). Not e tha t w e could writ e (d) a s X
4q,
an d thus :
(30.2) ^ / = (1,20(2)AV
However, this is not written as a linear combination of 1,/, and A
2g,
as it should. T o
do so, one has to use the identities of Theorem 29.4 , which tell us that X
l(q
2) = 0
for i = 3,4 (sinc e UJ
2
= (2 , 2) = (1,1 ) = 2) . Tha t is ,
(30.3) X
3q
- 2\
2q
+ 3q - 4 = 0
(30.4) \
Aq
- 2X
3q
+ 3X
2q
- Aq + 5 = 0 .
More generally , i f X i s a finite etal e coverin g o f rank n o f a scheme S, on e ca n associat e wit h
it i n a functoria l wa y a finite etal e schem e X^ o f ran k N. [Indeed , X define s a n SV^-torso r
Px ove r S, an d X^ i s the fiber spac e associate d wit h Px an d th e 0-ac t ion o f S
n
o n th e finite se t
{!,..., AT}.]
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