APPENDIX A
A lette r fro m M . Ros t t o J-P . Serr e
Regensburg, de n 3.3.9 1
Dear Professo r Serre ,
Thank yo u ver y muc h fo r you r letter . You r invariant a i s interesting i n fac t
I foun d i t a t firs t somewha t curiou s an d i t concerne d m e som e time . A s fa r a s I
know i t i s th e onl y cohomologica l invarian t vanishin g o n isotropi c forms . Le t m e
make th e followin g remarks .
I) Relatio n wit h exterio r algebr a
Roughly said , I view you r constructio n a s the natura l wa y t o giv e a decompo -
sition o f the exterio r algebr a o f a form wit h respec t t o th e *-operator .
To b e mor e precis e le t m e firs t introduc e fo r a n n-dimensiona l for m p ~
(ai,..., a n) wit h 5 = a\ ... a
n
th e followin g invariant s (takin g value s sa y i n th e
set o f isometry classe s o f Pfister forms ; henc e the y yiel d als o cohomological invari -
ants):13
*(?) = ((-ai,...,-a
n
) )
P(f) = ((-ai,...,-a
n
_i) ) mo d ((£)) , n 3
7((/?) = ((-aia
2
,..., -aia
n
_i)) mo d ((5)), n eve n
Here ((—&i,... , bk)) = (1, &i) ... (1,&&). You r invarian t i s th e cohomologica l ver -
sion o f P(—(f).
To interpre t thes e invariant s i t i s appropriat e t o conside r th e exterio r power s
Aktp: A kV ^ F o f a for m p: V - * F. The n
Akp(v1 A A vk) = p(v
±
) (p(vk)
for Vi _L Vj and on e find s fo r a G D(p) = ip(V) \ {0}:
a{ip) = ((-6))/3((p) ~Ap = ®
kAk(p
/3(p) ~ A
everV
= ®
k
A
2k
^ i f n i s odd, S = 1
7(p) ~ 7(y ) := e ^ A
2k
p if n = 2 (mo d 4) , 5 = 1
a7(p) ~
7
(y,)
:
= e ^ 1
A
2fc+1 y ? i f n = 0 (mo d 4) , 5 = 1
/?(/?) = ((—a))^{p) i f n 4 , n even , 5 = 1.
The forms 7(y?) are multiples of Pfister form s which are for n 2 (!) hyperboli c
if (p is isotropic . Sinc e fo r a , 6 G -D(^) th e produc t ab i s a nor m o f a quadrati c
aRost i s referring t o th e invarian t from Sectio n 20 , which Serr e ha d communicate d t o hi m
in a letter .
In th e followin g equations , S denotes th e determinan t d((p).
93
http://dx.doi.org/10.1090/ulect/028/12
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