98
B. A LETTE R FRO M J-P . SERR E T O R.S . GARIBALD I
Now, abou t cohomologica l invariants .
The notio n seem s reasonabl e enough . I t bring s som e orde r t o al l th e man y
known invariants . I t explain s fo r instanc e wh y nobod y eve r cam e u p wit h a non -
constant invarian t (mo d 3 ) fo r etal e algebras . O n th e othe r hand , mos t o f th e
cases treate d i n th e presen t note s ar e basicall y trivial.
a
The y ar e relativ e t o case s
where there is a versal object ove r a purely transcendental functio n field: henc e an y
invariant i s characterized b y it s residues . On e the n use s th e fac t tha t on e alread y
knows enoug h suc h invariant s t o fill in al l possibilitie s fo r th e residues . W e rarel y
have to construct ne w invariants fro m scratc h ( a few exceptions: 20.1, 25.16, 27.19,
27.22). Thi s i s very differen t fro m Merkurjev' s part , wher e th e Ros t invarian t ha s
to b e constructe d th e har d way . Incidentally , I had als o tried i n 1992 to construc t
the sam e invarian t b y a simila r metho d (makin g us e o f th e etal e cohomolog y o f
SL,n/G); however , I was unabl e t o sho w tha t "my " invarian t wa s no t identicall y 0
(even fo r th e simpl e cas e o f G2 , ho w sa d ...) . I corresponde d wit h Ros t o n thi s i n
December 1992 an d foun d tha t h e ha d gon e muc h further ; no t onl y di d h e kno w
his invarian t i s no t alway s 0 , bu t h e coul d defin e i t a s takin g value s i n Q/Z(2) ,
and h e coul d relat e i t t o hi s spac e U Q(G)" o f quadratic forms . I t wa s particularl y
satisfying t o se e a ne w invarian t takin g value s i n H
3,
instea d o f th e stal e H
1
an d
H2
o f th e previou s decades . A s Ev a Baye r explaine d t o m e o n severa l occasions ,
the groun d ha d bee n well-prepare d b y th e dee p wor k o f Pfister , Merkurjev-Suslin ,
and others , wh o definitely wen t beyon d H
2.
I also benefited fro m th e wor k o f (an d
my correspondenc e with ) Petersson-Racin e o n F4 .
With bes t wishe s and man y thank s fo r you r enormou s hel p (withou t whic h
these note s woul d neve r hav e bee n written) ,
Yours,
J-P.Serre
a The mai n exceptio n i s the on e I treate d first , namel y tha t o f etal e algebras . Her e w e us e a
subfunctor o f Et
n
(namel y th e multiquadrati c algebras ) whic h i s stron g enoug h t o detec t al l th e
invariants, withou t bein g equa l t o Et
n
itself .
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