Introduction
As wa s alread y sai d i n th e preface , thes e note s explai n ho w t o determin e th e
cohomological invariant s fo r variou s kind s o f objects , especiall y quadrati c form s
and etal e algebras . W e hav e als o include d "Wit t invariants" , wher e th e Wit t rin g
replaces cohomology .
The conten t o f th e differen t chapter s i s a s follows :
Chapter I introduces th e differen t functor s whic h ar e use d later , suc h a s Quad
n
,
Et
n
, an d H(*,C).
The nex t tw o chapter s contai n basi c materia l o n Galoi s cohomology , fo r in -
stance th e notio n o f residu e (i n th e loca l case , cf . Chapte r II ) an d th e elementar y
properties (residu e formula ) o f th e cohomolog y o f purel y transcendenta l extension s
(Chapter III) . Ther e i s littl e her e whic h i s new , bu t sinc e i t wa s no t possibl e t o
give precis e reference s especiall y abou t sig n convention s w e hav e include d
essentially complet e proofs .
Chapter I V give s severa l specializatio n propertie s o f cohomologica l invariant s
(the mai n on e bein g a n unpublishe d resul t o f Rost) .
Chapter V contain s th e basi c propertie s o f th e restrictio n an d corestrictio n
of invariants , i n perfec t analog y wit h th e cas e o f grou p cohomology , cf . Cartan -
Eilenberg [CE56 , Chap . XII] .
Chapter V I applie s th e technique s o f Chapter s II I an d I V t o th e determinatio n
of th e invariant s fo r quadrati c forms , octonions , Alber t algebras , etc . I n eac h case ,
a ver y helpfu l fac t i s that ther e i s a "versal " objec t whos e fiel d o f definitio n i s purel y
transcendental ove r th e groun d field.
The las t thre e chapter s ar e mainl y concerne d wit h th e cas e o f etal e algebras .
This cas e wa s th e startin g poin t o f th e theory : se e th e lette r o f Serr e t o Garibald i
appended a t th e en d o f thes e notes . Th e mai n result s wer e give n i n a serie s o f
lectures a t th e Colleg e d e France : se e th e resum e [S e 94] an d als o E . Bayer' s repor t
[Bay 94]. U p t o now , th e proof s ha d no t bee n published . Amon g th e result s are :
th e fac t tha t al l cohomologica l an d Wit t invariant s o f etal e algebra s ar e
derived fro m jus t on e invariant , th e trac e form ;
a n explici t descriptio n o f al l possibl e trac e form s o f ran k 7 ;
a proof tha t Noether' s proble m (o n the rationalit y o f the field o f invariants )
has a negativ e answe r ove r Q fo r group s suc h a s SL(2,F7) , 2.AQ, o r th e
quaternion grou p QIQ o f orde r 16.
This portio n o f th e boo k end s wit h letter s fro m Rost , Serre , an d Totaro .
A fe w complement s hav e bee n give n throughou t th e tex t i n th e for m o f "exer -
cises" . W e hop e th e reade r wil l find the m interestin g (o r challenging ) enough .
5
http://dx.doi.org/10.1090/ulect/028/02
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