APPENDIX C
A lette r fro m B . Totaro t o J-P . Serr e
11 December 200 2
Dear Serre ,
Here i s a geometri c descriptio n o f the grou p o f cohomologica l invariants . T o
me, i t seem s lik e a natural companio n t o the Compatibility Theore m 11. 1 in your
notes.
LEMMA.
Let G be a smooth algebraic group over a field ko, and let C be a
finite Y
ko
-module of order invertible in ko. Let V be a representation of G over
ko, and suppose that V has a G-invariant Zariski-open subset U
7
with V U of
codimension 2,a such that U is a G-torsor. Then the group Inv(G, Hl(C)) of co-
homological invariants is isomorphic to the subgroup of Hl(ko(U/G),C) consisting
of the elements which are unramified along each irreducible divisor in U/G.
PROOF.
You r notes show that Inv(G , H^C)) i s a subgroup o f Hl (ko(U / G), C)
(since th e G-torso r ove r th e functio n field ko(U/G) i s versal , 5.3 , together wit h
12.3), and that ever y element o f this subgrou p i s unramified alon g each irreducibl e
divisor i n U/G (b y the Compatibility Theore m 11.1).
It remain s t o sho w tha t a n element x o f H
l(ko(U/G),C)
whic h i s unramifie d
along eac h diviso r i n U/G belong s t o the subgroup Inv(G,iJ
2(G)).
So , given any
G-torsor E ove r a n extensio n field k o f /co , we need t o construc t a n elemen t o f
Hl{k,C)
associate d wit h x. First , "extendin g scalars" , x determine s a n element of
Hl(k(Uk/Gk),C), whic h I will als o cal l x. I t i s unramified alon g eac h irreducibl e
divisor i n U
k
/Gk. (T o check this , on e can use that th e morphism Uk/Gk * U/G
is flat, s o that ever y diviso r i n Uk/Gk eithe r dominate s U/G o r dominate s som e
divisor i n U/G.)
Consider th e diagram o f ^-varieties:
Spec k = E/G
k
^-{Ex U
k
)/Gk - U
k
/Gk.
We ca n pul l x bac k fro m th e cohomolog y o f th e functio n field o f U
k
/Gk t o th e
cohomology of the function field of(ExUk)/Gk. Th e resulting element is unramified
along eac h irreducibl e diviso r i n (E x U
k
)/Gk. (Again , thi s follow s fro m th e fac t
that th e morphism (E x U k)/Gk— U k/Gk i s flat.)
Since V U has codimension 2 in V, we have an element of the cohomology of
the function field of (E x Vk)/Gk whic h is unramified alon g each irreducible divisor ;
the divisor s her e ar e the sam e a s those i n (E x U k)/Gk. No w (E x V k)/Gk i s a
vector bundl e ove r E/G
k
Specfc . An d your note s show (10.1) tha t a n element of
the cohomolog y o f the function field o f a vector spac e ove r k whic h i s unramifie d
a
This hypothesi s i s harmless ; i f a give n representatio n V doe s no t satisf y it , the n on e may
replace V wit h V x V.
99
http://dx.doi.org/10.1090/ulect/028/14
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