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