100
C. A LETTE R FRO M B . TOTAR O T O J- R SERR E
along eac h diviso r belong s t o th e cohomolog y o f k itself . Thu s w e hav e produce d
an elemen t o f H l(k, C) associate d wit h a n elemen t x a s abov e an d a G-torso r ove r
k.
Thus, ever y unramifie d elemen t x a s abov e give s an elemen t o f Inv(G, H l(C));
the necessar y functorialit y propert y a s k varie s i s automati c fro m th e abov e con -
struction. T o complet e th e proof , w e wil l sho w tha t applyin g thi s cohomologica l
invariant t o th e natura l G-torso r ove r th e field ko(U/G) give s th e elemen t x i n
Hl(ko(U/G),
C). B y going through th e construction , w e see that i t suffice s t o show
that th e tw o projectio n map s fro m (17 x U)/G t o U/G induc e th e sam e pullbac k
homomorphism fro m unramifie d cohomolog y classe s o n U/G (meanin g element s
of H l(ko(U/G),C) whic h ar e unramifie d o n eac h diviso r i n U/G) t o unramifie d
cohomology classe s o n (U x U)/G.
To prove this , w e can argu e tha t th e tw o projectio n map s fro m (U x U)/G t o
U/G ar e homotopi c i n a suitable sense . T o be precise , le t
W = {(t,x,y) e Aff 1 x(U x U)/G :tx + (l- t)y e U}.
Then W i s an open subset o f Aff x {U x U)/G whos e complement ha s codimensio n
at leas t 2 . W e have a flat morphis m W U/G define d b y
{t,x,y) i-t a + ( l -t)y.
So w e hav e a pullbac k homomorphis m fro m th e unramifie d cohomolog y o f U/G
to tha t o f W , o r equivalentl y o f Aff
1
x(U x U)/G. B y 10.1i n you r notes , ever y
unramified cohomolog y class on
Aff1
x (U x U)/G i s "constant" , tha t is , pulled bac k
from a clas s o n (U x U)/G. I n particular , th e pullbac k t o W o f an y unramifie d
cohomology clas s on U/G ha s the same restriction t o t = 0 as to t 1. Thi s mean s
that th e pullback homomorphism s o n unramified cohomolog y associate d t o the tw o
projection map s fro m (U x U)/G t o U/G ar e equal .
B. Totar o
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