32 IV . SPECIALIZATION PROPERTIE S O F COHOMOLOGICA L INVARIANT S
Now suppose that k is finite. Th e torso r T define s a n element T
t
G if1 (/c(t), G)
by th e bas e extensio n k fc(t), and by the cas e wher e k i s infinite , w e have
a(Tt) 0. So a(T) i s in the kernel of the ma p H(k, G)— H(k(t), G) , which is zero
by 9.2 . Henc e a(T ) = 0 .
COROLLARY
12.4. Assume ko is algebraically closed. Then Inv * (G,G) 0 for
every i ed(G).
PROOF .
B y the definition o f ed(G), ther e exist s a versal G-torso r P define d
over a field K wit h transcendenc e degre e ed(G ) ove r ko. B y a well-known resul t of
Grothendieck-Tate (se e e.g. [Se65 , §11.4.2]) one has
Hl{K,
C) = 0, hence W ( G, C)
is 0 by Th. 12.3.
Hence an y non-zero invarian t give s a lowe r boun d fo r ed(G) . Fo r instance,
assume th e characteristic i s ^ 2 , and take G = O(n) ; the non-vanishing o f the
Stiefel-Whitney invarian t w
n
(se e §17) implies tha t ed(G ) n; since i t is easy to
construct a versal torso r wit h ti.deg(K) = n , thi s show s tha t ed(0(n) ) = n. A
similar argumen t give s the values o f ed(SO(n)) an d ed(G2) mentione d i n 5.7, se e
[Re 00]. I n the case of the symmetric grou p S n, thi s metho d give s only the bound
ed(Sn) [Vi/2], which is probably no t sharp fo r large n's.
REMARK
12.5. Th . 12.3 show s tha t th e map Inv(G, G) - H(K,C) give n by
a i— a(P) i s an injection. W e may then vie w Inv(G , G) a s a subgroup of H(K, G) .
By Th . 11.7, the elements o f that subgrou p ar e "unramified ove r X" i n the sense
that the y have residue 0 for all the discrete valuations of K defined by the irreducible
divisors o f X. Ther e ar e important case s wher e th e converse i s true, i.e. , where
Inv(G, G) i s equal to the subgroup o f H(K, C) made u p of the element s whic h are
unramified ove r X, cf. B. Totaro's lette r reproduce d i n Appendix C.
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