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.