12. SPECIALIZATIO N PROPERTIE S

31

THEOREM. For a e Invfc

0

(G, C) and a = CL(TK), we have:

(1) The residue of a at v is 0.

(2) The value of a at v (in the sense of 7.13) is a(Tk).

PROOF.

W e may assume tha t R i s complete, i n which cas e thi s follow s fro m

Th. 11.1•.

APPLICATION

11.8 . Suppos e we have a G-torsor T ove r X wher e X i s a normal

irreducible k- variety (an d k i s an extension o f ko). Le t K b e the function fiel d of

X, an d write TK for the torsor ove r K obtaine d fro m T. Th e following i s a direc t

consequence o f 11.7: Fo r every invarian t a ,

CL(TK)

i s "unramifie d a t al l irreducible

divisors o f X" , i.e. , i t i s unramifie d (7.13) a t th e valuatio n define d b y suc h a n

irreducible divisor .

12. Specializatio n propertie s

12.1. Her e and in the following theorem , le t R b e a noetherian domai n whic h

contains ko. Le t K b e the quotient fiel d o f R, whic h i s an extension o f ko. Le t m

be a maximal idea l of R with residu e fiel d k. A n i?-G-torsor TR defines torsor s TK

over K an d Tk ove r k.

SPECIALIZATION THEORE M

12.2. Suppose that R

m

is a regular local ring. Let

TR be an R-G-torsor, and denote by Tk and TK the k-G- and K-G-torsors defined

by

TR.

Let a be an invariant in Inv(G, C). If

CL(TK)

= 0 , then a(T

k

) = 0.

PROOF.

W e may replace R wit h R

m

an d so assume tha t R i s local. W e us e

induction o n the dimension o f R.

If R ha s dimension 1, then i t is a discrete valuatio n ring . B y Theorem 11.7.2,

a(Tk) i s the value of a(T^), whic h i s 0.

If n — dimi? i s 1, m is generated b y n elements ti , ..., tn. Le t Rf — R/t\R.

This rin g ha s dimension n — 1 and is also regular ; le t K

r

b e its quotient field . B y

the dimensio n 1 case, we have

CL(TK')

= 0 . By the induction hypothesi s (applie d to

R' an d K'), w e get a(Tk) = 0 . •

1 2 . 3 . COHOMOLOGICA L INVARIANT S AN D VERSAL TORSORS . Le t P € ^(K, G)

be a versal elemen t i n the sense o f 5.1. Suc h a n element P detect s th e equality of

cohomological invariants , i.e.:

THEOREM.

Let a,b be two invariants in Inv/ ^ (G,C). If a(P) = b(P) in

H(K,C), then a = b.

PROOF.

B y replacing a and b with a — b and 0 respectively, w e may assume

that 6 = 0.

We have to show that a(T) = 0 for every k/ko an d every k-G-tovsoi T. Suppos e

first tha t k i s infinite . B y the versa l property , T i s isomorphi c t o Q

x

fo r som e

x G X(k) wit h th e notatio n o f 5.1. Le t R b e th e loca l rin g o f Xj

k

a t x. It s

residue fiel d i s k. Le t K

x

b e its field o f fractions, i.e. , the function fiel d o f X/

k

.

Since X i s smooth, th e ring R i s regular; le t QR be the pullback o f Q by the map

Spec(i?)— X. Th e fiber o f QR at x (resp . at the generic poin t rj) is T (resp . Q^).

We hav e a(Q v) — 0 because Q

v

i s obtained fro m P b y the base chang e K — Kx.

Hence, by 12.2, we have a(T) = 0.