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.
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