27. PROPERTIE S O F TH E WIT T RIN G

65

Let K b e a field wit h a discret e valuatio n v, rin g o f integer s R, uniformizin g

element 7r , and residu e field k(v) =

R/TTR;

le t K

v

b e it s completion . I f q G W{K),

its residu e d 7V(q) G W(k(v)) i s the imag e o f q under th e ma p

W(K) - W(K V) ^ W(k(v)).

If i t i s 0 , q is said t o b e unramified at v; it s imag e i n W(K V) belong s t o W(k(v))

and i s called th e value of q at v.

27.7.

W I T T RIN G OF

K = k(t). Wit h th e sam e notatio n a s i n Sectio n 9 , w e

have a n exac t sequenc e (du e t o Tat e an d Milnor , cf . [M i 70, Th . 2.3] )

0 - W(k) - W{K) ^ ®

v

^ooW{k{v)) -+ 0

where d i s given b y th e residue s (th e choic e o f the uniformizin g element s doe s no t

matter).

In particular , a n elemen t o f W{K) i s constant (i.e. , belong s t o W(k)) i f an d

only i f its residue s a t al l v ^ o o are 0 .

Moreover, a n elemen t q o f W(K) whic h i s unramified outside 0 and o o (i.e. ,

with zer o residues fo r v ^ 0 , oo) ca n b e written uniquel y a s q = qo + qi(t), wit h qo,

Qi £ W(k); o f course , q± is the residu e o f q at 0 with respec t t o th e choic e o f t a s

uniformizing element .

27.8. K = Ar(£i,.. . ,tn ). B y inductio n o n n an d startin g wit h 27.7 , on e see s

that i f q G W(K) ha s zero residues for al l the discret e valuations o f K comin g fro m

irreducible hypersurface s o f Aff n,k, the n q is constant

27.9.

COMPATIBILIT Y AN D SPECIALIZATIO N THEOREMS .

Al l th e statement s

of Chapte r I V exten d wit h essentiall y th e sam e proofs . Mor e precisely , fo r 27.10

through 27.13, le t G b e a smoot h algebrai c grou p ove r ko, an d le t a be a n elemen t

of Inv(G, W) = Inv(Torsors G, W). Then :

THEOREM

27.10. If K is local as in §11, the diagram

H^k^G) H^K.G)

ai

[

a

W(k) W{K)

is commutative.

PROOF.

Sam e a s fo r Theore m 11.1•.

THEOREM

27.11 . Let (K,v) be as in 27. 6 above (K not necessarily complete),

with ko C R, and let T be an R-G-torsor. Write TK and T k(v) for the K-G-torsor

and the k(v)-G-torsor deduced from T by the base changes R — K and R — • k(v).

Put a = a(T K) in W(K). Then:

(1) a is unramified at v.

(2) The value of a at v is a(Tk).

This follow s fro m 27.10 by replacin g K wit h it s completio n K

v

a t v.

THEOREM

27.12. Let R

m?

K, k, and T be as in 12.1and 12.2. Then a{T K) = 0

implies a(Tk) = 0 .

PROOF.

Sam e a s for Th . 12.2: b y induction o n dimi?

m

, startin g fro m th e cas e

dim#

m

= 1, which i s 27.11. •