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 , 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.
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