64 VIII. WIT T INVARIANT S
EXERCISE 27.2 . (1) Sho w tha t th e two propertie s
\t(q + q') = \t(q)MQ)
Xt(q) = 1 + tq i f q = (a) with a G k*,
characterize X
t
(q).
(2) Le t n G Z be the image o f g under th e rank ma p VFGr(/c)— Z. Us e (1) to
prove tha t
d(\t(g))/dt
=
q-nt 2
Xt(q) ~~ 1 - t 2 '
Deduce fro m thi s formul a tha t th e power s q % of q can be writte n a s linea r
combinations (wit h integra l coefficient s dependin g o n n) of the X z~2d(q), e.g. :
q2 = 2X 2(q)+n
q3 = 6X3(q) + (3n-2)q
(3) I f u, v are indeterminates, sho w tha t
Xu(q) -X
v
(q) = (1 + uv)n -X
(u+v)/{1+uv)
(q)
in the rin g WGr(k)[[u, v]]. Us e this identit y to express the products X l(q) -X j(q)
as Z-linea r combination s o f the X t+i~2a(q), 0 a sup(z,j).
27.3. W I T T INVARIANTS . A morphis m o f functor s Torsors o W i s calle d a
Witt invariant (o f G-torsors). Almos t al l of what w e have don e fo r cohomolog y
and cohomologica l invariant s work s fo r W an d Witt invariants . Fo r example:
27.4. LOCA L CASE . Le t K b e complete wit h respec t t o a discret e valuatio n v
with residu e field k (o f char ^ 2) , and let n b e a uniformizin g elemen t o f K.
There i s a natura l embeddin g W(k) W(K), whic h map s ( a i , . . . , a
n
) (o r
rather it s class), wit h ai G &*, to (ui,.. ., ^
n
) , wher e th e Ui are units o f X liftin g
the ai. I f on e identifies th e ring W(k) wit h it s imag e i n W(K), the n on e has a
direct su m decomposition:
T H E O R E M 27.5 . (Springer , se e e.g. [Lam 73, VI.1.5]) The ring W{K) is a free
W(k)-module of rank 2, with basis {1, (n)}. In particular, one has
W(K) = W(k) ©
(TT)
W(k).
If q = q\ + (n) -q^ is an element o f W(K) wit h #i , #2 £ W^(^) ? the n g i is calle d
the "firs t residue " o f g and q^ the "second residue " o f q. I n what follows , th e second
residue wil l b e called simpl y th e residue; i t wil l b e denoted b y d
n
(q). W e have a n
exact sequenc e
0 - W(k) - W ( K) - ^ ^(jfc ) - 0.
(Although th e residu e ma p 9^ depend s o n th e choic e o f TT, thi s choic e ha s no
influence o n the applications w e give later. a)
An elemen t o f W(K) i s called unramified i f its (second ) residu e i s 0, i.e., if it
belongs t o the subring W(k) o f W(K).
27.6. T H E N O N - C O M P L E TE CASE . Th e definition s o f 27. 4 ar e extende d t o th e
non-complete cas e a s follows :
a
Note th e similarity an d the differences wit h th e cohomological residu e o f §7 . I n the coho -
mology situation , w e got an intrinsi c residue , bu t at th e cost o f a Tat e twistin g o n the module .
Here, w e have a residu e dependin g o n the choice of n; it coul d als o b e interpreted a s a canonica l
residue wit h value s i n a suitable twis t o f W(k).
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