26 III . COHOMOLOGICAL PRELIMINARIES : TH E FUNCTION FIEL D CAS E
By wha t w e have see n abov e (se e (9.14)), j
w
is the i-t h componen t map . Thi s
proves (9.8 ) and completes th e proof o f Th. 9. 2 when k i s perfect .
9.18. G E N E R A L CASE . Le t k' = k
p
° ° b e th e perfec t closur e o f k. W e hav e
seen tha t Theore m 9. 2 holds fo r kf; w e are going t o deduce i t for k.
Note first tha t th e place s V ar e the sam e fo r k(t) an d k'{t): th e canonica l
map Pi/fc / Pi/fc i s a homeomorphism . I f we denote b y v' th e valuation o f k'(t)
extending th e valuation v of h(t), then :
(1) th e ramification inde x e(v f jv) i s equal t o qv = [k(v) : k\i\
(2) th e residue field k'{v') o f v' i s k'k(v); it s degree ove r k' i s [k(v) : k]/q
v
.
Consider no w th e diagra m
H{k,C) H{k{t),C) e
v
iJ(fc(v),C(-l) ) ff(fe,C(-l))
(9.19) a | . J
7
| , J
#(*:', C) * H{k'{t),C) ®
v
H{kf(v), C ( - l ) ) —- ff(fc',C(-l)),
where a , /? , (5 are the natural isomorphism s associate d wit h th e purely inseparabl e
extensions &///c , k'(t)/k(t), an d fc'/fc, and 7 = 0
V
7
V
wher e 7 ^ is the natural isomor -
phism H(k(v),C(—l)) » if(A/(i;),C( —1)) multiplie d b y gv. Al l the vertical map s
are isomorphism s (becaus e g
v
is prime t o the order o f C) .
To prov e Th . 9.2 for /c, it is sufficient t o prove tha t Diagra m (9.19) commutes ,
since Th . 9.2 holds fo r k'. Th e commutativity o f the a(3 square i s clear. Tha t o f
the /? 7 square follow s fro m th e commutativity o f 8.4. A s for commutativity o f the
jS square , i t follow s from :
L E M M A 9.20 . Letki be a finite extension ofk; let q = [k\ : k]i and let k[ = k'k\.
If A is any T^-module, the diagram
H(kuA) - ^ - H(k,A)
H(k[,A) - ^ - H(k',A)
is commutative, where cp is the natural isomorphism H(k,A) » H(k\A) and p\ is
the natural isomorphism H{k\,A) H(k[^A) multiplied by q.
P R O O F . Thi s i s easily checke d whe n k\jk i s separable (i.e. , whe n q 1) or is
purely inseparabl e (i.e. , whe n k[ k'). Th e general cas e follow s b y devissage. D
This complete s th e proof o f Theorem 9.2.
E X A M P L E 9.2 1 (th e Brauer group , se e [Fa51]). Le t C = / i
n
for n no t divisibl e
by th e characteristic, s o that C ( - l ) = Z / n Z . W e have H 2(K,C) = Br
n
(if), a th e
kernel o f multiplication b y n o n the Brauer grou p Br(K). Th e exact sequenc e i n
9.2 ca n be written :
0 - Br
n
(fc) - Brn{K) - » e
veV
H1(k(v), Z/nZ) - H l(k, Z/nZ) - 0.
EXERCISE 9.22 . Le t K b e a field containing a primitive n-t h roo t o f unity.
(1) Sho w tha t Hom(rx,M
n
) i s a projective Z/nZ-modul e (an d eve n a free modul e
if n is a power o f a prime).
a
There ar e two natural identification s o f H
2{K,
fi n) wit h Br n(K), differin g b y a sign , se e
[KMRT98, pp . 396, 397]. Whic h on e is chosen her e make s n o difference t o the statement.
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