9. GALOI S COHOMOLOG Y O F k(t)
25
and le t D be th e grou p o f divisor s o f Pi ove r k, i.e. , th e fre e Z-modul e wit h basi s
V. Let Do be the subgroup o f D mad e up of divisors of degree 0. W e have an exac t
sequence
(9.10) 0 - D 0/nD0 - + D/nD - ^ Z/nZ - » 0
If / i s an element of fc(t)*, the divisor div(/) o f / belong s to Do; the map / i- » div(/ )
induces a n isomorphis m
(9.11) k(ty/k(ty n ^ Do/nDo.
By Kumme r theory , w e also have a natural isomorphis m
(9.12) Hom(iV ,
M
n) = k{t)*/k{ty
n.
By combinin g (9.11 ) an d (9.12), we obtain :
(9.13) Hom(7V , /zn) = D
0
/nD0.
Now le t t ; be a n elemen t o f V, and le t w be a n extensio n o f v to ifs. Th e
restriction o f w t o k{t) define s a n elemen t v of V. Th e ma p z
w
: Iw—•TV of (9.6 )
gives
(9.14) C : Hom(iV, /xn) - + Hom(/
w
/in) = Z/nZ.
Its interpretatio n i n term s o f (9.13) i s the obviou s one : t o a n elemen t o f Do/nDo,
it associate s it s ^-coefficient i n Z/nZ . (Thi s i s easily checke d usin g Remar k 7.2. )
9.15.
EN D O F TH E PROO F FO R
k
PERFECT .
Not e tha t (9.13) implie s tha t
Hom(iV, fjLn) is a free Z/nZ-module . I t follows se e als o Exerc . 9.2 2 tha t
Hom(iV, C) Hom(7V, /xn) 0 C(—1), and tha t w e have a n exac t sequenc e
(9.16) 0 - Hom(iV, C) - D/nD ® C ( - l) - C ( - l ) - 0.
This exac t sequenc e i s compatible wit h th e actio n o f IV Th e grou p Tk act s o n V
with finite orbits , eac h orbit consistin g o f the element s lyin g abov e a given elemen t
v o f V. Th e subgrou p o f T ^ fixing a n elemen t v of V with imag e v in V is T ^ ) ,
and thi s describe s L as
D = 0
v G
yindr"
( t ; )
Z.
By th e Shapiro-Faddee v Lemma , H l~1 (k,D 0 C(—1)) ma y b e identifie d wit h
®
v
fP - 1 (&(?;) , C(—1)), and unde r thi s identificatio n th e ma p ont o H %~l{k1C(— 1))
induced b y D 0 C(—l)— C(—1) i s given b y th e corestriction , whic h prove s (9.7) .
It remain s t o prov e (9.8) . I f v, v, and w ar e a s above , w e ma y appl y Th . 6. 1
and Remar k 6. 4 t o th e diagra m (9.6) . W e thu s obtai n th e commutativit y o f th e
diagram
Hl{K,C) F-H^Hom^C) )
i i
i P p e c ^ C ) y if
i-1(/c(v),Hom(7w,C)),
which w e may als o rewrite as :
Hl(K, C) —?— ke r ( © H^ikiy), C(-l) ) - H l-\k, C(-l)) )
(9-i7) | y
w
HHVecw,C) ^
J
ffi-1.(fc(^),C(-l))
Previous Page Next Page