10. GALOI S COHOMOLOG Y O F k(t
1:
...,tn) 27
(2) Le t C be a T^-module wit h nC 0. Prov e tha t th e natural map
Hom(r
K
, / x J 8 C ( - l) - Hom(rx , C)
is an isomorphism.
[Hint: Fo r (1), use the following criterio n fo r a Z/nZ-modul e E t o be projective (or
injective): fo r every m dividin g n and every x £ E wit h m x = 0 , there exist s y 6 E wit h
(n/m)-y x. On e applies thi s wit h E 1 = K*/K* n. Not e tha t "projective " i s equivalent
to "free " whe n n is a power o f a prime. A s for (2), it follows fro m (1).]
EXERCISE 9.2 3 (valu e o f a cohomolog y clas s a t a rationa l point) . Fi x some a £
Hr(k(t),C) an d x G Pi(&), x / oo . Assum e tha t a i s unramified a t infinit y an d at x.
Prove tha t
a(x) - a(oo ) = ] P Cor^ (u) [(x - t
v
) -r
v
(a)] .
vev
(In the expressio n (x tv), x is viewed as an element o f k, and t
v
denote s the image of
t G k[t] in fc(v). Then (x tv) i s the correspondin g clas s in i71 (/c(v), /xn) an d (x tv)-r,v(a)
is its cup produc t wit h r
v
(a) G Ht~1(k(v)1 C(— 1)).)
9.24. GENERALIZATIO N T O ALGEBRAIC CURVES . Par t o f wha t ha s bee n don e
above fo r th e projective lin e ca n be extende d t o a n arbitrar y connecte d smoot h
projective curv e X ove r k. W e limit ourselve s t o stating th e results. Le t V b e the
set o f closed point s o f X. W e writ e K fo r the function fiel d o f X ove r k. A n element
v G V ca n be identifie d wit h a discret e valuatio n o f K whic h i s trivia l o n k; we
write k(v) fo r the corresponding residu e field , whic h i s a finit e extensio n o f k.
It i s still tru e tha t ever y elemen t i n Hl(K, C) has only a finite numbe r o f poles,
hence th e sequence occurrin g i n 9.2 is still defined . Th e residue formul a als o holds ;
it ca n be proved b y writing th e curve a s a (ramified ) coverin g o f the projectiv e lin e
and usin g a corestrictio n argument .
Let u s assume tha t X ha s a rational /c-poin t (or , more generally , a closed poin t
whose degre e i s prim e t o n , wher e nC = 0) . I n tha t case , th e map Hl(k,C)—
Hl(K, C) i s injective, an d one has an exact sequenc e
(9.25) H l-X(k, J
n
0 C ( - l ) ) - H\K, C)/H l(k, C) A
^ QlevH'-^Hv), C ( - l ) ) - + H\k, J
n
0 C ( - l ) ) - ,
where:
J
n
i s the group o f ?i-divisio n points b o f the Jacobia n o f X; i t i s a fre e
Z/nZ-module o f rank 2# , where g is the genus o f X;
© ^ y i T - 1 ^ ) , C ( - l ) ) i s the subgroup o f e ^ v f P -1 ^ ) , C ( - l ) ) mad e
up o f the elements whos e su m of corestrictions i n H l~l(k, C( 1)) i s 0;
th e map r i s given b y the residues relativ e t o the elements v of V.
Note tha t th e kernel o f r i s usually ^ 0 . If , fo r instance, k i s algebraically closed ,
C = /x
n
, an d i = 1, this kerne l i s isomorphic t o J
n
.
10. Galoi s cohomolog y o f k(ti, ..., t
n
)
As before , C i s a finit e T^-modul e wit h orde r prim e t o the characteristic .
There i s a canonical isomorphis m A
2
Jn = [in. Henc e th e Tfe-module J
n
(g ) C(—1) ma y also
be writte n a s J^(8 ) C o r as Hom(Jn , C).
Previous Page Next Page