28 III . COHOMOLOGICA L PRELIMINARIES : TH E FUNCTIO N FIEL D CAS E

THEOREM 10.1 . Let n 1 and K = k(ti,. .. , £n) where the ti are independent

indeterminates. Then

(1) The natural map

Hl(k,C)—

» H

l{K,C)

is injective.

(2) Le t a 6 e a n element of H

l(K,C),

whose residues are 0 at all discrete

valuations of K/k which correspond to irreducible hypersurfaces of the

n-dimensional affine space Aff

n.

Then a is constant, i.e., belongs to

Hl(k,C).

PROOF .

Bot h part s ar e proved b y induction o n n, an d part (1) is immediat e

from th e case n = 1, cf. Theore m 9.2.

Set K' = fc(ti,..., t

n

_i), th e function field o f Affn-1, an d consider th e projec-

tion ma p 7r:

Affn—

» Aff

n_

give n b y forgettin g th e las t coordinate . Th e generi c

fiber i s a n affin e lin e Af f ,K, ove r K' whos e coordinat e rin g i s K'[t n\] it s functio n

field is K = K'(t n).

The irreducibl e divisor s o n

Affn

ar e of two kinds:

(a) Th e vertical ones , i.e. , thos e whic h ar e the invers e imag e unde r TT o f an

irreducible diviso r o f Affn_ ; they correspon d t o irreducibl e polynomial s

P(t\,..., t n) whic h ar e independent o f t n.

(b) Th e non-vertical ones , which correspon d t o the closed point s o f Aff , K,.

Since th e residue s o f a ar e zer o fo r al l place s o f typ e (b) , 9. 3 show s tha t a

belongs t o H^K'.C).

Let u s now show that a is constant (whe n viewed a s an element o f

H%(K'',

C)).

Let D' be an irreducible diviso r of

Affn_1

wit h function field F'. Th e inverse image

of D' i n

Affn

i s a divisor D whose functio n field i s F = F

f(tn).

B y 8.2.1 applied t o

K' an d K (instea d o f K an d

Kf)

wit h e = 1, the diagra m

Hl(K',C)

- ^ U W-^F^Ci-l))

i i

W(K,C) —^^^ f P - ^ C C - l ) )

commutes, where the horizontal arrows are the residue maps and the vertical arrow s

are injection s (9.3) . Sinc e a G Hl(K,C) ha s residue zer o a t D , it als o ha s residue

zero at D' when viewed as an element of

Hl(K',

C). Thi s shows that al l the residues

of a are 0; by the induction assumptio n applie d to K' — k(t\^... , t

n

_i), thi s implies

that a i s constant. •

REMARK

10.2. Th e conditions of 10.1.2 refer only to the behavior of a along the

divisors of

Affn.

Hence , if a "behave s well" in codimension 1, it does so everywhere.

One should compare this "purity " statemen t wit h the analogous (bu t deeper) resul t

of Bloch , Ogus , an d Gabber fo r etale cohomolog y (se e e.g. [CHK97]) .