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]) .
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