DUALITY THEOREMS IN GALOIS
COHOMOLOGY OVER NUMBER FIELDS
By JOHN TATE
1.
Notation and terminology
Let
X
be a Dedekind ring with
field
of fractions
k
and let
G
be a commu
tative group scheme over
X.
Except in the special case
X=T&
or C (real or
complex field) we put, for all r€Z,
H'(X,C)=iimH
r
(G
Klk
!C
Y
),
~K
the direct limit taken over all finite Galois extensions
K
of
k
in which the
integral closure
Y
of
X
is unramified over
X,
where
G
KJk
denotes the Galois
group of such an extension, and where
G
Y
denotes the group of points of
G
with coordinates in
Y.
For example, if
X=k,
our notation coincides with
that of [10]. For any
X,
the group
H
r
(X,C)
is the rth cohomology group of
the profinite group
G
t
jr
=lim
G
Klk
(fundamental group of Spec
X)
with
coefficients in the
ö
x
module
lim
C
Y
of points of
C
with coordinates in the
maximal unramified extension of
X;
a general discussion of the cohomology
theory of profinite groups can be found in [5]. In the special case
X
™
R orC
we put
H
r
(B,C)=Ê
r
(G
m
,C)
and
H
r
(G,C)=E
r
(G
mg
C)~0,
where
Ê
denotes the complete cohomology sequence of the finite group
G,
in general non trivial in negative dimensions ([2], Ch. 12).
In our applications,
X
will be a ring associated with an algebraic number
field,
or with an algebraic function
field
in one variable over a finite constant
field, and the group scheme
G
will be one of two special types, which we will
denote by
M
and
A,
respectively. By
M
we shall always understand (the
group scheme of relative dimension zero over
X
associated with) a finite
6r
z
module
whose order,

M
\ =
card
M,
is prime to the characteristics of
the residue class fields of
X.
By
A
we shall denote an abelian scheme over
X
(i.e.,
an abelian variety defined over
k
having "nondegenerate reduction"
at every prime of
X).
Underlying our whole theory is the cohomology of the multiplicative
group,
€r
m
,
as determined by class field theory. For any
M,
we put
M'
=
Horn
(M,G
m
).
By our assumption that
\M

is invertible in
X,
we see that
M'
is a group scheme of the same type as
M,
namely the one associated with
the
6rjrmodule
M'
=Hom(Jf
,/Lt)
where
/u,
denotes the group of roots of unity.
Moreover we have
\M\
=\M'\,
and
M^(M')'.
For any
A
we put
A' =
T£xt(A,(x
m
),
the dual abelian variety; then
A^(A')
f
by
"biduality".
Our
aim is to discuss dualities between the cohomology of
G
and
C
in both
cases,
C=M
and
G=A.
Notice that
Ext(Jf,6
m
)
=0 and
Hom(^4,6
m
)
=0, so
that in each case,
0'
denotes the only
nonvanishing
group in the sequence
Ext
r
((7,G
m
).
Thus our results are presumably special cases of a vastly more
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DUALITY THEOREMS IN GALOIS COHOMOLOGY
2 8 9
general hyperduality theorem
for
commutative algebraic groups envisaged
by Grothendieck, involving all the
Ext
r
(C,G
m
)
simultaneously.
Finally, for any locally compact abelian group
H,
we let
Ë*
denote its
Pontrjagin character group.
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 Winter '11
 PhanThuongCang
 Class field theory, John Tate

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