2. THE TRIPOD
Theorem A is going to be proved by /-completing and 5-truncating the construction of a Tate
sequence. As a reference for this construction we refer to [We, Chapter 5] (or [RWl]).
In this chapter the set S is larger in the sense that it is large and moreover satisfies G
UpeS^p'
w
^ h Gp denoting the decomposition group of p
10.
We also fix, once and for all, a
set 5* of G-orbit representatives of S.
2.1 Tate sequences
For any prime p of K let Kpi respectively &p, denote the completion of K, respectively of k1
at p. Moreover, when p is finite, let Up be the group of units in Kp. The local fundamental
class of K/k at p gives rise to a short exact sequence K* - Vp AGP (see [GW1, p.279]),
in which Vp has finite projective dimension over ZG
p
. Analogously, the global fundamental
class of K/k gives CK —» %$ AG, where CK is the idele class group of K and where 93 is
a cohomologically trivial ZG-module. Denoting the group of 5-ideles of K by J$ and setting
Vi = 0 indgpyp ®l[Up, = 0 indgp AGP
pes* p£S pes*
we get a commutative diagram
E --
I
A
I
is - K
1
C K
--
i
2J
L
I
A ,
4
AG
with exact rows and columns: compare the first diagram on p.24 in [We], to which we have
added the kernel sequence. Note that A* AG is surjective, because S is larger.
2.2 l-completion
Let C denote the /-completion functor, i.e.,
C(X) = i™X/lnX for ZG-modules X .
We wish to apply C to diagram (D2.1). For that we use
LEMMA 2.1. 1. C(X) = Z/ 0 X if X is finitely generated over Z.
2. C is right exact; in fact, exact on finitely generated modules.
3. If X' X X" is an exact sequence of ZG'-modules in which X" has no non-
trivial l-torsion, then C{X') - C(X) C(X") is exact.
10There is no real need to have S larger. A large S would suffice but require more references.
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