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, A» = 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.