14

JURGEN RITTER AND ALFRED WEISS

L E M MA 2.5. Let K be totally real. Then

1. \[s C(K*)

and

%iS have the same Zi-rank.

2. If K satisfies Leopoldt's conjecture, then G(M/K) has Zi-rank 1.

With respect to the first claim look at the exact sequence C(UP) —• £(Kp) -» Zj, which arises

from /-completing the valuation sequence Up — Kp -» Z. Because of Up — Up -» K

p

, with

* , denoting the residue field of K„ we have C(UP) = ( f f ^ * . { ' , and therefore,

rk{C(K?))-^ ^rVTTl= _

1 if p f /

l + ikU^ l + [Kp:Qi\ ifp|/.

AS ~

I

ZS -

i

z -

£

n*1

P

x

1

G(M/K)

Summing up, and denoting the set of infinite primes of K by SQOJ we get

rktf l W ) ) = r k ( J ] W ) ) = | 5 \ 5 o o | + B ^ P

:

«M = 1^1 " l^ool + [* : Q] = |5 |

since C(Kp) is finite for infinite p, and since |5oo| = [K : Q] by assumption.

The second claim of the lemma now follows from the left column in (D2.3) and Dirichlet's

unit theorem. Indeed, rk(Z/g)£) + ik(G(M/K)) = \S\ and rk(Zj g £ ) = | 5 | - 1 . This finishes

the proof of the lemma.

We now suppose that we are given a G-equivariant homomorphism Z 5 -» TlpeS Kp which

maps AS injectively into E. This fits into a commutative diagram

(D2.5)

in which E — Yis Kp

ls

the canonical map and \[s

Kpx

—* G(M/K) arises from Yls Kp —

]\SC(KX) -+ G{M/K). Recall that E is sent to zero in G(M/K) by the reciprocity law,

since every Up, p 0 5, is in the kernel.

We will also require that Z —• G(M/K) is injective.

The /-completed diagram of (D2.5) is

Z; ® AS - Zj g E

I i

(D2.5«) z 5 ~ nA^i

PES'

i I

Zi ~ G(M/K)

with right column exact by Lemma 2.3. We stick to this notation.

Notice that the above maps are G-monomorphisms from the right ends of the exact 4-term

sequences in (D2.4) into their respective left ends. Therefore we are in a position to perform