6

JURGEN RITTER AND ALFRED WEISS

Our first goal is to move the sequence

G(M/K) ~ Y -» %i ® A G

up the cyclotomic Iwasawa tower at the prime /

K = K0C ...CKnC ...cKoo

in order to arrive at a governing sequence

(4.oo) GiM^/K^) ^Yoo^ AGoc .

We use the following notation.

For 0 n oo, Mn is the maximal abelian 5-ramified

4

/-extension of Kn\

Gn = G(Kn/k);

ZIGQQ is the completed group ring of G ^ over Z/

5

and AGoo its augmentation ideal.

The analogue of (4.oo) at level n oo is

(4.n) G(Mn/Kn) ^Yn-oli® AGn .

However, the set of primes of Kn above 5 , for n 0, need not be large anymore. Nevertheless,

we prove that Yn has finite projective dimension over Z{Gn (Lemma 4.1).

By definition, (4.oo) is the limit of the sequences (4.n). We show in Proposition 4.2 that

(4.oo) is an exact sequence and, moreover, that Y^ is a finitely generated ZiGoo-module with

projective dimension 1 °.

Our second goal is to construct a map \£ : Z/GQO — ^oo which, when passing to the zero

level, becomes a map "of type I". Recall from section 1.4 that $ : Z/G —» Y may be obtained

from $ : Zz - G(M/K) and the endomorphisms a = f3 = \G\ of Zj 0 AG. In particular, 4

and $ are related by the commutative diagram

Zj - Z,G -» Z , G / ( G )

M i i |G|(|G|-6)i , G ^ E ^ G P -

G(M/K) - y -» Zt®AG

As a matter of fact, $ can be recovered from the right half of the diagram.

To describe ^ we need some more notation.

rk =

G(fcoo/fc)6,

rK =

G(KX/K)

7& is a generator of T^ and % is a lift of 7^ in Goo

H = GiKoo/koo), e = j^ j S/ieH ^

R= {r G Z/Tx : r has regular image in the group ring of every finite quotient of TK}.

4

more precisely, Sn -ramified, where Sn is the set of primes of Kn above S

Zi[[Coo]] in the standard notation which soon becomes tedious.

0

We have recently learned from T. Nguyen Quang Do that this result, even in a more general form, is

already contained in his paper [Springer LNM 1068 (1983), 167-185].

6koo/k

is of course the cyclotomic /-extension of k