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
/-extension of Kn\
Gn = G(Kn/k);
ZIGQQ is the completed group ring of G ^ over Z/
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 =
rK =
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}.
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.
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].
is of course the cyclotomic /-extension of k
Previous Page Next Page