8 JURGEN RITTER AND ALFRED WEISS
basic objects, or also tensoring with M. over Z/GQO, shifts these to the .M-level where much
is known. In Lemma 5.2 we prove that 8
M ® G{M00/K00) - M 0 coker # X ® coker ^
is exact with a finite kernel on the left.
From these preparations we obtain that the image of Us G KoR(ZiGoo) in K$R{Ai) equals
[M ® G{M00/K00)) G K0R(M) (Proposition 5.3).
At this stage we bring in the Deligne-Ribet power series Gx^s(T) G Z/[x][[T]] (see e.g. [Wi,
p.494]), which is defined for each abelian character x of ^oo having finite order and taking
values in (Q/
C
)
x
. This power series depends on 7&. The 7^ being fixed, we prove the existence
of a unique element Qs M satisfying x(®s)
=
Gx^s(0) for all \ (Proposition 5.4). As a
matter of fact, Os G (i?-
1
Z
/
G
?
00
)
x
-
(R~lM)x
and can hence be regarded in K\{R
1A4)
(Lemma 5.5). We also discuss the dependence of d(Ss) G KQR(ZIGOQ) on 7^ (Lemma 5.6).
The relationship between Qs and Us is given in
THEOREM C. If K/k is an abelian extension of totally real fields and I is odd
9,
then
Us - d(Qs) is in the kernel of pM : K0R(ZiGoo) - K0R(M) .
The proof of the theorem is due to the validity of the Main Conjecture of Iwasawa theory by
which GXJS(T), up to a unit in Z/[x][[T]], is the characteristic polynomial of ex 0Z/Goo ^00
over the Iwasawa algebra Z/[x]T. The l-part of Gx^s(T) requires some extra effort to control
(see section 5.2).
A consequence of Theorems B and C is that there is a unit T in X x in terms of which
a representing homomorphism for Q$ may be expressed. In other words, determining 0 $
amounts to proving a "Main Conjecture" of equivariant Iwasawa theory.
The next two chapters, 6 and 7, are concerned with constructing a representing homomorphism
for £lipc. Chapter 6 is local preparation for this and is fairly complete in the tame case.
Chapter 7 starts with a realization of the diagram (*) in terms of a global unit £ G K when
K is a Galois extension of Q. This realization is made in such a way that the f^
£
-class splits
into two classes
VtipL = fi(Joo) + ft(_) .
Theorem D of chapter 6, in the later notation of Proposition 7.3, now says that Q(_) is
represented by
x
_ JT [tdimV/*Gpdet(Fr„ - Npk | VxG°)det(Frp - 1 | V?'/V?')-1]
s-,
inHom
e i
(fl,(G),F, x ) .
Turning to O ^ ) we need the assumption that K/k is tame above /, since we are going to
apply the main results of additive Galois module structure. In order to lift the strategy of
§ 12 in [RW2] we need to generalize Frohlich's concept of resolvents. This will lead us to
computing two determinants
/dk/qdimVx
Y[NSpkp/Ql(spbp
pk\l
ix;
8
with ® short for (g^Goo
9For
I = 2, the present status of the Main Conjecture should permit only a partial result, as in [RW2].
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