LIFTED ROOT NUMBER CONJECTURE AND IWASAWA THEORY 7
Also, from now on we assume that not only K but also all Kn , 0 n oo , satisfy Leopoldt's
conjecture. This has the consequence that G(M00/K00) is i?-torsion (Proposition 4.4).
Now we take any element Coo G
(R~1ZIGOG)X
such that
doo = c00((7/e - l)e + (1 - e)) belongs to AGoo ,
choose, in (4.oo), a preimage ?/oo of doo and set
# : Z/Goo - Foo , 1 ^
y o o
.
Proposition 4.5 states that ^ is injective and that coker \I/ is .R-torsion and has projective
dimension 1. It is related to the "Iwasawa module" G(M00/K00) by the short exact sequence
G{M00/K00) -+ coker^ coker ^
of .R-torsion Z/Goo-modules with no finite submodule. Here, ip is the map Z/Goo —» AG?oo ,
so we can regard its cokernel to be known from the construction of doo-
The aim of "equivariant Iwasawa theory" is to describe coker \I/. More precisely, let KQR(ZIG OQ)
be the Grothendieck group of all finitely generated i?-torsion Z/Goo-modules of finite projec-
tive dimension. It is part of the localization sequence
JTifaGoo) - tfi^ZjGoo) ^ ^ofi(ZjGoo) - ^o(Z;Goo).
The map ^ gives rise to the element [coker ^J G KQR(ZIGOO). We define
Us = [coker tf] - d f / T ^ G o o , Coo] G KQR{ZiGoo).
As a matter of fact, then Us depends only on the choice of 7^. (and on S) (Proposition 4.6).
The connection to fi$n, for any £n : Z/ G(Mn/Kn), is given in Theorem B below, in which
0"n denotes the image of Us under the deflation map KQR(ZIGOO) KoT(ZiGn) of Lemma
4B.1.
THEOREM B. 15n-d[(Qi®AGn)e,%-i\ = £lqn-d{Qi, |$
n
|] , where | $
n
| G Z/ zs determined
by $
n
( l ) having image yk
n |
under G(Mn/Kn) —• G^^/Kn) —• G(kOQ/k).
In chapter 5 we assume that G? is abelian. Then (700 = # X T with T ~ T^. We define
JW = 0 Z
z
G o o - e
x
x
with x running through the abelian (Q/C)x-valued characters of H, modulo Galois conjuga-
tion over Q/. Above, ex = J^ ex r and ex r is the primitive central idempotent
reG(Qi(x)/Qi)
associated to
\a•
Hence e = e\H. Also, M. is the maximal ZjT-order in Quot(Z/T)[if].
Since ZjGoo e
x
~ Z/[x]T is the Iwasawa algebra of Y with respect to the base ring Z/[x]
7
,
we can exploit the theory of finitely generated modules over such algebras. Note that each
such has finite projective dimension, since Z/[x]T is regular.
We introduce in Lemma 5.1 the map KQR^IGOQ) KQR(M.) and, moreover, show that
finite .M-modules become 0 in KQR(M). Applying the map KQR{Z[G00) —• KQR(M) to our
7Z,[X] = ^[
X
(/i):hetf ]
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