LIFTED ROOT NUMBER CONJECTURE AND IWASAWA THEORY
3
in the Grothendieck group of all finitely generated ZG-modules having finite projective di-
mension, and conjectures that ft is the root number class of K/k.
1.4 The lifted-Q construction
Let Mi,M2 be finitely generated Z/G-modules and / : M
2
- M\ a Z/G-monomorphism
inducing an isomorphism Q/ g M
2
^ Qj g M\. Let K, E x t |
G
(M2, Mi) have a realization
Mi A i? M2 in which A, B have finite projective dimension over Z\G and B has no
Zj-torsion. Following [GRW2, §2] we associate to this data an element
« ( / , K) in K0T(ZtG),
the Grothendieck group of finite Z/G-modules of finite projective dimension. In order to do
this first break any such realization of K, into two parts
M i ^ A ^ L , L^B ^ M
2
.
Then take Z/G-monomorphisms a,/3 : L - L which induce 0 in Ext
1
. The pull-back along
a and push-out along /3 give
Mi ^ L 0 M i ^ L L - B M
2
II
5J,
a I /U
/5|
||
Mi -+ A L L - L 0 M
2
M
2
The composite map
/ : £ A L 0 M
2
1
- ^ / L 0 Mi - ^ 4
is injective with finite cokernel and so [coker/] G KQT(Z[G). We define
Q(/, «) = [coker /] - d[Qt ®z, L, a(3]
with d : i^i(Q/G) KQT(ZIG) from the localization sequence of X-theory
1TI(Z
Z
G) - ATi(QjG) - ^ KoTiZtG) - K0{ZiG) - K
0
(QiG).
The proofs of Lemma 1 and 2 in [GRW2] show that 0 ( / , « ) is independent of the choice of
a,a,/3,/3 and the realization of /c.
1.5 The Lifted Root Number Conjecture
If in the preceding section Z\ is replaced by Z and K is the Tate canonical class r, then the
construction gives
^ =
f
% , r ) G K
0
T ( Z G )
for any ZG-monomorphism y : A S ~ E, see [GRW2, §2]. Since K0T(ZG) = ®K0T(ZpG),
p
with p ranging over all prime numbers, we can speak of the /-component Q^/ of Vt^. Clearly
Oj/ = il((fi, 77) where ipi : Z\ g) A S ^ Z/ (g) £ is induced by (/? and 77 = Z/ (8 r.
The local Horn description is the isomorphism Det in
KifaG) - #i(Qz3) - KoTiZtG)
Det J
Hom^(fl
z
(G),F
z
x
)
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