LIFTED ROOT NUMBER CONJECTURE AND IWASAWA THEORY
13
and look at the two cokernel sequences of the (injective) southwest arrows. Because of (D2.1/)
and Lemma 2.3 they yield the diagram
G(K^l/K)
~ C(vo) Aj3
(D2.2) / / S=
G(M/K) - Y A/G
defining Y. This information is summarized in the commutative diagram with exact rows and
columns
Z/ ® E - Zi® A Z/ (8) L
I I I
(D2.3) r W £ ( # p X ) ^ I l p
6
s ^ p ) ^ Z * ® A ,
G{M/K) - y A,G
Collecting everything so far, we have almost established
L E M MA 2.4. Assume that K satifies LeopoldVs conjecture and that either I is odd or K/k is
split at all infinite primes. Then
1. Y has finite projective dimension over Z/(7;
2. the translation functor transforms G(M/K) -* G(M/k) G into
G(M/K) Y Z, g AG.
PROOF. Claim 1. is an immediate consequence of the middle column of (D2.3) and of 4. in
Lemma 2.1.
Claim 2. is due to the fact that (D2.2) is a push-out diagram for Y. Moreover, in
H2(G,CK) - H2(G,G{K^l/K)) - H2(G,G(M/K))
II II II
ExtJ
G
(AG,CA:) -+ E x t ^ ( A 2 , G ( t f a b ' 7 * 0 ) -* ExtJ
G
(AG,G(M/A') )
II II
ExtlaiAtGiGiK^/K)) - E x t ^
G
( A ^ , G ( M / ^ ) ) ,
with the bottom equalities because
G(Kah'l/K)
and G(M/K) are Z/-modules, the fundamen-
tal class in H2(G, CK) is sent to the extension class of G(M/K) G/M/k) G in the top
line, by Safarevic-Weil.
2.4 The totally real case
Remembering Tate's canonical class Z{ g) r and observing that A* fits into
A , ~ 0 indg
p
Z«G
p
- 0 indg
p
Z, ,
pes* pes*
we enlarge each row in (D2.3) to a 2-extension
ZiS)E - Zi®A - Z/(g)B A/ 5
(D2.4) I W W ) ^ I l p e S ^ ) - (Z,G)™ - ZZS
G(M/A-) ~ Y -+ ZiG ^ Zt ,
with m = 15* |, having middle terms of finite projective dimension and the third term without
Z/-torsion. This will be exploited below assuming that Leopold^s conjecture holds for K and
that K is totally real.
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