12
JURGEN RITTER AND ALFRED WEISS
2.3 S-truncation
Let M = Ms denote the maximal abelian 5-ramified ^-extension of K. In particular,
G{K^l/M)
is the ^-completion of
G(K*h/M).
The canonical map
Js C/c G{K&h/K) sends the subgroup Flpgs ^ P o f
j
s
= n K x n f/p
pes pgs
onto
G(K&b/M).
This is because the local reciprocity map identifies Up with the inertia group
at p. We /-complete and arrive at £(Js) YlpeS
£(KpX) x
Ilp^5^(^p)
a s w e n a s a^
the
canonical surjection it\ : IIpgs£(^p)
G(K^1
jM).
This information is brought into the commutative diagram
kerTrx ~ npts£(UP) - G(K-b'l/M)
I I I
Z, ® £ - - £ ( J
S
) -* G{K&h'l/K)
ker7r2 ~ I l p e s W ) - G(M/tf )
in which the middle row is a copy of the left column of (D2.1/), in view of 5. of Lemma 2.1.
The map 7T2 is the induced one; it is necessarily surjective. The snake lemma ensures the
surjectivity of Z/ 0 E k e r ^ .
LEMMA 2.3. If Leopoldt's conjecture holds for K, then the composite map
Zi^E^C(Js)^l[C(K^)
pes
in the above diagram is injective and has cokernel G(M/K).
In particular, ker7Ti = 0 and Tip&s £(Up) —* G(if
a b
'*/M) is an isomorphism.
Recall that Leopoldt's conjecture asserts the injectivity of the natural map Zi^E^ YlpU Up
(with EQQ denoting the group of global units of K and
Upl
the group of 1-units in Kp). Now,
C(UP) =
Upl
for p|Z. Consequently, Z/ (g) -Eoo + YlpeS £(Up)
1S
injective, as all p|Z belong to S.
This yields the left part of the commutative diagram
1i ® ^oo ~ Zi®E Z, g E/Eoo
w i t h t h e b o t t o m r o w a n d r i g h t
col_
* J- ^ umn induced by the valuations tD for
pes pes s\sx v c w ° °
Since the vertical map on the right is injective, the middle one is so as well, and the lemma
follows.
Next, start out from the commutative diagram
Z( ® E -• Z
(
® A Zj ® L
/ I I / I I /
C(JS)
~ my,)
-
Z,®A ,
||
II II
I Z; ® E - | - Z( ® .4 - || Z( ® L
/ / /
I\s£(Kpx) ~ Us£(vp) Z
(
® A *
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