Invent. math. 87, 253-302 (1987)
/n/2e////o~/8s
mathematicae
Springer-Verlag 1987
SK I
of finite abelian groups, II
Roger C. Alperin 1., R. Keith Dennis 2., R. Oliver 3, and Michael R. Stein 4.
t Department of Mathematics, The University of Oklahoma, Norman, Oklahoma 73019, USA
2 Department of Mathematics, Cornell University, Ithaca, New York 14853, USA
3 Mathematisk Institut, Aarhus Universitet, Ny Munkegade, DK-8000 Aarhus C, Denmark
4 Department of Mathematics, Northwestern University, Evanston, Illinois 60201, USA
Introduction
This paper is a continuation of [A-D-S] in which we laid the theoretical
foundation for computing the torsion,
SKI(ZIG]),
in the Whitehead group of a
finite abelian group G. In that paper we showed how
SKI(Z[G])
can be
computed from a knowledge of
SKI(Z[Gv])
for the Sylow p-subgroups of G,
with some complications for p=2. In this paper we eliminate those com-
plications, showing that when H is an abelian 2-group and (9 is a totally
imaginary ring of integers in which 2 is unramified, then
SK, ((9
[HI) ~
SK,
(Z [H] ~
SK, ((9 [H/H2]),
and we calculate the second summand.
A second foundational result in [A-D-S] described
SKI(TI[G])
as the
cokernel of a map between the
Kz's
of certain finite rings. More specifically,
for a finite abelian p-group
G, SKI(Z[G])
is the cokernel of the homomor-
phism
q~s: K z (Z [G]/(pk)) ~ n K 2 (Z Ix]/(pk))
where the product is taken over certain equivalence classes of rational charac-
ters Z of G, Z[X] is the ring of cyclotomic integers generated by Im(x), and k is
large enough (e.g., pk> IG[). (This description requires some minor modification
when p=2.) Now it follows from [D-S] that
K2(Z[z]/(pk))
is isomorphic to
Im(x ) under a homomorphism induced by the norm residue symbol, and this
isomorphism is given explicitly by a formula of Artin and Hasse [A-H]
involving the p-adic logarithm. The key idea of this paper, which is developed
in w 1, is to use the integral logarithm of Taylor and Oliver to give a simple
combinatorial description of Im(~os) as a subgroup of//Im(x) (Propositions 1.5
and 1.7).
* Research supported by the National Science Foundation