UNRAVELING THE INTEGRAL KNOT CONCORDANCE GROUP
7
e-symmetric isometric structures whose ^ [X]-module structure can be
reduced to S . Therefore trace induces a natural isomorphism
H€(A~ (S/^)) with C^(^) . We note that we only require that the under-
lying S-module structure be torsion-free. This differs from the usual
requirement that the underlying module be projective.
We then develop an appropriate analog of the localization exact
sequence
0 - H€(A~1(S/S)) - HE(E) - H€(E/A"~1(S/2G))
for any order S . The restriction mapping from the maximal order D
induces a mapping of the localization short exact sequences which is an
injection H (A~ (D/Z)) - » H (A~ (s/Z)) on S and is an isomorphism on
the quotient field. In a series of technical lemmas we relate the
boundary for D with that of S (At each maximal ideal M the relation
on the residue fields is ^^nc
= tr*
° ^#? where tr is the appropriate
map induced by trace for finite fields.) and finally compute the boun-
dary for D . (Lemma 4.9). Utilizing Landherr 's classical computation
for H (E) , we can then compute H (A" (D/Z)) for the maximal order
D . This is detailed in Section Four, Theorem 4.11 and 4.14. In the
case of the maximal order D , J. Lannes [La] has proven similar results
in the symmetric bilinear and quadratic cases. An interesting algebraic
feature (which does not occur for the orders S naturally arising in
the knot concordance group (by Lemma 5.1)) is a distinction between two
types of dyadic primes which are ramified in the quadratic extension of
E over the fixed field under the involution, F .
For the maximal order D , torsion-free D-modules are projective
and the question arose concerning the relation of H (A (D/z)) with
H (D) . We first define a certain arithmetic invariant SC(-l) of
order two of the field E . If this invariant is trivial, then P. E.
-1 —a
Conner has shown us how to prove that H (A (D/Z))— H " (D) . The
interchange of symmetry is intimately connected with the structure of
the inverse different. This Theorem is given in (4.15) and can be
viewed as a generalization of the classical theorem of Hecke on the
class of the different in the ideal class group to the Hermitian case.
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