We call N a metabolic submodule of M or metabolizer, for short.
Proposition 1.3. (M/ , /t) is metabolic over R if and only if
(M, , ,t) & K is metabolic.
The proof is standard. Utilizing this fact, we may weaken the con-
ditions of Definition 1.2 to the following:
Proposition 1.4. (M, , /t) is metabolic if and only if there is
an R-submodule N satisfying:
i) rank M = 2 rank N
ii) N c
iii) N is invariant under t
Proof: We may assume that our metabolic submodule is pure (that is,
if nx is in N for some n in R and x ^ 0 in M then x is in
x ad|N
N) . Then we have the following exact sequence: 0 ~ * N - M Horn (N,R)
"- * 0 since any form on the pure submodule N can be lifted to M and
by the non-singularity of the bilinear form represented as the adjoint of
some element in M . Now, by the additivity of rank over short exact
sequences and condition i), rank N = rank N""" . Since N
N by ii)
and both are pure submodules of the same rank , N = N
The orthogonal direct sum of two isometric structures, (Mo#^ ' ' ntn^
and (M]_, , 1»t1) is (MQ S Mx, , © , 1»tQ © t^ . Finally we
define the appropriate equivalence relation:
Definition 1.5 . Two isometric structures, MQ and M-, , are Witt-
equivalent (or concordant) if there are metabolic isometric structures,
H^ and H, , so t h a t IVL © H„ i s i s o m o r p h i c t o M, © H, .
U 1 u u l l
The resulting set of equivalence classes, C*~ (R) , form a group
under (orthogonal) direct sum with the inverse of {M, , ,t} given by
{M,- , ,t} (where { } will denote the appropriate equivalence class.)
Proposition 1.6. {M, , ,t} = 0 in C"(R) if and only if
(M, , #t) is metabolic.
Proof: The assumption gives a metabolic structure H such that
M © H is metabolic with metabolizers Hn for H and N for M © H .
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