12 NEAL W. STOLTZFUS
By Proposition 1.3 it suffices to consider the case when R is a field.
First we will rechoose the metabolizer N so that Hn is contained in
N . Let L c H„ be a invariant subspace of M © H which is maximal
with respect to the property L c L . Consider the subspace L + (N DL ).
This is invariant and easily shown to be self annihilating. Since
L c: L + (N f l LX ) , by maximality equality holds. Therefore N D L =
N f l L = (N + L) c L and by a standard property of annihilators
L c N + L . Hence L c L C L + ( N H L ) = L yields the necessary meta-
bolizer containing Hn .
Let Nn be the projection of L onto M . We will verify the
conditions of (1.4) for Nn . By exactness of the sequence 0 - Hn - » N
N„ - 0 , and the fact 2 rank H = rank H , 2 rank N = rank (M © H) ,
the rank condition is satisfied by additivity of the rank over exact
sequences. Next consider (x,y) in N for x in M and y in H .
Since N = N and H = N , y is in H = H so (x,0) is in N
since it is a submodule. Therefore Nn c N is self-annihilating and if
z is in NQ in M then (z#0) annihilates both Nn and Hn and
consequently N also. But N = N hence Nn ^ Nn . The last condition
of (1.4) follows since the projection commutes with the endomorphism of
the direct sum.
Stated in words, a stably metabolic isometric structure is actually
metabolic.
Definition 1.7. An isometric structure is anisotropic if, for any
invariant R-submodule L , L ( 1 L = 0 .
Proposition 1.8. Every Witt equivalence class has an anisotropic
representative.
Proof: Let L be an invariant pure R-submodule of M (i.e. M/L
J.
j.
is torsion free) with L c L . Then L /L inherits a quotient isome-
tric structure (L /L, , ,tQ) ( , is well-defined since L is
self-annihilating.) We claim that M © -L /L is a metabolic isometric
structure. Consider N = { (x,x+L): x is in L} . Then- ( , ©- , n)
((x,x+L),(y#y+L)) = x,y - x+L,y+L0 = 0 so N is self annihilating.
Now rank L + rank L1 = rank M , so 2 rank L = rank M + rank L /L .
Previous Page Next Page