Observing that N is invariant, by Proposition 1.4 M 3 (-L /L) is
metabolic and therefore M is Witt equivalent to L /L .
The existence of an anisotropic representative follows since we have
shown that an isometric structure with a non-trivial invariant self-
annihilating subspace is Witt equivalent to a structure with strictly
smaller rank, a process that must eventually terminate in an anisotropic
representative by the finite generation of M .
Corollary 1.9. If L is an invariant pure submodule of an isome-
tric structure (M, , ,t) and L ^ L then M is Witt equivalent to
(L /L, , n»t0) where the form , _ and the endomorphism tn are
induced from their restrictions to L
Corollary 1.10. Every isometric structure is equivalent to a struc-
ture with t injective.
Proof: Let L = kernel t . Then L c L since for x and y in
L , 0 = t(x),y = x,(1-t)(y) = x,y .
For the field K , an isometric structure with t injective is
equivalent to a more familiar object. For t is now an isomorphism and
s = 1 - t is defined. But sx,sy = x,sy - t x,sy = x,x -
x,t y - t x,t y + t x,t y = x,y since by defining relation
ii), t(t_1x) ,t_1y = t_1x, (1-t)(t~1y) . This is the traditional
notion of an isometry of a bilinear form. We further note that the
original structure is metabolic if and only if there is a self-annihila-
ting subspace of one-half the dimension of M invariant under the
isometry s .
By the usual algebraic trick of defining the action of an indeter-
minate X to be the same as that of t , we may view M as a module
over the polynomial ring R[X] . From the identity tx,y = x,(l-t)y
and bilinearity the formula f(t)x,y = x,f(l-t)y holds for any
integral polynomial f in R[X] . By the nonsingularity of the form,
if f(X) annihilates the module M , if and only if f(1-X) does so.
Therefore if CD (X) is the minimal polynomial of t , cp(l-x)| c p (x) and
by comparing the highest coefficients, we have (-1)
* cp(l-x) = c p (x) .
For any polynomial f(X) , we may define the dual polynomial f (X) =
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