On the non-trace-valued forms

Abstract

Let { + f be an involutory antiautomorphism of the divison ring k and E a k-vector space. A hermitian form @ on E is said to be trace-valued if, for all x E E, @(x, x) is a trace, i.e ., of the form < + f for some <E k [ 11. If the characteristic is not 2 or if the center of k is not fixed under the involution then all @ are trace-valued. Non-trace-valued forms are known as rather unmanageable-even in finite dimensions: nondegenerate isotropic planes may fail to be hyperbolic, the cancellation theorem is not valid and so Witt’s theorem does not hold either. This last fact seems to entail a particularly unfavourable verdict on the nontrace-valued forms as the classic theory of quadratic forms pivots on this theorem (see also the contention supported in [6, p. 2491). It is the purpose of this paper to demonstrate that matters may be looked at differentently and that the classification problem on subspaces in the nontrace-valued situation may be successfully attacked by what might be called the lattice method. The ground for this method has been laid in [2]. In order to give perspicuousness to the constituent parts we have stripped the problem of all unnecessary complications. In particular, we have made the following assumptions throughout the paper: