Cancellation of quadratic forms over principal ideal domains

Abstract

Let R be a commutative ring of dimension one in which 2 is invertible. It was proved in [8, Theorem 7.21 that if 4 is a quadratic space (i.e. a non-singular quadratic form on a finitely generated projective module) over R of Witt-index at least two, cancellation holds for q. The aim of this note is to prove a refinement of this result if R is a principal ideal domain. More precisely, we prove that if R is a principal ideal domain and 4 an isotropic quadratic space over R, then cancellation holds for 4.’ (We note that a quadratic form q over a Dedekind domain is isotropic if and only if its Witt-index is at least one.) We give examples to show that, in general, cancellation fails for quadratic spaces of Witt-index one over arbitrary Dedekind domains and for anisotropic spaces over principal ideal domains, thereby showing that our result is in some sense the best possible. I am thankful to M.A. Knus for the interesting discussions I had with him during the preparation of the paper.