Rationally trivial quadratic spaces are locally trivial:III

TL;DR

This paper proves that quadratic spaces over certain regular semi-local domains are locally trivial if they are isotropic over the fraction field, extending classical results to new algebraic settings.

Contribution

It establishes local triviality of quadratic spaces over regular semi-local domains under isotropy conditions, including cases with characteristic 2 and specific regularity assumptions.

Findings

Quadratic spaces over regular semi-local domains are locally trivial if isotropic over the fraction field.

The result applies to domains with infinite residue fields and smooth projective quadrics.

It extends classical local triviality results to cases with characteristic 2 and semi-regular spaces.

Abstract

Let R be a regular semi-local domain containing a field such that all the residue fields are infinite. Let K be the fraction field of R. Let q be a quadratic space over R on a free rank n R-module P such that the projective quadric q=0 is smooth over R. It is proved that if the quadratic space q is isotropic over K, then there is a unimodular vector v in the free rank n R-module P such that q(v)=0. If characteristic of R is 2, then in the case of even n our assumption on q is equivalent to the one that q is a non-singular space in the sense of \cite{Kn} and in the case of odd n > 2 our assumption on q is equivalent to the one that q is a semi-regular in the sense of \cite{Kn}.