1. Rationally isotropic quadratic spaces are locally isotropic
- Author
-
Ivan Panin
- Subjects
Discrete mathematics ,Pure mathematics ,Lemma (mathematics) ,Quadratic equation ,General Mathematics ,Isotropy ,Zero (complex analysis) ,Field of fractions ,Field (mathematics) ,Regular local ring ,Isotropic quadratic form ,Mathematics - Abstract
Let R be a regular local ring, K its field of fractions and (V,ϕ) a quadratic space over R. Assume that R contains a field of characteristic zero we show that if (V,ϕ)⊗ R K is isotropic over K, then (V,ϕ) is isotropic over R. This solves the characteristic zero case of a question raised by J.-L. Colliot-Thelene in [3]. The proof is based on a variant of a moving lemma from [7]. A purity theorem for quadratic spaces is proved as well. It generalizes in the charactersitic zero case the main purity result from [9] and it is used to prove the main result in [2].
- Published
- 2008