Rationally isotropic quadratic spaces are locally isotropic

• 成果类型: Article
• 署名作者: Panin, Ivan
• 署名单位: University of Bielefeld
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-008-0168-0
• 发表日期: 2009
• 页码: 397-403
• 关键词: purity

摘要

Let R be a regular local ring, K its field of fractions and (V,phi) a quadratic space over R. Assume that R contains a field of characteristic zero we show that if (V,phi)aS- (R) K is isotropic over K, then (V,phi) is isotropic over R. This solves the characteristic zero case of a question raised by J.-L. Colliot-Th,lSne 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