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Rationally isotropic quadratic spaces are locally isotropic

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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-Thélène 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].

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Authors and Affiliations

  1. POMI, Fontanka 27, St. Petersburg, 191023, Russia

    Ivan Panin

  2. Fakultät für Mathematik der Universität Bielefeld, SFB-701, Bielefeld, Germany

    Ivan Panin

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  1. Ivan Panin
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Panin, I. Rationally isotropic quadratic spaces are locally isotropic. Invent. math. 176, 397–403 (2009). https://doi.org/10.1007/s00222-008-0168-0

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  • DOI: https://doi.org/10.1007/s00222-008-0168-0

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