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Rationally isotropic quadratic spaces are locally isotropic. III

Authors: I. Panin and K. Pimenov
Original publication: Algebra i Analiz, tom 27 (2015), nomer 6.
Journal: St. Petersburg Math. J. 27 (2016), 1029-1034
MSC (2010): Primary 13H05
Published electronically: September 30, 2016
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Abstract: Let $ R$ be a regular semilocal domain containing a field such that all the residue fields are infinite. Let $ K$ be the fraction field of $ R$. Let $ (R^n,q\colon R^n \to R)$ be a quadratic space over $ R$ such that the quadric $ \{q=0\}$ is smooth over $ R$. If the quadratic space $ (R^n,q\colon R^n \to R)$ over $ R$ is isotropic over $ K$, then there is a unimodular vector $ v \in R^n$ such that $ q(v)=0$. If $ \mathrm {char}(R)=2$, then in the case of even $ n$ the assumption on $ q$ is equivalent to the fact that $ q$ is a nonsingular quadratic space and in the case of odd $ n > 2$ this assumption on $ q$ is equivalent to the fact that $ q$ is a semiregular quadratic space.

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Additional Information

I. Panin
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia

K. Pimenov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Petergof, 198504 St. Petersburg, Russia

Keywords: Quadratic form, regular local ring, isotropic vector, Grothendieck--Serre conjecture
Received by editor(s): June 15, 2015
Published electronically: September 30, 2016
Additional Notes: Theorem 3 was proved with the support of the Russian Science Foundation (grant no. 14-11-00456). The research of the second author was partially supported by RFBR grant 12-01-33057 “Motivic homotopic cohomology theories on algebraic varieties” and by RFBR grant 13-01-00429 “Cohomological, classical, and motivic approach to algebraic numbers and algebraic varieties”
Dedicated: Dedicated to Professor S. V. Vostokov with great respect
Article copyright: © Copyright 2016 American Mathematical Society